Equadiff 9: Conference on Differential Equations and their Applications : Proceedings

CONFERENCE ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, BRNO, AUGUST 25 – 29, 1997 Equadiff 9 Papers edited by Z. ...

Author: Z. Dosla | J. Kuben | J.Vosmansky

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CONFERENCE ON DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, BRNO, AUGUST 25 – 29, 1997

Equadiff 9 Papers edited by Z. Doˇ sl´ a, J. Kuben, J. Vosmansk´ y

EQUADIFF 9 • MASARYK UNIVERSITY RETURN

BRNO 1998

Preface

The Conference on Differential Equations and Their Applications (EQUADIFF 9) was held in Brno, August 25–29, 1997. It was organized by the Masaryk University, Brno in cooperation with Mathematical Institute of the Czech Academy of Sciences, Technical University Brno, Union of Czech Mathematicians and Physicists, Union of Slovak Mathematicians and Physicists and other Czech scientific institutions with support of the International Mathematical Union. EQUADIFF 9 was attended by 269 participants from 32 countries and more than 50 accompanying persons and other guests. The scientific program comprised 8 plenary lectures and 34 main lectures in sections. In addition 208 papers were presented as communications, at the poster session and in the form of enlarged abstracts. This volume contains 31 papers accepted for the presentation at the Equadiff 9 Conference and submitted for this type of publication by the authors. A great part of them is in the final form and will not be published elsewhere, the others are preliminary version or overview articles. This volume is published in the electronic form, both on Equadiff 9 CD ROM and on the Internet (http://www.math.muni.cz/Equadiff9CDROM/). The authors are provided with the hardcopies of their papers. Our aim was to harness the possibilities of new computer technologies and for this reason all Equadiff 9 publications were prepared in the hypertext PDF form. We hope that this form of publication will be accepted favourably. Brno, May 1998

Editors

Table of Contents

Stability Theorems for Nonlinear FDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oleg Anashkin (Simferopol State University)

1

Analysis of Equations in the Phase-Field Model . . . . . . . . . . . . . . . . . . . . . . . . 17 Michal Beneˇs (Czech Technical University) Summation of Polyparametrical Functional Series . . . . . . . . . . . . . . . . . . . . . . 37 Andriy Blazhievskiy (Technological University of Podillia, Ukraine) System of Differential Equations with Unstable Turning Point . . . . . . . . . . . 43 V. N. Bobochko and I. I. Markush (Ukraine) On the Symmetric Solutions to a Class of Nonlinear PDEs . . . . . . . . . . . . . . 53 Gabriella Bogn´ ar (University of Miskolc) The Abstract Cauchy Problem in Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Igor A. Brigadnov (North-West Polytechnical Institute St. Petersburg) Thermoelastic Far-field Patterns for the Vector Thermoelastic Equation . . . 73 Fioralba Cakoni (University of Tirana) Transformations of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Jan Chrastina (Masaryk University Brno) Abstract Differential Equations of Arbitrary (Fractional) Orders . . . . . . . . 93 Ahmed M. A. El-Sayed (Alexandria University) Parametric Representation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Peter L. Simon and Henrik Farkas (Technical University of Budapest) Homogenization of Scalar Hysteresis Operators . . . . . . . . . . . . . . . . . . . . . . . . 111 Jan Franc˚ u (Technical University Brno) Global Qualitative Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Valery Gaiko (Belarus State University of Informatics and Radioelectronics) The Existence of Global Solutions to the Elliptic-Hyperbolic DSI System . . 131 Nakao Hayashi (Science University of Tokyo) and Hitoshi Hirata (Sophia University) Scaling in Nonlinear Parabolic Equations : Locality versus Globality . . . . . . 137 Grzegorz Karch (Uniwersytet Wroclawski) Almost Sharp Conditions for the Existence of Smooth Inertial Manifolds . . 139 Norbert Koksch (Technical University Dresden)

The Property (A) for a Certain Class of the Third Order ODE . . . . . . . . . . 167 Monika Kov´ aˇcov´ a (Slovak Technical University Bratislava) On Factorization of Fefferman’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Miroslav Krbec (Acad. Sci. Prague), Thomas Schott (FSU Jena) A Time Periodic Solution of Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . 193 Petr Kuˇcera (Czech Technical University) Numerical Solution of Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 M´ aria Luk´ aˇcov´ a-Medvid’ov´ a (Technical University Brno) Structure of Distribution Null-Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Takeshi Mandai (Gifu University, Gifu) A Functional Differential Equation in Banach Spaces . . . . . . . . . . . . . . . . . . . 223 Nasr Mostafa (Suez Canal University) On the Limit Cycle of the van der Pol Equation . . . . . . . . . . . . . . . . . . . . . . . 229 Kenzi Odani (Aichi University of Education) Lp Solutions of Non-linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Alejandro Om´ on and Manuel Pinto (Universidad de Chile) Rothe’s Method for Degenerate Quasilinear Parabolic Equations . . . . . . . . . 247 Volker Pluschke (University Halle) A Posteriori Error Estimates for a Nonlinear Parabolic Equation . . . . . . . . . 255 Karel Segeth (Academy of Sciences Praha) The Solvability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 G. M. Sklyar (Ukraine) and I. L. Velkovsky (Regensburg) Elliptic Equations with Decreasing Nonlinearity I . . . . . . . . . . . . . . . . . . . . . . 269 Tadie (Matematisk Institut Copenhagen) Elliptic Equations with Decreasing Nonlinearity II . . . . . . . . . . . . . . . . . . . . . 275 Tadie (Matematisk Institut Copenhagen) Mathematical Models of Suspension Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Gabriela Tajˇcov´ a (University of West Bohemia, Pilsen) Numerical Analysis of High-Temperature Strains . . . . . . . . . . . . . . . . . . . . . . . 307 Jiˇr´ı Vala (Brno) Asymptotic Behavior of Solutions of Partial Difference Inequalities . . . . . . . 319 Patricia J. Y. Wong (Nanyang Technological University) Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

iii

EQUADIFF 9 CD ROM, Brno 1997 Masaryk University

PAPERS pp. 1–15

Stability Theorems for Nonlinear Functional Differential Equations Oleg Anashkin Department of Mathematics, Simferopol State University 4, Yaltinskaya St., 333007 Simferopol, Ukraine Email: [email protected] WWW: http://www.ccssu.crimea.ua/~anashkin

Abstract. New approach in stability theory for a class of retarded nonlinear functional differential equations is discussed. The problem of stability of the zero solution is considered under assumption that the system of interest has a trivial linearization, i.e. it is essentially nonlinear. Sufficient conditions for uniform asymptotic stability and instability are given by auxiliary functionals of Lyapunov-Krasovskii type. The method is also applicable to linear systems with a small parameter in standard form. Some examples are given. AMS Subject Classification. 34K20 Keywords. asymptotic stability, instability, Lyapunov functionals, parametric resonance

1

Introduction

The paper is devoted to the problem of stability of the zero solution of nonlinear system of functional differential equations (FDE) of retarded type x˙ = F (t, xt ).

(1.ana )

There is no need to elaborate on the role that Lyapunov-Krasovskii functionals play in the analysis of asymptotic properties of solutions of FDE. In particular, a suitable functional may ensure the uniform asymptotic stability of a trivial solution of (1.ana ). Due to classical results [1], [2]. suitable here may mean uniformly positive definite on state space Ch = C([−h, 0]), Rn ), which strictly and uniformly decreases along nontrivial solutions. There is the celebrated converse theorem on Lyapunov-Krasovskii functional [2]. But to find actually such functionals in concrete examples is not an easy problem. In this paper we establish new sufficient conditions on stability for a class of FDE in terms of functionals which satisfy less strong restrictions. We suggest a new approach in context of generalized Lyapunov’s direct method [3] – [6]. Unlike the well-known theorems on stability for FDE [1], [2] suitable functionals satisfy main restrictions only in This is the preliminary version of the paper.

2

Oleg Anashkin

some cone AhR ⊂ Ch not in the whole space Ch , moreover, they are nonmonotone along nontrivial solutions of (1.ana ). It gives a possibility to use a simple procedure to construct suitable functionals. The paper is organized as follows. In the second section we give the statement of the problem, some definitions and mathematical facts. Theorems on uniform asymptotic stability and instability are stated and proved in sections 3 and 4, respectively. In the last section we consider two examples. The first one is an example of nonlinear scalar equation with deviating argument which has unstable zero solution for all values of the constant delay h ∈ [a, b] but, from the other hand, it is shown that there exist a time-varying delay h0 (t) with the same range of values, h0 (t) ∈ [a, b] for all t, such that this equation with delay h0 (t) already have uniformly asymptotically stable trivial solution. We also study the parametric resonance in linear equation with small parameter of Mathieu type. It is shown that the delay being introduced may damp the demultiplicative parametric resonances and make the equation either unstable or asymptotically stable.

2

Preliminaries

Consider a system of nonlinear functional differential equations with finite delay written as x(t) ˙ = F (t, xt ),

(2.ana )

h h where F : GhH → Rn , GhH = R+ × ΩH , R+ = [0, ∞), ΩH = {ϕ ∈ Ch : kϕk < H} is the open H – ball in the Banach space Ch = C([−h, 0], Rn ) of continuous functions ϕ : [−h, 0] → Rn with the supremum norm kϕk = max{|ϕ(s)|: −h ≤ s ≤ 0}, | · | is a norm in Rn . For a given function x(t) we denote by xt the element in Ch defined by xt (s) = x(t + s), −h ≤ s ≤ 0. In the context of FDE the element xt is called the state at time t. Denote by UI(R+ ) a set of all functions L : R+ → R+ which are integrable on any finite segment [t0 , t0 +∆] ⊂ R+ and for any ∆ > 0 there exists a constant L∆ > 0 such that tZ 0 +∆

L(t) dt ≤ L∆

for any t0 ∈ R+ .

(3.ana )

t0

We assume that there are exist functions L, M 0 ∈ UI(R+ ) and a constant h d0 > 1 such that for any t ∈ R+ and ϕ, ψ ∈ ΩH |F (t, ϕ) − F (t, ψ)| ≤ L(t)kϕ − ψk,

(4.ana )

|F (t, ϕ)| ≤ M 0 (t)kϕkd0

(5.ana )

h for any t ∈ R+ and ϕ, ψ ∈ ΩH .

3

Stability Theorems for Nonlinear FDE’s

h will be denoted by x(t0 , ϕ) : A solution of (2.ana ) through (t0 , ϕ) ∈ R+ × ΩH R+ → Rn , t 7→ x(t; t0 , ϕ), so that xt0 (t0 , ϕ) = ϕ. It is known that x(t0 , ϕ) satisfies the integral equation

Zt x(t; t0 , ϕ) = ϕ(0) +

F (τ, xτ ) dτ,

xt0 = ϕ,

t ≥ t0 .

(6.ana )

t0

Using this representation and Gronwall’s lemma it is easy to get the following results [7]. h Lemma 1. Let t0 ∈ R+ and ϕ ∈ ΩH be given and the functional F satisfies Lipschitz inequality (4.ana ). Then until x(t0 + ∆; t0 , ϕ) ∈ Ωhh the following inequality holds

kxt0 +∆ (t0 , ϕ)k ≤ kϕk exp(L∆ ),

(7.ana )

where L∆ is a constant from the estimate (3.ana ). Note that the right-hand part of inequality (7.ana ) does not depend on t0 . h Lemma 2. Let t0 ∈ R+ and ϕ ∈ ΩH be given and the functional F satisfies inequalities (4.ana ) and (5.ana ) then

|x(t0 + ∆; t0 , ϕ) − ϕ(0)| ≤ kϕkd0 E∆ ,

(8.ana )

0 0 where E∆ = M∆ exp(d0 L∆ ), M∆ and L∆ are the constants from the estimates of the type (3.ana ) for functions M 0 (t) and L(t) respectively.

Lemma 3. Let x : [t0 − h, ∞) → Rn be a continuous function and there exist a constant R > 1 such that |x(t)| ≤ kxt k/R ≡ (1/R) max{|x(t + s)| : −h ≤ s ≤ 0} for all t ≥ t0 . Then lim |x(t)| = 0, and t→∞

|x(t)| ≤ kxt0 k/RN +1

for t ≥ t0 + N h, N = 0, 1, 2, . . . .

In this paper we use the known definition of stability. Definition 4. The zero solution x = 0 of the system (2.ana ) is said to be stable if for each σ ≥ 0, α > 0 there is β = β(α, σ) > 0 such that ϕ ∈ Ωβh implies that xt (σ, ϕ) ∈ Ωαh for any t ≥ σ; uniformly stable if it is stable and β is independent of σ; asymptotically stable, if it is stable and for each σ ≥ 0 there is β0 = β0 (σ) > 0 such that ϕ ∈ Ωβh0 implies xt (t; σ, ϕ) → 0 as t → ∞; uniformly asymptotically stable if it is uniformly stable and if there is a β0 > 0 and for each η > 0 there exists t0 (η) > 0 such that for any σ ≥ 0 xσ = ϕ ∈ Ωβh0 implies xt (σ, ϕ) ∈ Ωηh for t ≥ σ + t0 (η).

4

Oleg Anashkin

For given h0 > h and R ≥ 1 consider the set AhR0 = {ϕ ∈ Ch0 : kϕk ≤ R|ϕ(0)|}. It is easy to see that AhR0 6= ∅ for R > 1, AhR01 ⊂ AhR02 for R1 < R2 , the boundary ∂AhR0 of the set AhR0 is defined as ∂AhR0 = {ϕ ∈ Ch0 : kϕk = R|ϕ(0)|}, Ah1 0 ≡ ∂Ah1 0 and Ah1 0 ⊂ AhR0 for any R > 1, AhR0 is a nonconvex cone in Ch0 . The cone AhR0 plays a crucial role in our approach. The fact is that the norm kxt (σ, ϕ)k may increase if and only if xt (σ, ϕ) ∈ Ah1 0 and |x(t; σ, ϕ)| tends to zero when xt (σ, ϕ) 6∈ AhR0 for some R > 1. Therefore in the context of the stability problem it is enough to investigate a behavior of the state xt (σ, ϕ) only in the h0 cone AhR0 not in the whole neighborhood ΩH .

3

Sufficient conditions on asymptotic stability

In this section we present sufficient conditions for uniform asymptotic stability of the zero solution of the system (2.ana ) in terms of Lyapunov’s functionals v(t, ϕ) which can be nonmonotone along the solutions. It means that the derivative v| ˙ (2.ana ) can change the sign. This ) (σ, ϕ) of the functional v along the solution of (2.ana derivative is defined as v(σ + ∆t, xσ+∆t (σ, ϕ)) − v(σ, ϕ) . ∆t→+0 ∆t

v| ˙ (2.ana ) (σ, ϕ) = lim

If v is differentiable v| ˙ (2.ana ) (σ, ϕ) is obtained using the chain rule. We start with the following technical lemma. Lemma 5. Let h0 ≥ h and R > 1 be given. Assume that for some τ0 ≥ 0 and ) is defined for τ0 ≤ t ≤ ψ0 ∈ AhR0 ∩ Ωηh0 a solution x(τ0 , ψ0 ) of the system (2.ana τ0 + 2h0 and η < ηR , ηR =

R−1 (R + 1)Rd0 E0

1/(d0 −1) ,

0 E0 = M2h exp(d0 L2h0 ). 0

(9.ana )

Then xt (τ0 , ψ0 ) ∈ AhR0 for τ0 + h0 ≤ t ≤ τ0 + 2h0 . If, in addition, kxt (τ0 , ψ0 )k < ηR for all t ≥ τ0 + h0 , then xt (τ0 , ψ0 ) ∈ AhR0 for all t ≥ τ0 + h0 . Proof. According to Lemma 2 the solution x(τ0 , ψ0 ) satisfies the inequality |x(t; τ0 , ψ0 ) − x(τ0 )| ≤ kxτ0 kd0 E0

(10.ana )

5


), we obtain the for any t ∈ [τ0 , τ0 + 2h0 ], here xτ0 = ψ0 , x(τ0 ) = ψ0 (0). Using (10.ana following upper and lower estimates for the norm |x(t; τ0 , ψ0 )| on the segment [τ0 , τ0 + 2h0 ]: |x(t; τ0 , ψ0 )| ≤ |x(t; τ0 , ψ0 ) − x(τ0 ) + x(τ0 )| ≤ |x(τ0 )| + kxτ0 kd0 E0 ≤ |x(τ0 )|(1 + Rd0 |xτ0 |d0 −1 E0 ),

(11.ana )

|x(t; τ0 , ψ0 )| ≥ |x(τ0 )| − kxτ0 kd0 E0 ≥ |x(τ0 )|(1 − Rd0 |xτ0 |d0 −1 E0 ).

(12.ana )

Hence for t ∈ [τ0 + h0 , τ0 + 2h0 ] we have max{|x(t + s), t − h0 ≤ s ≤ t} 1 + Rd0 E0 |x(τ0 )|d0 −1 kxt k ≤ ≤ |x(t)| min{|x(s)|, τ0 + h0 ≤ s ≤ τ0 + 2h0 } 1 − Rd0 E0 |x(τ0 )|d0 −1 ≤

d0 −1 1 + Rd0 E0 ηR 1 + Rd0 E0 η d0 −1 < = R. (13.ana ) d0 −1 d d −1 1 − R 0 E0 η 0 1 − Rd0 E0 ηR

It means, that kxt k < R|x(t)|, i.e. xt ∈ AhR0 , for t ∈ [τ0 + h0 , τ0 + 2h0 ]. If kxt (τ0 , ψ0 )k < ηR for all t ≥ τ0 + h0 , then |x(τ0 + kh0 ; σ, ψ0 )| < ηR for any k = 1, 2, . . . . The constant E0 does not depend on xτ0 , consequently, the estimates (11.ana )–(13.ana ) hold for τ0 + kh0 ≤ t ≤ τ0 + (k + 1)h0 , k = 1, 2, . . . . It means that xt ∈ AhR0 for any t ≥ τ0 . Lemma is proved. Let K denotes the Hahn class, i.e. the set of all continuous strictly increasing functions a : R+ → R+ such that a(0) = 0. Theorem 6. Suppose that for some h0 ≥ h and R > 1 the following assumptions hold: 1) there exist functionals v, Φ : GhH0 → R and functions a, b ∈ K such that a) v| ˙ (2.ana ) (t, ϕ) ≤ Φ(t, ϕ), b) v(t, ϕ) ≤ b(kϕk) for (t, ϕ) ∈ GhH0 , h0 ; c) v(t, ϕ) ≥ a(kϕk) for t ≥ 0 and ϕ ∈ AhR0 ∩ ΩH 2) there exist constants d > 1, m > 0 and a function M ∈ UI(R+ ) such that |Φ(t, ϕ)| ≤ M (t)kϕkd0

for (t, ϕ) ∈ GhH0 ,

|Φ(t, ϕ) − Φ(t, ψ)| ≤ M (t)rd−1 kϕ − ψk for ∀t ≥ 0 and ϕ, ψ ∈ Ωrh0 , 0 < r < H; 3) there exist constants T > 0, β > 0 and δ > 0 such that for any t0 ≥ 0, x0 ∈ Bβ and ∆t ≥ T t0Z+∆t

I(∆t, t0 , x0 ) =

Φ(t, x0 ) dt ≤ −2δ|x0 |d ∆t. t0

Then the zero solution of (1) is uniformly asymptotically stable.

6

Oleg Anashkin

Proof. First of all we will show that xt (t0 , xt0 ) ∈ AhR0 for all t ≥ 0 if xt0 ∈ AhR0 at some moment t0 and kxt0 k is small enough. Then we prove that the zero solution of the system (2.ana ) is uniformly stable and kxt k → 0 as t → ∞. Uniformity of the asymptotic stability is guaranteed by the properties of the functional v in the cone AhR0 . Let us fix an arbitrary small ε ∈ (0, ηR ), where ηR is defined as in Lemma 5 by given h0 ≥ h and R > 1. Put ε1 = ε/2. Denote by h0 : v(τ, ϕ) < γ} {v < γ}τ = {ϕ ∈ ΩH

a cut for t = τ of the region {v < γ} = {(t, ϕ) ∈ GhH0 : v(t, ϕ) < γ}. h0 Due to the positive definiteness of the functional v in the region R+ ×(ΩH ∩AhR0 ) there exists a constant γ > 0 such that for all t ≥ 0

{v < γ}t ∩ AhR0 ⊂ Ωεh10 . It is enough to take γ = a(ε1 ). Choose a value η0 > 0 such that η0 < η0 (1 + η0d0 −1 Eh0 ) < b−1 (γ),

(14.ana )

) implies that η0 < ηR . where Eh0 = Mh00 exp[Lh0 (d0 + 1)]. Note that (14.ana ) implies that v(σ, ψ) < γ. If Let σ ≥ 0 and ψ ∈ Ωηh00 be given. Inequality (14.ana h0 ψ∈ / AR , then kxt (σ, ψ)k will decrease while xt ∈ / AhR0 . The rate of decreasing is given by Lemma 3. Suppose that xτ0 ∈ AhR0 for some τ0 ≥ σ. According to the choice of η0 (see the inequality (14.ana )) v(t, xt ) < γ for t ∈ [τ0 , τ0 + h0 ]. Due to Lemma 5 xτ0 +h0 ∈ AhR0 , hence, a(kxτ0 +h0 k) ≤ v(τ0 + h0 , xτ0 +h0 ) < γ = a(ε1 ), therefore kxτ0 +h0 k < ε1 < ηR and xt ∈ AhR0 at least for t ∈ [τ0 + h0 , τ0 + 3h0 ]. xt (σ, ψ) will be in AhR0 while kxt k < ηR . Remember that kxt k may increase only if xt ∈ Ah1 0 ⊂ AhR0 . According to the choice of γ we see that kxt k ≤ ηR /2 till v(t, xt ) ≤ γ. Suppose that v(t0 , xt0 ) = γ for some t0 ≥ τ0 + h0 the point xt leaves the domain {v < γ}. Note that xt0 ∈ AhR0 , i.e. kxt0 k ≤ R|x(t0 )|. According to condition 2a) of the theorem v| ˙ (2.ana ) (t, xt ) ≤ Φ(t, xt ). Thus

Z

t

v(t, xt (σ, ψ)) − v(t0 , xt0 (σ, ψ)) ≤

Φ(τ, xτ ) dτ.

(15.ana )

t0

Define a function z : R → Rn such that z(t) = x(t; σ, ψ) for t0 − h0 ≤ t ≤ t0 Rt and z(t) = x(t0 ) for t ≥ t0 Adding and subtracting t0 Φ(τ, zτ ) dτ , we get Zt

Zt Φ(τ, xτ ) dτ =

t0

Zt [Φ(τ, xτ ) − Φ(τ, zτ )] dτ.

Φ(τ, zτ ) dτ + t0

t0

(16.ana )

7


According to Lemma 2 kxτ − zτ k ≤ M1 kxt0 kd0 , for τ ∈ [t0 , t0 + T1 ], where the constant T1 ≥ T will be selected below, the constant M1 depends only on T1 . Using Lemma 1 and the Lipschitz condition we obtain t0Z+T1

[Φ(τ, xτ ) − Φ(τ, zτ )] dτ ≤ C0 T1 kxt0 kd+d0 −1 .

(17.ana )

t0 h0 , i.e. the The estimate (17.ana ) is uniform with respect to t0 ≥ 0 and xt0 ∈ ΩH constant C0 > 0 depends only on T1 . To estimate first integral in the right-hand side of (16.ana ) we represent it in the form

Zt

t0Z+h0

Φ(τ, zτ ) dτ =

Zt Φ(τ, yτ ) dτ −

Φ(τ, zτ ) dτ +

t0

t0

t0Z+h0

t0

Φ(τ, x(t0 )) dτ,

(18.ana )

t0

Due to construction of the function z, condition 2) of the theorem and Lemma 1, we obtain t +h 0Z 0 ≤ h0 Mh0 kxt0 kd exp(dLh0 ), Φ(τ, z ) dτ (19.ana ) τ t0 t +h 0Z 0 Φ(τ, yτ ) dτ ≤ h0 Mh0 |x(t0 )|d exp(dLh0 ). (20.ana ) t0

Choose T1 ≥ T such that Mh0 (Rd + 1)h0 exp(dLh0 )/T1 ≤ δ/2,

(21.ana )

where δ is a constant from condition 3) of the theorem. Using condition 3) and taking into account that kxt0 k < R|x(t0 )|, from (18.ana )–(21.ana ) we get t0Z+T1

t0Z+T1

Φ(τ, x(t0 )) dτ −

Φ(τ, zτ ) dτ = t0

t0Z+h0

t0

t0Z+h0

Φ(τ, x(t0 )) dτ + t0

Φ(τ, zτ ) dτ t0

t +h t +h 0Z 0 0Z 0 Φ(τ, x(t0 )) dτ + Φ(τ, x(t0 )) dτ + Φ(τ, zτ ) dτ

t0Z+T1

≤ t0

t0

t0

≤ −2δ|x(t0 )| T1 + h0 Mh0 exp(dLh0 )(|x(t0 )| + kxt0 kd ) Mh0 (Rd + 1)h0 exp(dLh0 ) 3 ≤ − δ|x(t0 )|d T1 . ≤ |x(t0 )|d T1 −2δ + T1 2 d

d

8

Oleg Anashkin

Thus t0Z+T1

3 Φ(τ, zτ ) dτ ≤ − δ|x(t0 )|d T1 . 2

(22.ana )

t0

Suppose that ε is small enough to be true the following inequality C0 Rd+d0 −1 (ε/2)d0 −1 ≤ δ/2.

(23.ana )

Then from (15.ana )–(17.ana ) and (22.ana ) we have Zt v(t0 + T1 , xt0 +T1 ) ≤ v(t0 , xt0 ) +

Zt [Φ(τ, xτ ) − Φ(τ, zτ )] dτ

Φ(τ, zτ ) dτ + t0

t0

3 ≤ v(t0 , xt0 ) − δ|x(t0 )|d T1 + C0 T1 Rd+d0 −1 |x(t0 )|d+d0 −1 2 3 ≤ v(t0 , xt0 ) + |x(t0 )|d T1 (− δ + C0 T1 Rd+d0 −1 |x(t0 )|d0 −1 ) 2 ≤ v(t0 , xt0 ) − δ|x(t0 )|d T1 < v(t0 , xt0 ). It gives us the main inequality v(t0 + T1 , xt0 +T1 ) ≤ v(t0 , xt0 ) − δ|x(t0 )|d T1 .

(24.ana )

We emphasize that the estimate (24.ana ) is uniform with respect to t0 ≥ 0 and ) means that the state xt has returned into the xt0 ∈ AhR0 ∩ Ωεh10 . Inequality (24.ana domain {v < γ}t , moreover, we can choose ε > 0 so small that xt does not leave the ball Ωεh0 . Indeed, according to Lemma 2 for t ∈ [t0 , t0 + T1 ] |x(t; t0 , xt0 ) − x(t0 )| ≤ kxt0 kd0 MT1 exp(d0 LT1 ) ≤ |xt0 |d0 Rd0 MT1 exp(d0 LT1 ), therefore |x(t; t0 , xt0 )| ≤ |x(t0 )| + |x(t) − x(t0 )| ≤ ε, if ε is small enough to ensure (ε/2)d0 Rd0 MT1 exp(d0 LT1 ) ≤ ε/2. According to Lemma 5 xt ∈ AhR0 for t ∈ [t0 , t0 + T1 ]. It has been shown that at the moment t1 = t0 + T1 xt1 belongs to the domain {v < γ}t1 ⊂ Ωεh10 . If the point xt will leave the domain {v < γ}t again at some moment t0 0 > t1 then it will return back in finite time less than T1 because all estimates we employed above to obtain the main inequality are uniform with respect to t ≥ t0 and xt0 ∈ AhR0 ∩ Ωεh10 . Consequently, we have proved that xt ∈ AhR0 ∩ Ωεh10 for all t ≥ t0 . Since ε is arbitrary small and η0 does not depend on initial moment σ, the uniform stability of the zero solution of the system (2.ana ) is proved.

9


To prove the asymptotic stability we note that in virtue of the uniformity ) of all estimates derived above with respect to t ≥ t0 and xt0 ∈ AhR0 ∩ Ωεh10 (24.ana yields 0 < v(tk + T1 , xtk +T1 ) ≤ v(t0 , xt0 ) − δT1 (|x(t0 )|d + |x(t1 )|d + . . . + |x(tk )|d ), (25.ana ) where tk = t0 + kT1 , for any integer k ≥ 1. By condition 1c) of the theorem v(t0 , xt0 ) ≥ a(kxt0 k) > 0, and (25.ana ) means that |x(tk )| → 0 as k → ∞, therefore kxtk k → 0 as k → ∞, because kxtk k < R|x(tk )|. / AhR0 and Uniformity of the asymptotic stability follows from Lemma 3 if xt ∈ h0 from conditions 1b) and 1c) if xt ∈ AR . The theorem is proved.

4

Sufficient conditions for instability

Theorem 7. Suppose that for some h0 ≥ h, β > 0, σ > 0 and R > 1 there exist functionals v, Φ : [σ, ∞) × Ωβh0 → R such that the following conditions satisfied: 1) v| ˙ (1) (t, ϕ) ≥ Φ(t, ϕ) for (t, ϕ) ∈ [σ, ∞) × Ωβh0 ; 2) for each t ≥ σ and η, 0 < η < β, there exists ϕ ∈ AhR0 ∩ Ωηh0 such that v(t, ϕ) > 0; 3) there exists a function b ∈ K such that v(t, ϕ) ≤ b(kϕk) for each (t, ϕ) ∈ [σ, ∞) × AhR0 ∩ Ωβh0 ; 4) the functional Φ satisfies condition 2) of Theorem 6 for (t, ϕ) ∈ [σ, ∞)×Ωβh0 and 0 < r < β; 5) there exist constants T > 0 and δ > 0 such that for any t0 ≥ σ, x0 ∈ Bβ and ∆t ≥ T I(∆t, t0 , x0 ) ≥ 2δ|x0 |d ∆t. Then the zero solution of the system (1.ana ) is unstable. Proof. By way of contradiction, assume that the zero solution of (2.ana ) is stable. By the conditions of the theorem choose a small enough value ε < min{β, ηR } and a constant T1 ≥ h0 such that the inequalities (21.ana ) and (23.ana ) hold. According to our assumption there exists η0 > 0 such that for any initial function ψ0 ∈ Ωηh00 xt (σ, ψ0 ) ∈ Ωεh0 for all t ≥ σ. Let us fix arbitrary small η ∈ (0, ε) and choose ψ0 ∈ AhR0 ∩ Ωηh0 such that α = v(σ, ψ0 ) > 0. Lemma 5 implies that xt (σ, ψ0 ) ∈ AhR0 for all t ≥ σ + h0 . It means that xt ∈ AhR0 ∩ Ωεh0 for t ≥ σ + h0 . By condition 3) of the theorem the functional v is bounded along the given solution x(σ, ψ0 ) of the system (2.ana ). Denote tk = σ + kT1 , k = 0, 1, . . . . Since ψ0 ∈ AhR0 , then kψ0 k < R|ψ0 (0)|. According to condition 1) Z t Φ(τ, xτ ) dτ. v(t, xt (σ, ψ)) − v(t0 , xt0 (σ, ψ)) ≥ t0

10

Oleg Anashkin

By the same way as in the proof of Theorem 6 we obtain the main inequality v(t0 + T1 , xt0 +T1 ) ≥ v(t0 , xt0 ) − δT1 |x(t0 )|d .

(26.ana )

This inequality is valid for all (t0 , xt0 ) ∈ [σ, ∞) × (AhR0 ∩ Ωεh0 ). Denote tk = σ + kT1 , k = 0, 1, . . . . Since kxt (σ, ψ0 )k < ε for all t ≥ σ, we obtain from (26.ana ) that for any integer k = 0, 1, 2, . . . v(tk + T1 , xtk +T1 (σ, ψ0 )) ≥ v(σ, ψ0 ) + δT1 (|x(t0 )|d + |x(t1 )|d + . . . + |x(tk )|d ). (27.ana ) Note that for any integer k |x(tk )| > (1/R)kxtk k ≥ (1/R)b−1 (v(tk , xtk )) > (1/R)b−1 (α) > 0. Hence the right-hand side of (27.ana ) tends to +∞ as k → +∞. It contradicts the boundedness of the functional v in the region [σ, ∞)×(AhR0 ∩Ωεh0 ). The theorem is proved.

5

Remarks

Remark 8. Theorems 6 and 7 are valid also for the systems in the standard form of the type x˙ = µL(t, xt ),

(28.ana )

where µ is a positive small parameter, L is linear in xt and |L(t, ϕ)| ≤ M (t)kϕk h for any t ≥ 0 and ϕ ∈ ΩH with some function M ∈ UI(R+ ). Remark 9. Condition 3) of Theorem 6 (respectively, condition 5) of Theorem 7) will be fulfilled if there exists the average 1 {Φ}(t0 , x0 ) = lim ∆t→∞ ∆t

t0Z+∆t

Φ(t, x0 ) dt

(29.ana )

t0

and a constant δ0 > 0 such that for all t0 ≥ 0 {Φ}(t0 , x0 ) ≤ 2δ0 |x0 |d (respectively, {Φ}(t0 , x0 ) ≥ 2δ0 |x0 |d ).

6

Examples

By simple examples we present an algorithm for construction of functionals which satisfy all conditions of new theorems on stability given in the previous sections. Example 10. Consider a nonlinear equation with a time-varying delay x˙ = b(t)x3 (ρ(t)),

(30.ana )

where ρ is a differentiable function, t − h ≤ ρ(t) ≤ t for all t > 0 and some positive constant h > 0. Suppose that the function b has a zero average {b(t)}


11

and a bounded antiderivative B, B 0 (t) = b(t), on t ∈ [0, ∞). To construct an appropriate functional v we take a positive definite function v0 (x) = x2 /2. Then 3 v˙ 0 |(30.ana ) = b(t)x(t)x (ρ(t)) = Φ0 (t, x(t), x(ρ(t))).

Following to Remark 8, we have to evaluate the average {Φ0 }(t0 , x0 ) along the constant solution x0 of the trivial system. The average {Φ0 (t, x0 , x0 )} ≡ 0, since {b(t)} = 0. Consider a functional v(t, xt ) = v0 (x(t)) + u(t, x(t), x(ρ(t))), where the function u(t, p, q) is a bounded solution of the equation ∂u/∂t = −Φ0 (t, p, q) = −b(t)pq 3 . Putting u(t, p, q) = −B(t)pq 3 , we obtain a functional v = v0 + u = x(t)2 /2 − B(t)x(t)x3 (ρ(t)).

(31.ana )

Calculating a full derivative of this functional in virtue of the system (30.ana ), we get v| ˙ (30.ana ) = Φ1 (t, x(t), x(ρ(t)), x(ρ(ρ(t)))) = −B(t)[b(t)x6 (ρ(t)) + 3x(t)x2 (ρ(t))x3 (ρ(ρ(t)))b(ρ(t))ρ0 (t)]. A sign of the average {Φ1 (t, x0 , x0 , x0 )} is defined by a sign of the average δ0 − {B(t)(b(t) + 3ρ0 (t)b(ρ(t)))}.

(32.ana )

According to Theorems 6 and 7 the zero solution of the system (30.ana ) is uniformly asymptotically stable if δ0 < 0 and it is unstable if δ0 > 0. Let b(t) = cos t, ρ(t) = t − β + α sin ωt, where α, β and ω are some constants. ), we Then B(t) = sin t, ρ0 (t) = 1 + αω cos ωt. Substituting given functions to (30.ana obtain the equation x˙ = cos tx3 (t − h(t)),

(33.ana )

where h(t) = β − α sin ωt. The index of stability (32.ana ) now has the form δ0 = −{sin t(1 + αω cos ωt) cos(t − β + α sin ωt)},

(34.ana )

since the average {sin t cos t} = 0. If α = 0 the system (33.ana ) takes the form x˙ = cos tx3 (t − β)

(35.ana )

with the constant delay β. In this case (34.ana ) gives δ0 = − sin β. Thus the zero solution of the equation (35.ana ) is unstable for any β ∈ (π, 2π), since sin β < 0 and all conditions of Theorem 7 are fulfilled.

12

Oleg Anashkin

It is interesting that it is possible to choose the values of the parameters α, β and ω such that the zero solution of the equation (33.ana ) with time-varying delay becomes already uniformly asymptotically stable although h(t) = β − α sin ωt ∈ (π, 2π), i.e. for every t the value of the delay h(t) lies in the domain of instability of the trivial solution of the same equation but with a constant delay (35.ana ). Indeed, put β = 1.5π, α = 1.55 < π/2 and ω = 2. Then (34.ana ) gives δ0 = −0.04033 < 0 and all conditions of Theorem 6 on asymptotic stability are fulfilled, but sin h(t) < 0 for all t ∈ (−∞, ∞) because h(t) = 1.5π − 1.55 cos 2t ∈ [3.16, 6.27] ⊂ (π, 2π) This phenomenon of changing of the type of stability after replacing of the constant parameter by the continuous function with the same range of values is well-known for ordinary differential equations. It have been first demonstrated for linear equation with deviating argument by A. D. Myshkis [8]. Example 11. [9] Consider the linear equation of Mathieu type with time delay x¨ + ω 2 [x(t) − µ(2 cos νt)x(t − h)] = 0,

(36.ana )

where µ is small parameter. This equation turns into the well-known Mathieu equation at h = 0: x ¨ + ω 2 [x(t) − µ(2 cos νt)x(t)] = 0.

(37.ana )

There are infinite sequence of the so-called regions of dynamical instability for the Mathieu equation (37.ana ) at the critical values ν = 2ω/m, m = 1, 2, 3, . . . . This phenomenon is called parametric resonance. By Theorems 6 and 7 we show that the main resonance ν = 2ω also appears in the equation (36.ana ) for any delay h. At ν 6= 2ω the type of stability depends greatly from h. The delay being introduced may damp the demultiplicative parametric resonances and make the equation either unstable or asymptotically stable. Introducing complex conjugate variables ζ and ζ¯ x˙ ζ¯ exp(−iωt) = x + i , ω

x˙ ζ exp(iωt) = x − i , ω

(38.ana )

and using more short notations for the variables with deviating argument: ζh = ζ(t − h),

ζ2h = ζ(t − 2h),

... ,

we reduce (36.ana ) to the linear system in standard form ζ˙ = µZ(t, ζh , ζ¯h ),

¯ ζh , ζ¯h ), ζ¯˙ = µZ(t,

(39.ana )

where Z(t, ζh , ζ¯h ) = −0, 5iω[ζhe−iωh (eiνt + e−iνt ) + ζ¯h eiωh (e−i(2ω+ν)t + e−i(2ω−ν)t )].

13


To construct a suitable functional we start with the positive definite function ¯ = ζ ζ. ¯ Differentiating it in virtue of (39.ana ), we obtain v0 (ζ, ζ) ∂v0 ¯ Z v˙ 0 |(39.ana ) = µΦ0 (t, ζh , ζh ) = µ2Re ∂ζ ¯ h e−iωh (eiνt + e−iνt ) = µRe[(−iω)(ζζ + ζ¯ζ¯h eiωh (e−i(2ω+ν)t + e−i(2ω−ν)t )].

(40.ana )

It is obvious that average (29.ana ) {Φ0 } = {Φ0 (t, ζ0 , ζ¯0 )} = 0, if ν 6= 2ω. Let ν = 2ω, then {Φ0 } = Re[(−iω)ζ¯02 eiωh ] = iω(ζ02 e−iωh − ζ¯02 eiωh ) Denoting

¯ = i(ζ 2 e−iωh − ζ¯2 eiωh ), w(ζ, ζ)

we have that ¯ w| ˙ (39.ana ) = µΨ (t, ζh , ζh ) = 2µRe

∂w Z ∂ζ

= 2µRe[ω(ζζh e−i2ωh (ei2ωt + e−i2ωt ) + ζ ζ¯h (e−i4ωt + 1))]. The average {Ψ (t, ζ0 , ζ¯0 )} = 2ω|ζ0 |2 is positive definite and for small enough |µ| all conditions of Theorem 7 on instability are fulfilled, moreover, {Ψ } does not depend on h. Suppose now that ν 6= 2ω, then {Φ0 } ≡ 0. Following to the theory of the generalized Lyapunov functions [5] we have to construct the so-called perturbed functional V1 = v0 + µv1 ,

(41.ana )

where a perturbation v1 of the functional (function) v0 is calculated by a bounded solution of the equation ∂v1 /∂t = −Φ0 . Since iνt Z −iνt ∂v0 ¯ h e−iωh e − e Z dt = (ω/2) ζζ − ∂ζ ν −i(2ω+ν)t e−i(2ω−ν)t e iωh ¯ ¯ + + const, − ζ ζh e 2ω + ν 2ω − ν we can take the functional v1 in the form iνt −iνt ¯ ζh , ζ¯h ) = ωRe ζζ ¯ h e−iωh e − e v1 (t, ζ, ζ, ν −i(2ω+ν)t e−i(2ω−ν)t e + . − ζ¯ζ¯h eiωh 2ω + ν 2ω − ν

14

Oleg Anashkin

Then V˙ 1

∂v1 + µ2 2Re = v˙ 0 |(39.ana )+µ ∂t (39.ana )

∂v1 ∂v1 Z+ Zh ∂ζ ∂ζh

= µ2 Φ1 .

Making necessary calculations, we obtain ∂v1 ∂v1 Z+ Zh = ∂ζ ∂ζh = (−iω 2 /4) ζ¯h ζh (e−i2νt − ei2νt )/ν −

e−i2νt 4ω ei2νt − − 2 2 4ω − ν 2ω + ν 2ω − ν

+ ζ¯h2 ei2ωh (e−i(2ω+2ν)t − e−i(2ω−2ν)t )/ν i(2ω+2ν)t ei(2ω−2ν)t + ei2ωt + ei2ωt e 2 −i2ωh + − ζh e 2ω + ν 2ω − ν −i2ωh −i2νt −iνh −iνh iνh ¯ 2h e (e e −e + e − e−i2νt eiνh )/ν + ζζ + ζ¯ζ¯2h (1/ν)(e−i2ωt ei(2ω+ν)h + e−i(2ω−2ν)t ei(2ω−ν)h − e−i(2ω+2ν)t ei(2ω+ν)h − e−i2ωt ei(2ω−ν)h ) − ζζ2h e−i2ωh (e−iνh ei(2ω+2ν)t + ei2ωt eiνh )/(2ω + ν) + (ei(2ω−2ν)t eiνh + e−iνh ei2ωt )/(2ω − ν) − ζ ζ¯2h (ei(2ω+ν)h + ei2νt ei(2ω−ν)h )/(2ω + ν) + (e

i(2ω+ν)h i2νt

e

+e

i(2ω−ν)h

)/(2ω − ν) .

Here we use a notation: Zh = Z(t − h, ζ2h , ζ¯2h ). Consequently, the average of the functional Φ1 (t, ζh , ζ¯h , ζ2h , ζ¯2h ) for ν 6= 2ω, ν 6= ω and ζ = ζ0 has the form ω3 sin(2ω + ν)h sin(2ω − ν)h 2 |ζ0 | − . {Φ1 } = 2ν 2ω + ν 2ω − ν It is not equal to zero for all values of h, in spite of the zeros of a function σν (ωh) =

sin(2 + ν/ω)ωh sin(2 − ν/ω)ωh − . 2 + ν/ω 2 − ν/ω

Since the functional (41.ana ) is positively defined in the cone AR ⊂ Ch . R > 1, for small enough µ, the equation (36.ana ) is uniformly asymptotically stable for σν (ωh) < 0 and it is unstable for σν (ωh) > 0.

References [1] N. N. Krasovskii, Some Problems in the Stability Theory of Motions. Fizmatgiz, Moscow, 1959. [2] J. Hale, Functional Differential Equations. Springer Verlag, New York-HeidelbergBerlin, 1971.


15

[3] M. M. Khapaev, Theorem of Lyapunov Type, Soviet Math. Dokl., 8 (1967), 1329– 1333. [4] M. M. Khapaev, Instability for Constantly Operating Perturbations, Soviet Math. Dokl., 9 (1968), 43–47. [5] M. M. Khapaev, Averaging in Stability Theory, Kluwer, Dordrecht-Boston-London 1993. [6] O. V. Anashkin, Asymptotic Stability in Nonlinear Systems, Differential Equations, 14 (1978), 1061–1063. [7] O. V. Anashkin, Averaging in the Theory of Stability for Functional Differential Equations, Differents. Uravneniya, 33 (1997), 448–457 (in Russian). [8] A. D. Myshkis, Linear Differential Equations with Deviating Argument, Moscow, Nauka, 1972. [9] O. V. Anashkin, Parametric Resonance in Linear System with Time Delay, Transactions of RANS, series MMMIC, 1 (1997), 3–18 (in Russian).


PAPERS pp. 17–35

Analysis of Equations in the Phase-Field Model Michal Beneˇs Department of Mathematics, Czech technical University, Trojanova 13, 120 00 Prague, Czech Republic Email: [email protected] WWW: http://kmdec.fjfi.cvut.cz/~benes

Abstract. The article presents basic numerical analysis of equations in the phase-field model which is performed using a FDM semi-discrete scheme. The compactness technique allows to prove convergence of the scheme. Simultaneously, existence and uniqueness of weak solution to the original system is shown. Additionally, the asymptotical behaviour of the solution with respect to the small parameter ξ is studied. Both temperature and phase fields converge in certain sense if ξ → 0. The phase field gives rise to a step-wise function indicating the presence of different phases. AMS Subject Classification. 80A22, 82C26, 35A40 Keywords. phase-field model, method of lines, compactness method

1

Introduction

The paper contains several remarks concerning basic analysis of the standard form of phase-field model. The system of equations in question reads as follows: ∂p ∂u = ∇2 u + L , ∂t ∂t ∂p αξ 2 = ξ 2 ∇2 p + f0 (p) − βuξ ∂t

(1.ben ) ,

with initial conditions u |t=0 = u0 ,

p |t=0 = p0

,

and with boundary conditions of Dirichlet type u |∂Ω = uΩ ,

p |∂Ω = pΩ

,

where L, α, β, ξ are positive constants, Ω is a bounded domain in Rn and f0 derivative of a quartic potential. For the sake of simplicity, we will consider rectangular form of Ω in 2D, f0 (p) = ap(1 − p)(p − 12 ) with a > 0 and homogeneous boundary conditions. This is the final form of the paper.

18

Michal Beneˇs

Such a system of equations has been studied by many authors throughout last decade (see, e.g. [5], [5], [1], [8], [16], [13], [10]). In the physical context, the system (1.ben ) is treated as a regularization of the modified Stefan problem describing microstructure formation in solidification of a pure substance if ξ → 0, see [7], [2]: ∂u = ∇2 u in Ωs and Ωl ∂t u |∂Ω = uΩ , u |t=0 = u0 , ∂u ∂u |s − |l = −LvΓ , ∂n ∂n r 2 βu = −κ + αvΓ , 6 a Ωs (t) |t=0 = Ωso ,

,

(2.ben ) (3.ben ) (4.ben ) (5.ben ) (6.ben ) (7.ben )

where Ωs , Ωl are solid and liquid phases, respectively, L is latent heat per unit volume, melting point is 0, u temperature field. Discontinuity of heat flux on Γ (t) is described by the Stefan condition (5.ben ), the formula (6.ben ) is the GibbsThompson relation on Γ (t). The parameter α is the coefficient of attachment kinetics. Following [2], the relation of (1.ben ) and (2.ben )–(7.ben ) is studied using asymptotical analysis. The article presents the following results: convergence of the semi-discrete scheme, existence and uniqueness of the original system of equations, and convergence towards the sharp-interface state.

2

Interpolation theory for grid functions

The analysis of the system (1.ben ) concerning the existence and uniqueness of the weak solution is performed using a semi-discrete scheme based on finite differences. The following notations are introduced (see [15]): L1 L2 , h2 = , xij = [x1ij , x2ij ], uij = u(xij ), N1 N2 ωh = {[ih1 , jh2 ] | i = 1, . . . , N1 − 1; j = 1, . . . , N2 − 1} ,

h = (h1 , h2 ) , h1 =

ω ¯ h = {[ih1 , jh2 ] | i = 0, . . . , N1 ; j = 0, . . . , N2 }

,

¯ h − ωh , γh = ω uij − ui−1,j ui+1,j − uij , ux1 ,ij = ux¯1 ,ij = h1 h1 uij − ui,j−1 ui,j+1 − uij ux¯2 ,ij = , ux2 ,ij = h2 h2 1 ux¯1 x1 ,ij = 2 (ui+1,j − 2uij + ui−1,j ) , h1

(8.ben ) (9.ben ) (10.ben ) (11.ben )

,

(12.ben )

,

(13.ben ) (14.ben )

19

Phase-Field equations

and ¯ h u = [ux¯1 , ux¯2 ], ∇

∇h u = [ux1 , ux2 ],

∆h u = ux¯1 x1 + ux¯2 x2

,

(15.ben )

¯ h → R} is a set of grid functions, the following notations will If Hh = {f | f : ω be used (f, g ∈ Hh ) : ! p1 N1 −1,N X 2 −1 p h1 h2 |fij | for p > 1 , (16.ben ) kf kph = i,j=1

(f, g)h =

N1 −1,N X 2 −1

h1 h2 fij gij

,

kf k2h = (f, f )h

,

(17.ben )

i,j=1

(f , g c = 1

1

N1X ,N2 −1

1 h1 h2 fij1 gij

,

kf 1 c|2 = (f 1 , f 1 c

,

(18.ben )

2 h1 h2 fij2 gij

,

kf 2 e|2 = (f 2 , f 2 e

,

(19.ben )

i=1,j=1

(f 2 , g 2 e =

N1X −1,N2 i=1,j=1

(f , g] = (f 1 , g 1 c + (f 2 , g 2 e ,

kf ]|2 = (f , f ]

,

(20.ben )

where f = [f 1 , f 2 ] and g = [g 1 , g 2 ]. Referring to [2], we recall the following formulas – Green formulas (f, gx¯1 x1 )h = −(fx¯1 , gx¯1 c +

NX 2 −1

(f gx¯1 |N1 ,j −f gx1 |0,j )h2 ,

(21.ben )

j=1

and (f, gx¯2 x2 )h = −(fx¯2 , gx¯2 e +

NX 1 −1

(f gx¯2 |i,N2 −f gx2 |i,0 )h1 ,

(22.ben )

i=1

In a natural way, we define the space lp (ωh ) = {Hh | k · kph } .

(23.ben )

– Poincaré inequality. Let u ∈ l2 (ωh ) and u |γh = 0. Then kuk2h ≤ C(Ω)[ kux¯1 c|2 + kux¯2 e|2 ]

.

(24.ben )

We continue by introducing an extension of grid functions, so that they are defined almost everywhere on Ω. Such extensions are studied by the usual technique of Lp and Hk spaces. The approach of [14] is adopted for the equations in question. The limiting process requires a refinement of the FDM grid ω ¯ h , if h → 0. For this purpose, a proper metric should be chosen. If we intent to use ¯h → R the compactness technique, a mapping converting a grid function fh : ω into a function f : Ω → R is needed. Then, the norm of Lp spaces will serve as a metric for convergence of the numerical scheme.

20

Michal Beneˇs

Definition 1. Be ω ¯ h an uniform rectangular grid imposed on a domain Ω ⊂ R2 . Let h = [h1 , h2 ] is the mesh size. Then, the dual grid is a set n ¯ | Σij = ω ¯ h∗ = Σij ⊂ Ω o h1 h1 2 h2 2 h2 ¯ × xj − , xj + ∩ Ω for [x1i , x2j ] ∈ ω ¯h . x1i − , x1i + 2 2 2 2 The dual simplicial grid is a set ¯ h∗/ ∪ ω ¯ h∗. ω ¯ h∗s = ω

,

(25.ben )

with / ¯ | Σ / = [xi,j , xi−1,j , xi,j−1 ]κ ∩ Ω ¯ for [x1 , x2 ] ∈ ω ⊂Ω ¯ h} , ω ¯ h∗/ = {Σij ij i j ∗. . . 1 ¯ ¯ ω ¯ h = {Σij ⊂ Ω | Σij = [xi−1,j−1 , xi−1,j , xi,j−1 ]κ ∩ Ω for [xi , x2j ] ∈ ω ¯ h}

,

where [ ]κ denotes the convex hull. S ¯ - the system ω ¯ h∗ covers the domain Ω. Remark 2. Consequently, Σ∈¯ω∗ Σ = Ω h ∗ 1 2 Each (rectangular) set Σ ∈ ω ¯ h has the point [xi , xj ] in its center. Similarly, the ¯ system ω ¯ ∗s also covers Ω. h

¯ h . Define the following Definition 3. Let Hh be a set of grid functions on ω mappings: ¯ such that for each u ∈ Hh – Qh : Hh → C(Ω) (Qh u)(x1 , x2 ) = ui−1,j−1 + ∇h uh,i−1,j−1 · [x1 − x1i−1,j−1 , x2 − x2i−1,j−1 ]

,

. . , Σij ∈ω ¯ h∗. ; if [x1 , x2 ] ∈ Σij

¯ h uij · [x1 − x1 , x2 − x2 ] (Qh u)(x1 , x2 ) = uij + ∇ ij ij

,

/ / , Σij ∈ω ¯ h∗/ . if [x1 , x2 ] ∈ Σij – Sh : Hh → L1 (Ω) such that for each u ∈ Hh

(Sh u)(x1 , x2 ) = uij

,

¯ h∗ ; if [x1 , x2 ] ∈ Σij , Σij ∈ ω ¯ ¯ – Ph : C(Ω) → Hh such that for each u ∈ C(Ω) (Ph u)ij = u(xij ) , ¯h. if xij ∈ ω ¯ to Hh , and can Remark 4. The operator Ph is linear and continuous from C(Ω) 1 be extended to H (Ω) via density argument. Qh u is a continuous piecewise linear function, ∇(Qh u) exists a.e. in Ω. We proceed by determining basic properties of the above defined maps as proven in [2]:

21


1. If u, v |γh = 0 the scalar product coincides with the scalar product in l2 (ωh ) Z Sh uSh vdx = (u, v)h . (26.ben ) Ω

2. Let ωh is a grid on the domain Ω with the mesh h, let u, v ∈ Hh is such that u, v |γh = 0. Then ¯ h u, ∇ ¯ h v] (∇(Qh u), ∇(Qh v)) = (∇

.

(27.ben )

3. Let ωh is a grid on the domain Ω with the mesh h, let u ∈ Hh . Then kQh ukL2 (Ω) ≤ kSh ukL2 (Ω)

.

(28.ben )

4. Let ωh is a grid on the domain Ω with the mesh h, let u ∈ Hh , u |γh = 0. Then Z |h|2 ¯ k∇h u]|2 , |Qh u − Sh u|2 dx ≤ (29.ben ) 6 Ω if u |γh = 0. 5. Let p ∈ C 0,ν (Ω), ν ∈ (0, 1). Then, Sh (Ph p) → p in Ls (Ω), if h → 0

,

(30.ben )

for s > 1. 6. Let u ∈ H10 (Ω) ∩ H2 (Ω). Then Qh (Ph u) → u

(31.ben )

in H1 (Ω), if h → 0. 7. Let p ∈ C 2 (Ω) and p |∂Ω = 0. Then ∇(Qh (Ph p)) → ∇p

,

(32.ben )

in L2 (Ω), if h → 0.

3

Main result

In this section, we give a proof of existence and uniqueness of the solution to (1.ben ) regardless on values of coefficients. Compared to [5], we get a more general result. Similar procedure has been presented in [3]. Definition 5. Consider a bounded domain Ω ⊂ R2 , T > 0. The classical solution of the system of phase-field equations is a couple of functions ¯ → R2 [u, p] : h0, T i × Ω

,

22

Michal Beneˇs

satisfying the equations ∂p ∂u = ∇2 u + L in (0, T ) × Ω ∂t ∂t u |∂Ω = 0 , t ∈ (0, T ) , u |t=0 ∂p αξ 2 ∂t p |∂Ω p |t=0

= u0

in Ω

,

(33.ben )

= ξ 2 ∇2 p + f0 (p) − βξu

in (0, T ) × Ω

,

= 0 , t ∈ (0, T ) , = p0 in Ω .

Remark 6. The form of the phase-field equations is referred to [5]. For the sake of simplicity, we consider a 2-D rectangular domain and homogeneous boundary condition. Obviously, the extension to higher dimensions, and to other boundary conditions is possible. Let [u, p] is a classical solution such that u, p ∈ C2 (h0, T i× ¯ and let v, q ∈ D(Ω). Multiplying the first one of equations (1.ben Ω) ) by v and the second one by q (scalar product in L2 (Ω)), and using the Green formula, we get d d (u, v) + (∇u, ∇v) = L (p, v) dt dt u(0) = u0 ,

a.e. in (0, T ) ,

d (p, q) + ξ 2 (∇p, ∇q) = (f0 (p), q) − βξ(u, q) a.e. in (0, T ) , αξ dt p(0) = p0 .

(34.ben )

2

This leads to the next definition: Definition 7. Weak solution of the boundary-value problem for the phase-field equations is a couple of functions [u, p] from (0, T ) to [H10 (Ω)]2 such that it satisfies (34.ben ) for each q, v ∈ H10 (Ω). The term f0 (p) requires that p ∈ L4 (Ω) for almost all t ∈ (0, T ). As Ω ⊂ R2 , it suffices to take p ∈ H10 (Ω) for almost all t ∈ (0, T ) due to the continuous imbedding into Lq (Ω) for each q ∈ (1, +∞). If [u, p] ∈ [L∞ (0, T ; H10 (Ω))]2

,

[u, p] is continuous mapping from h0, T i to H−1 (Ω), as shown in [11]). Next statement gives an information about the existence and uniqueness of the solution to (34.ben ); the proof by its virtue contains the investigation of convergence of a semi-discrete scheme based on method of lines. Theorem 8. Consider the problem (34.ben ) in a rectangular domain Ω = (0, L1 ) × (0, L2 ), where u0 , p0 ∈ H2 (Ω) ∩ H10 (Ω) .

23


Then, there is a unique solution of the problem (34.ben ) satisfying u, p ∈ L∞ (0, T ; H10 (Ω) ∩ H2 (Ω)) ∂t u, ∂t p ∈ L2 (0, T ; L2 (Ω))

,

.

Proof. The proof is constructive. Cover Ω by an uniform grid with the mesh h = [h1 , h2 ], use the previously introduced notations. Consider the semi-discrete scheme u˙ h = ∆h uh + Lp˙ h u |γ h = 0 h

on (0, T ) × ωh

,

u |t=0 = Ph u0 h

on ω ¯h

,

αξ p˙ = ξ ∆h p + f0 (p ) − βξuh 2 h

2

p h |γ h = 0

,

h

h

in (0, T ) × ωh

(35.ben )

,

,

p |t=0 = Ph p0 h

on ω ¯h

.

where dot denotes the time derivative. In the proof, the major role is played by the a priori estimate for both equations in question. Multiply the first one of equations (35.ben ) by u˙ h , and the second one by p˙ h ; sum over ωh . ku˙ h k2h +

1 d ¯ h 2 k∇h u ]| = L(p˙ h , u˙ h )h 2 dt

,

1 d ¯ h 2 k∇h p ]| = (f0 (ph ), p˙ h )h − βξ(uh , p˙ h )h 2 dt Using Schwarz and Young inequalities, we get αξ 2 kp˙ h k2h + ξ 2

1 1 h 2 1 d ¯ h 2 ku˙ kh + k∇h u ]| ≤ L2 kp˙ h k2h , 2 2 dt 2 2 d 1 1 2 h 2 d ¯ h ph ]|2 ≤ − (w0 (ph ), 1)h + β kuh k2h αξ kp˙ kh + ξ 2 k∇ 2 2 dt dt 2α Combining these estimates, we have

(36.ben ) .

(37.ben ) .

1 d ¯ h 2 1 2 h 2 αξ 2 h 2 αξ 2 d ¯ h 2 αξ kp˙ kh + k∇h u ]| + ξ 2 k∇h p ]| + ku˙ kh + 4 4L2 4L2 dt 2 dt β2 h 2 d ku kh + (w0 (ph ), 1)h ≤ dt 2α Using the discrete Poincaré inequality (24.ben ) h 2 ¯ h uh ]|2 , ku kh ≤ C(Ω)k∇

.

(38.ben )

.

(39.ben )

and adding non-negative terms on the right-hand side, 1 2 h 2 αξ 2 h 2 αξ 2 d ¯ h 2 αξ kp˙ kh + k∇h u ]| + ku˙ kh + 4 4L2 4L2 dt d 1 d ¯ h 2 k∇h p ]| + (w0 (ph ), 1)h ≤ + ξ2 2 dt dt o n αξ 2 2β 2 L2 ¯ h ph ]|2 + (w0 (ph ), 1)h ¯ h uh ]|2 + ξ 2 1 k∇ k ∇ ≤ 2 2 C(Ω) α ξ 4L2 2

24

Michal Beneˇs

Integrating over (0, t), we have o ¯ h ph ]|2 + (w0 (ph ), 1)h (t) ≤ ¯ h uh ]|2 + ξ 2 1 k∇ k ∇ 4L2 2 n αξ 2 o n 2 2 o 1 h 2 2 ¯ h u ]| + ξ k∇ ¯ h ph ]|2 + (w0 (ph ), 1)h (0) exp 2β L C(Ω)t k∇ 2 2 4L 2 α ξ n αξ 2

(40.ben ) ,

which implies ¯ h ph ∈ L∞ (0, T ; l2 (ωh )) , ¯ h uh , ∇ ∇ ph ∈ L∞ (0, T ; l4 (ωh )) . Integrating the preceding result over (0, T ) again, we get Z

Tn 0

o αξ 2 ¯ h 2 21 ¯ h 2 h k ∇ k ∇ u ]| + ξ p ]| + (w (p ), 1) h h 0 h (t)dt ≤ 4L2 2 n αξ 2 ¯ h uh ]|2 + ξ 2 1 k∇ ¯ h ph ]|2 + ≤ k∇ 4L 2 o o n 2β 2 L2 1 C(Ω)T − 1 , exp + (w0 (ph ), 1)h (0) 2β 2 L2 α2 ξ 2 2 2 C(Ω)

(41.ben )

α ξ

which implies ¯ h ph ∈ L2 (0, T ; l2 (ωh )) ¯ h uh , ∇ ∇ p ∈ L2 (0, T ; l4 (ωh )) h

,

.

Extending these results into the continuum of Ω, we see that ∇Qh (Ph p0 ) and ∇Qh (Ph u0 ) are bounded in L2 (Ω) (by (31.ben )), and Sh (Ph p0 ) is bounded in L4 (Ω) (by (30.ben )). Therefore ∇Qh uh , ∇Qh ph ∈ L∞ (0, T ; L2 (Ω)) Sh ph ∈ L∞ (0, T ; L4 (Ω))

,

,

from which, ∇Qh uh , ∇Qh ph ∈ L2 (0, T ; L2(Ω)) Sh p ∈ L2 (0, T ; L4(Ω)) h

,

,

are bounded independently on h. Moreover, we obtain that Sh u˙ h , Sh p˙ h ∈ L2 (0, T ; L2 (Ω))

,

are bounded independently on h as follows from (39.ben ). We conclude that Qh uh , Qh ph ∈ L∞ (0, T ; H10 (Ω))

,

Qh uh , Qh ph ∈ L2 (0, T ; H10 (Ω))

,

are bounded independently on h. According to (28.ben ), Qh u˙ h , Qh p˙ h ∈ L2 (0, T ; L2(Ω)) Passing to a subsequence, we have

.

25


– – – – –

Qhn uhn , Qhn phn *∗ u, p in L∞ (0, T ; H10 (Ω)); Qhn uhn , Qhn phn * u, p in L2 (0, T ; H10 (Ω)); Shn p˙hn , Qhn p˙hn * ∂t u, ∂t p in L2 (0, T ; H−1 (Ω)); Shn u˙ hn , Qhn u˙ hn * ∂t u, ∂t p in L2 (0, T ; H−1 (Ω)); Shn uhn , Shn phn * u, p in L2 (0, T ; L2 (Ω)).

The non-linear terms in the equation (1.ben ) require stronger convergence result. Using the lemma on the compact imbedding, we conclude that Qhn phn converges ) implies the same result for Shn phn . strongly in L2 (0, T ; L2(Ω)). Relation (29.ben Denote their common limit as p and the weak limit of Shn p˙ hn in L2 (0, T ; L2 (Ω)) as q1 . The estimate kf0 (Sh ph )kL4/3 (Ω) ≤ i h1 3 ≤ a kSh ph kL4/3 (Ω) + k(Sh ph )2 kL4/3 (Ω) + k(Sh ph )3 kL4/3 (Ω) = 2 2 i h1 3 h , = a kSh p kL4/3 (Ω) + kSh ph k2L8/3 (Ω) + kSh ph k3L4 (Ω) 2 2

(42.ben )

justifies the existence of weak limit of f0 (Shn phn ) in L2 (0, T ; L4/3 (Ω)) denoted by q2 (dual space). These limits exist as a consequence of the a priori estimate and of (29.ben ), (28.ben ). We prove that q1 = ∂t p, q2 = f0 (p). First relation is implied by the uniqueness of the limit in D0 (0, T ), as Z T Z T hn hn ˙ (Shn p˙ − Qhn p˙ , q)ψ(t)dt = − (Shn phn − Qhn phn , q)ψ(t)dt , 0

0

where q ∈ D(Ω), ψ ∈ D(0, T ). The remaining equality is proven in the following lemma. Lemma 9. If p denotes the weak limit of Shn phn in L2 (0, T ; L2 (Ω)), then f0 (Shn phn ) → f0 (p) weakly in L 43 (0, T ; L 34 (Ω))

.

Proof. According to the compact imbedding, we have that Shn phn converges strongly in L2 (0, T ; L2 (Ω)) and it can be considered to converge a.e. in this space (see [9]). Furthermore, we observe that as Shn phn was bounded in L∞ (0,T ;L4 (Ω)) (see (42.ben ), f0 (Shn phn ) is bounded in L∞ (0, T ; L 34 (Ω)). These two facts together with the Aubin lemma [2] give the final result. t u Before proceeding in the proof, we show more about regularity of p. Lemma 10. Under the assumptions of the theorem, the function p belongs to H10 (Ω) ∩ H2 (Ω). Proof. Multiply the equation of phase by a function Phn q, where q ∈ D(Ω). ¯ hn p hn , ∇ ¯ hn Phn q] = αξ 2 (p˙hn , Phn q)h + ξ 2 (∇ = (f0 (phn ), Phn q)h − βξ(uhn , Phn q)h

.

(43.ben )

26

Michal Beneˇs

In terms of L2 (Ω), this means that αξ 2 (Shn p˙hn , Shn (Phn q)) + ξ 2 (∇(Qhn phn ), ∇Qhn (Phn q)) = = (f0 (Shn phn ), Shn (Phn q)) − βξ(Shn uhn , Shn Phn q)

. (44.ben )

n→∞

According to (32.ben ), we realize that Qhn (Phn q) → q in H10 (Ω), and similarly n→∞ )). We can pass to the limit in the sense of Shn (Phn q) → q in L2 (Ω) (see (30.ben D0 (0, T ) obtaining αξ 2 (∂t p, q) + ξ 2 (∇p, ∇q) = (q2 , q) − βξ(u, q) .

(45.ben )

Consequently, the function p is continuous from h0, T i into L2 (Ω). We rewrite the previous equality in the sense of D0 (Ω), αξ 2 ∂t p = ξ 2 ∆p + q2 − βξu

.

(46.ben )

Note that q2 = f0 (p) and p ∈ L∞ (0, T, Ls (Ω)) for any s > 1. Consequently, q2 ∈ L2 (0, T, L2 (Ω)). As ∂t p, q2 belong to L2 (Ω), this means that ∆p ∈ L2 (Ω) for each t ∈ (0, T ). Consequently, we find that p must be in the domain of ∆ — see [11], [2]: p(t) ∈ D(∆) = H2 (Ω) ∩ H10 (Ω) for t ∈ (0, T ) .

t u

Next statement investigates the convergence of gradient. Lemma 11. The sequence ∇Qhn phn converges strongly to ∇p in L2 ((0, T )×Ω). Proof. Following the technique of [12], the statement of the lemma is shown. Multiply the equation of phase in (35.ben ) by phn − Phn p and sum over ωh . ¯ hn p hn , ∇ ¯ hn (phn − Phn p)] = αξ 2 (p˙hn , phn − Phn p)h + ξ 2 (∇ = (f0 (phn ), phn − Phn p)h − βξ(uhn , phn − Phn p)h

.

(47.ben )

(uhn , Shn (phn − Phn p))dt.

(48.ben )

Rewrite this equality in terms of L2 (Ω), and integrate over (0, T ). Z αξ

T

(Shn p˙ hn , Shn (phn − Phn p))dt +

2 0

Z

T

(∇(Qhn phn ), ∇Qhn (phn − Phn p))dt =

+ ξ2 0

Z

T

(f0 (Shn phn ), Shn (phn − Phn p))dt −

= 0

Z − βξ 0

T

27


), it means that p(t) ∈ As we have shown that p ∈ L2 (0, T ; H2 (Ω)) satisfies (45.ben C 0,1 (Ω), t ∈ (0, T ), and consequently, Shn (Phn p) → p, and ∇Qhn (Phn p) → ∇p in L2 (0, T ; L2 (Ω)) (see (30.ben ), (31.ben )). We add and subtract a term Z T 2 (∇(Qhn (Phn p)), ∇Qhn (phn − Phn p))dt ξ 0

to the equality (48.ben ) knowing that it tends to 0 as ∇Qhn (phn − Phn p) → 0 , weakly in L2 (0, T ; L2 (Ω)), if n → ∞. Then, we have Z ξ

T

(∇(Qhn phn − Phn p), ∇Qhn (phn − Phn p))dt =

2 0

Z = − αξ Z

T

(Shn p˙ hn , Shn (phn − Phn p))dt +

2 0

T

(f0 (Shn phn ), Shn (phn − Phn p))dt +

+ 0

Z

T

(uhn , Shn (phn − Phn p))dt +

− βξ 0 Z T

(∇(Qhn (Phn p)), ∇Qhn (phn − Phn p))dt

+ ξ2

. (49.ben )

0

As all terms in the right hand side tend to 0 if n → ∞, we see that ∇(Qhn (phn − ) gives the desired result. Phn p)) → 0 in L2 (0, T ; L2 (Ω)), which together with (32.ben t u Passage to the limit. Take the system of (35.ben ) into the consideration, multiply by test functions Phn w, Phn q where w, q ∈ D(Ω). Integrate it over ωh . Then, we have, in terms of L2 (Ω), (Shn u˙ hn , Shn Phn w) + (∇Qhn uhn , ∇Qhn Phn w) = L(Shn p˙ hn , Shn Phn w) 2

αξ (Shn p˙

hn

, Shn Phn q) + ξ (∇Qhn p 2

= (f0 (Shn p

hn

hn

,

, ∇Qhn Phn q) =

), Shn Phn q) − βξ(Shn uhn , Shn Phn q)

. (50.ben )

Knowing that 1. Shn p˙hn , Shn u˙ hn converge weakly in L2 (0, T ; L2 (Ω)) to ∂t p, ∂t u; 2. ∇Qhn phn , ∇Qhn uhn converge strongly in L2 (0, T ; L2(Ω)) to ∇p, ∇u; 3. Shn Phn p0 , Shn Phn u0 converges strongly to p0 , u0 in H10 (Ω), multiply (50.ben ) by a scalar function ψ(t) ∈ C 1 h0, T i, for which ψ(T ) = 0. We integrate by parts. Taking into account all previous results, the fact that Shn phn (0) = Shn Phn p0 ,

Shn uhn (0) = Shn Phn u0

28

Michal Beneˇs

and the Lebesgue theorem, we are able to pass to the limit. Z

Z

T

0

Z

T

˙ + (u − Lp, w)ψdt

(u0 − Lp0 , w)ψ(0) −

Z

T

˙ + αξ 2 (p, q)ψdt

αξ 2 (p0 , q)ψ(0) − 0

ψ[(∇u, ∇w) = 0

,

0 T

ψ[ξ 2 (∇p, ∇q) − 0

− (f0 (p), q) + βξ(u, q)]dt = 0

.

(51.ben )

If ψ ∈ D(0, T ), we have d (u − Lp, w) + (∇u, ∇w) = 0 , dt αξ 2

(52.ben )

d (p, q) + ξ 2 (∇p, ∇q) = (f0 (p), q) − βξ(u, q) . dt

It remains to show that the weak solution satisfies the initial condition. Multiplying (51.ben ) by a scalar function ψ(t) ∈ C 1 h0, T i, for which ψ(T ) = 0, and integrating by parts, we obtain Z Z

Z

T

T

˙ + (u − Lp, w)ψdt

(u(0) − Lp(0), w)ψ(0) − 0

ψ[(∇u, ∇w) = 0

,

0

T

˙ + αξ 2 (p, q)ψdt

αξ (p(0), q)ψ(0) − 2

0

Z

T

ψ[ξ 2 (∇p, ∇q) − (f0 (p), q) + βξ(u, q)]dt = 0

+

.

(53.ben )

0

Subtracting this equation from (51.ben ), we get (u0 − Lp0 − u(0) + Lp(0), w)ψ(0) = 0,

(p0 − p(0), q)ψ(0) = 0 .

From this we see that u(0) = u0 , p(0) = p0 in L2 (Ω). To prove uniqueness, consider two solutions of the problem (34.ben ), denoted as [u, p] and [v, q]. Subtracting two systems of equations and denoting [w, r] = [u − v, p − q], multiplying the first equation by w and the second equation by r˙ via the semi-discrete scheme, we have 1 d kwk2 + (∇w, ∇w) = (r, ˙ w) in (0, T ) , 2 dt w(0) = 0 , 1 d (∇r, ∇r) = (f0 (p) − f0 (q), r) αξ 2 krk ˙ 2 + ξ2 ˙ − βξ(w, r) ˙ 2 dt r(0) = 0 .

(54.ben )

in (0, T ) , (55.ben )

Denote Ψ (p, q) = − 12 a+ 23 a(p+q)−a(p2 +pq+q 2 ). The existence proof guarantees that there is a constant C˜ such that kΨ (p, q)kL4 (Ω) ≤ C˜

,

29


(as implied by the continuous imbedding H10 (Ω) ⊂> Lq (Ω) for q ∈< 0, +∞)). Therefore, we have ˜ ˙ L2 (Ω) ≤ Ckrk ˙ L2 (Ω) |(Ψ (p, q)r, r)| ˙ ≤ kΨ (p, q)kL4 (Ω) krkL4 (Ω) krk L4 (Ω) krk

.

Using the Poincaré and Schwarz inequalities, we get C(Ω) 1 d kwk2 ≤ krk ˙ 2 2 dt 4 1 d ˜ k∇rk2 ≤ Ckrk αξ 2 krk ˙ 2 + ξ2 ˙ L2 (Ω) + L4 (Ω) krk 2 dt 1 β kwk2 + αξ 2 krk ˙ 2+ 2 2αξ

,

(56.ben )

.

(57.ben )

or, considering the fact, that there is a constant C4 > 0 such that krkL4 (Ω) ≤ C4 k∇rk

,

we obtain C(Ω) 1 d kwk2 ≤ krk ˙ 2 2 dt 4 1 C˜ 2 1 d k∇rk2 ≤ αξ 2 krk αξ 2 krk ˙ 2 + ξ2 ˙ 2 + 2 C42 k∇rk2 + 2 dt 4 αξ β 1 kwk2 ˙ 2+ + αξ 2 krk 2 2αξ Combining these inequalities, we have β 1 d 1 d 1 C˜ 2 2 αξ 2 kwk2 + ξ 2 k∇rk2 ≤ kwk2 C k∇rk2 + C(Ω) 2 dt 2 dt αξ 2 4 2αξ

.

Such inequality implies, together with the initial conditions, that r(t) = w(t) = 0

∀t ∈ (0, T ) in L2 (Ω) . t u

4

Convergence towards the sharp interface model

This paragraph deals with the relation of the phase-field model to a sharpinterface formulation of the Stefan problem. It uses estimates derived above to show certain compactness statements leading to the existence of a step function defining the position of solid domain in time. Consider the weak formulation of the standard phase-field model. d d (u, v) + (∇u, ∇v) = L (p, v) dt dt u(0) = u0 , αξ 2

in (0, T ) ,

d (p, q) + ξ 2 (∇p, ∇q) = (f0 (p), q) − βξ(u, q) in dt p(0) = p0 .

(58.ben )

(0, T ) , (59.ben )

30

Michal Beneˇs

Main purpose of next investigation will be the dependence on ξ. Consider the solution of the semidiscrete scheme (35.ben ). We multiply the first equation by uh and the second one by p˙h . 1 d h 2 ¯ h uh ]|2 = L(p˙ h , uh )h , ku kh + k∇ 2 dt d 1 d ¯ h 2 k∇h p ]| = − (w0 (ph ), 1)h − βξ(uh , p˙ h )h αξ 2 kp˙ h k2h + ξ 2 2 dt dt

(60.ben ) .

(61.ben )

Combining previous equalities, we get αξ 2 kp˙ h k2h + ξ 2

1 d ¯ h 2 k∇h p ]| = 2 dt o d βξ n 1 d h 2 ¯ h uh ]|2 ku kh + k∇ = − (w0 (ph ), 1)h − dt L 2 dt

,

(62.ben )

or αξ 2 kp˙ h k2h + ξ 2

d βξ 1 d h 2 1 d ¯ h 2 k∇h p ]| + (w0 (ph ), 1)h + ku kh = 0. 2 dt dt L 2 dt

We integrate over (0, t), which gives n

βξ 1 h 2 o 1 ¯ h 2 h ku kh (t) ≤ ξ 2 k∇ h p ]| + (w0 (p ), 1)h + 2 L 2 n 1 o ¯ h ph ]|2 + (w0 (ph ), 1)h + βξ 1 kuh k2h (0) ≤ ξ 2 k∇ 2 L 2

. (63.ben )

Passing to the limit, if h → 0, which is justified by the proof of the Theorem 8, we get 1β 1β kuξ (t)k2 + Eξ [pξ ](t) ≤ kuξ (0)k2 + Eξ [pξ ](0) t ∈ (0, T ) , 2L 2L

(64.ben )

h→0

where we denoted (ph −−−→ pξ ), Z 1 1 Eξ [pξ ](t) = [ξ |∇pξ |2E + w0 (pξ )]dx ξ Ω 2

.

Additionally, there is an estimate for the time derivative, if we integrate (62.ben ) over (0, T ) and pass to the limit for h → 0. Z

T

k∂t pξ k2 dt + Eξ [pξ ](T ) − Eξ [pξ ](0) +

αξ 0

β (ku(T )k2 − ku(0)k2 ) = 0 (65.ben ) 2L

Consequently, there is a constant C1 such that 1 αξ 2

Z

T

k∂t pk2h dt + Eξ [pξ ](T ) ≤ Eξ [pξ ](0) + C1 0

.

(66.ben )

31


These estimates allow to use the method proposed by [4] and used in [2]. Define the following monotone function Z s |1 − (1 − 2r)2 |dr . (67.ben ) G(s) = 0

We prove next lemma ) where Eξ [pξ ](0) ≤ M0 independently on Lemma 12. Be pξ the solution of (34.ben ξ. Then there are constants M > 0 and M1 > 0 such that Z (68.ben ) sup{ |∇G(pξ )|dx | t ∈ h0, T i} ≤ M Ω

and, for 0 ≤ t1 < t2 , Z

t2

Z

t1

|∂t G(pξ )|dxdt ≤ M1 (t2 − t1 )0.5

.

(69.ben )

Ω

Proof. We have shown that Eξ [pξ ](t) ≤ M0 + C1

,

on h0, T i. We write Z Eξ [p](t) =

1 1 [ξ |∇p|2E + w0 (p)]dx ≥ ξ Ω 2 Z √ q √ Z 2|∇pξ | w0 (pξ )dx = 2 |∇G(pξ )|E dx ≥ Ω √1 (M0 2

which shows (68.ben ) by setting M = Z

Z

t2

dt t1

Z Z ≤

Z

t2

(70.ben )

,

(71.ben )

+ C1 ). Furthermore, if

dx|p˙ξ ||G0 (pξ )| ≤

dt t1

dx|p˙ξ |

dt t1

Z

t2

dx|∂t G(pξ )| = Ω

,

Ω

Ω

12 Z

dt

Ω

t1

q then (69.ben ) is shown, if setting M1 =

Z

t2

2

0

dx|G (pξ )|

12 2

≤

Ω

1 1 2 ≤ ( (C1 + M0 )2 ) 2 (t2 − t1 ) 2 α

2 α (C1

+ M0 ).

t u

The previous statement leads to the existence of a step function as expected. ) with the initial data Theorem 13. Let uξ , pξ is the solution of the problem (34.ben satisfying Eξ [pξ ](0) < M0 and uξ , pξ ∈ H2 (Ω) ∩ H1 (Ω), and let Z |pξ (0, x) − v0 (x)|dx → 0, kuξ (0)kL2 (Ω) ≤ C2 , Ω

32

Michal Beneˇs

as ξ → 0, for a function v0 ∈ L1 (Ω). Then for any sequence tending to 0 there is a subsequence ξn0 such that uξn0 (t, x) * u(t, x) in L2 ((0, T ) × Ω),

lim pξn0 (t, x) = v(t, x),

ξn0 →0

and u, v are defined a.e. in (0, T ) × Ω. The function v reaches values 0 and 1, and satisfies Z 1 |v(t1 , x) − v(t2 , x)|dx ≤ C|t2 − t1 | 2 , Ω

where C > 0 is a constant, and Z |∇v|E dx ≤ C1

sup t∈h0,T i

,

Ω

in the sense of BV (Ω), where C1 > 0 is a constant. The initial condition is lim v(t, x) = v0 (x)

t→0

,

a.e. Proof. The proof follows steps presented in [4]. We find that 4 G(s) = 2s2 − s3 for s ∈ h0, 1i , 3 4 4 3 for s ∈ (1, +∞) . G(s) = s − 2s2 + 3 3 Consequently, a direct computation justifies that |G(s)| ≤

4 + [1 − (1 − 2s)2 ]2 3

.

Then, we are able to obtain the upper bounds for the function G and its spatiotemporal gradient. According to (64.ben ), we have Z TZ w0 (p)dxdt ≤ M2 ξ . (72.ben ) 0

Ω

Putting (72.ben ), (68.ben ) and (69.ben ) together, we conclude, that G(pξ ) is in BV ((0, T )×Ω) regardless the value ξ > 0. Following [6], BV ((0, T ) × Ω) ⊂>⊂> L1 ((0, T ) × Ω)

.

Consequently, there is a sequence G(pξn ) converging to an element G∗ in the space L1 ((0, T ) × Ω). According to [9], there is a further subsequence G(pξn0 ) converging to G∗ almost everywhere in (0, T ) × Ω. The function G :< 0, +∞) →< 0, +∞) is monotone, which implies existence of the unique function v such that G∗ = G(v)

,

33


and pξn0 → v

a.e. in (0, T ) × Ω

According to (72.ben ) and by the Fatou lemma, we obtain Z

T

Z w0 (v)dxdt = 0

0

,

(73.ben )

Ω

from which follows that the function p takes only the values 0, 1. Now, we prove that G is Hölder-continuous in the time variable. The function pξn0 satisfies Z |G(pξn0 (t1 , x)) − G(pξn0 (t2 , x))| ≤

t2

t1

|∂t G(pξn0 (t, x))dt

for 0 ≤ t1 ≤ t2 ≤ T . Integrating over Ω, Z |G(pξn0 (t1 , x)) − G(pξn0 (t2 , x))|dx ≤ M1 |t1 − t2 |0.5

,

,

(74.ben )

Ω

according to the Lemma 12. Passing to the limit for n0 → ∞, we find Z |G∗ (t1 , x) − G∗ (t2 , x)|dx ≤ M1 |t1 − t2 |0.5 , Ω

for almost all t1 , t2 ∈ (0, T ). The statement of theorem is obtained by the fact that |G∗ (t1 , x) − G∗ (t2 , x)| = (G(1) − G(0))|v(t1 , x) − v(t2 , x)| . The a.e. argument makes from the function v a continuous map from h0, T i to ) and according to the assumption L1 (Ω). Taking t1 = 0 in (74.ben Z |pξ (0, x) − v0 (x)|dx → 0 , Ω

as ξ → 0, (similarly for G-valued function) we have Z |G(v0 (x)) − G(v(t2 , x))|dx ≤ M1 t0.5 2

,

Ω

from which

Z |v0 (x) − v(t2 , x)|dx ≤ Ω

M1 t0.5 G(1) − G(0) 2

.

This concludes the proof. It remains to show the boundedness of the total variation of v ([6]). The lower semicontinuity of the total variation in L1 -space together with the Lemma 12 yields Z ess sup |∇G∗ |dx ≤ M0 , 0 0, t > 0 + ∂t ∂r2 r ∂r on boundary conditions ∂T , ∂r r=0

T |r=R = S+ (t)

and zero initial condition. Solution of this problem is (λ2n + χ2 )t J (λ r) ∞ 1 − 1 X 1 n 2 λn . T (t, r) = S+ (t) 2 2 2 2 R λ +χ [J0 (λn R) + J1 (λn R)] n=1 n The stationary state is described by function TCT. =

∞ I0 (χr) 2 X λn J1 (λn R)J0 (λn r) = ≡ S2 (r, χ), 2 2 2 2 R n=1 λn + χ J0 (λn R) + J1 (λn R) I0 (χR)

where Iν (x) – modify Bessel function of the first kind of order ν. Notice that S2 (r, χ) is finite on [0, R] solution of the boundary problem 2 1 d dS2 d 2 − χ S2 = 0, + , S2 |r=R = 1. dr2 r dr dr r=0 Problem 3. Let us consider the finite thin plate Π1 = {r : r ∈ (0, R1 ) ∪ (R1 , R2 ), R2 < ∞} with continuous heat characteristics on the first part and linear heat characteristics on the second part. The structure problem of non-stationary heat field in this plate lead to mathematical construction in region D1 = {(t, r) : t > 0, r ∈ Π1 } of finite solution for separate system of equations ∂ 2 T1 1 ∂T1 2 + χ T − = 0, t > 0, r ∈ (0, R1 ), 1 1 a21 ∂t ∂r2 2 1 ∂ 1 ∂T2 ∂ 2 + χ T2 = 0, χ2 > 0, t > 0, r ∈ (R1 , R2 ) T − + 2 2 a22 ∂t ∂r2 r ∂r on zero initial condition, boundary conditions T1 |r=0 = S+ (t),

∂T2 =0 ∂r r=R2

40

Andriy Blazhievskiy

and conditions of non-ideal heat contact ∂ + 1 T1 − T2 = 0, R0 ∂r r=R1 ∂T2 ∂T1 − γ1 = 0. ∂r ∂r r=R1

Solution of this problem is T1 (t, r) = S+ (t)

2 ∞ X 2 2 2 β1n β2n ω12 (λn ) sin β1n r 1 − 1−(λn + a2 χ2 )t , 2 2 2 2 (λn + a2 χ2 )kV (r, λn )k n=1

T2 (t, r) = S+ (t)

2 ∞ γ1 X β1n β2n ω1 (λn ) −(λ2n + a22 χ22 )t × 1 − 1 R1 n=1 (λ2n + a22 χ22 )kV (r, λn )k2

Notice that β1n

× [ω3 (λn )N0 (β2n r) − ω2 (λn )J0 (β2n r)]. p = λ2n + a22 χ22 − a21 χ21 , β2n = λn , βjn = a−1 j βjn , j = 1, 2,

ω1 (λn ) = J1 (β2n R2 )N0 (β2n R1 ) − N1 (β2n R2 )J0 (β2n R1 ), ω2 (λn ) = (R0 β1n cos β1n R1 + sin β1n R1 )N1 (β2n R2 ),

If

(a21 χ21

ω3 (λn ) = (R0 β1n R1 + sin β1n R1 )J1 (β2n R2 ). p − a22 χ22 ) ≥ 0 then β1n = λn , β2n = λ2n + a21 χ21 − a22 χ22 . In this case

T1,CT. (t, r) =

2 ∞ X ∆1 (χ sh χ1 r β1n β2n ω12 (λn ) sin β1n r = ch χ1 r + = S3 (r, χ), 2 2 2 2 (λn + a2 χ2 )kV (r, λn )k ∆(χ) χ1 n=1

2 ∞ γ1 X β1n β2n ω1 (λn )[ω3 (λn )N0 (β2n r) − ω2 (λn )J0 (β2n r)] = T2,CT. (t, r) = R1 n=1 (λ2n + a22 χ22 )|V (r, λn )k2

=−

∆(χ)=γ1 χ22

χ2 [K1 (χ2 R2 )I0 (χ2 r) + I1 (χ2 R2 )K0 (χ2 r)] ≡S4 (r, χ), ∆(χ) χ={χ1 , χ2 }, χj =aj χj ,

shχ1 R1 R0 chχ1 R1 + (I1 (χ2 R1 )K1 (χ2 R2 )−I1 (χ2 R2 )K1 (χ2 R1 )− χ1 − χ2 ch χ1 R1 (I0 (χ2 R1 )K1 (χ2 R2 ) + I1 (χ2 R2 )K0 (χ2 R1 )) ,

∆1 (χ) = −ν1 χ22 (R0 χ1 sh χ1 R1 + ch χ1 R1 )(I1 (χ2 R1 )K1 (χ2 R2 ) − I1 (χ2 R2 ) × × K1 (χ1 R1 )) + χ1 χ2 ch χ1 R1 (I0 (χ2 R1 )K1 (χ2 R2 ) − I1 (χ2 R2 )K0 (χ2 R1 )), kV (r, λn )k2 =

ω12 (λn ) 2 sin 2β1n R1 2 R12 (R1 − ω4 (λn ), + − 2 2 β1n (πβ1n )2

ω4 (λn ) = ω22 (λn )(J0 (β2n R1 ) + J1 (β2n R1 )) − 2ω2 (λn )ω3 (λn )[J0 (β2n R2 ) × × N0 (β2n R1 ) + J1 (β2n R1 )N1 (β2n R1 )] + ω32 (λn )[N02 (β2n R1 ) + N12 (β2n R1 )].

Summation of Polyparametrical Functional Series

41

Notice that S3 (r, χ) and S4 (r, χ) is finite on Π1 solution of the boundary problem 2 d 2 S3 (r, χ) = 0, r ∈ (0, R1 ), − χ 1 dr2 2 1 d d 2 − S4 (r, χ) = 0, χ2 > 0, r ∈ (R1 , R2 ) + χ 2 dr2 r dr on boundary conditions S3 |r=0 = 1, and contact conditions

dS4 =0 dr r=R2

d + 1 S3 − S4 R0 = 0, dr r=R1 dS4 dS3 − γ1 = 0. dr dr r=R1

My research is devoted to the summation of just such series by the method of finite hybrid integral transforms. By the Cauchy’s method for the separate systems of the ordinary differential equations we have constructed the solution of the corresponding boundary problem in the case of general assumption on the differential and connected operators. The condition of the non-limited solving and the structure of the general solution for the boundary problem have been written in the explicit form. On the other hand solution of this problems has been constructed by the method of the finite hybrid integral transforms. Since this problems has one and only one solution we may compare the first solution with the second and, as a result, get the sums of functional series.

References [1] A. Blazhievskiy, Summation of polyparametrical functional series by the method of finite hybrid integral transforms, Thesis for a degree of Doctor of Philosophy (Ph.D.) in Physics and Mathematics, Chernivtsy, Ukraine 1996.


PAPERS pp. 43–52

System of Differential Equations with Unstable Turning Point and Multiple Element of Spectrum of Degenerate Operator V. N. Bobochko1 and I. I. Markush2 1

St. Gerojev Stalingrada, 12-1-101, 316 031 Kirovograd, Ukraine Email: [email protected] 2 Square Theatralnaya 17/7, 294000 Uzhgorod, Ukraine Email: [email protected]

Abstract. A uniform asymptotics of a solution is constructed for a system of singularly perturbed differential equations with a turning point. The paper investigates the case when the spectrum of a phase operator consists of multiple elements. AMS Subject Classification. 34B05, 34E05, 34E20 Keywords. Operator, spectrum, turning point, differential equation, parameter, system, singular solution, manifolds, asymptotic

1

Introduction

The systems of singular perturbation differential equations (SSPDE) play the great role in many mathematical models of biological problems and in medicine. This can be seen from monograph [1] in which kinetic rule of Michaelis-Menten is described by Tikhonov SSPDE (see equations (1.20), (1.21)). One of the known problems for investigation such problems is as follows: Is there a biological process stable or not? If the spectrum of the degenerate operator is stable, then such a system is known pretty well. Remembering classical results (Vasilieva’s method, Lomov’s method and others) which give the answer of the discussed problems concerning bounded solutions of the adequate SSPDE. If the system contains turning points, i.e. some elements of the spectrum of the degenerate operator are unstable, then the general theory of investigating such problems is not constructed yet, however some special problems have the solution (see [2,3,4]). In the presented article we consider the problem:  Lε W (x, ε) ≡ ε2 W 00 (x, ε) − AW (x, ε) = h(x) ,     −2 c E1 W (m, ε) = E1 (µ αm. + Wm ) , (1.mar )   dW (m, ε)  −4 −3 c  = E2 (µ αm. + µ Wm ) , E2 dx This is the final form of the paper.

44

V. N. Bobochko and I. I. Markush

where ε → +0,

x ∈ I = [0, 1],

m = 0, 1,

µ=

√ 3

ε.

cm are given vectors, Ek Here A denotes a linear operator on Rn , αm and W (k = 1, 2) — diagonal matrices of the nth order of the form E1 = diag{1, 0, . . . , 0}, E2 = diag{0, 1, . . . , 1), h(x) — given vector-function, W (x, ε) — a sought vectorfunction. We shall consider the problem (1.mar ) under the following conditions: 1◦ A(x), h(x) ∈ C ∞ [I] 2◦ Spectrum of the degenerate operator A is real and fulfills the following condition e1 (x) < λ2 (x) < · · · < λp. (x) ≡ · · · ≡ λn (x), 0 ≤ λ1 (x) ≡ xλ

(2.mar )

e1 (x) > 0 for all x ∈ I. where λ One can see from (2.mar ) that the point x = 0 is a turning point for equation (1.mar ), and moreover of λ1 (x) ≥ 0, then it is unstable turning point. Thus it is necessary to construct a solution of (1.mar ) in the case when the simplified equation −Aω(x) = h(x)

(3.mar )

has, in general case, a point of discontinuity at 0. System (1.mar ) with several conditions for the spectrum of degenerate operator A was consider in [2,3,4,5] and in other articles of authors. In the present article we shall prove, that ignoring the unstability of the turning point some partial solutions of the vector equation (1.mar ) can be bounded in the domain.

2

Extension of the perturbation problem

One of the main problem of asymptotics of solution of singular perturbed problem (SPP) (1.mar ) is: choice, description and conserve them as a whole all essentially singular manifolds (ESM) contained in the solutions SPP (1.mar ). For this purpose, together with the independent variable x, we shall consider vector-variable  t = {tik } , i = 1, p , k = 1, 2 according to    Z x 23   p  3 −2 −2  λ1 (x)dx ≡ µ ϕ1 (x) ≡ φ1 (x, ε),  t1k ≡ t1 = µ  2 0 (4.mar ) Z x q   −3 k −3  tjk = µ (−1) λj (x)dx ≡ µ ϕjk (x) ≡ φjk (x, ε),    (k−1)l    k = 1, 2, j = 2, p

45

System of Differential Equations

Then instead of vector-function W (x, ε) we shall consider a new “extended” vecf (x, t, ε), where in view of the regularization method (see [5]), the tor-function W extension is taken in such a way that f (x, t, ε) W ≡ W (x, ε), (5.mar ) t=φ(x,ε)

where φ(x, ε) = {φ1 (x, ε), φjk (x, e), k = 1, 2; j = 2, p} . Differentiating twice the identity (5.mar ) and substituting the derivative of the f f (x, t, ε) second order of W (x, t, ε) to equation (1.mar ), we obtain, for the extension W the following “extended” problem:  eεW f (x, t, ε) = h(x) , L Mm = (m, t(m)) ,     −2 c f E1 W (Mm , ε) = E1 (µ αm + Wm ) , (6.mar )   dW (Mm , ε)   cm ) . = E2 (µ−4 αm + µ−3 W E2 dx Here ∂2 ∂ ∂2 e ε ≡ µ2 ϕ02 L + µ4 d1 + µ6 2 − 1 (x) 2 ∂t1 ∂t1 ∂x −A+

p 2 X X k=1 j=2

[ϕ02 jk (x)

∂ ∂ + µ3 djk ] + Yε1 , ∂tjk ∂tjk

(7.mar )

where djk ≡ 2ϕ0jk (x)

∂ + ϕ00jk (x) . ∂x

(8.mar )

Yε1 — operator that plays the role of annihilator. Then there is no idea to give its full form.

3

Spaces of nonresonance solutions

We shall describe the sets (subspaces) of functions in which we shall solve the extended problem (6.mar ). We have  Yrilk = {bi (x)[Vrik (x)Uk (t1 ) + Qrik (x)Uk0 (t1 )]} ,      Yrijk = {bi (x)αrijk (x) exp(tjk )}, j = 2, p , (9.mar )  Vri = {bi (x)[fri (x)ν(t1 ) + gri (x)ν 0 (t1 )]} ,     X = {b (x)ω (x)} , i = 1, n; k = 1, 2, ri

i

ri

46


where the coefficients in ESM and the functions ωri (x) are arbitrary sufficiently smooth functions for x in I, and bi (x) (i = 1, n) — a complete system of proper vectors for proper values λi (x) (i = 1, p). Since the operator A is operator of a simple structure, then for a multiple proper value λp (x) ≡ · · · ≡ λn (x) there exists a system of linearly independent vectors (fundamental solutions) bi (x) (i = p, n). By b∗i (x) (i = 1, n) we shall denote the complete system of proper vectors of the conjugate operator A∗ , where those vectors are chosen in such a way that together the vectors bi (x) they form a biorthogonal system of vectors. The existence of such a system is proved since A is an operator of a simple structure (see [6, p. 218]). Later on U1 (t1 ) ≡ Ai(t1 ), U2 (t1 ) ≡ Bi(t1 ) — are the Eiry-functions, which properties are described in monograph [7, chapter 1.1]. Essentially singular manifolds ν(t1 ) ≡ −Gi (t1 ) — are Skorera functions (see [7, p. 412]). From the subspaces (9.mar ) we shall construct the space of the form: Yr =

n M

Yri ≡

i=1

p n M 2 M M i=1

Yrijk ⊕ Vri ⊕ Xri ,

(10.mar )

k=1 j=1

which in view of the known terminology [5] will be called the space of nonresonance solutions (SNS). The element Wr (x, t) ∈ Yr has the form Wr (x, t) =

n X i=1

bi (x)Wri (x, t) ≡

n X

fri (x, t) , W

(11.mar )

i=1

where Wri (x, t) ≡

2 h X

Vrik (x)Uk (t1 ) + Qrik (x)Uk0 (t1 ) +

k=1

+

p X

i αrijk (x) exp(ttj ) + fri (x)ν(t1 ) + gri (x)ν 0 (t1 ) + ωri (x) .

j=2

4

Regularization of the singular perturbed problem

e ε on elements from We have to find the properties of the extended operator L SNS (10.mar ). Method of obtaining such procedure is described in articles [2,3,4,5]. That is why we shall write only the final result of that property. We have: e ε Wr (x, t) ≡ [R0 + µ2 R2 + µ3 R3 + µ4 R4 + µ6 R6 ]Wr (x, t) . L

(12.mar )

47


Operators Rs can be written in the form R0 Wr (x, t) ≡

n X

2 n hX bi (x) (λ1 − λi ) [Vrik (x)Uk (t1 ) + Qrik (x)Uk0 (t1 )] +

i=1

k=1

i + fri (x)ν(t1 ) + gri (x)ν 0 (t1 ) + +

p X

(λj − λi )

j=2

R2 Wr (x, t) ≡

n X

o αrijk (x) exp(tjk ) − λi (x)ωri (x) ,

(13.mar )

k=1

n i nhX e i1 × bi (x) ϕ1 · Tsi1 + D s=1

i=1

×

2 X

2 hX

i o Qrik (x)Uk (t1 ) + gri (x)ν(t1 ) − π −1 ϕ02 (x)f (x) , ri 1

(14.mar )

k=1

R3 Wr (x, t) ≡

n X

p X 2 h n nX i o X Dijk αrijk (x) + bi (x) Tsijk αrsjk (x) exp(tjk ) ,

i=1

j=2 k=1

s6=i s=1

(15.mar ) R4 Wr (x, t) ≡

n X i=1

n nhX

bi (x)

i Tsi1 + Di1 ·

s6=i

2 hX

Vrik (x)Uk0 (t1 ) +

k=1

s=1

io + fri (x)ν 0 (t1 ) + π −1 gri (x) , R6 Wr (x, t) ≡

(16.mar )

fr (x, t) ∂2W . ∂x2

(17.mar )

Here Dijk ≡ 2ϕ0jk (x)

∂ + (b0i (x), b∗i (x)) , ∂x

e i1 ≡ ϕ1 (x)Di1 + ϕ02 (x) ≡ ϕ1 (x) · 2ϕ0 (x) D 1 1 Tsi1 ≡ 2ϕ01 (x)(b0s (x), b∗i (x)),

∂ + (b0i (x), b∗i (x)) + ϕ02 1 (x) , ∂x (18.mar )

Tsijk ≡ 2ϕ0jk (x)(b0s (x), b∗i (x)) .

Analyzing the obtained identities we can make the following implications. 1. Spaces of nonresonance solutions Yr are invariant with respect to operators eε R0 , R2 , R3 , R6 and consequently with respect to the extended operator L which is represented in the form (12.mar ). e ε in SNS (10.mar 2. Operator R0 is a main operator of the extended operator L ). 3. Extended problem (6.mar ) depends regularly on a small parameter µ > 0 in SNS (10.mar ).

48

5


Formalism of construction of solution of extended problem

Since the extended problem (6.mar ) is regularly dependent on a small parameter µ > 0 in SNS (10.mar ), then the asymptotic solution of that problem can be found in the form of a series f (x, t, ε) = W

+∞ X

µr Wr (x, t) ,

(19.mar )

r=−2

where

Wr (x, t) ∈ Yr .

Let us substitute (19.mar ) to the extended problem (6.mar ) and compare the coefficients by the parameter µ > 0. Then for defining of the coefficients of the series (19.mar ) we shall get the following recurrence system of problems: R0 W−2 (x, t) = 0, Gm W−2 (x, t) ≡ E2

E1 W−2 (Mm ) = E1 αm , p 2 X X k=1 j=2

ϕ0jk (m)

∂W−2 (Mm ) = 0, ∂tjk

(20.mar )

R0 W−1 (x, t) = 0,

E1 W−1 (Mm ) = 0 , h ∂W−2 (Mm ) i , Gm W−1 (x, t) = E2 αm − ϕ01 (m) ∂t1

(21.mar )

cm , R0 W0 (x, t) = h(x) − R2 W−2 , E1 W0 (Mm ) = E1 W h i cm − ϕ0 (m) ∂W−1 (Mm ) , Gm W0 (x, t) = E2 W 1 ∂t1

(22.mar )

In this way we obtain a series of recurrence problems with point boundary conditions which is in the general case insufficient for uniqueness of solutions of each of those separate problems. However, further considerations show that in SNS (10.mar ) the series of problems (20.mar )–(22.mar ) is asymptotically correct, i.e. each of the problems (20.mar )–(22.mar ) has the unique solution in SNS if we consider those problems step by step. Further on we should ask a question of the existence of a solution in SNS (10.mar ) of the iteration equation R0 Wr (x, t) = Hr (x, t) .

(23.mar )

From identity (13.mar ) one can describe the structure of the kernel of the operator R0 . We have KerR0 = bj (x)αrjjk (x) exp(tjk ), j = 2, p − 1, bj (x)αrjjk (x) exp(tpk ), j = p, n , b1 (x)[Vr1k (x)Uk (t1 ) + Qr1k (x)Uk0 (t1 )], b1 (x)[fr1 (x)ν(t1 ) + gr1 (x)ν 0 (t1 )], k = 1, 2

(24.mar )

49


We introduce the following notations: bi (x)Sri (x), i = 1, n projection of the vector-function Hr (x, t) to the subspace Xri . Then we have the following Theorem 1. For equation (1.mar ) let a) the conditions 1◦ and 2◦ hold, b) right side of the equation (23.mar ) belongs to SNS (10.mar ) and contains no element of the kernel of the operator R0 , c) Sr1 (0) = 0. Then there exists a solution of (23.mar ) in the space Yr and it can be represented in the form Wr (x, t) = Zr (x, t) + yr (x, t) .

(25.mar )

Here 2 hX Zr (x, t) = b1 (x) [Vr1k (x)Uk (t1 ) + Qr1k (x)Uk0 (t1 )] + k=1 2 X n i X bj (x)αrjjk (x) exp(tjk ) , + fr1 (x)ν(t1 ) + gr1 (x)ν 0 (t1 ) +

(26.mar )

k=1 j=2

where the coefficients of ESM are arbitrary of the sufficiently smooth functions with x ∈ I and yr (x, t) uniquely defined and sufficiently smooth functions for all x ∈ I, in particular for x = 0. Remark 2. Since the point x = 0 is unstable, then ESM U2 (t1 ) ≡ Bi(t1 ) and its derivative is unboundedly increasing when t1 → +∞. However in spite of that in ESM the ν(t) and ν 0 (t) contain unboundedly increasing functions Bi(t1 ) and B 0 i(t1 ), they all are still bounded functions for t ≥ 0 i.e. for t ∈ [0, µ−2 ϕ1 (1)]. Because of the shortness of the article we are not able to construct the full solution of at least three equations (20.mar )–(22.mar ). We are bound to give the only remark. Remark 3. Coefficients αrjjk (x), j = 2, p − 1 derived from the simple elements λj (x) can be defined by scalar linear differential equations of the first order. Multiplicity of the element λp (x) ≡ · · · ≡ λn (x) introduces in the construction of asymptotic solutions of the extended problem (6.mar ) the following changes. Coefficients αrjjk (x) (j = p, n) derived from multiple elements of the spectrum can be determined by the system of (n − p + 1) differential equations.

50

6


Asymptotic correctness of iterated problems

Applying Theorem 1 consequently we can obtain solutions of the iterated equations (20.mar )–(22.mar ) and so on. Each of those solutions contains 2n arbitrary constants which were obtained by integrations of differential equations and systems of differential equations with respect to unknown functions Vrlk (x), αrjjk (x), j = 2, n; k = 1, 2. Substituting the obtained solutions Wr (x, t) to the adequate boundary conditions one can obtain the system of 2n algebraic equations from which one can find unknown constants: ∆(ε)Cr = Γr ,

(27.mar )

where 0 0 , V0112 , α00221 , . . . , α00nn1 , α00222 , . . . , α00nn2 ) Cr = (V0111

is an unknown vector and Γr a given vector. One can show that the asymptotic equality holds: ∆(ε) = KBi t1 (l) 1 + O Bi−1 (t1 (l)) ,

(28.mar )

where K = 2− 3 Γ −1 (2/3) 2

2 Y

b11 ((k − 1)l) ×

k=1

×

p Y

e1 ((k − 1)l) · B e2 ((k − 1)l) , ϕ0jk ((k − 1)l) · B

(29.mar )

j=2

e1 (x) = (bij (x))p−1 , B i,j=2

e2 (x) = (bij (x))n , B i,j=p

where bij (x) is ith coordinate of the vector bj (x). Lemma 4. Let Bs ((k − 1)l) 6= 0,

k, s = 1, 2

(30.mar )

Then for sufficiently small values of the parameter ε > 0 the determinant of the matrix ∆(ε) is distinct from zero. Thus if the conditions (30.mar ) holds, then each of the systems (19.mar ), where r ≥ −2, has a unique solution, i.e. each of the functions Wr (x, t) is uniquely determined as a solution of the adequate iterated problem.

51


7

Estimation of the last part of the asymptotic of a solution

Let us write the formal solution of the extended problem (6.mar ) in the form: f (x, t, µ) ≡ Wεq (x, t, µ) + µq+1 ξeq+1 (x, t, ε) , W

(31.mar )

where Wεq (x, t, µ) ≡

q X

µr Wr (x, t) — q-partial sum of the series (7.mar ).

r=−2

If in the equality (31.mar ) we realized the narrowing by t = φ(x, ε) then we obtain the equality W (x, ε) ≡ W (x, φ(x, ε), ε) ≡ Wεq (x, φ, ε) + ε

q+1 3

ξq+1 (x, φ, ε) .

(32.mar )

Lemma 5. If the conditions 1◦ and 2◦ hold then for sufficiently small values of the parameter ε > 0 : a) the series (19.mar ) is an asymptotic series for the solution of the extended problem (6.mar ) b) the narrowing of the series (19.mar ) by t = φ(x, ε), i.e. the series (32.mar ), is asymptotic series for a solution SSPDE (1.mar ). Applying Lemma 5 one can prove that the following asymptotic equality holds: 1 3 ξq+1 (x, φ, ε) ∼ = O(µ− 2 exp{µ−3 (2/3)ϕ 2 (x)}) .

(33.mar )

The results obtained in the article can be formulated in the form of the following theorem: Theorem 6. If the conditions 1◦ and 2◦ hold then for sufficiently small values of the parameter ε > 0: a) one can construct (applying the above described methods) a unique asymptotic series (19.mar ) as a solution of the extended problem (6.mar ) in SNS; b) the narrowing of the series (19.mar ) for t = φ(x, ε) (32.mar ) is asymptotic series for a solution SSPDE (1.mar ); c) the last part of the asymptotic series of the solution (1.mar ) has the estimation (33.mar ). Remark 7. Let (h(0), b∗1 (0)) = 0. Then 1) a solution of the degenerate equation (3.mar ) is a sufficiently smooth function for each x ∈ [0, 1]; 2) a solution SSPDE (1.mar ) contains no negative degrees of a small parameter µ > 0; ), i.e. they 3) αm = 0 (m = 0, 1) in the boundary conditions for SSPDE (1.mar represent the form like in the problems with stable spectrum of degenerate operator.

52


References [1] Marry J., Nonlinear differential equations in biology (lectures on models), Moscow Mir, 1983, 400 p. [2] Bobochko V. N., Asymptotic integration of systems of differential equations with a turning point, Diff. Equat. 1991, t. 27, No 9, 1505–1518. [3] Bobochko V. N., Turning point in a system of differential equations with analytic operator, Ukrainian Math. J. 1996, t. 48, No 2, 147–160. [4] Bobochko V. N., Asymptotics solutions of a system differential equations with a multiple turning point, Diff. Equat. 1996, t. 32, No 9, 1153–1155. [5] Lomov S. A., Introduction to the general theory of singular perturbations, Moscow, Nauka, 1981, 400 p. [6] Gantmacher F. R., Theory of matrices, Moscow, 1953, 492 p. [7] Olver F. W. J., Asymptotic and special functions, Moscow, Nauka, 1990, 528 p.


PAPERS pp. 53–60

On the Symmetric Solutions to a Class of Nonlinear PDEs Gabriella Bognár Mathematical Institute, University of Miskolc, 3515 Miskolc-Egyetemv´ aros, Hungary Email: [email protected]

Abstract. The existence and uniqueness of symmetric solutions to the boundary value problem of nonlinear partial differential equations are established. The Dirichlet boundary condition is given on the ball in RN .

AMS Subject Classification. 35A05, 34B15

Keywords. Boundary value problem, symmetric solutions

1

Introduction

We consider the following boundary value problem p∗ N X ∂ ∂u + f (u, gradp u p ) = 0 ∂xi ∂xi i=1 u=0

on

in

Bp ,

∂Bp ,

(1.bog ) (2.bog )

∗

where 0 < p < ∞ and the function up is defined as follows: ∗

p

u = |u|p−1 u, and the domain Bp ∈ RN is an open unit “ball” centered at the origin and ∂Bp means the boundary of the domain Bp. In (1.bog ) gradp u denotes the expression ∗

∗

p

∗

p

p

gradp u = (ux1 , ux2 , .., uxN ), and

|(x1 , x2 , . . . , xN )|p =

If p = 1, the operator

N P

N P i=1

i=1 ∂ ∂xi

|xi |

This is the final form of the paper.

1 p +1

∂u ∂xi

u = u(x1 , x2 , . . . , xN )

p p+1

.

p∗ in the equation (1.bog ) is reduced to ∆u.

54

Gabriella Bogn´ ar

For the problem (1.bog )–(2.bog ) we shall define the distance ρ between the point and the origin in RN as follows: 1

ρ p +1 =

N X

1

|xi | p +1 .

(3.bog )

i=1

In the case ρ = 1 the equation (3.bog ) gives the equation of the unit “ball” Bp in RN . We mention that the curve ρ = 1 in R2 is a central symmetric convex curve which plays the same role in the case of nonlinear differential equation (1.bog ) as the unit circle in the case of linear (p = 1) partial differential equation. For the “ball” Bp we introduce now instead of rectangular coordinates x1 , x2 , x3 , . . . , xN a new type of polar coordinates ρ, ϕ1 , . . . , ϕN −1 as follows N −1 Y

x1 = ρ

[S 0 (ϕi )] ,

i=1

xk = ρ [S(ϕk−1 )]

N −1 Y

(4.bog )

0

[S (ϕi )]

1 < k ≤ N,

if

i=k

where S = S(ϕi ), 1 < i ≤ N − 1 is the generalized sine function given by ´ Elbert [6]. The Pythagorean relation for this generalized sine function has A. the form 1

1

|S| p+1 + |S 0 | p+1 = 1,

where

S0 =

dS(ϕ) dϕ .

The unit “ball” Bp in RN is defined by N n o X 1 +1 |xi | p ≤ 1 , Bp = (x1 , x2 , . . . , xN ) :

0 < p < ∞.

i=1

When we study the radially symmetric solution u(x) = ν(ρ) of the nonlinear boundary value problem (1.bog )–(2.bog ) the nonlinear partial differential equation (1.bog ) is reduced to the following nonlinear ordinary differential equation (5.bog ) ∂ ∂ρ

∂ν ∂ρ

p∗

N −1 + ρ

∂ν ∂ρ

p∗

+ f (ν, |ν 0 |) = 0,

ρ ∈ (0, 1) ,

(5.bog )

p2 where f (u, gradp u p ) = f (ν, |ν 0 |) since gradp u p = |ν 0 | p+1 . We note that the equation (5.bog ) can be written also in the form ∗

(ρN −1 ν 0 (ρ))0 + ρN −1 f (ν, |ν 0 |) = 0, p

ρ ∈ (0, 1) .

Instead of the boundary condition (2.bog ) we shall consider the conditions ν(1) = 0,

(6.bog )

55

On the Symmetric Solutions

ν 0 (0) = 0.

(7.bog )

Now let us take the boundary value problem of another nonlinear partial differential equation instead of (1.bog )–(2.bog ) N i X ∂ h p−1 |∇u| ∇u + f (u, |grad u|) = 0 ∂xi i=1

u=0

on

in

B,

∂B,

(8.bog ) (9.bog )

where the unit ball B in RN is defined by N n o X B = (x1 , x2 , . . . , xN ) : x2i ≤ 1 , i=1

as it is usual in the Euclidean metric and |grad u| =

N P i=1

12 u2xi

(p = 1). The

expression N i X ∂ h p−1 |∇u| ∇u ∂xi i=1

in (8.bog ) is used to call p-Laplacian.This operator appears in many contexts in physics: non-Newtonian fluids, reaction-diffusion problems, non-linear elasticity, and glaceology, just to mention a few applications( see [3], [9], [10], [11], [12]). If p = 1 the equation (8.bog ) is also reduced to the semilinear problem ∆u + f (u, |grad u|) = 0, the existence of these problems are investigated in [1], [2], [7].The radially symmetric solutions of the Dirichlet problem of ∆u + f (u) = 0 were examined by Gidas, Wei-Ming Ni, and Nirenberg for the ball B [8]. If p > 0 then, applying the usual spherical transformation, the radially symmetric solutions of equation (8.bog ) has to satisfy formally the same equation as (5.bog ). So, if we examine the solutions of (5.bog ) we get results on the radially symmetric solutions both for the nonlinear partial differential equation (1.bog ) in the “ball” Bp and also for the nonlinear partial differential equation (8.bog ) in the Euclidean ball B. Here our aim is to show existence and uniqueness results of symmetric solutions for the problem ∂ ∂ρ

∂ν ∂ρ

p∗

N −1 + ρ

∂ν ∂ρ

p∗

ν(1) = 0,

0 + eλν+κ|ν | = 0,

ν 0 (0) = 0,

ρ ∈ (0, 1) ,

56

Gabriella Bogn´ ar

where λ, κ are negative real numbers. In the case p = 1 the existence and uniqueness results of the problem 0 N − 1 ∂ν ∂ ∂ν + + eλν+κ|ν | = 0, ρ ∈ (0, 1) , ∂ρ ∂ρ ρ ∂ρ ν 0 (0) = 0,

ν(1) = 0, are established in [4].

2

Results

Let us consider the following boundary value problem ∂ ∂ρ

∂ν ∂ρ

p∗

N −1 + ρ

ν(1) = a,

∂ν ∂ρ

p∗

0 + eλν+κ|ν | = 0,

a∈R , +

ρ ∈ (0, 1)

(10.bog )

0

ν (0) = 0.

We shall say that the function is the positive solution of problem (10.bog ) if i) ν (ρ) is continuous on [0, 1] and ν (ρ) > 0 in the interval (0, 1]; ii) ν 0 (ρ) exists and is continuous, moreover ν 0 (ρ) ≤ 0 in the interval [0, 1]; iii) ν (ρ) satisfies the boundary conditions: ν (1) = a, for a ≥ 0, ν 0 (0) = 0; iv) ν 00 (ρ) exists almost everywhere and locally Lebesgue integrable in the interval [0, 1]; v) ν (ρ) satisfies the differential equation ∂ ∂ρ

∂ν ∂ρ

p∗

N −1 + ρ

∂ν ∂ρ

p∗

0 + eλν+κ|ν | = 0, ρ ∈ (0, 1) .

Theorem 1. If a ≥ 0 then the boundary value problem (10.bog ) has at most one positive radial solution. Proof. Let us denote by ν1 (ρ) and ν2 (ρ) two different positive solutions to the boundary value problem (10.bog ). Without loss of generality we may suppose, that there exists a point ρ = γ, γ ∈ [0, 1) such that ν1 (ρ) ≥ ν2 (ρ). If ν1 (ρ) − ν2 (ρ) < 0 in the interval [0, 1) then we change the notations of ν1 (ρ) and ν2 (ρ) for the opposite. Let us denote by δ ∈ (γ, 1], the first zero of the function ν1 (ρ) − ν2 (ρ) which lays to the right from γ. By the Lagrange’s theorem there exists β ∈ (γ, δ) for which ν10 (β) − ν20 (β) < 0 and ν1 (β) − ν2 (β) > 0. We shall denote by α ∈ [0, β) the zero of the function ν10 (ρ) − ν20 (ρ). If there are more zeroes α1 , α2 , . . . , αk of ν10 (ρ) − ν20 (ρ) in the interval [0, β) then let us take the notation α = max(α1 , α2 , . . . , αk ). In this case we can summarize that ν1 (ρ) − ν2 (ρ) > 0 and ν10 (ρ) − ν20 (ρ) < 0, ρ ∈ (α, β], ν10 (α) − ν20 (α) = 0.

57


Since the functions ν1 (ρ) and ν2 (ρ) satisfy the nonlinear differential equation in (10.bog ) therefore substituting them into the differential equation and subtracting the two equations we get the equation ∗

∗

[ρN −1 (ν10 − ν20 )]0 + ρN −1 [eλν1 +κ|ν1 | − eλν2 +κ|ν2 | ] = 0. p

0

p

0

(11.bog )

We introduce the following notations ∗

∗

p

p

J(ρ) = ν1 (ρ) − ν2 (ρ) , ∗

∗

K(ρ) = ν10 (ρ) − ν20 (ρ) , p

p

moreover J(ρ) and K(ρ) have the properties J(1) = 0, J(γ) > 0,

K(0) = 0, K(β) < 0,

J(ρ) > 0, ρ ∈ (α, β],

K(α) = 0, K(ρ) < 0, ρ ∈ (α, β].

(12.bog )

Rearranging the differential equation (11.bog ) we obtain 0 N −1 K(ρ) + ρN −1 K(ρ)A(ρ) − ρN −1 J(ρ)B(ρ) = 0 , ρ where the expressions A(ρ) and B(ρ) have the forms eλν1 +κ|ν1 | − eλν1 +κ|ν2 | 0

A(ρ) =

B(ρ) =

∗

0

∗

ν10p (ρ) − ν20p (ρ) 0 0 eλν2 +κ|ν2 | − eλν1 +κ|ν2 | ∗

∗

ν1p (ρ) − ν2p (ρ)

,

.

0 Using the properties of the function eλν+κ|ν | we get that A(ρ) ≥ 0 and B(ρ) ≥ 0 when ρ ∈ (α, β]. Thus from the equation (11.bog ) we obtain the inequality

N −1 0 eλν1 +κ|ν1 | − eλν1 +κ|ν2 | ρ K(ρ) + ρN −1 K(ρ) ≥ 0, ∗ ∗ ν10p (ρ) − ν20p (ρ) 0

0

ρ ∈ (α, β].

(13.bog )

If we multiply the inequality in (13.bog ) by the expression ) ( Z 0 0 b eλν1 +κ|ν1 | − eλν1 +κ|ν2 | eta dτ , exp − ∗ ∗ ρ ν10p (τ ) − ν20p (τ ) and take the integral on the interval [δ, β] where δ ∈ (α, β) we get the inequality ( Z ) 0 γ λν1 +κ|ν10 | − eλν1 +κ|ν2 | e N −1 N −1 K(β) − δ K(δ) exp − dτ ≥ 0. β ∗ ∗ δ ν10p (τ ) − ν20p (τ )

58

Gabriella Bogn´ ar

If we take δ → α then we get that K(β) ≥ 0, since K(α) = 0. This is contradiction with (12.bog ). In the next theorem we establish the existence result: Theorem 2. If a ≥ 0 then the boundary value problem (10.bog ) has a unique positive solution. In the following we need some subsidiary statements. Lemma 3. If a ≥ 0 then there is a positive solution to problem (10.bog ). Proof. Let us define the mappings Z

1

µ(τ )dτ,

(Φµ) (t) = a + t

p1 Z t τ N −1 λ(Φµ)(τ )+κµ(τ,a) e dτ , (Ψ µ) (t) = t 0 λa p1 o n e , t ∈ (0, 1) , µ (0, a) = 0 . H = µ (τ, a) ∈ C[0, 1), 0 ≤ µ (τ, a) ≤ N The functions which belong to the set ΦH are uniformly bounded and equicontinuous functions therefore H is compact. Since every Cauchy sequence being contained in the set H converges in H then H is closed. Thus the set H is bounded, convex, closed and compact in the Banach space C[0, 1). The mapping Ψ is a continuous mapping from H to H. Applying the Schauder fixed point theorem the mapping Ψ has a fixed point. ) has Using notation µ (ρ, a) = −ν 0 (ρ, a) the positive solution to problem (10.bog the form p1 Z 1 Z τ N −1 Z 1 ρ µ(τ )dτ = a + eλν(ρ,a)+κµ(ρ,a) dρ dτ. (14.bog ) ν (t, a) = a + τ t t 0 Lemma 4. Let ν (t, a) be the unique positive solution to the problem (10.bog ). If 0 ≤ a2 < a1 , then ν (t, a1 ) ≥ ν (t, a2 ) and ν 0 (t, a1 ) ≥ ν 0 (t, a2 ) for all t ∈ [0, 1). Proof. Let ν (t, a1 ) and ν (t, a2 ) be the unique positive solution to the problem (10.bog ) for a1 and a2 , respectively. Let us take the notation ∗

∗

∗

∗

j(t) = ν (t, a1 ) − ν (t, a2 ) and k(t) = ν 0 (t, a1 ) − ν 0 (t, a2 ) . p

∗

p

∗

p

p

Clearly j(1) = ap1 − ap2 > 0 and k(0) = 0. Hence there exists at least one point t = α, α ∈ [0, 1) such that k(α) = 0 and j(t) > 0 in the interval (α, 1]. If

59


there are more values of α (α1 , α2 , . . . , αk ) for which k(α) = 0 in the interval [0, 1) then let us take the notation α = max(α1 , α2 , . . . , αk ). We may assume that there exists a point β ∈ [α, 1) where ν (t, a1 ) > ν (t, a2 ) and ν 0 (t, a1 ) < ν 0 (t, a2 ), that is j(t) > 0, and k(t) < 0 in the interval (α, β]. In an analogous way as in the proof of Theorem 1 one can obtain that k(β) ≥ 0. It is contradiction since we supposed that k(β) < 0. The inequality ν (t, a1 ) ≥ ν (t, a2 ) we get in a similar way as in the proof of Theorem 1. Proof of Theorem 2. When a → 0 we get that ν (t, a) and ν 0 (t, a) converges uniformly to ν (t, 0) and ν 0 (t, 0) in the interval [0, 1], respectively. Taking a → 0 in the expression (14.bog ) we shall get the positive solution to the problem ∂ ∂ρ

∂ν ∂ρ

p∗

N −1 + ρ

∂ν ∂ρ

p∗

0 + eλν+κ|ν | = 0,

ν(1) = 0,

ρ ∈ (0, 1) ,

ν 0 (0) = 0,

in the following form Z

1

Z

ν(t, 0) = t

0

τ

ρ N −1 τ

e

λν(ρ,0)−κν 0 (ρ,0)

p1 dρ

dτ.

Supported by the Grant No. OTKA 019095 (Hungary).

References [1] F. V. Atkinson, L. A. Peletier, Ground states of ∆u = f (u) and the Emden-Fowler equation, Archs. Ration. Mech. Analysis, 93 (1986), 103–107. [2] J. V. Baxley, Some singular nonlinear boundary value problems, SIAM J. Math. Analysis, 22 (1991), 463–479. [3] T. Bhattacharya, Radial symmetry of the first eigenfunction for the p-Laplacian in the ball, Proc. of the Amer. Math. Soc., 104 (1988), 169–174. [4] G. Bognar, On the radially symmetric solutions to a nonlinear PDE, Publ. Univ. of Miskolc, Series D. Natural Sciences. 36 No.2. Mathematics (1996), 13–20. [5] G. Bognar, On the radial symmetric solutions of a nonlinear partial differential equation, Publ. Univ. of Miskolc, Series D. Natural Sciences. 36 No.1. Mathematics (1995), 13–22. ´ Elbert, A half-linear second order differential equation, Coll. Math. Soc. J´ [6] A. anos Bolyai, 30. Qualitative theory of differential equations, Szeged, (1979), 153–179. [7] A. M. Fink, J. A. Gattica, G. E. Hernandez, P. Waltman, Approximation of solutions of singular second order boundary value problems, SIAM J. Math. Analysis, 22 (1991), 440–462. [8] B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related properties via Maximum Principle, Commun. Math. Phys., 68 (1979), 209–243. [9] B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math. Mech., to appear.

60

Gabriella Bogn´ ar

[10] P. Lindqvist, Note on a nonlinear eigenvalue problem, Rocky Mountain J. of Math., 23 (1993), 281–288. [11] M. Otani, A remark on certain nonlinear elliptic equations, Proc. of Faculty of Science, Tokai Univ., 19 (1984), 23–28. [12] F. de Thelin, Quelques résultats d’existence et de non-existence pour une E.D.P. elliptique non linéaire, C. R. Acad. Sci. Par., 299 Serie I. Math. (1986), 911–914.


PAPERS pp. 61–72

The Abstract Cauchy Problem in Plasticity Igor A. Brigadnov North-West Polytechnical Institute Millionnaya Str. 5, St. Petersburg, 191186, Russia Email: [email protected]

Abstract. The boundary-value problem of plasticity is formulated as the evolution variational problem (EVP) over the parameter of external loading for the displacement in the framework of the small deformations theory. The questions of the mathematical correctness of the plasticity EVP are discussed. The general existence and uniqueness theorem is formulated. The main necessary and sufficient condition has the simplest algebraic form and does not coincide with the classic Drucker’s hypothesis and similar thermodynamical postulates. By means of finite element approximation the plasticity EVP transforms into the Cauchy problem for a non-linear system of ordinary differential equations unsolved regarding derivative. Moreover, this system can be stiff. Therefore, for the numerical solution the implicit Euler scheme with the decomposition method of adaptive block relaxation (ABR) is used. The numerical results show that, for finding the displacement and the time of calculation, the ABR method has advantages over the standard method.

AMS Subject Classification. 73E05, 73V20, 35J55

Keywords. Plasticity BVP, evolution variational equation, mathematical correctness, stiff system, adaptive finite element method

1

Introduction

The solution of plasticity boundary-value problems (BVPs) is of particular interest in both theory and practice. At present there are many models of plasticity in the framework of the small deformations theory [1,2,3]. Adequacy and the field of application of every model must be found only by correlation between experimental data and solutions of appropriate BVPs. Therefore, the analysis of mathematical correctness and the treatment of numerical methods for these problems is very important [4,5,6,7]. In this paper the plasticity BVP is formulated as the evolution variational problem (EVP) (i.e. as the abstract Cauchy problem in the weak form) for the displacement in the Hilbert space [8]. For this reason the parameter of external loading in the interval [0, 1] is used. The general existence and uniqueness theorem for the plasticity EVP is formulated. The proof of this theorem is based on This is the final form of the paper.

62

Igor A. Brigadnov

the monotonous operators theory and the theory of the abstract Cauchy problem in the Hilbert space [7]. The main necessary and sufficient condition has the simplest algebraic form and does not coincide with the classic Drucker’s hypothesis and similar thermodynamical postulates [2,3,9,10]. This condition is the general criterion of mathematical correctness for plasticity models. Its independence is illustrated for the plasticity model of linear isotropic-kinematic hardening with ideal Bauschinger’s effect, dilatation and internal friction [11,12]. For the numerical solution of the plasticity EVP the standard spatial piecewise linear finite element approximation is used [13]. For some models the appropriate finite dimensional Cauchy problem can be stiff [14,15]. The main cause of this phenomenon consists of the following: the global shear stiffness matrix has lines with significantly different factors (it is badly determined). Moreover, for real plasticity models both initial continuum and discrete Cauchy problems are principally unsolved regarding derivative [1,2,12]. Therefore, for the numerical solution the implicit Euler scheme with the decomposition method of adaptive block relaxation (ABR) is used [4,5,6,7]. The main idea of this method consists of iterative improvement of zones with ”proportional” deformation by special decomposition of domain (variables), and separate calculation in these zones (on these variables). The global convergence theorem for the ABR method is formulated. The proof of this theorem is based on the monotonous operators theory [4,5]. The numerical results show that, for finding the displacement and the time of calculation, the ABR method has advantages over the standard method.

2

Evolution formulation of the plasticity BVP

Let a homogeneous rigid body in the undeformed reference configuration occupy a domain Ω ⊂ R3 with boundary Γ . In the deformed configuration each point x ∈ Ω moves into a position x+u(x) ∈ R3 , where u : Ω → R3 is the displacement. In the framework of the small deformations theory the strain Cauchy tensor ε = ε(u) = 12 ∂ i uj + ∂ j ui : Ω → S 3 is used as the measure of deformation, where ∂ i = ∂/∂xi ; i, j = 1, 2, 3. The symbol S 3 denotes the subspace of real symmetrical 3 × 3 matrices. In the mathematical theory of plasticity the isotropic material is described by the constitutive relation for speeds [1,2,3,7,10,12] ˙ = Cijkm ε˙km − P˙km (ε, ε) ˙ , σ˙ ij = Sij (ε, ε) (1.bri ) 2 Cijkm = 2µ δik δjm + k0 − 3 µ δij δkm , where σ : Ω → S 3 is the Cauchy stress tensor, P : Ω → S 3 is the plastic part of the Cauchy strain tensor, Cijkm are the components of elasticity acoustic tensor [2,3], µ > 0 and k0 > 0 are the shear and bulk moduli, respectively; δij is the Kronecker symbol, the above point is d/dt and t ∈ [0, 1] is the parameter of external loading. Here and in what follows we use the rule of summing over

63

The Abstract Cauchy Problem in Plasticity 1/2

for modulus of repeated indices and the designation |A| = |Akm | = (Aij Aij ) matrix A. We consider the following boundary-value problem. The quasi-static influences acting on the body are: a mass force with density f in Ω, a surface force with density F on a portion Γ 2 of the boundary, and a surface displacement uγ on a portion Γ 1 of the boundary is also given. Here Γ 1 ∪ Γ 2 = Γ , Γ 1 ∩ Γ 2 = ∅ and area(Γ 1 ) > 0. According to the evolution description [8] the external influences, internal displacement and stress tensor are taken as continuous and piecewise smooth abstract functions acting from interval [0, 1] to appropriate Banach spaces, supposing that Γ 1 = const(t) and f = 0, F = 0, uγ = 0 for t = 0. The plasticity BVP is formulated as the evolution variational problem (EVP) (i.e. as the abstract Cauchy problem in the weak form): the sought displacement corresponds to the abstract function u∗ (t) = u0 (t) + u(t), where the piecewise smooth abstract function u0 (t) with u0 (0) = 0 corresponds to the surface displacement uγ , and unknown abstract function u : [0, 1] → V 0 must satisfy the initial condition u(0) = 0 and the differential equation for every v ∈ V 0 and almost every t ∈ (0, 1) Z Sij ε(u0 + u), ε(u˙ 0 + u) ˙ ∂ j vi dx = L(t, v), Ω

Z

Z f˙i (t)vi dx +

L(t, v) = Ω

(2.bri ) F˙i (t)vi dγ.

Γ2

Here V 0 = {v : Ω → R3 ; v(x) = 0, x ∈ Γ 1 } — is the set of kinematically admissible variations of the displacement. For real plasticity models this equation is principally unsolved regarding u˙ [2,7,12]. Concerning the constitutive relation S, the domain Ω and the functions f , F , uγ we make the following hypotheses: (H1) Matrix function S(A, B) is the continuous and strongly monotonous in B, i.e. there exists a constant m0 > 0 such that for every A, B 1 , B 2 ∈ S 3 the following estimate is true 2 1 2 Sij (A, B 1 ) − Sij (A, B 2 ) Bij − Bij ≥ m0 B 1 − B 2 . (H2) Matrix function S(A, B) is the Lipschitz continuous in A, i.e. there exists a scalar function M0 : S 3 → (0, +∞) such that for every A1 , A2 , B ∈ S 3 the following estimate is true S(A1 , B) − S(A2 , B) ≤ M0 (B) A1 − A2 . (H3) Matrix function S(A, B) has the growth in A and B no above linear, i.e. there exists a constant M1 > 0 such that for every A, B ∈ S 3 the following estimate is true |S(A, B)| ≤ M1 (|A| + |B|).

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(H4) Ω is a connected bounded domain in R3 with a Lipschitz boundary Γ . (H5) f ∈ C 0,1 [0, 1], L6/5 (Ω, R3 ) . (H6) F ∈ C 0,1 [0, 1], L4/3 (Γ 2 , R3 ) . (H7) uγ ∈ C 0,1 [0, 1], L2 (Γ 1 , R3 ) . We define the set of kinematically admissible variations of the displacement in the following way: V 0 = v ∈ H 1 : v(x) = 0, x ∈ Γ 1 , where H 1 := W 1,2 (Ω, R3 ) is the Hilbert space. Theorem 1 (was proved in [7]). In the framework of the hypotheses (H1)– (H7) the following statements are true: (i) The unique strict solution of the EVP (2.bri ) exists, i.e. the absolutely continuous function u ∈ C 0,1 ([0, 1], V 0 ), u(0) = 0 with the strong derivative u, ˙ satisfying the equation (2.bri ) for a.e. t ∈ (0, 1). (ii) The map (f, F, uγ ) 7→ u is continuous. Remark 2. For the constitutive relation S the main condition (H1) is necessary and sufficient. It is the general criterion of mathematical correctness for plasticity models. This question is in detail discussed in [7]. Therefore, we rewrite this condition for the matrix function P˙ (ε, ε), ˙ usually used in the modern theory of plasticity [2,7,12]. (H1) Matrix function P˙ (A, B) is continuous in B and satisfies the following estimate for every A, B 1 , B 2 ∈ S 3 2 1 2 < 2 µ B 1 − B 2 . (3.bri Cijkm P˙km (A, B 1 ) − P˙km (A, B 2 ) Bij − Bij ) This condition does not coincide with the Lipschitz condition of the matrix function P˙ (A, B) over second matrix argument. It is easily to get convinced that the Lipschitz condition is stronger than the condition (3.bri ). In the following section we show that this condition is independent and does not coincide with the classic Drucker’s hypothesis based on the thermodynamical postulates [2,3,9,10].

3

Example of analysis of plasticity models

The independence of the main necessary and sufficient condition (3.bri ) of mathematical correctness of plasticity EVP (2.bri ) we illustrate for the generalized model of plasticity with linear isotropic-kinematic hardening, ideal Bauschinger’s effect,

65

The Abstract Cauchy Problem in Plasticity

dilatation and internal friction [12] P˙km = (1 + h0 + 3λΛ)−1 H (ρe − ε∗ + λ tr(ε − P )) × ˙ ρ−2 × H cos γ + λε˙−1 e tr(ε) e (ρkm + Λρe δkm )(ρpq + λρe δpq )ε˙pq , D ρkm = εD km − (1 + h0 )Pkm ,

ρe = |ρij | , H(q) = 0 f or q < 0

and

cos γ = (ρe ε˙e )−1 ρij ε˙D ij , D ε˙e = ε˙ij ,

(4.bri )

H(q) = 1 f or q ≥ 0,

where h0 is the parameter of plastic hardening, λ ≥ 0 and Λ ≥ 0 are the parameters of dilatation and internal friction, respectively; ε∗ ≥ 0 is the limit of 1 elastic strain, AD ij = Aij − 3 tr(A)δij are the components of deviatoric part and tr(A) = δij Aij is the trace (first invariant) of matrix A. For λ = Λ = 0 model (4.bri ) equals the classic model of plasticity with linear isotropic-kinematic hardening and ideal Bauschinger’s effect [1,2,3]. In this case tr(P ) = 0 and the constitutive relation (4.bri ) is associated with the Mises yield surface ρe − ε∗ = 0 [2,3]. For λ = Λ 6= 0 the constitutive relation (4.bri ) is associated with the yield surface for strain ρe − ε∗ + λ tr(ε − P ) = 0. This for h0 = 0 corresponds to Dsurface the Mises-Schleiher yield surface for stress σ + c−1 λ tr(σ) − 2µ ε∗ = 0, where c = 3k0 /(2µ) [11]. For λ 6= Λ the constitutive relation (4.bri ) is non-associated with some yield surface. In both cases tr(P ) 6= 0 what is a well known experimental phenomenon of dilatation [11,12]. Let matrices A, B 1 , B 2 ∈ S 3 be arbitrary. Then from condition (3.bri ) for model (4.bri ) we have 1 2 Cijkm P˙km (A, B 1 ) − P˙ km (A, B 2 ) Bij − Bij ≤ ≤ (1 + h0 + 3λΛ)−1 B 1 − B 2 + λ tr B 1 − B 2 × × 2µ B 1 − B 2 + 3k0 Λ tr B 1 − B 2 ≤ 2 ≤ 2µ Ψ (λ, Λ, h0 ) B 1 − B 2 , where Ψ=

(1 +

√ √ 3 λ)(1 + 3 cΛ) . 1 + h0 + 3λΛ

The constant c = (1 + ν)/(1 − 2ν) ≥ 1, because for real materials the Poisson ratio 0 ≤ ν < 1/2 [1,2,3]. Therefore, for parameters λ, Λ ≥ 0 the condition (3.bri ) is true (Ψ < 1) only for the positive parameter of plastic hardening, satisfying the following estimate √ h0 > 3(c − 1)λΛ + 3 (λ + cΛ) ≥ 0. (5.bri ) If this condition is disturbed then the effects of bifurcation and internal instability exist in the plasticity EVP (2.bri ) [9,12]. The classic Drucker’s hypothesis σ˙ ij P˙ ij ≥ 0 is the only necessary condition for the uniqueness of solution of EVP (2.bri ). For the model (4.bri ) it has the following

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form h0 ≥ 3 (cΛ − λ)Λ.

(6.bri )

For example, if Λ = 0, λ > 0 then the condition (5.bri ) is carried out only √ ) is fulfilled for h0 ≥ 0. This simple for h0 > 3 λ > 0, but the condition (6.bri example proves that the classic Drucker’s hypothesis, based on thermodynamical postulates [2,3,9,10], does not provide the existence of solution of the plasticity EVP (2.bri ).

4

Computational method

For the numerical solution of the plasticity EVP (2.bri ) the standard spatial piecewise linear finite element approximation is here used: Ωh = ∪Th , Γh = ∂Ωh and vol(Ω\Ωh ) → 0, area(Γ \Γh ) → 0 for h → 0 regularity, where Th is the simplest simplex and h is the step of approximation [13]. For the displacement the following piecewise linear approximation is used uh (t, x) = U γ (t)Φγ (x)

(γ = 1, 2, . . . , m),

where U γ ∈ R3 is the displacement in the node xγ , Φγ : Ωh → R is the standard continuous piecewise linear function such that Φγ (xα ) = δαγ (α, γ = 1, 2, . . . , m), m is the number of nodes. In this case the subspace V 0 ⊂ H 1 is approximated by the subspace Vh0 ⊂ R3m Vh0 = U ∈ R3m : U α = 0, xα ∈ Γh1 . The plasticity EVP (2.bri ) is approximated by the Cauchy problem for nonlinear system of ordinary differential equations: vector function U : [0, 1] → Vh0 must satisfy the initial condition U (0) = 0 and the following differential equation for almost every t ∈ (0, 1) Apq (U, U˙ )U˙ q = Bp ,

(7.bri )

where U is the global vector of free nodal displacements, A is the global shear stiffness matrix and in the end B is the global vector of nodal speeds of influences; p, q = 1, 2, . . . , 3m. Here Up = Uiγ with index p = 3(γ − 1) + i. Due to the properties of the real plasticity models this equation is principally unsolved regarding U˙ in the explicit form. For some plasticity models the differential system (7.bri ) can be stiff. The main cause of this phenomenon consists of the following: matrix A has lines with significantly different factors (it is badly determined) for the small parameter of plastic hardening h0 1 [4,5,6,7]. Example 3. Let the bounded rigid body Ω ⊂ R3 with the regular boundary Γ consist of incompressible material describing by the model (4.bri ) with parameters λ = Λ = 0. The body is fastened on a portion Γ 1 of the boundary (i.e. uγ ≡ 0)

67


and deformed by the external forces. In this case the set of kinematically admissible displacements is 0 = u ∈ V 0 : div(u(x)) = 0, x ∈ Ω . Vdiv We use the following approximation for unknown displacement uN (t, x) = Yr (t)wr (x)

(r = 1, . . . , N ),

0 where {wr }N r=1 ⊂ Vdiv are the basic functions. In this case the plasticity EVP (2.bri ) is approximated by the Cauchy problem for nonlinear system of ordinary differential equations: vector function Y : [0, 1] → RN must satisfy the initial condition Y (0) = 0 and the following differential equation for almost every t ∈ (0, 1)

Aqr (Y, Y˙ )Y˙ r = Bq where

(q, r = 1, 2, . . . , N ),

(8.bri )

Z Ψqr (Y, Y˙ )|ε(wq )| |ε(wr )| dx,

Aqr (Y, Y˙ ) = Ω

Ψqr = cos γqr − (1 − ψ)H(ρe − ε∗ )H(cos γ) cos γq cos γr , Bq = (2µ)−1 L(t, wq ). Here and in what follows the summing over indices q, r, s does not used, ρe = ρe (uN ), γ = γ(uN , u˙ N ) from (4.bri ), the parameter ψ = h0 /(1 + h0 ) and cos γs = (ρe |ε(ws )|)

−1

ρij εij (ws ) (s = q, r),

cos γqr = (|ε(wq )| |ε(wr )|)

−1

εij (wq )εij (wr ).

Due to the properties of the finite element approximation the matrix A is symmetrical and has the largest elements on the main diagonal Z Aqq (Y, Y˙ ) = Ψqq (Y, Y˙ )|ε(wq )|2 dx, Ω

Ψqq = 1 − (1 − ψ)H(ρe − ε∗ )H(cos γ) cos2 γq . If the solution of problem (8.bri ) has the zone of active deformation with a small curvature trajectory (γ ∼ 0) then for basic functions wq with cos γq ≈ 1 the factors Ψqq ≈ ψ. In the zone of passive deformation, or for a large curvature trajectory (γ ∼ π/2), or for basic functions wr with cos γr ≈ 0 the factors Ψrr ≈ 1. It is easily seen that for the small parameter of plastic hardening (h0 1) the global shear stiffness matrix A has lines with significantly different factors (it is badly determined). As a result, the following estimate was proved in [4,6] νmax ≥ C N 2 h−1 cond(A) := 0 1, νmin where cond(A) is the condition number of matrix A; νmax and νmin are the largest and smallest eigenvalues of the matrix A, respectively, and C = const(N, h0 ).

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Igor A. Brigadnov

According to standard technique [14,15] for the solution of the unsolved regarding derivative and stiff problem (7.bri ) the implicit Euler scheme is used V ∈ Vh0 ,

Apq (U k + τ V, V )Vq = Bpk+1 , U k+1 = U k + τ V,

U 0 = 0,

(9.bri )

where index k corresponds to the parameter tk = kτ , k = 0, 1, . . . , K − 1; τ = 1/K and K 1. Here and in what follows the summing over index k does not used. For the numerical solution of algebraic system (9.bri ) for every k = 0, 1, . . . , K −1 the decomposition method of adaptive block relaxation (ABR) is used. This method disregards the condition number of the matrix A and has the following description [4,5,6,7]. Step 1. As the initial approach the explicit solution is used (here O is the zero vector) k k+1 Yq(0) = A−1 . pq (U , O)Bp Step 2. Due to the properties of the finite element approximation the matrix A has the largest elements on the main diagonal. Therefore, by the current approach Y (m) variables are separated on quick and slow ones by the (m) = proximity criterion of appropriate diagonal elements of the matrix A k (m) (m) A U + τY ,Y o n (m) /d(m) < ∆s/L , Is(m) = p = 1, 2, . . . , N : ∆(s−1)/L ≤ App (m)

IL

= {1, 2, . . . , N }\

L−1 [

Is(m) ,

s=1

where s = 1, 2, . . . , L − 1; ∆ = D(m) /d(m) ; D(m) and d(m) are the largest and smallest diagonal elements of the matrix A(m) , respectively, L = int(ω lg ∆) + 1 is the number of blocks (1 ≤ L ≤ N ), ω ≥ 0 is the decomposition parameter. Step 3. The block version of the Seidel method is used [16]. In practice one step of this method is enough (here the summing over index s does not used) T Y (m+1) = w1 ⊕ w2 ⊕ · · · ⊕ wL , s−1 L X X s st t st t Ξ − Λ w − Λ v wis = [Λss ]−1 j jr r jr r , ij t=1

t=s+1

o n (m) (m) , : p ∈ Is(m) , q ∈ It Λst = Apq o o n n (m) Ξ s = Bpk+1 : p ∈ Is(m) , v t = Yq(m) : q ∈ It . It is easily seen that the ABR method practically disregards the condition number of the matrix A(m) because cond (Λss ) ∼ cond1/L A(m) cond A(m)

69


for every s = 1, 2, . . . , L even if L = 2. By the new approach Y (m+1) , variables are separated on quick and slow ones too, etc. Step 4. For termination of the iteration process the following condition is used (m) (m) (10.bri ) Apq Yq − Bpk+1 < ξ, where ξ is the prescribed precision. Theorem 4. In the framework of the hypotheses (H1)–(H7) the following statements are true: (i) The solutions of systems (7.bri ) and (9.bri ) exist. (ii) The ABR method converges: lim Y (m) = V . m→∞

Proof. According to the properties of the finite element approximation for the constitutive relation satisfying the conditions (H1)–(H3) the vector function {Apq (U + τ Y, Y )Yq } : R3m → R3m is strongly monotonous in Y for every U ∈ R3m and τ ∈ [0, 1] [8,13]. Therefore, according to the classic results of the theory of ordinary differential equations [15] and algebra [16] the statements (i) and (ii) are true. Remark 5. In the computational mathematics the Schwarz decomposition methods are well known [17]. But they are used only for linear BVPs without the main idea of adaptiveness (see References in [17]).

5

Numerical results

The numerical analysis was realized on series of BVPs with model (4.bri ) for the axisymmetrical kinematic deformation of long round tube fastened on the internal radius ρ = a. The complicated plane deformation was given by different regimes of the displacement on the external radius ρ = b [4,6,7]: (here the summing over indices ϕ and ρ does not used) u0ϕ (t) = Cϕ Zϕ (t),

u0ρ (t) = Cρ Zρ (t) √

where t ∈ [0, 1], Cϕ = ε∗ b(1 − a2 /b2 ) and Cρ = 23 Cϕ are the maximum external displacements for which the clearly elastic deformation is realized in the framework of the classic model of plasticity (i.e. for the model (4.bri ) with parameters λ = Λ = 0) [6]. In the computer experiments the following data were used: a = 10, b = 20 (mm), k0 = 105 , µ = 7.5 · 104 (MPa), ε∗ = 5 · 10−3, h0 = 0.001 and λ = Λ = 0 in the model (4.bri ). The radius [a, b] was discretized by 50 segments and the standard piecewise linear approximation was used for unknown functions uϕ (t, ρ) and uρ (t, ρ) such that uϕ ≡ 0, uρ ≡ 0 for ρ = a and uϕ ≡ u0ϕ (t), uρ ≡ u0ρ (t) for ρ = b.

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Igor A. Brigadnov

Fig. 1. The tangential displacement in the end of the simplest radial regime

The ABR method with the decomposition parameter ω = 0.5 was compared with the standard method of simple iterations which equals the ABR method with parameter ω = 0. For the simplest radial regime of clear twisting Zϕ (t) = 10t, Zρ (t) ≡ 0 in the implicit Euler scheme (9.bri ) K = 100 steps over the parameter of loading were used. In figure 1 the following solutions in the end of process are shown: curves 1 and 2 correspond to the standard method with the single ξ = 10−3 and double ), respectively; curve 3 corresponds to the ξ = 10−5 precision in the criterion (10.bri ABR method with the single precision. The last numerical solution (curve 3) practically equals the analytical solution which was built in [4,5]. For the complicated cyclic regime Zϕ (t) = 10 sin(4πt), Zρ (t) = 10 sin(2πt) in the scheme (9.bri ) K = 800 steps over parameter t ∈ [0, 1] were used. In figure 2 the following solutions in the end of process are shown: curves 1 and 2 correspond to the standard method with the single and double precision, respectively; curve 3 corresponds to the ABR method with the single precision. In all experiments the time of calculation with single precision was approximately equal for both methods; whereas with double precision, the time of calculation was longer for the standard method than for the ABR method. It is easily seen that, for finding the displacement and the time of calculation, the ABR method has advantages over the standard method.


71

Fig. 2. The tangential displacement in the end of the cyclic regime

6

Conclusion

The questions of mathematical correctness and effective numerical solution for the plasticity BVP have been discussed. By using the evolution variational method: 1) the general algebraic criterion of mathematical correctness for plasticity models has been constructed; 2) the effective qualitative FE analysis has been realized. As a result, an original implicit adaptive strategy has been presented for the numerical simulation of practically important plastic and similar effects in the Mechanics of Solids.

Acknowledgement I would like to thank Professor Yu. I. Kadashevich for consultations and the Organizing Committee and sponsors of the conference on Differential Equations and their Applications (EQUADIFF 9) for their support of my visit to Brno, Czech Republic.

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References [1] Novozhilov, V. V., Kadashevich, Yu. I.: Microstresses in Technical Materials. Mashinostrojenije, Leningrad (1990) [2] Hill, R.: Classical plasticity: a retrospective view and a new proposal. J. Mech. and Phys. Solids. 42(11) (1994) 1803–1816 [3] Kljushnikov, V. D.: Mathematical Theory of Plasticity. Moscow State Univ. press., Moscow (1979) [4] Brigadnov, I. A.: Methods of solution of elastoplastic boundary value problems for small hardening materials. Ph.D. Thesis. Leningrad Polytech. Inst., Leningrad (1990) [5] Brigadnov, I. A., Repin, S. I.: Numerical solution of plasticity problems for materials with small strain hardening. Mekh. Tverd. Tela 4 (1990) 73–79; English transl. in Mech. of Solids 4 (1990) [6] Brigadnov, I. A.: On the numerical solution of boundary value problems for elastoplastic flow. Mekh. Tverd. Tela 3 (1992) 157–162; English transl. in Mech. of Solids 3 (1992) [7] Brigadnov, I. A.: Mathematical correctness and numerical methods for solution of the plasticity initial-boundary value problems. Mekh. Tverd. Tela 4 (1996) 62–74; English transl. in Mech. of Solids 4 (1996) [8] Gajewski, H., Gr¨ oger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademia-Verlag, Berlin (1974) [9] Noll, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Arch. Rat. Mech. Anal. 52 (1973) 62–92 [10] Pal’mov, V. A.: Reological models in the non-linear mechanics of solids. Advances in Mech. 3(3) (1980) 75–115 [11] Novozhilov, V. V.: On the plastic loosening. Prikl. Mat. Mekh. 29(4) (1965) 681– 689; English transl. in J. Appl. Math. Mech. 29(4) (1965) [12] Garagash, I. A., Nikolajewskiy, V. N.: Non-associated laws of flow and localization of plastic strains. Advances in Mech. 12(1) (1989) 131–183 [13] Ciarlet, Ph. G.: The Finite Element Method for Elliptic Problems. North-Holland publ. co., Amsterdam etc. (1980) [14] Rakitskiy, Yu. V., Ustinov, S. M., Chernorutskiy, I. G.: Numerical Methods for Solution of Stiff Systems. Nauka, Moscow (1979) [15] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. Pt.2. Stiff Differential Algebraic Problems. Springer, Berlin etc. (1991) [16] Collatz, L.: Funktionalanalysis und Numerische Mathematik. Springer, Berlin (1964) [17] Dryja, M., Widlund, O. B.: Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Communic. Pure & Appl. Math. 48 (1995) 121–155


PAPERS pp. 73–82

Dense Sets and Far-field Patterns for the Vector Thermoelastic Equation Fioralba Cakoni Department of Mathematics, Faculty of Natural Sciences, University of Tirana Tirana, Albania Email: [email protected] [email protected]

Abstract. We study the set of far-field patterns which are generated by entire incident thermoelastic fields scattered by a boundary nopenetrable obstacle. Necessary and sufficient conditions are given for the set to be dense in the set of all square integrable vector fields defined on the unit sphere. The method of Herglotz thermoelastic function is utilized to prove the dense properties of the asymptotic fields.

AMS Subject Classification. 35L, 73D, 47A

Keywords. Scattering theory, thermoelastic equation, far-field pattern

1

Introduction

A basic problem in inverse scattering theory is the classification of far field patterns corresponding to the scattering of a time harmonic thermoelastic incident wave by a bounded, connected obstacle. Indeed, if T denotes the operator mapping the incident field and scattering obstacle onto the far field patterns, then the inverse scattering problem is to construct T −1 defined on the range of T , and the determination of this range is nothing more than the description of the class of far field patterns. It is easily verifiable that the class of functions that can be far field patterns is a subset of the class of the entire functions for each positive fixed value of the wave numbers. The crucial point is the question if the far field patterns for a fixed obstacle and all incident plane wave are complete in a production of L2 (Ω). Colton, Kress and Kirsch [2,3] gave an answer of this question for some acoustic and electromagnetic scattering problems. Dassios [4,5] has investigated the case of elastic rigid scattering problems where the situation becomes much harder since in elasticity, besides the vectorial (displacement, surface traction) as well as the tensorial (stress, strain) characteristics of the fields, there are two separate wave solutions propagating at different phase velocities. The purpose of this work is to extend the mentioned results to coupled thermoelasticity. In this case, there are five types of waves present, two of which This is the preliminary version of the paper.

74

Fioralba Cakoni

are longitudinal elastic, one is transverse elastic, and two are thermal waves [7]. Consequently, there are four complex dimensionless parameters by means of which the different wave numbers are connected. The situation is much more complicated. In particular we shall show that the set of thermoelastic far field patterns corresponding to the scattering of the entire incident fields by a bounded thermoelastic rigid at zero temperature obstacle is dense in the production space of square integrable function on ∂Ω if and only if does not exist a eigenfunction of an eigenvalue problem which is a thermoelastic Herglotz function. This result will be established by first constructing an appropriate complete set of the function defined on the boundary of the scattering obstacle and then establishing an integral representation for the displacement part of the thermoelastic Herglotz function. In order to avoid the difficulties come from the existence of polarization of the transverse displacement wave, we have to raise the rank of the tensorial character of the fields involved by one. This idea is used by Twersky in electromagnetic scattering and after by Dassios in elastic case.

2

Scattering Problems

The direct scattering problem asks: given an open domain V ⊂ R3 with connected C2 boundary S and V e = R3 \ V , given a plane incident wave of time harmonic dependence e−iωt ˆ : V e → R4 Ui (r, k)

(1.cak )

ˆ the direction of propagation), determine in V e a solution (k ˆ = Ui (r, k) ˆ + Us (r, k) ˆ U(r, k) of the equation ˆ = ˜ r )U(r, k) L(∂

(µ∆ + ρω 2 )˜I3 + (λ + µ)∇∇· −γ∇ qκη∇· ∆+q

(2.cak )

ˆ u(r, k) ˆ Θ(r, k)

=0

(3.cak )

such that ˆ =0 ˜ k (∂r , n ˆ )U(r, k) B

(4.cak )

˜ k (∂r , n ˆ ) is expressed via the on S, where the boundary differential operator B thermoelastic surface traction operator. ˆ has to satisfy the asymptotic Kupradze condition as r → ∞ More over U(r, k) [7]. In the theory of direct problems in thermoelasticity [1,4], it is shown how the solution of a boundary value problem and related far-field corresponding to an incident field and to a given obstacle can be calculated. We call the set of vectors j P0 : Ω → C3 ; j = 1, 2, s = ˆ r, P 2 (ˆr, k)ˆ ˆ r, P s (ˆr, k) ˆ θˆ + P s (ˆr, k) ˆ φˆ = P 1 (ˆr, k)ˆ (5.cak ) r0

r0

θ0

φ0

75

Thermoelastic Far-field Patterns

defined on the unite sphere Ω, as set of thermoelastic far-field patterns corresponding to the thermoelastic radiation solution U(r), propagating in the diˆ The thermal component is cancelled in this definition because of linrection k. 1 ˆ (ˆr, k), ear dependence between the module of asymptotic displacement fields Pr0 2 1 2 ˆ ˆ ˆ Pr0 (ˆr, k) and the asymptotic thermal fields t0 (ˆr, k), t0 (ˆr, k) [1,4] with known coefficient. Let us do the following symbolic notation U∞ = Pj0 , j = 1, 2, s ∈ [L2 (Ω)]3 . In terms of the mapping F : U → U∞

(6.cak )

we want to solve the equation F U = U∞ We have proved in [1] by means of Atkinson expansion theorem the following result known as the correspondence theorem. There exists one to one correspondence between: 1 and elastothermal 4-dimensional part of radielastothermal far-field pattern Pr0 ation solution U1 ; 2 thermoelastic far-field pattern Pr0 and thermoelastic 4-dimensional part of radiation solution U2 ; s s transverse far-field patterns Pθ0 , Pφ0 and elastothermal 3-dimensional part of ras s diation solution U = (u , 0) . In other words, the mapping (6.cak ) is an one to one in its range. Theorem 1. The regular solution of thermoelastic equation (3.cak ) allows the following representation of the displacement part and the temperature part h λ + 2µ i γ λ + 2µ γ u(r) = ∇ − Φ + Θ − Φ + Θ 1 1 2 2 ρω 2 ρω 2 ρω 2 ρω 2 i h 1 ) + ∇ × rΨ (r) + ∇ × (rχ(r)) , (7.cak ks (8.cak ) Θ(r) = Θ1 (r) + Θ2 (r) , where the potentials Φ1 , Φ2 , Ψ, χ, Θ1 , Θ2 solve the scalar Helmholtz equation (∆ + k12 )Φ1 = (∆ + k12 )Θ1 = 0, (∆ + k22 )Φ2 (∆ + ks2 )Ψ

= (∆ + = (∆ +

k22 )Θ2 = 0, ks2 )χ = 0,

(9.cak ) (10.cak ) (11.cak )

where k1 , k2 , k3 are wave numbers. The prove is a consequence of Kupradze decomposition [7] by straightforward calculation. We write again the above relations as U = U1 + U2 + Us γ ∇[− λ+2µ ρω 2 Φ1 + ρω 2 Θ1 ] , (12.cak ) U1 = Θ1 γ ∇[− λ+2µ 2 Φ2 + ρω 2 Θ2 ] ρω U2 = , (13.cak ) Θ2 ∇ × [rΨ (r) + k1s ∇ × (rχ(r))] , (14.cak ) Us = 0

76

Fioralba Cakoni

where the irrotational and solenoidal part of the displacement fields are distinguished.

3

Herglotz thermoelastic function

We restrict so far to the rigid at zero temperature scattering thermoelastic problem. Definition 2. An thermoelastic Herglotz function is defined to be a classical solution of the thermoelastic equation (3.cak ) U(r) in all of R3 , which satisfies the growth condition Z 1 lim sup kU(r0 )k2 dv(r0 ) < ∞ . (15.cak ) r→∞ r B(o,r) Using orthogonality of the vector spherical harmonics we can easily verify Proposition 3. [L2 (Ω)]3 = [L2r (Ω)]3 ⊕ [L2t (Ω)]3 ,

(16.cak )

2 3 where [L2r (Ω)]3 is the subspace spanned by the set {Pm n } and [Lt (Ω)] is the m m subspace spanned by the set {Bn } ∪ {Cn }.

This implies that for every f ∈ [L2r (Ω)]3 we have the unique expansion in L2 sense f (ˆr) = fr (ˆr) + ft (ˆr) =

n ∞ X X

m [αm r)] + [βnm Bm r) + γnm Cm r)] . n Pn (ˆ n (ˆ n (ˆ

(17.cak )

n=0 m=−n

Moreover U(r) = fn (kr)Yn (ˆr) where fn (r) denotes any spherical Bessel function, is a solution of the Helmholtz equation (∆ + k 2 )U(r) = 0. Obviously, from these considerations we get Proposition 4. Every solution of thermoelastic equation in a regular domain satisfies the following unique decomposition. u(r) = u1 (r) + u2 (r) + us (r) =

∞ X n X

m [αm 1n L1n (r) +

n=0 m=−n m m m m m + αm 2n L2n (r) + βn Mn (r) + γn Nn (r)] ,

(18.cak )

Θ(r) = Θ1 (r) + Θ2 (r) = ∞ X n X m m δ1n fn (k1 r)Yn (ˆr) + δ2n fn (k2 r)Yn (ˆr) . =

(19.cak )

n=0 m=−n

The convergence is considered to be in the L2 sense.

77


The Navier elastostatic eigenvectors are given by Lm (1,2)n (r) =

1 k(1,2)

∇ fn (k(1,2) r)Yn (ˆr) = h =

d d(k(1,2) r)

i fn (k(1,2) r) Pm r) + n (ˆ +

p fn (k(1,2) r) m Bn (ˆr) , n(n + 1) k(1,2) r

p r) = n(n + 1)fn (ks r)Cm r) , Mm n (r) = ∇ × rfn (ks r)Yn (ˆ n (ˆ Nm n (r) =

1 ∇ × ∇ × rfn (ks r)Yn (ˆr) = ks fn (ks r) m Pn (ˆr) + = n(n + 1) ks r i p 1 h d (ks rfn (ks r)) Bm r) + n(n + 1) n (ˆ ks r d(ks r)

(20.cak )

(21.cak )

(22.cak )

and satisfy the vector Helmholtz equation 2 2 m 2 m )Lm (∆ + k(1,2) (1,2)n = (∆ + ks )Mn = (∆ + ks )Nn = 0 .

(23.cak )

We recall the result from [5] that there is a one to one correspondence between vector spherical harmonics and elastostatic eigenvectors. In the terminology that m m will be introduced latter, spherical harmonics Pm n , Cn , Bn are the vector Herm m m glotz kernels of the elastostatic eigenvectors Ln , Mn , Nn with fn = jn respectively. In other words, if the entire elastostatic eigenvectors are to be decomposed in plane waves propagating in all directions, then the corresponding vector spherical harmonics provide the distribution of the amplitudes over directions. The same between the solid harmonics and spherical harmonics. The behavior of Herglotz solution of thermoelastic equation (3.cak ) and in particular the connection of the its displacement part with the far-field patterns they generate are the main subject of the following. Let U : R3 → C4 an Herglotz function that satisfies the thermoelastic equation in the classical sense. Then, by the completeness of the elastostatic eigenvectors and spherical harmonics and the general theory of eigenfunction expansions we obtain for the displacement and temperature part the expansions (18.cak ), (19.cak ) respectively. The asymptotic analysis of the above expansions as r → ∞ gives for the far-field patterns generated by the Herglotz solution the following formulas L1 (ˆr) =

n ∞ ik1 X X −n m m i α1n Pn (ˆr) , 2 n=0 m=−n

(24.cak )

L2 (ˆr) =

n ∞ ik2 X X −n m m i α2n Pn (ˆr) , 2 n=0 m=−n

(25.cak )

78

Fioralba Cakoni

T(ˆr) =

∞ n 1 X X np i n(n + 1)[βnm Cm r) + iγnm Bm r)] n (ˆ n (ˆ 2 n=0 m=−n

(26.cak )

and for the temperature part l1 (ˆr) =

∞ ∞ n n ik1 X X −n m m ik2 X X −n m m i δ1n Yn (ˆr), l2 (ˆr) = i δ2n Yn (ˆr) . (27.cak ) 2 n=0 m=−n 2 n=0 m=−n

By virtue of the Riesz-Fisher theorem and the relations involving the Fourier coefficients of (18.cak ), (19.cak ) we claim that the far field patterns L(1,2) , T are well defined in the L2 -sense. The coefficient of the linear dependence between the qκηik see [1]. displacement far fields and temperature far fields is k2 (1,2) −q (1,2)

The most important result in the theory of Herglotz functions is given by the following representation theorem. Theorem 5 (Representation). If U is an Herglotz solution of the thermoelastic equation (3.cak ), then there are functions L1 , L2 , T : Ω → C4 which belongs to L2 (Ω) (i.e. the corresponding far field patterns), such that Z Z 1 ˆ ˆ ˆ k1 ik·r ˆ + 1 ˆ k2 ik·r ˆ + L1 (k)e ds(k) L2 (k)e ds(k) U(r) = 2π Ω 2π Ω Z 1 ˆ ˆ ks ik·r ˆ . T(k)e ds(k) (28.cak ) + 2π Ω Conversely, if U is given by L1 , L2 , T in L2 (Ω), then it is a thermoelastic Herglotz function. The L2 functions L1 , L2 , T are known as the Herglotz kernels. The proof argument of the first part is the interpretation of the series (24.cak ), (25.cak ), (26.cak ), (27.cak ) as the corresponding Herglotz kernels. Using orthogonality arguments of the considered eigenfunctions and the uniform convergence we obtain the growth condition of the Herglotz function provided the L2 Herglotz kernels exist. Theorem of the representation furnishes a proof that U is uniquely determined by L1 , L2 , T. This result is also obtainable from the unique determination of the Fourier coefficients of the expansions for U and L1 , L2 , T in the appropriate eigenvectors for both components, displacement and temperature.

4

Dense properties of the far field patterns

Let us turn back to the functional equation (6.cak ). The correspondence theorem provides the uniqueness of its solution. As we proved the thermoelastic far field patterns (which are defined as the displacement amplitudes) must satisfy the expantion (24.cak ), (25.cak ), (26.cak ). By using the technique of Colton, Kress [2] (theorem 2.15) and the fact that the elastostatic eigenvectors are expressed via spherical wave functions we easily verify that the existence of a solution requires a kind of growth condition of the Fourier coefficients (it is analytically complicated that is

79


why we do not present it here)to be satisfied, for a given function U∞ ∈ [L2 (Ω)]3 . So, the solution of equation (6.cak ) will, in general, not exist. The argument of [2, p. 36] is valid here to claim more over that, if a solution U does exist it will not depend continuously on U∞ in any reasonable norm. That is that the equation (3.cak ) is ill-posed. The image of the linear operator F is not equal to [L2 (Ω)]3 . But, is the far field patterns for a fixed rigid obstacle at zero temperature and all incident thermoelastic plane waves complete in [L2 (Ω)]3 ? In order to have an answer of this question we need the following dense result. To avoid long repetitions of requirements upon fields, we introduce the following spaces: – the space of incident Herglotz-type field n H(R3 ) = Φ : R3 → R4 Φ(r) = Z o ˆ ˆ ˆ ˆ ik1 k·r ˆ ik2 k·r ˆ iks k·r = L1 (k)e + L2 (k)e + T(k)e ; L1 , L2 , T ∈ L2 (Ω)

(29.cak )

Ω

– the space of scattered fields e S(V e ) = U : V e → R4 U ∈ C 2 (V e ) ∩ C(V ) s.t. U satisfies the equation (3.cak ) and the Kupradze condition at ∞}

(30.cak )

– the space of rigid at zero temperature solutions P(V e ) = {Ψ = U + Φ : V e → R4 | U ∈ S, Φ ∈ H, r ∈ ∂V Ψ (r) = 0}

(31.cak )

– the space of traction traces ˜ ˜ : ∂V → R4 ; Ψ ∈ P RP(∂V ) = RΨ

(32.cak )

˜ Theorem 6. The space RP(∂V ) is dense in L2 (∂V ) Proof. From [6, Theorem 2] we have that if g ∈ L2 (∂V ) and Z ˜ r, n ˜ ˆ )Ψ (r)ds(r) = 0 g · T(∂ (T Ψ, g) = ∂V

˜ for every T˜Ψ ∈ RP(∂V ) · ˜I3 , then g = 0 almost everywhere on ∂V . The same for the operator ∂n . Now, let us consider a 4-dimensional vector G ∈ L2 (∂V ). ˜ ˆ ) −γ n ˆ R(∂r , n ˜ r, n ˆ) = implies that G = 0 if for The shape of operator R(∂ 0 ∂n ˜ ∈ RP(∂V ˜ every RΨ ), Z ˜ r, n ˜ G) = ˆ )Ψ (r)ds(r) = 0 G · R(∂ (RΨ, ∂V

which ends the proof of the theorem.

80

Fioralba Cakoni

It is well known that there are three types of plane displacement fields and tow types of plane temperature fields that can propagate in a thermoelastic medium. The displacement waves depend on two orthogonal vectors which are ˆ and p ˆ respectively for two types of longitudinal waves and one type ˆ ⊥ k k of transversal wave. These reflects the same vectorial nature of the far field patterns (5.cak ). The existence of two perpendicular vector complicates the study of the far field patterns. To avoid this difficulty we raise the rank of the tensorial character of the fields involved by one. Then the incident displacement fields depend only on one vector, the direction of propagation, while the transverse polarization vector is now replaced by 2-dimensional complement of the direction of propagation. Our tensorial thermoelastic model is as following. The scatterer is exited by a tensorial 4 × 3 thermoelastic time harmonic wave ˆ ˆ ˆ = A1 (k ˆ ⊗ k, ˆ β1 k)e ˆ ik1 k·r ˆ ⊗ k, ˆ k)e ˆ ik2 k·r ˜ i (r; k) + A2 (β2 k + U ˆ

ˆ ⊗ k, ˆ 0)eiks k·r . + As (˜I3 − k

(33.cak )

The tensorial incident field can be interpreted as a tensor superposition of three vector fields which appear as the first vectors of tensors, while the second vectors ˆ θˆk , ϕˆk }. This tensorial character are provided by the incident orthogonal base {k, ˜ s = (us , Θs ) it generates of the incident field is inherited in the scattered field U ˆ = us (r; k, ˆ k) ˆ ⊗k ˆ + us (r; k, ˆ k) ˆ ⊗k ˆ+ ˜ s (r; k) u 1 2 s s ˆ θˆk ) ⊗ θˆk + u (r; k, ˆ ϕˆk ) ⊗ ϕˆk , + u (r; k, ˆ = Θ (r; k) s

s ˆ k ˆ Θ1s (r; k)

s

+

(34.cak )

ˆ k ˆ. Θ2s (r; k)

Then the total tensorial field ˆ +U ˜ s (r; k) ˆ ˜ k) ˆ =U ˜ i (r; k) U(r;

(35.cak )

solve the tensorial thermoelastic coupled system ˆ + (λ + µ)∇ ⊗ ∇ · u ˆ = γ∇ ⊗ Θ(r; k) ˆ , ˜ (r; k) µ∆˜ u(r; k) ˆ = −iω∇ · u ˆ ˆ + iω Θ(r; k) ˜ (r; k) ∆Θ(r; k) κ

(36.cak ) (37.cak )

ˆ =0 ˜ k) ˜ and the same Kupradze asymptotic conditions as r → ∞. More over U(r; on S. The asymptotic analysis uniform over Ω leads to the tensorial shape of far field patterns j ˜∞ = P ˜ , j = 1, 2, s ∈ [L2 (Ω)]9 . U (38.cak ) 0 ˜ 2 are the lon˜ 1, P Note that in accord with known results the radial patterns P 0 0 s ˜ gitudinal wave of displacement part propagating along ˆr and P0 is transversal spherical wave propagating along ˆr and polarized orthogonally to ˆr. Obviously,


81

the definition and the results of thermoelastic Herglotz function may be translated to the terms of tensors. Everything remains true provided the vectors are replaced by tensors. An answer of the question whether is the set of far field patterns complete in [L2 (Ω)]3 is given by the following theorem ˆ n ) be a sequence of unit vectors that is dense on Ω. Theorem 7. Let (k ˆn) ˜ ∞ (ˆr, k [L2 (Ω)]9 = spanU if and only if there does not exist a Herglotz thermoelastic function 1 u u + u2 + us U= = Θ1 + Θ2 Θ

(39.cak )

(40.cak )

such that ˆ + u2 ⊗ k ˆ + us ⊗ θˆ + us ⊗ ϕˆ ˜ = u1 ⊗ k u is an eigenfunction of the interior eigenvalue problem ∗ u=0 (∆ + ω 2 )˜ in V ˜=0 ∇·u ˜ = 0 on S u

(41.cak )

(42.cak )

where ∆∗ is elastostatic operator. Proof. Recall that W is complete in the Hilbert space X if and only if (w, ϕ) = 0 for all w ∈ W implies that ϕ = 0. Let us write the dual relation in the space [L2 (Ω)]9 for every ˜l1 , ˜l2 , ˜t ∈ [L2 (Ω)]9 Z Ω

ˆn) + P ˆn) + P ˆn ) : ˜ 2 (ˆr; k ˜ s (ˆr; k ˜ 1 (ˆr; k P r0 r0 t0 ˆ n ) + ˜l2 (ˆr; k ˆ n ) + ˜t(ˆr; k ˆ n ) ds(ˆr) = 0 ˜, : ˜l1 (ˆr; k

which in a simpler symbolic way should be written Z ˆ n )K(ˆ ˜ r; k ˆ n )ds(ˆr) = 0 ˜∞ (ˆr; k ˜, U

(43.cak )

(44.cak )

Ω

where ˆr ∈ Ω and n = 1, 2, . . . The relation (43.cak ) implies that exists a nontrivial ˜ i , (with kerthermoelastic Herglotz function, which we take as incident wave U 2 9 ˜ ˜ ˜ nels for displacement part l1 , l2 , t ∈ [L (Ω)] ) for which the far field patterns of ˜ ∞ = 0. By the one to one correspon˜ s is U the corresponding scattered wave U dence between the radial solution and corresponding far field patterns we have ˜ s = 0 in V e . By the ˜ ∞ = 0 on Ω is equivalent to U that the vanishing far field U s i ˜ ˜ boundary condition U + U = 0 on S and the uniqueness Dirichlet thermoelastic ˜ i = 0 on S. Again by Kupradze eigenvector problem [7], this is equivalent to U

82

Fioralba Cakoni

[7], the first thermoelastic eigenvalue problem is equivalent with the eigenvalue problem (42.cak ). The above result gives the basic tool to construct a Colton-Monk type algorithm for the thermoelastic inverse scattering problem corresponding to a rigid scatterer at zero temperature of the reconstruction the structures. This inverse frame means: given the thermoelastic incident field Ui ∈ H and by the knowledge of the far field patterns U∞ correspond to the scattered field Us ∈ S(V e ), Dirichlet boundary condition Ψ = 0, Ψ ∈ P(V e ) and the equations (3.cak ) which govern the phenomena, one determines the geometrical shape of the boundary ∂V . Apriori conditions are given about the boundary, for example to be star shape.

References [1] F. Cakoni, G.Dassios, The Atkinson-Wilcox expansion theorem for thermoelastic waves, (accepted to be published in Quart. Appl. Math.). [2] D. Colton, R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, (1992). [3] D. Colton, A. Kirsch, Dense sets and far-field patterns in acoustic wave propagation, SIAM J. Appl. Math., 15 (1984), 996–1006. [4] G. Dassios, V. Kostopoulos, The scattering amplitudes and cross sections in the theory of thermoelasticity, SIAM J. Appl. Math., 48, 1 (1988), 1283–1284. [5] G. Dassios, Z.Rigou, Elastic Herglotz function, SIAM J. Appl. Math., (in print). [6] G. Dassios, Z.Rigou, On the density of traction traces in scattering of elastic waves, SIAM J. Appl. Math., 53/1 (1993), 141–153. [7] V. Kupradze, Tree-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland, New York. (1979).


PAPERS pp. 83–92

Transformations of Differential Equations Jan Chrastina Department of Mathematics, Faculty of Science, Masaryk University, Jan´ aˇckovo n´ am. 2a, 662 95 Brno, Czech Republic

Abstract. The article concerns the second order differential equations with one unknown function and the aim is twofold: to compare some results for the well-known linear with the more complicated nonlinear case, and to point out some distinctions between ordinary and partial differential equations. We shall mention automorphisms permuting the conjugate points, moving frames for particular fiber-preserving mappings, the Darboux transformations of ordinary differential equation, and the Laplace series for the hyperbolic case of two independent variables.

AMS Subject Classification. 34K05, 35A30, 35L10

Keywords. Dispersions, contact transformation, moving coframe, diffiety, Laplace series

Our reasoning will be developed in the real smooth category. As a rule, we shall not specify the definition domains and our primary aim is to outline some new ideas and methods rather than to derive certain definite theorems. For the convenience of a possible reader, let us outline the contents. We begin with the family of all equations (2.chr ). It is preserved if transformations (1.chr ) are performed, and certain self-transformations of this kind (so called central dispersions) of a given equation (2.chr ) are determined by the location of roots of solutions: they permute the roots. This result can be easily adapted for the nonhomogeneous family (5.chr ) subjected to a broader class (6.chr ) of transformations, then the intersections of solutions undertake the previous role of roots. These wellknown results can be verified by a manner which can be carried over the class of all nonlinear equations (7.chr ) subjected to contact transformations. In particular, certain automorphisms of a given equation (7.chr ) exist which permute the intersection points of infinitesimally near couples of solutions. They may be regarded for nonlinear generalization of dispersions. Our next aim is to determine some subfamilies of the family of all equations (7.chr ) which are preserved if all transformations of the kind (6.chr ) are applied. We use the moving frames. On the other hand, a given equation (2.chr ) can be transformed into the family of all equations (2.chr ) by many rather peculiar transformations, even if the independent variable x is kept fixed. They can be explicitly found and the famous This is the final form of the paper.

84

Jan Chrastina

Darboux transformation of eigenvalue problems appears as a very particular subcase. As yet the transformations did not much change the order of derivatives. However passing to partial differential equations, already the classical Laplace series has quite other properties: invertible transformations of not too special equations (16.chr ) exist where the new unknown function U (x, y) may depend on derivatives of arbitrarily high order of the primary unknown u(x, y). This is a well-known result but we again pass to a nonlinear generalization: the Laplace coframes permit to determine all invertible mappings of a given equation (18.chr ) into the class of all equations (18.chr ), at least in principle since the calculations are rather complicated. The cases when the independent variables need not be preserved are involved. We can state only a modest illustrative example of the equation ∂ 2 u/∂x∂y = g(∂u/∂x) + u here with new independent variables X = x − g 0 (∂u/∂x), Y = y and new unknown function (23.chr ). The method can be generalized and applied to higher order equations, as well.

1

The dispersion theory [1], [7]

We find ourselves in the plane x, y, where new variables X = X(x, y), Y = Y (x, y) can be introduced. In particular transformations of the kind X = X(x),

Y = c|X 0 (x)|

1/2

y

(c = const. 6= 0, X0 (x) 6= 0)

(1.chr )

are the most general ones which preserve the family of all equations d2 y/dx2 = q(x)y,

(2.chr )

i.e., which turn every equation (2.chr ) into certain d2 Y /dX 2 = Q(X)Y . It may be proved that under transformations (1.chr ), equations (2.chr ) are locally like each other. Roots of solutions y are obviously transformed into roots of solutions Y and this trivial remark can be developed to give the global theory. In particular, in the oscillatory subcase, there exist automorphisms (1.chr ) of equation (2.chr ) permuting the roots of solutions y, the so called central dispersions of (1.chr ). Since the transformations (1.chr ) between two mentioned equations can be determined as solutions of a certain nonlinear third order differential equation (depending on q,Q) for the function X, it follows in the particular case of automorphisms that the distribution of roots of solutions y is governed by a third order differential equation.

2

A note to proofs [3]

The shortest way to the mentioned results consists in introduction of function ζ = y¯/y, where y¯, y are two independent solutions of (1.chr ). The value ∞ at the roots of y with obvious rules of calculations should be admitted. Then 1 1/2 −1/2 00 |ζ 0 | , (3.chr ) ζ 0 = c/y 2 , y = (c/ζ 0 ) /2, q = y 00 /y = |ζ 0 |

85

Transformations of DE

where c = const. 6= 0 and the last expression is the familiar Schwarz derivative independent of the choice of y¯, y. Conversely, every such a function ζ(x) ) is with ζ 0 (x) 6= 0 can be arbitrarily chosen in advance. Then the equation (2.chr determined by (3.chr ), automorphisms (1.chr ) of (2.chr ) are (obviously) given by formula 3 ζ (X(x)) =

αζ(x) + β γζ(x) + δ

constants with αδ 6= βγ)

(α, β, γ, δ

which provides a third order equation for the function X(x) by applying the Schwarz derivative, and the central dispersions appear as a particular subcase ζ (X(x)) = ζ(x). Continuing in this way, analogous function Z = Y¯ /Y and the formula Z(X) =

αζ(x) + β γζ(x) + δ

(α, β, γ, δ

constants with

αδ 6= βγ)

(4.chr )

(obviously) provides all transformation into the equation d2 Y /dX 2 = Q(X)Y .

3

Nonlinear dispersions

The above results can be carried over the broader family of all equations d2 y/dx2 = q(x)y + r(x)

(5.chr )

subjected to the transformations of the kind X = X(x), Y = c|X 0 (x)|

1/2

y + Z(x) (c = const. 6= 0, X 0 (x) 6= 0).

(6.chr )

The previous role of the roots of solutions is undertaken by the points of intersection of pairs of solutions in this non-homogeneous case. We shall be however interested in still broader family of all nonlinear equations d2 y/dx2 = f (x, y, dy/dx).

(7.chr )

It may be easily seen that contact transformations X = X(x, y, y 0 ),

Y = Y (x, y, y 0 ),

Y 0 = Y 0 (x, y, y 0 )

(8.chr )

are the most general ones which preserve the family (7.chr ), that is, which turn every equation (6.chr ) into certain d2 Y /dX 2 = F (X, Y, dY /dX). (Indeed, owing to (8.chr ), differential form dY − Y 0 dX should be a linear combination of forms dy − y 0 dx and dy 0 − f dx with arbitrary f , hence a multiple of dy − y 0 dx.) It may proved that under contact transformations, equations (7.chr ) are locally like each other. Instead of common methods, we shall derive this well-known result by a geometrical reasoning which will be subsequently related to (nonlinear) dispersions.

86

Jan Chrastina

Let y = y(x, a, b) be a complete solution of (7.chr ). Keeping a, b fixed for a moment, choose a near solution with a common point x ¯, y¯ . In other terms, we suppose y¯ = y(¯ x, a, b) = y(¯ x, a + ε, b + δ),

y¯0 = yx (¯ x, a, b).

(9.chr )

Analogously let Y = Y (X, A, B) be a complete solution of the equation d2 Y /dX 2 = F . Choose fixed A = A(a, b), B = B(a, b) such that there exists a common ¯ Y¯ with the corresponding near solution, that means, we may write point X, ¯ A, B) = Y (X, ¯ A(a + ε, b + δ), B(a + ε, b + δ)), Y¯ = Y (X, ¯ A, B). Y¯ 0 = Y X(X,

(10.chr )

Keeping ε, δ fixed but a, b (hence x¯, y¯) variable, the invertible transformation ¯ Y¯ , Y¯ 0 ) appears. If ε, δ = δ(ε) −→ 0, we obtain even a con(¯ x, y¯, y¯0 ) −→ (X, tact transformation (as follows by simple geometrical arguments or by direct verification) implicitly given by formulae ¯ A, B), Y¯ 0 = Y X(X, ¯ A, B), Y¯ = Y (X, ¯ A, B)(Aa + λAb ) x, a, b) + λyb (¯ x, a, b) = 0 = YA (X, ya (¯ ¯ A, B)(Ba + λBb ), + YB (X,

y¯ = y(¯ x, a, b),

y¯0 = yx (¯ x, a, b),

where A = A(a, b), B = B(a, b), λ = δ 0 (0) and the parameters a, b, λ should be eliminated. Since every curve y = y(x, a, b) is (obviously) transformed into the curve Y = Y (X, A, B), the equation (7.chr ) turns into d2 Y /dX 2 = F . We shall mention two particular kinds of this construction. Assuming f (x, y, y 0 ) = F (x, y, y 0 ), we deal with automorphisms of equation (7.chr ). Since the functions A = A(a, b), B = B(a, b) can be (in principle) quite arbitrarily chosen, there is a huge family of them. In the case of oscillatory equation, the simple choice A = a and B = b gives (besides the identity) the automorphisms permuting the conjugated points: the common point x ¯, y¯ of two ) with ε, δ near to zero) can be transformed infinitesimally near solutions (cf. (9.chr 1 ¯ Y¯ of the same pair of solutions (cf. (10.chr into the next intersection point X, 1 ) with Y = y, A = a, B = b). So we have a nonlinear generalization of dispersions. Assuming f (x, y, y 0 ) = q(x)y, F (X, Y, Y 0 ) = Q(X)Y , we deal with the equation (2.chr ) and the above construction gives (besides the contact transformations) the point transformations (1.chr ) for a particular choice of functions A = A(a, b), B = B(a, b). In more detail, let y = a¯ y(x) + by(x),

Y = AY¯ (X) + BY (X)

be complete solutions in our linear case of equations. For our point transforma¯ Y¯ , Y¯ 0 ), the couple (X, ¯ Y¯ ) should depend only on (¯ tion (¯ x, y¯, y¯0 ) −→ (X, x, y¯) 0 , 10.chr ) in our particular case: and not on y¯ . Recall formulae (9.chr 1 1 y¯ = ay(¯ x + b¯ y(¯ x),

¯ + B Y¯ (X). ¯ Y¯ = AY (X)

(11.chr )

87


Choosing arbitrary A = αa − β, B = γa − δb, where αδ 6= βγ, and assuming y¯ = Y¯ = 0 for a moment, it follows ζ(¯ x) = −a/b hence ¯ =− Z(X)

αa − βb αζ(¯ x) + β A =− = B γa − δb γζ(¯ x) + δ

by using notation of Section 2. However this is just formula (4.chr ) and we already 1/2 ¯ = X(¯ ¯ x) completed by Y¯ = c|X ¯ 0 (¯ know that such functions X x)| y¯ provide 2 2 transformations into d Y /dX = Q(X)Y .

4

The moving frames method [2]

Our aim is to determine some kinds of the second order differential equations (7.chr ) which are preserved under the family of all transformations (6.chr ). For better clarity, we shall deal with the pseudogroup of all transformations (6.chr ), where X 0 (x) > 0. Then dX = u2 dx,

dY = vdx + cudy

(u2 = X 0 , v = cX 00 y/2u + Z 0 )

and it follows (from group composition properties) that two families of forms ω1 = u2 dx,

ω = vdx + cudy

(u, v are parameters)

(12.chr )

are preserved by mappings (6.chr ). One can verify that the converse is also true: transformations (8.chr ) preserving families (12.chr ) are just of the kind (6.chr ). On the other hand, the system dy − y 0 dx = dy 0 − f dx = 0

turns into

dY − Y 0 dX = dY 0 − F dX = 0

and it follows that two families of forms ω ¯ = λ(dy − y 0 dx),

¯ = µ(dy 0 − f dx) + ν¯(dy − y 0 dx), ω

where λ, µ, ν¯ are new variables make the intrinsical sense: they are transformed into the relevant “capital families”. Comparing ω with ω ¯ (hence cu = λ, v = −λy 0 ), we obtain better intrinsical family of forms ω2 = cu(dy − y 0 dx), ω3 = µ(dy 0 − f dx) + νω2 , where u, µ, ν = ν¯/cu are new variables (and c is an unknown constant). So we occur ourselves in the space x, y, y 0 , u, µ, ν, equipped with intrinsical families of forms ω1 , ω2 , ω3 . Exterior derivatives are intrinsical, too. However dω1 = 2ω4 ∧ ω1 with the most general factor ω4 =

du − ξω1 u

(ξ a new parameter)

88

Jan Chrastina

which provide still an intrinsical family. Analogously dω2 =

du dy 0 cν c ∧ ω2 − c ∧ ω1 = ω4 ∧ ω1 + (ξ − )ω1 ∧ ω2 + ω1 ∧ ω3 u u uµ uµ

and we may introduce intrinsical restrictions c/uµ = 1, ξ = cν/uµ hence uµ = c, ξ = ν (then dω2 = ω1 ∧ (ω3 − ω4 )). In the same manner dω3 = ω5 ∧ ω2 +

∂f /∂y 0 ω1 ∧ ω3 − ω4 ∧ ω3 , u2

where ω5 = dν + βω1 + γω2 + 2νω4

(13.chr )

ν ∂f µ ∂f 2 − 2 β=ν + 3 cu ∂y u ∂y 0

is intrinsical family with a new variable γ. However dω4 = −(dξ + 2ξω4 ) ∧ ω1 = (γω2 − ω5 ) ∧ ω1 owing to ξ = ν, and we may suppose γ = 0. Returning to (12.chr ), we have to distinguish two subcases A : ∂f /∂y 0 = 0, B = ∂f /∂y 0 6= 0. It follows that the family of all equations d2 y/dx2 = f (x, y) is preserved by transformations (6.chr ), and one can directly verify that other transformations do not have such property. Assuming A, then dω5 = dβ ∧ ω1 + 2βω4 ∧ ω1 + 2(dν ∧ ω4 − νω5 ∧ ω1 ) = 2ω5 ∧ ω4 + ζ ∧ ω1 , where ζ ∼ = dβ+4βω4 −2νω5 (mod ω1 ) is intrinsical form. However β = ν 2 +f y/u4 (use uµ = c) therefore ζ ∼ = fyy ω2 /cu5 after short calculation and we have to distinguish two subcases C : fyy = 0, D : fyy 6= 0 of our case A. Subcase B is the classical one f = q(x)y + r(x) mentioned above. Surveying the results, we have structural formulae dω1 = 2ω4 ∧ ω1 , dω2 − ω1 ∧ (ω3 − ω4 ) dω3 = ω5 ∧ ω2 − ω4 ∧ ω3 , dω4 = ω1 ∧ ω5 , dω5 = 2ω5 ∧ ω4 of a Lie group of automorphisms of an equation (5.chr ) and, since invariants are lacking, all equation (5.chr ) are (locally) like each other with respect to transformations (6.chr ) which is the already mentioned result. In subcase D, we may introduce the requirement cu5 = ∂ 2 f /∂y 2 which implies 5cu4 du = fyyx dx + fyyy dy or, in terms of intrinsical forms, 5cu5 (ω4 + νω1 ) =

∂ 3 f ω1 ∂ 3 f ω2 0 ω1 + y . + ∂x∂y 2 u2 ∂y 3 cu u2

It follows 5ω 4 = M ω1 + N ω2 with intrinsical coefficients. In particular ∂3f N= ∂y 3

∂2f ∂3f cu 2 = ∂y ∂y 3

2 6/5 4/5 ∂ f c ∂y 2

89


does not change after transformations (6.chr ). The constant c is not fixed here. Consequently if N is a conical (containing constant multiples)set of functions g(x, y), then the family of all equations d2 y/dx2 = f (x, y) such that N ∈ N 5 6 ) are (equivalently: (fyyy ) /(fyy ) ∈ N) is preserved when transformations (6.chr applied. Possibly some narrower families could be obtained by using coefficient M but we shall not continue further. Let us conclude with the remaining subcase B. Owing to (13.chr ), we may assume u2 = ∂f /∂y 0 and, analogously as above, one can obtain an identity of the kind ω4 = M ω1 + N ω2 + P ω3 with intrinsical coefficients. We shall mention only the simplest one 1/2 ∂2f ∂2f ∂f 2 P = 2µu 2c = , ∂y 02 ∂y 02 ∂y 0 which yields the following result: if P is a conical set of functions g(x, y, y 0 ) then the family of all equations d2 y/dx2 = f (x, y, y 0 ) such that P ∈ P (equivalently: ) are applied. fy20 y0 /fy0 ∈ P) is preserved when transformations (6.chr

5

On the Darboux transformation [4]

There exist many mappings (8.chr ) of the space x, y, y 0 which transform a given (single) equation (2.chr ) into an equation d2 Y /dX 2 = Q(X)Y . For the sake of brevity, we shall mention only the particular case q(x) = 0 and the x-preserving mappings (hence X = x) with Y = Y (x, y, y 0 ) arbitrary. One can then find Y 0 = ∂Y /∂x + y 0 ∂Y /∂y and the requirement Q(x)Y = ∂ 2 Y /∂x2 + 2y 0 ∂ 2 Y /∂x∂y + y 02 ∂ 2 Y /∂y 2

(14.chr )

for the function Y = Y (x, y, y 0 ). Denoting by Y = φ(x), Y = ψ(x) two linearly ) is satisfied if independent solutions of equation d2 Y /dX 2 = Q(X)Y , then (14.chr Y (x, y, y 0 ) = α(y − y 0 x, y 0 )φ(x) + β(y − y 0 x, y 0 )ψ(x), where α, β are arbitrary functions. On the other hand, let us suppose the conjecture Y = A(x)y + B(x)y 0 which leads to useful particular results. Then (14.chr ) gives Q(Ay + By 0 ) = A00 y + B 00 y 0 + 0 0 00 00 2y A and if follows QA = A , QB = B + 2A0 whence A00 B = A(B 00 + 2A0 )

(15.chr )

by elimination of function Q. Requirement (15.chr ) can be explicitly resolved. In particular, for the choice A = 1, one obtains the famous transformation of equation y 00 = 0 into the nontrivial equations with Q(x) = cos−2 x or Q(x) = cosh−2 x. The general case of the equation(2.chr ) can be investigated by the same manner, of course. Then the above conjecture leads to slight generalizations of the familiar Darboux transformation.

90

6

Jan Chrastina

The Laplace series [5], [6]

Turning to partial differential equations, we begin with the general linear hyperbolical equation ∂u ∂u ∂2u = a(x, y) + b(x, y) + c(x, y)u ∂x∂y ∂x ∂y

(16.chr )

to transparently illustrate the most important distinctive feature, the possibility of higher order invertible transformations. One can verify that the function U = uy − au satisfies a certain equation Uxy = AUx + BUy + CU of the same kind as (16.chr ), and the iteration provides an infinite series of higher order transformations in the family of equations of the kind (16.chr ). Moreover, if ax + ab 6= c then analogous substitution with variables x, y exchanged yields the inversion. (The exceptional case ax + ab = c is much easier and may be omitted: then (16.chr ) can be replaced by certain first order linear equations.) So we obtain an infinite in both direction series of invertible substitutions in the general case, the Laplace series. (The equation (16.chr ) moreover admits a change X = X(x), Y = Y (y) of independent variables and a linear change of function u; these are however well-known adaptations. In general, together with the Laplace series, invertible transformations do not exist, see below.) The existence of higher order substitutions is possible thanks to the fact that equation (16.chr ) is considered in the infinite-dimensional space with coordinates x, y, u, ur ≡ ∂ r u/∂xr , us ≡ ∂ s u/∂y s

(r, s > 0).

(17.chr )

Other derivatives usr ≡ ∂ r+s u/∂xr ∂y s (r, s > 0) can be expressed in terms of them by virtue of the equation (16.chr ) and its derivatives.

7

The Laplace coframe

Passing to the nonlinear case, we shall mention a hyperbolical equation 2 ∂2u ∂ u ∂ 2 u ∂u ∂u =f , , u, x, y , , ∂x∂y ∂x2 ∂y 2 ∂x ∂y

(18.chr )

again in the space of variables (17.chr ). Since then the contact transformations can be applied, it follows that the previous prominent role of variables x, y, u lost the sense: it is better to employ the contact form ω = du − u1 dx − u1 dy and in general the higher order contact forms should replace the functions usr . Quite analogously the characteristic vector fields Z+ = a+ ∂ + b+ δ, Z− = a− ∂ + b− δ, where a+ , b+ anda− , b− are (real and distinct) roots of the equation a2 ∂f /∂u2 + ab + b2 ∂f /∂u2 = 0, and ∂ = ∂/∂x + δ = ∂/∂y +

∞ X

X

ur+1 ∂/∂ur + X

r−1

∞ X

f ∂/∂ur +

Y s−1 f ∂/∂us,

∞ X

us+1 ∂/∂us

(19.chr )


91

are so called formal derivatives will replace the previous derivative operators ∂/∂x, ∂/∂y in equation (16.chr ). In full detail, (18.chr ) expressed in terms of contact forms means that ω11 =

∂f ∂f 2 ∂f ∂f 1 ∂f ω ω2 + ω + ω1 + ω + ∂u2 ∂u2 ∂u1 ∂u1 ∂u

(20.chr )

(we abbreviate ωr0 ≡ ωr , ω0s ≡ ωs ) and using the Lie derivatives L satisfying s L∂ ωrs ≡ ωr+1 , Lδ ωrs ≡ ωrs+1 , one can express the last identity in the manner LZ − LZ + ω = aLZ − ω + b LZ + ω + cω

(21.chr )

quite analogous to (16.chr ). (We shall not state rather clumsy formulae for coefficients a, b, c in terms of the function f .) Then the procedure of Section 6 can be simulated in terms of contact forms: the form Ω = LZ + ω − aω satisfies a certain identity LZ − LZ + Ω = ALZ − Ω + BLZ + Ω + CΩ

(22.chr )

and the procedure can be iterated. Moreover, if Z− a + ab 6= c, analogous substitution with Z+ , Z− exchanged yields the inversion. In general, one obtains an infinite in both direction series of certain differential forms which constitute a basis in the module of all contact forms, the Laplace coframe.

8

Applications

If the equation Ω = 0 can be expressed by five functions, that means, the PfaffDarboux shape is dU − P dX − QdY = 0 for appropriate U, X, Y, P, Q, then the functions X, Y may be regarded for independent variables and U for new unknown satisfying a certain second order equation (as follows from (22.chr )). The same conclusion can be made for any other term of the Laplace coframe and it may be proved that all possible invertible transformations into some second order equation arise only in this manner. Other applications as the Darboux method, representation of solution of an equation (18.chr ) by means solutions of another such equation, B¨ acklund correspondences, are also possible.

9

Example

One can easily investigate the Laplace coframes for the linear equations (16.chr ) and verify that they give the well-known Laplace series of transformations. In general, Laplace coframes are rather complicated. Therefore we shall mention only few results concerning the particular equation ∂ 2 u/∂x∂y = g(∂u/∂x) + u for a transparent example. Identity (20.chr ) means that ω11 = g 0 ω1 + ω and (since Z− = ∂, Z+ = δ are formal derivatives in our case and therefore a = g 0 , b = 0, c = 1 in identity (21.chr ) we have to introduce the form Ω = ω 1 − g 0 ω. One

92

Jan Chrastina

can find the Pfaff-Darboux shape Ω = dU − P dX − QdY of this form, where X = x − g 0 (u1 ), Y − y may be taken for new independent variables and Z U = u1 − ug 0 (u1 ) − G(u1 ) (G(u1 ) = (g − u1 g 0 )g 00 du1 ) (23.chr ) for new unknown function. Moreover coefficients P = ∂U/dX = g(u1 ) + u − u1 g 0 (u1 ),

Q = ∂U/∂Y = u2 − u1 g 0 (u1 )

(24.chr )

may be identified with new partial derivatives.Then, looking at the differential dP , one can find (surprisingly simple) formulae ∂ 2 U/dX 2 = u1 , ∂ 2 U/∂X∂Y = u1

(25.chr )

which means that functions (23.chr , 24.chr , 25.chr ) are related by the equation 2 2 ∂ 2 U 0 ∂ 2U ∂2U ∂U ∂2U ∂ U ∂ U 0 =U+ −g + g +G . g 2 2 2 2 ∂X∂Y ∂X ∂X ∂X ∂X ∂X dX 2 Analogous transformation with the role of x, y exchanged leads to the form Ω = ω1 , the independent variables are preserved, and the new unknown function U = u1 (obviously) satisfies ∂ 2 U/∂x∂y = g 0 (U )∂U/∂x + U . Modulo contact transformations, these are the only possible first order invertible transformations of the equation under consideration.

Acknowledgment This work was supported by the grant 201/96/0410 of Grant Agency of the Czech Republic.

References [1] O. Bor˚ uvka, Lineare Differentialgleichungen 2. Ordnung, VEB Verlag 1967. ´ [2] E. Cartan, Les Problèmes d’Equivalence, Séminaire de Math. exposé D, 1937. [3] J. Chrastina, On Dispersions of the 1st and 2nd Kind of Differential Equation y 00 = q(x)y, Spisy pˇr´ırod. fak. UJEP 580, 1969. [4] V. Chrastinov´ a, On the Darboux Transformation II, Arch. Math. 30, 1995. [5] G. Darboux, Le¸cons sur la Théorie Génerale des Surfaces II, Gauthier Villars 1894. ´ [6] E. Goursat, Le¸cons sur l’Intégration des Equations aux Dérivées Partielles du Second Order II, Hermann 1898 (second printing 1924). [7] F. Neuman, Global Properties of Linear Ordinary Diff. Equations, Kluwer Acad. Publ. 1991


PAPERS pp. 93–99

Abstract Differential Equations of Arbitrary (Fractional) Orders Ahmed M. A. El-Sayed Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt Email: [email protected]

Abstract. The arbitrary (fractional) order integral operator is a singular integral operator, and the arbitrary (fractional) order differential operator is a singular integro-differential operator. And they generalize (interpolate) the integral and differential operators of integer orders. The topic of fractional calculus ( derivative and integral of arbitrary orders) is enjoying growing interest not only among Mathematicians, but also among physicists and engineers (see [1]–[18]). Let α be a positive real number. LetX be a Banach space and A be a linear operator defined on X with domain D(A). In this lecture we are concerned with the different approaches of the definitions of the fractional differential operator Dα and then (see [5,6,7]) study the existence, uniqueness, and continuation (with respect to α) of the solution of the initial value problem of the abstract differential equation Dα u(t) = Au(t) + f (t),

D=

d , t > 0, dt

(1.els )

where A is either bounded or closed with domain dense in X. Fractional-order differential-difference equations, fractional-order diffusion-wave equation and fractional-order functional differential equations will be given as applications.

AMS Subject Classification. 34C10, 39A10

Keywords. Fractional calculus, abstract differential equations, differential-difference equations, nonlinear functional equations.

1

Introduction

Let X be a Banach space. Let L1 (I, X) be the class of (Lebesgue) integrable functions on the interval I = [a, b], 0 < a < b < ∞,


94

Ahmed M. A. El-Sayed

Definition 1. Let f (x) ∈ L1 (I, X), β ∈ R+ . The fractional (arbitrary order) integral of the function f (x) of order β is (see [1]–[11]) defined by Z x (x − s)β−1 β f (s)ds. (2.els ) Ia f (x) = Γ (β) a β−1

When a = 0 and X = R we can write I0β f (x) = f (x)∗φβ (x), where φβ (x) = xΓ (β) for x > 0, φβ (x) = 0 for x ≤ 0 and φβ → δ(x) (the delta function) as β → 0 (see [11]). Now the following lemma can be easily proved Lemma 2. Let β and γ ∈ R+ . Then we have (i) Iaβ : L1 (I, X) → L1 (I, X), and if f (x) ∈ L1 (I, X), then Iaγ Iaβ f (x) = Iaγ+β f (x). (ii) limβ→n Iaβ f (x) = Ian f (x), uniformly on L1 (I, X), n = 1, 2, 3, . . . , where Rx Ia1 f (x) = a f (s)ds. For the fractional order derivative we have (see [1]–[10] and [15]) mainly the following two definitions. Definition 3. The (Riemann-Liouville) fractional derivative of order α ∈ (0, 1) of f (x) is given by d 1−α dα f (x) I = f (x), (3.els ) dxα dx a Definition 4. The fractional derivative D α of order α ∈ (0, 1] of the function f (x) is given by d . (4.els ) dx This definition is more convenient in many applications in physics, engineering and applied sciences (see [15]). Moreover, it generalizes (interpolates) the definition of integer order derivative. The following lemma can be directly proved. Daα f (x) = Ia1−α Df (x),

D=

Lemma 5. Let α ∈ (0, 1). If f (x) is absolutely continuous on [a, b], then (i) Daα f (x) =

dα f (x) dxα

+

(x−a)−α Γ (1−α)

f (a) α

f (x) (ii) limα→1 Daα f (x) = Df (x) 6= limα→1 d dx α .

(iii) If f (x) = k, k is a constant, then Daα k = 0, but

dα k dxα

6= 0.

Definition 6. The finite Weyl fractional integral of order β ∈ R+ of f (t) is Z b 1 −β (s − t)β−1 f (s) ds , t ∈ (0, b), (5.els ) Wb f (t) = Γ (β) t and the finite Weyl fractional derivative of order α ∈ (n − 1, n) of f (t) is −(n−α)

Wbα f (t) = Wb

(−1)n Dn f (t),

Dn f (t) ∈ L1 (I, X).

(6.els )

95

Differential Equations of Fractional Orders

The author [6] stated this definition and proved that if f (t) ∈ C(I, X), then Z b p = 1, 2, . . . , Wb−1 f (t) = f (s)ds, (7.els ) lim Wb−β f (t) = Wb−p f (t),

β→p

t

and if g(t) ∈ C n (I, X) with g (j) (b) = 0, j = 0, 1, . . . , (n − 1), then q = 0, 1, . . . , (n − 1),

lim Wbα g(t) = (−1)q Dq g(t),

α→q

2

Wb0 g(t) = g(t). (8.els )

Ordinary Differential Equations

Let A be a bounded operator defined on X, consider the initial value problem α Da u(t) = Au(t) + f (t), t ∈ (a, b], α ∈ (0, 1], (9.els ) u(a) = uo . Definition 7. By a solution of (9.els ) we mean a function u(t) ∈ C(I, X) that satisfies (9.els ). Theorem 8. Let uo ∈ X and f (t) ∈ C 1 (I, X). If ||A|| ≤ the unique solution

Γ (1+α) , bα

uα (t) = Taα (t)uo + Iaα Taα (t)f (t) ∈ C 1 ((a, b], X),

then (9.els ) has (10.els )

where Taα g(t) =

∞ X

Iakα Ak g(t),

g(t) ∈ L1 (I, X).

(11.els )

e(t−s)A f (s) ds .

(12.els )

k=o

And (1) Taα (a)uo = uo , (2) Daα Taα (t)uo = ATaα (t)uo , (3) limα→1 Taα (t)uo = e(t−a)A uo . Moreover

Z lim uα (t) = e(t−a)A uo +

α→1

t

a

Proof. See [8]. As an application let 0 < β ≤ α ≤ 1 and consider the two (forward and backward) initial value problems of the fractional-order differential-difference equation ( Daα u(t) + CDaβ u(t − r) = Au(t) + Bu(t − r), t > a, (P ) (13.els ) u(t) = g(t), t ∈ [a − r, a], r > 0,

96


( (Q)

Wbα u(t) + CWbβ u(t + r) = Au(t) + Bu(t + r), u(t) = g(t), t ∈ [b, b + r], r > 0,

t < b,

α ≥ β,

(14.els )

where A , B and C are bounded operators defined on X. , then the problem (P) Theorem 9. Let g(t) ∈ C 1 ([a−r, a], X). If ||A|| ≤ Γ (1+α) bα α u(t) ∈ has a unique solution u(t) ∈ C((a, b], X), Du(t) ∈ C(Inr , X) and Da+nr C(Inr , X), where Inr = (a, a + nr]. Moreover if C = 0 then u(t) ∈ C 1 (I, X) and Daα u(t) ∈ C(I, X). Proof. See [8]. Theorem 10. Let u(t) be the solution of (P). If the assumptions of Theorem 9 are satisfied, then there exist two positive constants k1 and k2 such that ||u(t)|| ≤ k1 e(t−a)k2 ,

(15.els )

i.e., the solution of (P) is exponentially bounded. Proof. See [8]. The same results can be proved for the problem (Q) (see [8]).

3

Fractional-Order Functional Differential Equation

Consider the two initial value problems Daα x(t) = f (t, x(m(t))),

x(a) = xo ,

α ∈ (0, 1],

(16.els )

Wbα y(t)

y(b) = yo ,

α ∈ (0, 1],

(17.els )

= f (t, y(m(t))),

with the following assumptions (i) f : (a, b) × R+ → R+ = [0, ∞), satisfies Carathéodory conditions and there exists a function c ∈ L1 and a constant k ≥ 0 such that f (t, x(t)) ≤ c(t) + k|x|, for all t ∈ (a, b) and x ∈ R+ . Moreover, f (t, x(t)) is assumed to be nonincreasing (nondecreasing) on the set (a, b) × R+ with respect to t and nondecreasing with respect to x, (ii) m : (a, b) → (a, b) is increasing, absolutely continuous and there exists a constant M > 0 such that m0 ≥ M for almost all t ∈ (a, b), (iii) k/M < 1. Theorem 11. Let the assumptions (i)–(iii) be satisfied. If xo and yo are positive constants, then the problem (16.els ) has at least one solution x(t) ∈ L1 which is a.e. ) has at least one solution nondecreasing (and so Dx(t) ∈ L1 ) and the problem (17.els y(t) ∈ L1 which is a.e. nonincreasing (and so Dy(t) ∈ L1 ). Proof. See [9].

97


4

Fractional-Order Evolution Equations

Let A be a closed linear operator defined on X with domain D(A) dense in X and consider the two initial value problems γ D u(t) = Au(t), t ∈ (0, b], γ ∈ (0, 1], (18.els ) u(0) = uo , β D u(t) = Au(t), t ∈ (0, b], β ∈ (1, 2], (19.els ) u(0) = uo , ut (0) = u1 .

Remark 12. Some special cases of these two equations have been studied by some authors (see [12] and [16] e.g.). Definition 13. By a solution of the initial value problem (18.els ) we mean a function uγ (t) ∈ L1 (I, D(A)) for γ ∈ (0, 1] which satisfies the problem (18.els ). The solution uβ (t) of the problem (19.els ) is defined in a similar way. Consider now the following assumption (1) Let A generates an analytic semi-group {T (t), t > 0} on X. In particular Λ = {λ ∈ C : |argλ| < π/2 + δ1 }, 0 < δ1 < π/2 is contained in the resolvent set of A and ||(λI − A)−1 || ≤ M/|λ|, Reλ > 0 on Λ1 , for some constant M > 0, where C is the set of complex numbers. Theorem 14. Let u1 , uo ∈ D(A2 ). If A satisfies assumption (1), then there ) given by exists a unique solution uγ (t) ∈ L1 (I, D(A)) of (18.els Z

t

uγ (t) = uo −

rγ (s)es uo ds,

Duγ (t) ∈ D(A),

(20.els )

0

and a unique solution uβ (t) ∈ L1 (I, D(A)) of (19.els ) given by Z uβ (t) = uo + tu1 −

t

rβ (s)es (uo + (t − s)u1 )ds,

D2 uβ (t) ∈ D(A).

(21.els )

0

Here rγ (t) and rβ (t) are the resolvent operators of the the two integral equations Z

t

φγ (t − s)Auγ (s)ds, Z t φβ (t − s)Auβ (s)ds, uβ (t) = uo + tu1 + uγ (t) = uo +

(22.els )

0

0

respectively. Proof. See [6].

(23.els )

98


Now one of the main results in this paper is the following continuation theorem. To the best of my knowledge, this has not been studied before. Theorem 15. Let the assumptions of Theorem 14 be satisfied with u1 = 0, then lim uγ (t) = lim uβ (t) = T (t)uo ,

(24.els )

lim Dγ uγ (t) = lim+ Dβ uβ (t) = AT (t)uo = Du(t),

(25.els )

γ→1− γ→1−

β→1+

β→1

where {T (t), t ≥ 0} is the semigroup generated by the operator A and so u(t) = T (t)uo is the solution of the problem   du(t) = Au(t), t > 0 dt (26.els )  u(0) = uo . Proof. See [6].

5

Fractional-Order Diffusion-Wave Equation

Let X = Rn and u(t, x) : Rn × I → Rn , I = (0, T ]. Definition 16. The fractional D-W (diffusion-wave) equation is the equation (see [7]) ∂ α u(x, t) = Au(x, t), ∂tα

t > 0,

and the fractional diffusion-wave problem is the Cauchy problem  α  ∂ u(x, t) = Au(x, t), t > 0, x ∈ Rn , 0 < α ≤ 2, ∂tα (D-W)  u(x, 0) = uo (x), ut (x, 0) = 0, x ∈ Rn .

(27.els )

(28.els )

From the properties of the fractional calculus we can prove (see [7]) Theorem 17 (Continuation of the problem). If the solution of the (D-W) problem exists, then as α → 1 the (D-W) problem reduces to the diffusion problem   ∂u(x, t) = Au(x, t), t > 0, x ∈ Rn , ∂t (29.els )  u(x, 0) = uo (x), x ∈ Rn , and as α → 2 the (D-W) problem reduces to the wave problem  2  ∂ u(x, t) = Au(x, t), t > 0, x ∈ Rn , ∂t2  u(x, 0) = uo (x), ut (x, 0) = 0, x ∈ Rn .

(30.els )

99


Proof. See [7]. Theorem 18. Let uo ∈ D(A2 ). If A satisfies the condition (1) with X = Rn , then the (D-W) problem has a unique solution uα (x, t) ∈ L1 (I, D(A)) and this solution is continuous with respect to α ∈ (0, 2]. Moreover lim uα (x, t) = u1 (x, t)

α→1

and

lim uα (x, t) = u2 (x, t),

α→2−

(31.els )

) and (30.els ), respectively. where u1 (x, t) and u2 (x, t) are the solutions of (29.els Proof. See [7].

References [1] Ahmed M. A. El-Sayed, Fractional differential equations. Kyungpook Math. J. 28 (2) (1988), 18–22. [2] Ahmed M. A. El-Sayed, On the fractional differential equations. Appl. Math. and Comput. 49 (2–3) (1992). [3] Ahmed M. A. El-Sayed, Linear differential equations of fractional order. Appl. Math. and Comput. 55 (1993), 1–12. [4] Ahmed M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations. Apll. Math. and Compute. 68 (1) (1995), 15–25. [5] Ahmed M. A. El-Sayed, Fractional order evolution equations. J. of Frac. Calculus 7 (1995), 89–100. [6] Ahmed M. A. El-Sayed, Fractional-order diffusion-wave equation. Int. J. Theoretical Physics 35 (2) (1996), 311–322. [7] Ahmed, M. A. El-Sayed, Finite Weyl fractional calculus and abstract fractional differential equations. J. F. C. 9 (May 1996). [8] Ahmed M. A. El-Sayed, Fractional Differential-Difference equations, J. of Frac. Calculus 10 (Nov. 1996). [9] Ahmed M. A. El-Sayed, W. G. El-Sayed and O. L. Moustafa, On some fractional functional equations. PU. M. A. 6 (4) (1995), 321–332. [10] Ahmed M. A. El-Sayed, Nonlinear functional differential equations of arbitrary orders. Nonlinear Analysis: Theory, Method and Applications (to appear). [11] I. M. Gelfand and G. E. Shilov, Generalized functions, Vol. 1, Moscow 1958. [12] F. Mainardy, Fractional diffusive waves in viscoelastic solids. Wenger J. L. and Norwood F. R. (Editors), IUTAM Symposium-Nonlinear Waves in Solids, Fairfield NJ: ASME/AMR (1995). [13] F. Mainardy, Fractional relaxation in anelastic solids. J. Alloys and Compounds 211/212 (1994), 534–538. [14] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. Inc., New York (1993). [15] Igor Podlubny and Ahmed M. A. El-Sayed, On Two Definitions of Fractional Calculus. Slovak Academy Of Sciences, Institute of Experimental Physics, UEF-03-96 ISBN 80-7099-252-2 (1996). [16] W. R. Schneider and W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30 (134) (1989). [17] S. Westerlund, Dead matter has memory. Phisica Scripta 43 (1991), 174–179. [18] S. Westerlund and L. Ekstam, Capacitor theory. IEEE Trans. on Dielectrics and Electrical Insulation 1 (5) (Oct. 1994), 826–839.


PAPERS pp. 101–109

Some Examples for the Extended Use of the Parametric Representation Method Peter L. Simon1 and Henrik Farkas2 1

Department of Applied Analysis, E¨ otv¨ os Lor´ and University, Budapest Email: [email protected] 2 Institute of Physics, Department of Chemical Physics, Technical University of Budapest, Budapest H-1521, Hungary Email: [email protected]

Abstract. The Parametric Representation Method had been applied successfully to construct bifurcation diagrams relating to equilibria of dynamical systems whenever the equilibria are determined from a single equation containing two control parameters linearly. The Discriminantcurve (that is the saddle-node bifurcation curve parametrized by the state variable remained after the elimination) is the base of this method, as it had been shown. The number and even the value of the stationary state variables can be derived from that. Here we show some possible extensions of the method via two examples. 1. Nonlinear parameter dependence 2. Reaction-diffusion equations, Similarly to the above simple case, the PRM provides us with information about the stationary solutions. Although some features do not remain valid for these extensions.

AMS Subject Classification. 58F14, 34C23, 35B32,

Keywords. Bifurcation diagrams, multistationarity

1

Introduction

The parametric representation method is a geometric tool for the study of stationary solutions of differential equations depending on two parameters. There are well-known methods [12,20,22] giving the bifurcation parameter values (where the number or stability of stationary solutions changes), and serving with information on the stationary solutions if the parameters are in a small neighbourhood of the bifurcation values. Our aim is to divide the whole parameter space according to the number and the type of the stationary points. We shall call this separation the global bifurcation diagram;‘global’ refers here to the parameter space, while our investigation is local in the phase space. The first theoretical result in this direction was achieved by Rabinowitz [16]. He followed the changes of one stationary state This is the final form of the paper.

102

P. L. Simon and H. Farkas

varying a single parameter value. Details and other references can be found in [5]. In [3,4,9] there are methods for constructing global bifurcation diagrams which give the number of roots of polynomials. In many practical applications the construction of such bifurcation diagrams was carried out by ad hoc methods [11,13,15]. The Parametric Representation Method (PRM) [9] is a systematic approach, which is especially useful if the parameter dependence of the system is simpler than the dependence on the state variables. As an example, in chemical dynamical systems the parameter dependence is usually linear, therefore the PRM is easy to apply [2,14,17]. Some general features of the method together with a pictorial algorithm for determination of the exact number of stationary points can be found in [6,7]. PRM was also applied to study the root structure of polynomials and extended to study their complex roots [8]. This method is also a useful tool to reveal some relations between the saddle-node and Hopf bifurcation diagrams [17,19]. We summarize the main results concerning the case of linear parameter dependence in Section 2, the detailed study can be found in [18]. In Section 3 it is shown on an example how can be used the PRM, when the equation (determining the stationary states) contains parameters non-linearly. In Section 4 we illustrate that the PRM may be useful for the determination of stationary solutions of a scalar reaction-diffusion equation.

2

Linear parameter dependence

We want to give the number of the stationary points of the following ODE: x(t) ˙ = F (x(t), u), where F : Rn × Rk → Rn is a differentiable function, x(t) ∈ Rn is the vector of state variables and u ∈ Rk is the vector of parameters. The first step before executing the global bifurcation analysis is the reduction of the dimension of the system. There is no general method for that, the optimal one depends on the structure of the concrete system. The Liapunov-Schmidt reduction or — for polynomials — the Euclidean algorithm are often useful tools. In this section we assume that — the system of algebraic equations F (x, u) = 0 giving the stationary points is already reduced to a single equation and — we have two control parameters, u1 and u2 , which are involved in the right hand side of the reduced equation linearly. These control parameters may also be functions of the original parameters of the system. (Two control parameters are chosen regularly in practical applications, primarily because of the visualization.) With these assumptions the above general problem reduces to the following one: Problem. Let us divide the parameter plane (u1 , u2 ) with respect to the number of the solutions of equation f (x, u1 , u2 ) := f0 (x) + f1 (x)u1 + f2 (x)u2 = 0,

(1.far )

103

Parametric Representation Method

where fi ∈ C 2 and f12 (x) + f22 (x) 6= 0 for all x ∈ R; and give a geometric method determining the number and values of the solutions at a given parameter pair (u1 , u2 ). According to the implicit function theorem the number of solutions may change when the parameter values cross the singularity set: S = {(u1 , u2 ) ∈ R2 : ∃ x ∈ R , f (x, u1 , u2 ) = f 0 (x, u1 , u2 ) = 0}, where prime denotes differentiation with respect to x. The detailed study of singularities can be found in [1,10]. The PRM has the following advantages: 1. the singularity set can be easily constructed as a curve parametrized by x, called D-curve; 2. the solutions belonging to a given parameter pair can be determined by a simple geometric algorithm based on the tangential property; 3. the global bifurcation diagram, that divides the parameter plane according to the number of solutions can be geometrically constructed with the aid of the D-curve. Now let us see how to apply the PRM for (1.far ). Concerning the singularity set the determinant ∆(x) := f1 (x)f20 (x) − f10 (x)f2 (x) plays a crucial role. For simplicity we assume that ∆(x) 6= 0 for all x ∈ R (the general case, when ∆ may have zeros is considered in [18]). Then the system f0 (x) + f1 (x)u1 + f2 (x)u2 = 0,

(2.far )

f00 (x)

(3.far )

+

f10 (x)u1

+

f20 (x)u2

= 0,

has one and only one solution for (u1 , u2 ). These equations determine the D-curve for this case: Definition. The solution of the system (2.far ), (3.far ) for u1 and u2 is called D-curve (or discriminant curve). The point belonging to x will be denoted by D(x) = (D1 (x), D2 (x)), i.e. f2 (x)f00 (x) − f20 (x)f0 (x) =: D1 (x), ∆(x) f1 (x)f00 (x) − f10 (x)f0 (x) =: D2 (x). u2 = ∆(x)

u1 =

Thus we produced the singularity set as a curve parametrized by x. Let us introduce the straight lines: M (x) := {(u1 , u2 ) ∈ R2 : f (x, u1 , u2 ) = 0}, i.e. the set of parameter pairs for which a given number x is a solution of (1.far ). The main point of the PRM is the fact that the D-curve (the singularity set) is the envelope of these lines. This fact is involved in the following theorem, which will be referred to as tangential property.

104


u2

1 1 3 u1 -2

Fig. 1. Theorem 1. The line M (x) is tangential to the D-curve at the point D(x). For the proof see [18] Theorem 4. Corollary 1. The number of solutions of (1.far ) belonging to a given parameter pair (u1 , u2 ) is equal to the number of tangents, which can be drawn to the D-curve from the point (u1 , u2 ). Thus as a solution of our problem we got the following: Geometric algorithm. Draw the D-curve belonging to our equation. Given a parameter pair (u1 , u2 ) any tangent from this point to the D-curve gives a solution x of the equation; the value of x can be read on the D-curve at the tangential point. As an illustration let us consider the equation x3 + u1 x + u2 = 0. The D-curve is determined by the system x3 + u1 x + u2 = 0, 3x2 + u1 = 0. From this system we get D1 (x) = −3x2 ,

D2 (x) = 2x3 .

The D-curve is depicted in Fig. 1. If (u1 , u2 ) is on the left side of the D-curve, then the equation has three solutions, because we can draw three tangents from (u1 , u2 ) to the D-curve. If (u1 , u2 ) is on the right side of the D-curve, then there is one solution, because we can draw one tangent from (u1 , u2 ). The value of x on the D-curve is increasing with increasing u2 and it is zero at the origin. The determination of the number of the tangents is facilitated by the so-called convexity property: the D-curve consists of convex arcs that join together in cusp points. To be more formal we cite Theorem 5 from [18]:

105


Theorem 2. Suppose that the roots of the function B(x) := f000 (x) + f100 (x)D1 (x) + f200 (x)D2 (x) are isolated. (i) If B changes its sign at x0 then the D-curve has a cusp point at x0 . (ii) If B does not change its sign at x0 then the D-curve is locally on the left (right) side of its tangent belonging to x0 if ∆(x0 ) is positive (negative). The D-curve also gives the global bifurcation diagram (GBD) i.e. the curve (or system of curves) which divides the parameter plane into regions within which the number of solutions of (1.far ) is constant. The construction of the GBD is based on the fact that the number of roots of a function may change in two ways: 1. it has a multiple root (the derivative vanishes at a root), 2. a root goes to (or comes from) the infinity. The GBD consists of the D-curve and its tangents or asymptotes (if they exist) at the points belonging to x → ∞ and x → −∞. For the exact formulation see Theorem 6 in [18].

3

Nonlinear parameter dependence

In this section we apply the PRM for the special equation x2 + u21 x + u2 = 0.

(4.far )

Our aim is to divide the parameter plane (u1 , u2 ) according to the number of solutions (x ∈ R) of (4.far ). The singularity set is determined by (4.far ) and 2x + u21 = 0.

(5.far )

The solution of the system (4.far )–(5.far ) is u21 = −2x,

u2 = x2 .

(6.far )

Thus the singularity set can not be parametrized by x, but we can define the D-curve with two branches D+ and D− as follows (see Fig. 2.): Definition. The two solutions of (4.far )–(5.far ) for (u1 , u2 ) are called the two branches of the D-curve for x ≤ 0, i.e. √ D2+ (x) = x2 , D1+ (x) = −2x, √ D1− (x) = − −2x, D2− (x) = x2 .

106


Similarly as in Section 2 let us introduce M (x) := {(u1 , u2 ) ∈ R2 : x2 + u21 x + u2 = 0}, which is in this case a parabola for a given x ∈ R. The singularity set is the envelope of the parabolas belonging to x < 0, therefore the tangential property holds in the following form: Theorem 3. For a fixed x < 0 the parabola M (x) is tangential to the D + and D− curves at the points D+ (x) and D− (x). The tangential property does not refer to the values x > 0, however, it is obvious from (6.far ) that there is no singularity for these values. Therefore the parabolas M (x) belonging to x > 0 do not intersect each other, they form a one-fold cover of the lower half plane. Thus the number of solutions can be given by the following: Geometric algorithm. Given a parameter pair (u1 , u2 ) any tangential parabola of the form (4.far ) from this point to the D-curve gives a solution x of the equation (4.far ); the value of x can be read on the D-curve at the tangential point (the value of x is the same on the D+ and on the D− branches). If the parameter pair is in the upper half plane, then the number of the tangential parabolas is equal to the number of solutions. If the parameter pair is in the lower half plane, then the number of solutions is more by one than the number of tangential parabolas. Using this algorithm we get that the number of solutions of (4.far ) is 0 if (u1 , u2 ) is above the D-curve, and it is 2 if (u1 , u2 ) is below the D-curve, see Fig. 2. u2 D -

D + 0 10

x=-2 u1 2 2

Fig. 2.

x=1

107


4

Application of the PRM to a reaction-diffusion equation

Let us consider the reaction-diffusion equation ∂t u(t, x) = ∂xx u(t, x) + f (u(t, x), a, b) with the boundary condition u(t, 0) = u(t, 1) = 0, where f : R3 → R is a differentiable function. The stationary solutions are determined by the following boundary value problem: v 00 (x) + f (v(x), a, b) = 0, v(0) = v(1) = 0.

(7.far ) (8.far )

We will study the following: Problem. Divide the parameter plane (a, b) according to the number of the solutions of (7.far )–(8.far ). The boundary value problem (7.far )–(8.far ) is usually [20,21] reduced to the phase plane analysis of the system v 0 = w, w0 = −f (v, a, b).

(9.far ) (10.far )

If we have a p ∈ R, such that the trajectory t → (v(t), w(t)) of (9.far )–(10.far ) starting from (0, p) reaches the vertical axis (v = 0) at time 1 (i.e. v(1) = 0), then v is a solution of (7.far )–(8.far ). Therefore the time map T is defined that measures the time an orbit takes to get from the point (0, p) to the vertical axis. This time is the double of that one the orbit takes to get from the point (0, p) to the horizontal axis (say, at point (0, q)), because the flow of (9.far )–(10.far ) is symmetric to the horizontal axis. System (9.far )–(10.far ) has the first integral H(v, w) = where F (v) = explicitly:

Rv 0

w2 + F (v), 2

f (s) ds. This first integral enables us to calculate the time map Z T (p) = 2 0

q

1 p dv. 2(F (q) − F (v))

The relation between p and q is given by the first integral: F (q) = time map can be regarded as a function of q, a and b: Z q 1 p dv. S(q, a, b) = 2 2(F (q) − F (v)) 0

p2 2 .

Thus the

108


A solution q of S(q, a, b) = 1

(11.far )

gives a solution of (7.far )–(8.far ). However, several solutions of (7.far )–(8.far ) may give the same q as a solution of (11.far ). Therefore our problem partially reduces to the following one: Problem. Divide the parameter plane according to the number of solutions of (11.far ). This problem is similar to that one dealt with in Section 3 (the parameter dependence may be more complicated). Using the PRM we can define the D-curve (singularity set) belonging to equation (11.far ). It is determined by the equations: S(q, D1 (q), D2 (q)) = 1, ∂q S(q, D1 (q), D2 (q)) = 0. Solving these equations numerically one can get a curve on the parameter plane (a, b), which divides it into regions according to the number of solutions of (11.far ).

References [1] Arnol’d, V. I., The Theory of Singularities and its Applications, Cambridge University Press, Cambridge, 1991. [2] Balakotaiah, V. Luss, D., Global analysis of the multiplicity features of multireaction lumped-parameter systems, Chem. Engng. Sci. 39., 865–881, 1984. [3] Callahan, J., Singularities and plane maps, Am. Math. Monthly 81, 211–240, 1974. [4] Callahan, J., Singularities and plane maps II: Sketching catastrophes, Am. Math. Monthly 84, 765–803, 1977. [5] Chow, S. N., Hale, J. K., Methods of Bifurcation Theory, Springer-Verlag, New York, 1982. [6] Farkas, H., Gy¨ okér, S., Wittmann, M., Investigation of global equilibrium bifurcations by the method of parametric representation, Alk. Mat. Lapok, 14, 335–364, in Hungarian. 1989. [7] Farkas, H., Gy¨ okér, S., Wittmann, M., Use of the parametric representation method in bifurcation problems, In Nonlinear Vibration Problems, Vol. 25, p. 93, 1993. [8] Farkas, H., Simon, P. L., Use of the parametric representation method in revealing the root structure and Hopf bifurcation, J. Math. Chem. 9, 323–339, 1992. [9] Gilmore, R., Catastrophe Theory for Scientists and Engineers Wiley, New York, 1981. [10] Golubitsky, M., Schaeffer, D. G., Singularities and Groups in Bifurcation Theory Vol. I. Springer, New York, 1985. [11] Gray, P., Scott, S. K., Chemical Oscillations and Instabilities: Non-linear Chemical Kinetics, Clarendon Press, Oxford, 1994. [12] Guckenheimer, J. Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.


109

[13] De Kepper, P., Boissonade, J., From bistability to sustained oscillations in homogeneous chemical systems in a flow reactor mode, in Oscillations and Traveling Waves in Chemical Systems, eds.: R.J. Field and M. Burger, (Wiley, New York), p. 223, 1985. [14] Kertész, V., Farkas, H., Local investigation of bistability problems in physicochemical systems, Acta Chim. Hung. 126, 775, 1989. [15] Murray, J. D., Mathematical Biology, Springer-Verlag, New York, 1989. [16] Rabinowitz, P., Some global results for nonlinear eigenvalue problems, J. Func. Anal. 7, 487–513, 1971. [17] Simon, P. L., Nguyen Bich Thuy, Farkas, H., Noszticzius, Z., Application of the parametric representation method to construct bifurcation diagrams for highly non-linear chemical dynamical systems, J. Chem.Soc., Faraday Trans. 92 (16), 2865–2871, 1996. [18] Simon, P. L., Farkas, H., Wittmann, M., Constructing global bifurcation diagrams by the parametric representation method, submitted for publication. [19] Simon, P. L., Farkas, H., Relation between the saddle-node and Hopf bifurcation, work in progress. [20] Smoller, J. A, Shock Waves and Reaction Diffusion Equations Springer, 1983. [21] Smoller, J. A., Wasserman, A., Global bifurcation of steady-state solutions, J.Diff. Equ. 39, 269–290, 1981. [22] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.



Homogenization of Scalar Hysteresis Operators Jan Franc˚ u? Department of Mathematics. Technical University Brno Technick´ a 2, 616 96 Brno, Czech Republic Email: [email protected]

Abstract. Scalar hysteresis operators of Ishlinskii stop type are studied. Homogenization problem for hyperbolic equation with hysteresis operator is formulated. Formula for the homogenized operator is derived.

AMS Subject Classification. 35B27, 73B27, 73E05

Keywords. Scalar hysteresis operators, Ishlinskii operator, homogenization

Introduction The title of the contribution contains words homogenization and hysteresis. Homogenization is a mathematical method used in modelling composite materials with periodic structure. It consists in replacing the heterogeneous material modelled by equations with periodic coefficients with an equivalent homogeneous material modelled by constant coefficients, see e. g. [1], [2], [3], [4] and many others. Hysteresis is one of nonlinear phenomena that appears in evolutionary nonlinear problems of continuum mechanics, see e. g. [5], [6], [7], [8], [9], [10]. The basic feature of hysteresis behavior is a memory effect and irreversibility of the process, the response of the material by loading differs from the response by unloading. The behavior of the material is well characterized by the hysteresis loop. In mechanics hysteresis operators model plastic deformation. We shall deal with homogenization of a one-dimensional boundary value problem, that can be interpreted as a vibration of a plastic rod. We assume that all quantities depend on length variable x ∈ R only. The plasticity of the rod is modelled by Ishlinskii hysteresis operator based on the stop operators. Existence of the solutions of these problems was proved e. g. in [10]. The proof needs properties of stop and play operators and Ishlinskii operators of stop and play type studied in Section 2 and 3. ?

This research has been supported by grant No. 201/97/0153 of Grant Agency of Czech Republic. This is the preliminary version of the paper.

112

Jan Franc˚ u

The plan of the contribution is the following. In Section 1 we start with basic elements used in modelling of materials. Section 2 deals with the stop Sh and the play Ph operators and their properties. Combination in parallel of the stop or the play operators leads to Ishlinskii operator studied in Section 3. One-dimensional homogenization problem is introduced in Section 4. The last Section 5 deals with form of the homogenized operator. The complete proof and the convergence of the solutions is subject of future research.

1

Basic elements

In this section we briefly recall basic one-dimensional models of deformation of solid materials. We consider a one-dimensional homogeneous solid (a homogeneous rod of unit length and of uniform small cross-section). We assume a uniform (independent of place) strain e caused by loading. The strain corresponds to a displacement u(x) satisfying e = ux (≡ du/dx) or equivalently u(x) = e · x. Thus fixing one end u(0) = 0 the displacement in the second end x = 1 corresponds to the strain: e = u(1). The response of the material is the stress σ. It is also supposed to be uniform in the rod. The material is described by the constitutive relation between σ(t) and e(t) represented by the stress-strain σ-e (or strain-stress e-σ) diagram. The behavior is often modelled by a mechanical device (string, pipe, friction element etc.). Piston-in-cylinder element represents a geometric model. Elastic element E The elastic element means linear stress-strain relation σ = A ·e,

(1.fra )

where A is the elasticity constant (Young modulus of elasticity). In this model the stress depends on instant value of strain independently on preceding course of deformation. The elastic element is modelled by a string and its graph in the stress-strain diagram forms a line crossing the origin with slope A. Rigid-plastic element R The element is characterized by a parameter r (r > 0). Three cases occur: either the stress σ is inside the interval (−r, r), then the strain e does not change, or the stress σ reach r, then e can grow, or σ = −r then e may decrease. With cyclic loading and deloading we obtain a rectangular hysteresis loop. The element can be modelled by a mechanical system with friction, the parameter r of the model represents the friction coefficient of the Coulomb friction law. The behavior can be also described by a variational inequality σ(t) ∈ [−h, h] ,

such that

de(t) (σ(t) − σ e) ≥ 0 dt

∀σ e ∈ [−h, h] .

(2.fra )

113

Homogenization of Hysteresis Operators

Other basic elements In material modelling also other elements are used, e. g. viscous and brittle element, but they do not correspond to our framework of behaviour of plastic materials. Combination of elements The elements can be combined either in series or in parallel or in more complicated systems. Let us consider n elements and denote their strains by ei , their stresses by σi , i = 1, 2, . . . , n, the total strain by e and the total stress of the system by σ. Then in case of combination in parallel the deformation is common and the stresses are added: e = e1 = e2 = · · · = en ,

σ = σ1 + σ2 + · · · + σn

(3.fra )

while in case of combination in series the stress is common and the deformations are added: e = e1 + e2 + · · · + en ,

σ = σ1 = σ2 = · · · = σn .

(4.fra )

Many other combinations are used, we shall deal with the following serial elastoplastic combination. Elasto-plastic element. Combining elastic and rigid-plastic elements in series E − R we obtain an elastoplastic element. Its hysteresis loop is similar to that of plastic element, only its vertical segments are slanted, the slope reflects the elasticity constant. Geometric model: Piston-in-cylinder Piston-in-cylinder model represents a geometric equivalent of elasto-plastic element. Let us consider a cylinder of length 2h with a piston moving inside the cylinder. Denoting the input — the absolute (with respect to the coordinate system) position of the piston by u, the relative position of the piston with respect to the cylinder by Sh and the absolute position of the cylinder by Ph . Clearly, u = Ph + Sh . The relative position of the piston is proportional to the stress σ = η · Sh . Let us consider an increasing deformation e(t). The piston is moving in the cylinder, Sh (t) which is proportional to the stress σ(t) is increasing (elastic deformation). If the piston reaches the end of the cylinder and e(t) is still increasing, then Sh (t) and the stress σ(t) remain constant but the cylinder Ph (t) starts to move (plastic deformation). Now, if the e(t) starts decreasing, the piston starts moving backwards. In the beginning the piston is moving in the cylinder, Sh (t) (and the stress σ(t)) is decreasing (elastic deformation) down to the opposite

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end of the cylinder when again Sh (t) stops and Ph (t) starts to decrease (plastic deformation). The position of the cylinder Ph (t) describes an internal state yielding memory effects. Let us remark the degenerated cases. For h = ∞ the stop operator converts to identity: S∞ = I and for h = 0 the play operator becomes identity: P0 = I; in both cases the operators degenerate to the elastic element with A = 1.

2

Stop and Play operator

The introduced piston-in-cylinder model yields two operators Sh called the stop and Ph the play. They form the base for more complicated models. The operators are functional, they map a function to a function. They have a parameter h — it corresponds to the half length of the cylinder. Besides the input function u(t) we need to set the initial states sh , ph — the value of the operators in the initial time. The operators are complementary, i. e. they are connected by the relation (Sh u)(t) + (Ph u)(t) = u(t) ∀ u,

∀t

(5.fra )

and satisfy |(Sh u)(t)| ≤ h

and

|(Ph u)(t) − u(t)| ≤ h.

(6.fra )

The initial states sh , ph ∈ R are supposed to satisfy the compatibility conditions in the initial time t = a analogous to (5.fra ), (6.fra ): sh + ph = u(a)

and

|sh | ≤ h ,

|ph − u(a)| ≤ h , .

(7.fra )

If no initial values are stated we can assigns to the input u(t) the natural initial values sh for the stop operator Sh  if u(a) ≥ h h sh = u(a) if |u(a)| < h (8.fra )  −h if u(a) ≤ −h and similarly the natural initial value ph for the play operator Ph   u(a) − h if u(a) ≥ h if |u(a)| < h . ph = 0  u(a) + h if u(a) ≤ −h

(9.fra )

The operators are introduced by preceding geometric model, nevertheless let us give their exact definition. Max-min definition The definition, see e. g. [5], [8] starts with defining the operator for piecewise monotone functions, then it is extended to all continuous functions:

115


Let I = [a, b] be a time interval and u a continuous piecewise monotone function on I. Let us denote by ti the turning points of u i. e. a = t0 < t1 < · · · < tk = b and the continuous input u is monotone (non-decreasing or nonincreasing) on each subinterval I1 , . . . , In (Ii = (ti−1 , ti ]). Moreover we may assume that in each ti (0 < i < n) the function u changes from non-decreasing to non-increasing or the other way round. Then the value of the stop operator Sh is function Sh u : I → R defined by  sh for t = a ,      min{(Sh u) (ti−1 ) + u(t) − u(ti−1 ), h} for t ∈ Ii if u is nondecreasing on Ii , (10.fra ) (Sh u) (t) =    max{(Sh u) (ti−1 ) + u(t) − u(ti−1 ), −h} for t ∈ Ii if u is non  increasing on Ii . Since the play operator is complementary it can be defined either by a similar max-min definition or simply by (Ph u) (t) = u(t) − (Sh u) (t) .

(11.fra )

Since both operators are Lipschitz continuous, see Lemma 2, the definition can be extended to continuous functions by continuity. Definition by variational inequality The stop and play operator can be also introduced by a variational inequality, similar to (4.fra ), see [10], [9]. Let u ∈ W 1,1 (I) be an input function on the interval I = [a, b] and sh the initial state satisfying |sh | < h. Let x(t) be a solution of the following problem: Find x(t) ∈ W 1,1 (I) such that:

x(t) ∈ [−h, h] ,

x(a) = sh (a) ,

and for almost all t ∈ I the following inequality holds dx du (t) − (t) (x(t) − x e) ≥ 0 ∀ x e ∈ [−h, h] . dt dt

(12.fra )

Since the problem admits unique solution x(t) we use it to definition of the stop operator Sh and play operator Ph : (Sh u) (t) = x(t) ,

(Ph u) (t) = u(t) − x(t) .

(13.fra )

It can be proved that both ways of introducing the play and the stop operators define the same operators. Inverse operators Both operators S and P are not invertible, since they are not injective. Adding a multiple of identity denoted by I the operators become injective and invertible: inverse of the stop type operator is the play type operator and vice versa.

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Lemma 1. Let a, b, c, d, h, k be positive constants. Then we have the following relations −1

(a I + b Sh )

= c I + d Pk ,

(14.fra )

where c=

1 , a+b

d=

1 1 − , a a+b

k = (a + b)h ,

and conversely (c I + d Pk )

−1

= a I + b Sh ,

(15.fra )

where a=

1 , c+d

b=

1 1 − , c c+d

h = ck .

Scaling, continuity and monotony properties Due to definition we have the following dependence on the parameter h: u u , Ph u = h P1 . Sh u = h S1 h h

(16.fra )

The stop and the play operators are Lipschitz continuous, see e. g. [5], [8]: Lemma 2. Let u1 (t), u2 (t) be two inputs on I = [a, b] with the same initial values sh , ph . Then for all t ∈ (a, b] we have k(Sh u1 ) (t) − (Sh u2 ) (t)| ≤ 2 max |u1 (s) − u2 (s)|

(17.fra )

k(Ph u1 ) (t) − (Ph u2 ) (t)| ≤ max |u1 (s) − u2 (s)|

(18.fra )

s≤t

s≤t

The operators conserve monotony properties, non-decreasing inputs yields non-decreasing output and non-increasing input yields non-increasing output. The properties are formulated in [5], [8], [9].

3

Ishlinskii operators

Simple stop Sh and play Ph operators yield hysteresis loops consisting of segments with two slopes. By weighted parallel finite or infinite or continuum combination of stop (or play) operators (including a multiple of identity I) we obtain more general hysteresis loops.

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Stop type Ishlinskii operators Let us consider a parallel weighted combination of n different stop operators, in physical setting elasto-plastic elements. Each element has two parameters: hi half-length of the cylinder and ηi “stiffness” of the element, it describes stress contribution of the element σi = ηi ·fh . Thus the combination is determined by a family of n pairs (hi , ηi ) (i = 1, 2, . . . , n), ordered 0 < h1 < h2 < · · · < hn ≤ ∞, and ηi ≥ 0. Admitting the case hn = ∞ we include a multiple of identity ηn I — simple elastic element combined in parallel. Let us consider an input e(t). Then the output stress is σ(t) = (F (e)) (t) ≡

n X

ηi (Shi e) (t) .

(19.fra )

i=1

The operator F is called discrete Ishlinskii operator. It has to be completed by the initial values of stop operators shi . They can be given or set by (8.fra ) in case of intact initial state. The stress-strain diagram to the increasing loading e(t) from the intact state of this material forms a concave non-decreasing function Φ — another characterization of the hysteresis behavior called virgin curve. Using (19.fra ) we can compute the function Φ. Put the deformation e(r) = r for r ∈ [0, ∞) with shi = 0. Then we have X X X ηi Shi (r) = ηi Shi (r) + ηi Shi (r) . Φ(r) = hi ≤r

i

hi >r

Since in course of this loading Sh (r) = h for h ≤ r and Sh (r) = r for h > r we obtain X X Φ(r) = ηi hi + r ηi . (20.fra ) hi ≤r

hi >r

The function Φ(r) is clearly concave on [0, ∞). Considering the negative loading from the intact state the function Φ(r) can be extended to negative values to an odd function by Φ(r) = −Φ(−r). The Ishlinskii operator represents a generalization of parallel combination of elastic-plastic elements to the case of infinite elements with continuously increasing parameter h. The family of pairs (hi , ηi ) is replaced by a non-negative “stiffness” function η(h). Further we replace the sum in (21.fra ) by an integral (possible η∞ remains) and we obtain the Ishlinskii operator F Z ∞ η(h) (Sh e) (t)dh + η∞ e(t) . (21.fra ) σ(t) = (F (e)) (t) ≡ 0

The operator F (e) should be completed by the initial values sh that can be given or defined by (8.fra ). In the stress-strain diagram the cyclic loading has concave increasing arches and convex decreasing arches.

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Introducing a function µ(h) by Z h η(r)dr or µ(h) =

µ0 (h) = η(h) ,

µ(0) = 0

(22.fra )

0

we can replace the Riemann integral by Stieltjes integral Z ∞ (Sh e) (t)dµ(h) + η∞ e(t) . σ(t) = (F (e)) (t) ≡

(23.fra )

0

The advantage of the formula is the fact that it includes even the discrete case of the Ishlinskii operator (19.fra ) taking piecewise constant function X ηi . (24.fra ) µ(r) = hi r from (21.fra ) or (23.fra ) we can compute the function Φ by means of η∞ and η(h) or µ(h): Z r Z ∞ Φ(r) = hη(h)dh + r η(h)dh + η∞ r , (25.fra ) 0 r Z r Φ(r) = hdµ(h) + r [µ(∞) − µ(r) + η∞ ] . (26.fra ) 0 0

On the other hand since Φ (r) = µ(∞)− µ(r)+ η∞ we can express µ(r) by means of function Φ(r): η∞ = Φ0 (∞) ,

µ(r) = Φ0 (0) − Φ0 (r) .

(27.fra )

The function Φ(r) is continuous and concave on [0, ∞) with Φ(0) = 0. If η∞ > 0 or Φ(∞) = ∞, then the function Φ(r) is increasing and it has an inverse function Ψ (s) on [0, ∞) defined by Ψ (s) = r

iff

Φ(r) = s .

If Φ(∞) ≡ Φ∞ < ∞, then its inverse function Ψ is defined on [0, Φ∞ ] only. Moreover, it may be multivalued for s = Φ∞ . In both cases Ψ is a convex function. Play type Ishlinskii operators Replacing the stop operator Sh by the play operator Ph we obtain Ishlinskii operators of play type. Let us state the formula with input u(t), weight function ζ(h) and ζ0 multiple of identity: Z ∞ (G(u)) (t) = ζ(h) (Ph u) (t)dh + ζ0 u (28.fra ) 0

119


Rr

ζ(s)ds and Stieltjes integral Z ∞ (Ph u) (t)dν(h) + ζ0 u . (G(u)) (t) =

or with function ν(r) =

0

(29.fra )

0

The operator G can be characterized by a function Ψ (s) describing the response of the operator to monotone increasing load from the intact state. Using Ph (s) = s − h for h < s and Ph (s) = 0 for h ≥ s from (28.fra ) or (29.fra ) we obtain formula for the function Ψ (s): Z s Z s (s − h)ζ(h)dh + ζ0 s = sν(s) − hdν(h) + ζ0 s . (30.fra ) Ψ (s) = 0

0

On the other hand since Ψ 0 (s) = ν(s) + ζ0 the function Ψ (s) yields ν(s) and ζ0 : ζ0 = Ψ 0 (0) ,

ν(s) = Ψ 0 (s) − Ψ 0 (0) .

(31.fra )

The hysteresis loops have convex increasing arches and concave decreasing arches. Again if the function Ψ is defined on [0, ∞) it is invertible, its inverse is the Ishlinskii operator of stop type and vice versa. Thus the operators of the play type are used in modelling the inverse hysteresis constitutive relations i. e. dependence of strain on stress. Ishlinskii operators satisfy similar continuity and monotony properties like the stop and play operators.

4

Homogenization problem

Let us consider a two component periodically ordered layered material. We assume that both materials denoted as material A and B are characterized by Ishlinskii operators FA and FB with corresponding characteristics ηA , µA , ΦA and ηB , µB , ΦB . Material with a periodic layered structure The homogenization approach, see [1], [2], [3], [4] consists in considering a sequence of periodically ordered materials with a diminishing period ε. Let dA : dB (dA + dB = 1) be the proportion of the thicknesses of the components A and B. We define a space dependent Ishlinskii operator periodic in space Z ∞ F (y)(e)(t) = (Sh e) (t) dµ(y, h) + η∞ (y)e(t) , (32.fra ) 0

where µ(y, h) and η∞ (y) are functions periodic in y with period one defined by µA (h) for y ∈ [k, k + dA ) k ∈ Z , (33.fra ) µ(y, h) = µB (h) for y ∈ [k − dB , k) k ∈ Z . The function η∞ (y) is defined similarly. Now we define a sequence of operators F ε with diminishing period ε → 0 x (e) . (34.fra ) F ε (x)(e) = F ε

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Boundary value problems We shall consider a problem modelling longitudinal vibration of the rod. Let us consider a thin rod with space variable x ∈ [0, l] made of a periodically layered material. Let us denote the longitudinal displacement by u(x, t) and the stress by σ(x, t). We combine the equation of motion σx + f = ρ utt , where ρ is the density and f (x, t) the applied force, with the stress-strain relation σ = F (e), where F (x)(e) is the space dependent Ishlinskii operator and e(x, t) = ux (x, t). Since the material has periodic structure, the Ishlinskii operator F (x) is periodic with the period ε. We denote it by a superscript ε, see (34.fra ). We also denote the corresponding solution by uε . Thus for any ε > 0 we obtain the following equation [F ε (x)(uεx )]x + f = ρ uεtt

for x ∈ (0, l) ,

t > 0.

(35.fra )

We complete the equation (35.fra ) with suitable boundary conditions, e. g. fixed ends of the rod uε (0, t) = 0 ,

uε (l, t) = 0

for t > 0

(36.fra )

and initial conditions with a given initial displacement u0 and a given initial velocity u1 uε (x, 0) = u0 (x) ,

uεt (x, 0) = u1 (x)

for

x ∈ (0, l) .

(37.fra )

Finally we have to add the initial state of the stop operators Sh in the Ishlinskii operator sh (x)

for x ∈ (0, l) h > 0 ,

(38.fra )

satisfying |sh (x)| ≤ h. In case of the intact initial state we use (8.fra ). Homogenization problem Taking a sequence εi converging to zero (we shall write ε → 0 only) we obtain a sequence of boundary value problems (35.fra ) with a sequence of Ishlinskii operators (34.fra ) completed by boundary and initial conditions (36.fra ), (37.fra ) and the Ishlinskii operator initial state condition (38.fra ). Assuming the solvability of the problems, see [10], we have a sequence of the corresponding solutions uε . The following steps of homogenization are to be carried out. We have to show that the sequence {uε } converge to a function u0 which is a solution to the so called homogenized problem of the similar form. The homogenized problem consists of the equation 0 0 F ux x + f = ρ u0tt for x ∈ (0, l) t > 0 . (39.fra ) with the homogenized operator F 0 independent of the space variable x. The problem is completed by analogous boundary and initial conditions u0 (0, t) = 0 ,

u0 (l, t) = 0

for t > 0 ,

(40.fra )

121


u0 (x, 0) = u0 (x) ,

u0t (x, 0) = u1 (x)

for x ∈ (0, l) .

(41.fra )

Assuming that F 0 is also an Ishlinskii operator we add the same condition giving the initial state of its stop operators sh (x).

5

Homogenized operator

We try to derive the form of the homogenized operator F 0 . Let us consider the uniformly loaded layered rod. Both components described by Ishlinskii operators FA , FB can be considered as combination in series of the elements, namely the layers satisfy the rule (4.fra ). Denoting the deformation and stress in both materials A, B by eA , eB and σA , σB , the serial combination rule yields σ = σA = σB .

(42.fra )

for the total stress. Taking into account the ratio dA : dB of the thickness of the layers A, B (dA + dB = 1) we can write e = dA eA + dB eB .

(43.fra )

Inserting inverses eA = GA (σA ), eB = GB (σB ) of the constitutive relations σA = FA (eA ), σB = FB (eB ) into (43.fra ) we obtain e = dA GA (σ) + dB GB (σ) = (dA GA + dB GB ) (σ) , which implies the limit constitutive relation σ = F 0 (e) ≡ (dA GA + dB GB )

−1

(e) .

(44.fra )

The last equality gives little information on the homogenized operator F 0 . If we consider an increasing uniform loading from the intact state, we can replace the Ishlinskii operator by the corresponding functions ΦA , ΦB and rewrite the relation (44.fra ) to −1 −1 (r) = (dA ΨA + dB ΨB )−1 (r) . (45.fra ) Φ0 (r) = dA Φ−1 A + dB ΦB We have arrived to the following result: Theorem 3. Under assumptions that the homogenized operator F 0 of the homogenized problem (39.fra )–(41.fra ) is the Ishlinskii operator of stop type, it is defined by Z ∞ 0 (Sh e) (t)dµ0 (h) . (46.fra ) F (e)(t) = 0 0

The weight function µ (r) is given by 0 0 µ0 (r) = Φ0 (0) − Φ0 (r) ,

(47.fra )

0

). where Φ (r) is given by (45.fra Justification of the assumption and proof of the convergence uε → u0 is subject of the future research.

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References [1] I. Babuˇska: Homogenization approach in engineering. In Computing methods in applied sciences and engineering, Lecture notes in Econ. and Math. Systems 134, Springer, Berlin 1976, 137–153. [2] A. Bensoussan, J. L. Lions, G. Papanicolaou: Asymptotic analysis for periodic structure, North Holland, 1978. [3] E. Sanchez Palencia: Non-homogeneous media and vibration theory, Lecture Notes in Physics 127, Springer 1980. [4] N. S. Bakhvalov, G. P. Panasenko: Averaging of processes in periodic media (Russian), Nauka, Moscow 1984. [5] P. Krejˇc´ı: Methods of solving equations with hysteresis, Proceedings of 12 Seminar on P.D.E. (Czech), Lipovce 1987, 7–60, Union of Czechoslovak Mathematicians and Physicists Prague, Technical University Plzeˇ n, 1987. [6] M. A. Krasnosel’skii, A. V. Pokrovskii: Systems with hysteresis, Springer, Berlin 1989. Russian edition: Nauka, Moscow 1983. [7] P. Krejˇc´ı: Modelling of singularities in elastoplastic materials with fatigue, Appl. Math. 39 (1994), 137–160. [8] M. Brokate: Hysteresis Operators. In: Phase Transitions and Hysteresis, Lecture Notes of C.I.M.E., Montecatini Terme July 13–21, 1993, 1–38, Springer Berlin, 1994. [9] A. Visintin: Differential models of hysteresis, Springer, Berlin Heidelberg 1994. [10] P. Krejˇc´ı: Hysteresis, convexity and dissipation in hyperbolic equations, Gakuto Int. Series Math. Sci. & Appl., Vol. 8, Gakk¯ otosho, Tokyo 1996.



Global Qualitative Investigation, Limit Cycle Bifurcations and Applications of Polynomial Dynamical Systems Valery Gaiko Department of Mathematics University of Informatics and Radioelectronics Koltsov Str. 49-305, Minsk 220090, Belarus Email: [email protected]

Abstract. Two-dimensional polynomial dynamical systems are mainly considered. By means of Erugin’s two-isocline method we carry out the global qualitative investigation of such systems, construct canonical systems with field-rotation parameters and study limit cycle bifurcations. It is known, for example, that generic quadratic systems with limit cycles have three field-rotation parameters and bifurcation surfaces of multiplicity-two and three limit cycles are familiar saddle-node and cusp bifurcation surfaces respectively. We use the canonical systems, cyclicity results and apply Perko’s termination principle, stating that the boundary of a global limit cycle bifurcation surface typically consists of Hopf bifurcation surfaces and homoclinic (or heteroclinic) loop bifurcation surfaces, to prove the non-existence of swallow-tail bifurcation surface of multiplicity-four limit cycles for quadratic systems. We discuss also possibilities of application of obtained results to the study of higher-dimensional dynamical systems and systems with more complicated dynamics. AMS Subject Classification. 34C05, 34C23, 58F14, 58F21 Keywords. Hilbert’s 16th Problem, Erugin’s two-isocline method, Wintner’s principle of natural termination, Perko’s planar termination principle, field-rotation parameter, bifurcation, limit cycle, separatrix cycle

1

Introduction

This paper is connected with the development of a global bifurcation theory of dynamical systems and discussing possibilities of its application to more complicated systems. First of all, it is directed to the solution of Hilbert’s 16th Problem on the maximum number and relative position of limit cycles of two-dimensional dynamical systems given by the equations dx = Pn (x, y), dt This is the final form of the paper.

dy = Qn (x, y), dt

(1.gai )

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where Pn (x, y) and Qn (x, y) are polynomials of real variables x, y with real coefficients and not greater than n degree. This is the most difficult problem in the qualitative theory of systems (1.gai ). There are a lot of methods and results for the study of limit cycles [1], [2]. However, the Problem has not been solved completely even for the case of simplest (quadratic) systems. It is known only that a quadratic system has at least four limit cycles in (3:1) distribution [3]. Besides, we can finally state that a general polynomial system has at most a finite number of limit cycles [4]–[6]. A new impulse to the study of limit cycles was given by the introduction of ideas coming from Bifurcation Theory [7]–[10]. We know three principal bifurcations of limit cycles: 1) Andronov-Hopf bifurcation (from a singular point of centre or focus type); 2) separatrix cycle bifurcation (from a homoclinic or heteroclinic orbit); 3) multiple limit cycle bifurcation. The first bifurcation was studied completely only for quadratic systems: the number of limit cycles bifurcating from a singular point (its cyclicity) is equal to three [11]. For cubic systems the cyclicity of a singular point is not less than eleven [12]. The second bifurcation was studying in [13]–[15]. Now we have the classification of separatrix cycles and know the cyclicity of the most of them (of elementary graphics). The last bifurcation is the most complicated. Multiple limit cycles were considering, for example, in [16] and [17]–[19]. All mentioned bifurcations can be generalized for higher-dimensional dynamical systems [20]–[22] and can be used for the study of systems with more complicated dynamics [23]–[25]. But how to find a way to the solution of Hilbert’s 16th Problem? Unfortunately, all known limit cycle bifurcations are local. We consider only a neighborhood of either the point or the separatrix cycle, or the multiple limit cycle. We consider also local unfoldings in the parameter space. It needs a qualitative investigation on the whole (both on the whole phase plane and on the whole parameter space), i.e., it needs a global bifurcation theory. This is the first idea introduced for the first time by N. P. Erugin in [26]. Then we should connect all limit cycle bifurcations. This idea came from the theory of higher-dimensional dynamical systems. It was contained in Wintner’s principle of natural termination [27] and was used by L. M. Perko for the study of multiple limit cycles in two-dimensional case [17]–[19]. At last, we must understand how to control the limit cycle bifurcations. The best way to do it is to use field-rotation parameters considered for the first time by G. F. Duff in [28]. If we are able to realize these ideas we will answer many questions: 1) How to prove that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is (3:1)? 2) How to construct a cubic system with more than eleven limit cycles? 3) What is a strategy of the qualitative investigation on the whole for cubic and general polynomial systems? 4) How to generalize obtained results for the study of higher-dimensional dynamical systems and to use them for more complicated systems?

Global Qualitative Investigation

2

125

Methods and technical difficulties

Methods and approaches used for the study of limit cycles are very different: analytic, algebraic, geometric, etc. There are various combinations. In [4], for example, classical analytic methods are applied to the analysis of normal forms in special cases of polycycles. The techniques of [5] and [6] are much more sophisticated. However, by means of these methods we can prove only the finiteness of the number of limit cycles. In the particular case of quadratic (cubic) systems we do not need such powerful methods, since the number of possible situations is rather limited. It is enough to show that limit cycles cannot accumulate on any separatrix cycle. Other techniques are used in [13]–[15] where families of planar quadratic vector fields are considered and the cyclicity of unfoldings for various limit periodic sets is estimated. This is a new combination: of analytic and bifurcation methods. But it does not work for non-elementary (non-monodromic) limit periodic sets. It needs a global blow-up of some unfoldings. Even after such a desingularization we will have only an upper bound of the number of limit cycles. We must find the least upper bound of the number and estimate the relative position of limit cycles! Purely algebraic methods, for instance, are not able to solve even simpler problems: to distinguish centre and focus or to give the number of small amplitude limit cycles bifurcating from a singular point at least for cubic systems. These problems are algorithmically solvable. Nevertheless, it is still complicated to calculate all the Liapunov focus quantities and to estimate their independence. There are some types of integrable cubic systems: reversible, Hamiltonian, Darboux integrable. For the study of limit cycles we perturbate such systems, consider linear and higher order Abelian integrals (monodromy variations). But only eleven small amplitude limit cycles can be obtained in this way at present [12]. We will develop a geometric aspect of Bifurcation Theory. It will give a global approach to the qualitative investigation and will help to combine all other approaches, their methods and results. We will use the two-isocline method, which was developed by N. P. Erugin for two-dimensional systems [26] and then was generalized by his pupil V. A. Pliss for three-dimensional case [29]. An isocline portrait is the most natural construction in the corresponding polynomial equation. It is enough to have only two isoclines (of zero and infinity) to obtain a principal information on the original system, because these two isoclines are right-hand sides of the system. We know geometric properties of isoclines (conics, cubics, etc.) and can easily get all isoclines portraits. By means of them we can obtain all topologically different qualitative pictures of integral curves to within a number of limit cycles and distinguishing centre and focus. So, we will be able to carry out a rough topological classification of the phase portraits for the polynomial systems. It is the first application of Erugin’s two-isocline method. Studying contact and rotation properties of isoclines we can also construct the simplest (canonical) systems containing limit cycles. Two groups of parameters can be distinguished in such systems: static and dynamic. Static parameters determine a behavior of the phase trajectories in principle, since they control

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Valery Gaiko

the number, position and type of singular points in finite part of the plane (finite singularities). Parameters from the first group determine also a possible behavior of separatrices and singular points at infinity (infinite singularities) under the variation of parameters from the second group. Dynamic parameters are rotation parameters. They do not change the number, position and index of finite singularities and involve a directional rotation in the vector field. The rotation parameters allow to control infinite singularities, a behavior of limit cycles and separatrices. The cyclicity of singular points and separatrix cycles, the behavior of semistable and other multiple limit cycles are controled by these parameters as well. Thus, with the help of rotation parameters, we can control all limit cycle bifurcations, i.e., we can solve the most fine qualitative problems and carry out the global qualitative investigation of the polynomial systems. Of course, some technical difficulties may arise in such investigation. We have a good tool: rotation parameters. But we have no enough experience to use them. To control all limit cycle bifurcations (especially, of multiple limit cycles), we should know the properties and combine the effects of all rotation parameters. These difficulties can be overcome by means of the development of new methods based on Perko’s planar termination principle stating that the maximal oneparameter family of multiple limit cycles terminates either at a singular point, which is typically of the same multiplicity, or on a separatrix cycle, which is also typically of the same multiplicity [19]. This principle is a consequence of Wintner’s principle of natural termination, which was stated for higher-dimensional dynamical systems in [27] where A. Wintner studied one-parameter families of periodic orbits of the restricted three-body problem and used Puiseux series to show that in the analytic case any one-parameter family of periodic orbits can be uniquely continued through any bifurcation except a period-doubling bifurcation. Such a bifurcation can happen, for example, in a three-dimensional Lorenz system. Besides, the periods in a one-parameter family of a higher-dimensional system can become unbounded in strange ways: for example, the periodic orbits may belong to a strange invariant set (strange attractor) generated at a bifurcation value for which there is a homoclinic tangency of the stable and unstable manifolds of the Poincaré map [17]. This cannot happen for planar systems. It would be interesting (in the case of success) to generalize results on multiple limit cycle bifurcations to more complicated systems.

3

Aims and preliminary results

Global bifurcation theory of quadratic systems. It is known that the generic quadratic system with limit cycles has three rotation parameters and bifurcation surfaces of multiplicity-two and three limit cycles are familiar saddlenode and cusp bifurcation surfaces respectively. We will apply Perko’s termination principle to prove the non-existence of swallow-tail bifurcation surface of multiplicity-four limit cycles, i.e., using the data on the cyclicity of singular points and separatrix cycles we will prove by contradiction that a quadratic system cannot have more than four limit cycles, that the distributions (4:0), (2:2)


127

are impossible and the multiplicity of a limit cycle is not higher than three. Thus we intend to prove that for quadratic systems the maximum number of limit cycles is equal to four and the only possible their distribution is (3:1). Cubic and general polynomial systems. First of all, a general strategy of the qualitative investigation on the whole will be developed. The main results by analogy with quadratic systems will be obtained and systematized. We will apply Erugin’s two-isocline method to get all isocline portraits of cubic systems, to carry out the rough topological classification of their phase portraits and to construct the canonical systems with field-rotation parameters, which will be used for various aims: study of limit cycle bifurcations, classification of separatrix cycles, obtaining bifurcation diagrams. We will use Poincaré topographical systems and small parameter method, Abelian integrals and variational methods to construct a cubic system with more than eleven limit cycles. All these results will be generalized to develop a global bifurcation theory of planar polynomial systems. Higher-dimensional dynamical systems and applications. We will apply the theory of planar dynamical systems to the qualitative investigation of higher-dimensional systems. Various bifurcations in reversible, Hamiltonian and conservative systems will be considered: Hopf bifurcation, bifurcations of homoclinic and heteroclinic orbits (including degenerate cases). Multiple limit cycle bifurcations with the application of Wintner’s principle of natural termination will be studied as well. Since theory of dynamical systems and bifurcation methods can be used for the mathematical modelling natural systems with complicated dynamics, we will consider possibilities of the application of global bifurcation theory, for instance, to the study of Josephson junctions in forsed non-linear dynamical networks, non-linear evolution systems in Belousov-Zhabotinskii reaction, etc. Results. A particular preliminary work in this direction has already been carried out in [30]–[41]. By means of Erugin’s two-isocline method we carried out the global qualitative investigation of quadratic systems, constructed the canonical systems with field-rotation parameters and applied them for the study of limit cycle bifurcations: Andronov-Hopf bifurcation, bifurcations of homoclinic and heteroclinic orbits (separatrix cycles), multiple limit cycle bifurcations. We studied the bifurcations of various codimensions and introduced so-called a function of limit cycles: a cross-section of Andronov-Hopf surface formed by limit cycles and the corresponding values of rotation parameters. Using numerical and analytic methods, we constructed concrete examples of systems with different number and relative position of limit cycles. In particular, the following theorems have been proved: Theorem 1. A quadratic system has at least four limit cycles in (3:1) distribution. Theorem 2. . Any quadratic system with limit (separatrix) cycles can be reduced to one of the systems:

128

Valery Gaiko

dx = −(x + 1)y + αQ(x, y), dt

dy = Q(x, y) dt

(2.gai )

or dx = −y + αy 2 , dt

dy = Q(x, y), dt

(3.gai )

where Q(x, y) = x + λy + ax2 + b(x + 1)y + cy 2 . We developed a new approach to the classification of separatrix cycles. It is based on the application of canonical systems (2.gai ) and (3.gai ). The classification was carried out according to the number and type of finite singularities of the original reversible systems and with the help of the successive variation of rotation parameters. That approach allowed not only to define all possible types of separatrix cycles, but also to determine their cyclicity and relative position, to obtain both the corresponding phase portraits and the division of parameter space. By means of the field-rotation parameters and function of limit cycles we showed how to control semistable limit cycles: we were changing the rotation parameters so that to push the semistable cycles either to a singular point of focus (centre) type or to some separatrix cycle and to obtain the contradiction with their cyclicity. On the basis of reversible systems we constructed Poincaré topographical systems and with the help of small parameter method studied various periodic orbits: limit cycles, centre curves. We developed also a control theory of quadratic systems and considered possibilities of the application of our results to general polynomial systems.

References [1] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Progr. Scient. Transl., Jerusalem 1971. [2] A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Qualitative Theory of Second Order Dynamical Systems, Wiley, New York 1973. [3] Y. Ye et al., Theory of Limit Cycles, AMS Transl. Math. Monogr. 66, Providence, RI 1986. [4] H. Dulac, Sur les cycles limites, Bull. Soc. Math. France, 51 (1923), 45–188. [5] Yu. Iliashenko, Finiteness theorems for limit cycles, Russian Math. Surv., 40 (1990), 143–200. ´ [6] J. Ecalle, Finitude des cycles limites et accéléro-sommation de l’application de retour, Lect. Not. Math., 1455 (1990), 74–159. [7] S.-N. Chow, J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York 1982. [8] J. Guckenheimer, P. Holms, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York 1983.


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[9] M. Golubitsky, D. G. Shaeffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York 1985. [10] S.-N. Chow, C. Li, D. Wang, Normal Forms and Bifurcations of Planar Vector Fields, Cambridge Univ. Press, Cambridge 1994. [11] N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium point of focus or center type, Matem. Sbor., 30 (1952), 181–196. (in Russian) ˙ l¸adek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), [12] H. Zo 843–860. [13] F. Dumortier, R. Roussarie, C. Rousseau, Hilbert’s 16th problem for quadratic vector fields, J. Diff. Equations, 110 (1994), 86–133. [14] F. Dumortier, R. Roussarie, C. Rousseau, Elementary graphics of cyclicity 1 and 2, Nonlinearity, 7 (1994), 1001–1043. [15] F. Dumortier, M. El Morsalani, C. Rousseau, Hilbert’s 16th problem for quadratic systems and cyclicity of elementary graphics, Nonlinearity, 9 (1996), 1209–1261. [16] J.-P. Fran¸coise, C. C. Pough, Keeping track of limit cycles, J. Diff. Equations, 65 (1986), 139–157. [17] L. M. Perko, Global families of limit cycles of planar analytic systems, Transact. Amer. Math. Soc., 322 (1990), 627–656. [18] L. M. Perko, Homoclinic loop and multiple limit cycle bifurcation surfaces, Transact. Amer. Math. Soc., 344 (1994), 101–130. [19] L. M. Perko, Multiple limit cycle bifurcation surfaces and global families of multiple limit cycles, J. Diff. Equations, 122 (1995), 89–113. [20] S.-N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues, J. Dynam. Diff. Equations, 2 (1990), 177–244. [21] R. Roussarie, C. Rousseau, Almost planar homoclinic loops in R3 , J. Diff. Equations, 126 (1996), 1–47. [22] J. Knobloch, A. Vanderbauwhede, A general reduction method for periodic solutions in conservative and reversible systems, J. Dynam. Diff. Equations, 8 (1996), 71–102. [23] U. Schalk, J. Knobloch, Homoclinic points near degenerate homoclinics, Nonlinearity, 8 (1995), 1133–1141. [24] T. Bartsch, M. Kern, Bifurcation of steady states in a modified BelousovZhabotinskii reaction, Topol. Meth. Nonlin. Analysis, 2 (1993), 105–124. [25] T. Bartsch, Bifurcation of stationary and heteroclinic orbits for parametrized gradient-like flows, Topol. Nonlin. Analysis, 35 (1996), 9–27. [26] N. P. Erugin, Some questions of motion stability and qualitative theory of differential equations on the whole, Prikl. Mat. Mekh., 14 (1952), 459–512. (in Russian) [27] A. Wintner, Beweis des E. Stromgrenschen dynamischen Abschlusprinzips der periodischen Bahngruppen im restringierten Dreikorperproblem, Math. Z., 34 (1931), 321–349. [28] G. F. D. Duff, Limit cycles and rotated vector fields, Ann. Math, 67 (1953), 15–31. [29] V. A. Pliss, Nonlocal Problems of Oscillation Theory, Nauka, Moscow 1964. (in Russian) [30] L. A. Cherkas, V. A. Gaiko, Bifurcations of limit cycles of a quadratic system with two field-rotation parameters, Dokl. Akad. Nauk BSSR, 29 (1985), 694–696. (in Russian) [31] L. A. Cherkas, V. A. Gaiko, Bifurcations of limit cycles of a quadratic system with two critical points and two field-rotation parameters, Diff. Urav., 23 (1987), 1544–1553 (in Russian); Diff. Equations, 23 (1987), 1062–1069.

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[32] V. A. Gaiko, L. A. Cherkas, Bifurcations of limit cycles of the vector fields on C 2 and R2 , in Problems of Qualitative Theory of Differential Equations, Nauka, Novosibirsk 1988, 17–22. (in Russian) [33] V. A. Gaiko, Separatrix cycles of quadratic systems, Dokl. Akad. Nauk Belarusi, 37 (1993), 18–21. (in Russian) [34] V. A. Gaiko, The limit cycle bifurcations of quadratic autonomous systems, in Nonlinear Phenomena in Complex Systems, Inst. Phys. Press, St.Petersburg 1993, 60–65. [35] V. A. Gaiko, On application of two-isocline method to investigation of twodimensional dynamical systems, Adv. Synerg., 2 (1994), 104–109. [36] V. A. Gaiko, Bifurcations of limit cycles and classification of separatrix cycles of two-dimensional polynomial dynamical systems, Vest. Bel. Gos. Univ. Ser. 1 (1995), 69–70. (in Russian) [37] V. A. Gaiko, Geometric approaches to qualitative investigation of polynomial systems, Adv. Synerg., 6 (1996), 176–180. [38] V. A. Gaiko, On development of new approaches to investigation of limit cycle bifurcations, Adv. Synerg., 8 (1997), 162–164. [39] V. A. Gaiko, Application of topological methods to qualitative investigation of two-dimensional polynomial dynamical systems, to appear in Univ. Iagellon. Acta Math., 36 (1997). [40] V. A. Gaiko, Qualitative theory of two-dimensional polynomial dynamical systems: problems, approaches, conjectures, Nonlin. Analysis, Theory, Meth. Appl., 30 (1997), 1385–1394. [41] V. A. Gaiko, Control of multiple limit cycles, to appear in Adv. Synerg.



The Existence of Global Solutions to the Elliptic-Hyperbolic Davey-Stewartson System with Small Initial Data Nakao Hayashi1 and Hitoshi Hirata2 1

Department of Applied Mathematics, Science University of Tokyo 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162, Japan Email: [email protected] 2 Department of Mathematics, Sophia University 7-1, Kioicho, Chiyoda-ku, Tokyo 102, Japan Email: [email protected]

Abstract. We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson system. Our purpose in this paper is to prove global existence of small solutions of this system in the usual weighted Sobolev space H 3,0 ∩ H 0,3 . Furthermore we prove L∞ time decay estimates in L∞ of solutions u such that ku(t)kL∞ ≤ C(1 + |t|)−1 .

AMS Subject Classification. 35D05, 35E15, 35Q55, 76B15

Keywords. Davey-Stewartson system, commutator calculus

We study the initial value problem for the Davey-Stewartson systems  2 2 2 3  i∂t u + c0 ∂x1 u + ∂x2 u = c1 |u| u + c2 u∂x1 ϕ, (x, t) ∈ R , ∂x21 ϕ + c3 ∂x22 ϕ = ∂x1 |u|2 ,   u(x, 0) = φ(x),

(1.hir )

where c0 , c3 ∈ R, c1 , c2 ∈ C, u is a complex valued function and ϕ is a real valued function. The systems (1.hir ) for c3 > 0 were derived by Davey and Stewartson [4] and model the evolution equation of two-dimensional long waves over finite depth liquid. Djordjevic-Redekopp [5] showed that the parameter c3 can become negative when capillary effects are significant. When (c0 , c1 , c2 , c3 ) = (1, −1, 2, −1), (−1, −2, 1, 1) or (−1, 2, −1, 1) the system (1.hir ) is referred as the DSI, DSII defocusing and DSII focusing respectively in the inverse scattering literature. In [7], Ghidaglia and Saut classified (1.hir ) as elliptic-elliptic, elliptic- hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the respective sign of (c0 , c3 ) : (+, +), (+, −), (−, +) and (−, −). For the elliptic-elliptic and hyperbolic-elliptic cases, local and global properties of solutions were studied in [7] in the usual Sobolev spaces L2 , H 1 and H 2 . In this paper we consider The final version of this paper is published in Nonlinearity 9, which is cited in [10].

132

N. Hayashi and H. Hirata

the elliptic-hyperbolic case. In this case after a rotation in the x1 x2 -plane and rescaling, the system (1.hir ) can be written as ( i∂t u + ∆u = d1 |u|2 u + d2 u∂x1 ϕ + d3 u∂x2 ϕ, (2.hir ) ∂x1 ∂x2 ϕ = d4 ∂x1 |u|2 + d5 ∂x2 |u|2 , where ∆ = ∂x21 + ∂x22 , d1 , · · · , d5 are arbitrary constants. In order to solve the system of equations, one has to assume that ϕ(·) satisfies the radiation condition, namely, we assume that for given functions ϕ1 and ϕ2 lim ϕ(x1 , x2 , t) = ϕ1 (x1 , t) and

x2 →∞

lim ϕ(x1 , x2 , t) = ϕ2 (x2 , t).

x1 →∞

(3.hir )

Under the radiation condition (3.hir ), the system (2.hir ) can be written as Z ∞ i∂t u + ∆u = d1 |u|2 u + d2 u ∂x1 |u|2 (x1 , x2 0 , t)dx2 0 x2 Z ∞ + d3 u ∂x2 |u|2 (x1 0 , x2 , t)dx1 0 + d4 u∂x1 ϕ1 + d5 u∂x2 ϕ2

(4.hir )

x1

with the initial condition u(x, 0) = φ(x). In what follows we consider the equation (4.hir ). In this paper we use the following notations. Notations. We define the weighted Sobolev space as follows H m,l = {f ∈ L2 ; k(1 − ∂x21 − ∂x22 )m/2 (1 + x1 2 + x2 2 )l/2 f k < ∞}, H m,l (Rxj ) = {f ∈ L2 (Rxj ); k(1 − ∂x2j )m/2 (1 + xj 2 )l/2 f kL2 (Rj ) < ∞}, where k·k denotes the usual L2 norm. We let ∂ = (∂x1 , ∂x2 ), J = (Jx1 , Jx2 ), Jxj = xj +2it∂xj . For simplicity we write Lpxj = Lp (Rxj ), Lpx1 Lqx2 = Lp (Rx1 ; Lq (Rx2 )), P P Hxm,l = H m,l (Rxj ), k · kX m,l (t) = |α|≤m k∂ α · k + |α|≤l kJ α · k, where α = j (α1 , α2 ), |α| = α1 + α2 , α1 , α2 ∈ N ∪ {0}. Local existence of small solutions to (4.hir ) was shown when the initial function is in H m,l in [13] for H 12,0 ∩ H 0,6 , [8] for H m,0 ∩ H 0,l , (m, l > 1), [1] for H m,0 , (m is sufficiently large integer) and [9] for H m,0 , (m ≥ 5/2). Furthermore in [11] without smallness condition on the data local existence of solutions was proved in the analytic function space which consists of real analytic functions. Global existence of small solutions to (4.hir ) was also given in [11] when the data are real analytic and satisfy the exponential decay condition. Recently, H. Chihara [1] established the global existence of small solutions to (4.hir ) in higher order Sobolev spaces. Our purpose in this paper is to prove the global existence of small solutions to (4.hir ) in the usual weighted Sobolev spaces H 3,0 ∩ H 0,3 , which is considered as lower order Sobolev class compared to one used in [1], by the calculus of commutator of operators. We shall prove

133

The Existence of Global Solutions to DSI System

j+1 ∞ ϕ1 ∈ C(R; L∞ Theorem 1. Let φ ∈ H 3,0 ∩ H 0,3 , ∂xj+1 x1 ), ∂x2 ϕ2 ∈ C(R; Lx2 ), 1 (0 ≤ j ≤ 3), ε3 and δ3 be sufficiently small, where X εm = sup (1 + t)1+a k(t∂x1 )j ∂x1 ϕ1 (t)kL∞ + k∂xj+1 ϕ1 (t)kL∞ x1 x1 1 t∈R

0≤j≤m

δm

, + k(t∂x2 )j ∂x2 ϕ2 (t)kL∞ + k∂xj+1 ϕ2 (t)kL∞ x2 x2 2 1/2 X ≥ k∂xα11 ∂xα22 x1 β1 x2 β2 φk2 .

a > 0,

|α|+|β|≤m

Then there exists a unique global solution u of (4.hir ) such that sup t∈R

3,0 u ∈ L∞ ∩ H 0,3 ) ∩ C(R; H 2,0 ∩ H 0,2 ), local (R; H X X k∂ α J β u(t)k + (1 + t)−Cδ3 k∂ α J β u(t)k ≤ 4δ3 . |α|+|β|≤2

(5.hir ) (6.hir )

|α|+|β|≤3

Corollary 2. Let u be the solution constructed in Theorem 1. Then we have ku(t)kL∞ ≤ C(1 + |t|)−1 (kφkH 3,0 + kφkH 0,3 ). Moreover, for any φ ∈ H 3,0 ∩ H 0,3 there exist u± such that ku(t) − U (t)u± kH 2,0 → 0 where U (t) = e

it(∂x21 +∂x22 )

as

t → ±∞,

.

The rate of decay obtained in Corollary 2 is the same as that of solutions to linear Schr¨ odinger equations. Time decay of solutions for the Davey-Stewartson systems (1.hir ) was obtained in [3,7] when (c0 , c3 ) = (+, +) and (c0 , c3 ) = (−, +) and in [11] when (c0 , c3 ) = (+, −) and (c0 , c3 ) = (−, −) under exponential decay conditions on the data. We try to explain our strategy of the proof of Theorem 1. For simplicity we consider following equation Z ∞ i∂t u + ∆u = u ∂x1 |u|2 dx2 0 , x2

which have only main nonlinear term. We use following operator Kx1 and Kx2 , where Z x1 m ∞ X Am Dx1 kv(t, x1 0 )k2L2x dx1 0 Kx1 = Kx1 (v) = 2 m! hDx1 i −∞ m=0 and Kx2 = Kx2 (v) =

Z x2 m ∞ X Am Dx2 kv(t, x2 0 )k2L2x dx2 0 . 1 m! hDx2 i −∞ m=0

So, if we take A2 = 1/δ3 (for the definition of δ3 , see Theorem 1), by virtue of commutator estimates and following lemma,

134


Lemma 3. kgkL2x , k[hDx1 i1/2 , f ]gkL2x + k[hDx1 i, f ]gkL2x ≤ CkhDx1 if kL∞ x 1

1

1

1

which follows from Coifman and Meyer’s result (see [2, p. 154]), we have 1 d 2 dt

X

(kKx1 ∂ α J β u(t)k2 + kKx2 ∂ α J β u(t)k2 )

|α|+|β|≤3

+

1

X

1/2 4δ3 |α|+|β|≤3

ku(t)kL2 khDx1 i1/2 Kx1 ∂ α J β u(t)kL2 2 2 x x L 2

2

x1

2

1/2 + ku(t)kL2x khDx2 i Kx2 ∂ α J β u(t)kL2x L2 1

1

x2

−1

≤ C(1 + A) (1 + t) ku(t)k2X 2,2 (t) (1 + ku(t)k2X 2,2 (t) )ku(t)k2X 3,3 (t) Z ∞ X 0 α β 2 α β + ∂x1 |u| dx2 , Kx1 ∂ J u) Im(Kx1 ∂ J u x2 |α|+|β|≤3 2

Z + Im(Kx2 ∂ α J β u

∂x1 |u| dx2 , Kx2 ∂ J u) .

∞

0

2

x2

α β

(7.hir )

The second term of the left hand side of (7.hir ) means smoothing properties of solutions to the equation. So we have to estimate the term X

Z Im(Kx1 ∂ α J β u

∞

x2

|α|+|β|≤3

∂x1 |u|2 dx2 0 , Kx1 ∂ α J β u)

Z α β + Im(Kx2 ∂ J u

∞

x2

∂x1 |u| dx2 , Kx2 ∂ J u) . 2

0

α β

For this purpose, we pay attention to the special structure of the nonlinear term Z ∞ Z ∞ 1 u ∂x1 |u|2 dx2 0 = u u ¯Jx1 u − uJx1 udx2 0 . (8.hir ) 2it x2 x2 This deformation shows this nonlinear term has own time decay in some sence. Using this structure, we can estimate as following, X 1 d (kKx1 ∂ α J β u(t)k2 + kKx2 ∂ α J β u(t)k2 ) 2 dt |α|+|β|≤3 X

1 Cδ3

ku(t)kL2 khDx1 i1/2 Kx1 ∂ α J β u(t)kL2 2 2 − Ce + x2 x2 L x 1/2 1 4δ3 |α|+|β|≤3

2

1/2 + ku(t)kL2x khDx2 i Kx2 ∂ α J β u(t)kL2x L2 ≤ C(1 + t)−1 δ3 ku(t)k2X 3,3 (t) 1

1

x2

(9.hir )

135

The Existence of Global Solutions to DSI System

provided that δ3 is sufficiently small and sup

−T ≤t≤T

ku(t)k2X 2,2 (t) ≤ 4δ32 ,

(10.hir )

sup (1 + |t|)−Cδ3 ku(t)k2X 3,3 (t) ≤ 4δ32

(11.hir )

−T ≤t≤T

for some time T > 0. We choose δ3 satisfying 1 1/2 4δ3

− CeCδ3 ≥ 0.

Then we have

Z

ku(t)k2X 3,3 (t)

≤

eCδ3 δ32

+ Cδ3

t

(1 + s)−1 ku(s)k2X 3,3 (t) ds.

(12.hir )

0

Thus (9.hir ) shows that the nonliear term is controlled by the second term of the left hand side of (7.hir ) and the right hand side of (9.hir ). Global existence theorem is obtained by showing that (10.hir ) and (11.hir ) hold for any T . In order to prove (10.hir ) and (11.hir ) for any T > 0 we need (12.hir ) and the following inequality Z t ku(t)k2X 2,2 (t) ≤ eCδ3 δ32 + Cδ3 (1 + s)−1−2Cδ3 ku(s)k2X 3,3 (t) ds. (13.hir ) 0

The inequality (13.hir ) is obtained by the structure of nonlinear term (8.hir ) again. Theorem 1 is obtained by applying the Gronwall inequality to (12.hir ) and (13.hir ). It seems to be difficult to get the inequality (12.hir ) through the methods used in [8,9], because nonlinear terms are not taken into account to derive smoothing properties of solutions in [8,9]. On the other hand the operators Kx1 and Kx2 are made based on the nonlocal nonlinear terms (the second and the third terms on the right hand side of (4.hir )). The similar operators as those of Kx1 and Kx2 have been used in [1] to obtain Theorem 0.1 and the local existence theorem of small solutions to (4.hir ) in the usual order Sobolev space. Remark 4. We cannot apply above method to hyperbolic-hyerbolic Davey-Stewartson system. In fact, if we estimate similarly as above, we have 1 d 2 dt

X

(kKx1 ∂ α J β u(t)k2 + kKx2 ∂ α J β u(t)k2 )

|α|+|β|≤3

+

1

X

2

1/2 ˜ α β

( ku(t)kL2x khDx1 i K x1 ∂ J u(t)kL2x L2

1/2 4δ3 |α|+|β|≤3

2

2

x1

˜ x2 ∂ α J β u(t)kL2 2 2 ) + ku(t)kL2x khDx2 i1/2 K x L 1

1

x2

≤ C(1 + A)2 (1 + t)−1 ku(t)k2X 2,2 (t) (1 + ku(t)k2X 2,2 (t) )ku(t)k2X 3,3 (t) X

˜ x1 ∂ α J β u(t)kL2 2 2

ku(t)kL2 khDx1 i1/2 K + CeCδ3 x x L |α|+|β|≤3

2

2

x1

(14.hir )

136


under the condition (10.hir ) and (11.hir ), where Z x2 m R x2 ∞ Dx X A −∞ kv(t,x2 0 )k2L2 dx2 0 hDx1 i Am 0 2 0 Dx1 1 x1 ˜ kv(t, x2 )kL2x dx2 =e K x1 = 1 m! hDx1 i −∞ m=0 and ˜ x2 = K

Z x1 m R x1 ∞ Dx X A −∞ kv(t,x01 )k2L2 dx1 0 hDx2 i Am Dx2 2 x2 kv(t, x01 )k2L2x dx1 0 =e , 2 m! hDx2 i −∞ m=0

but we can easy to see that the last term of the right-hand side of (14.hir ) cannot controlled by the second term of the left-hand side of (14.hir ). Remark 5. For Davey-Stewartson systems, we can define formally the energy similar as the usual nonlinear Schrödinger equation if c1 , c2 ∈ R. But unfortunately, this energy is not conserved in elliptic-hyperbolic and hyperbolichyperbolic cases. So, we cannot use the usual H 1 a-priori estimate by energy. This is one reason that the global existence theorem of this system is difficult.

References [1] H. Chihara, The initial value problem for the elliptic-hyperbolic Davey-Stewartson equation, preprint (1995). [2] R. R. Coifman and Y. Meyer, Au del` a des opérateurs pseudodifférentieles, Astérique 57, Société Mathématique de France, 1978. [3] P. Constantin, Decay estimates of Schr¨ odinger equations, Comm. Math. Phys. 127 (1990), 101–108. [4] A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. A. 338 (1974), 101–110. [5] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillarygravity waves, J. Fluid Mech. 79 (1977), 703–714. [6] A. S. Fokas and L. Y. Sung, On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations, Inverse Problems 8 (1992), 673–708. [7] J. M. Ghidaglia and J. C. Saut, On the initial value problem for the DaveyStewartson systems, Nonlinearity 3 (1990), 475–506. [8] N. Hayashi, Local existence in time of small solutions to the Davey-Stewartson systems, Ann. Inst. Henri Poincaré, Physique théorique 65-4 (1996), 313–366. [9] N. Hayashi and H. Hirata, Local existence in time of small solutions to the elliptichyperbolic Davey-Stewartson system in the usual Sobolev spaces, Proc. Royal Soc. of Edinburgh Sec. A (1997) (to appear). [10] N. Hayashi and H. Hirata, Global existence and asymptotic behaviour in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity 9 (1996), 1387–1409. [11] N. Hayashi and J. C. Saut, Global existence of small solutions to the DaveyStewartson and the Ishimori systems, Diff. Integral Eqs. 8 (1995), 1657–1675. [12] H. Kumano-go, Pseudo-Differential Operators, The MIT press, 1974. [13] F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. Henri Poincaré, Anal. non linéaire 10 (1993), 523–548.



Scaling in Nonlinear Parabolic Equations : Locality versus Globality Grzegorz Karch Instytut Matematyczny, Uniwersytet Wroclawski pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland Email: [email protected] WWW: http://www.math.uni.wroc.pl/~karch

Abstract. The Cauchy problem for parabolic equations with quadratic nonlinearity is studied. We investigate the existence of global-in-time solutions and their large-time behavior assuming some scaling property of the equation as well as of the norm of the Banach space in which the solutions are constructed. AMS Subject Classification. 35K55, 35K15 Keywords. the Cauchy problem, self-similar solutions

We study the Cauchy problem for the parabolic equation ut = ∆u + B(u, u) supplemented by the initial condition u(x, 0) = u0 (x). Here u = u(x, t), x ∈ Rn , and t ∈ [0, T ) for some T ∈ (0, ∞]. We assume that the nonlinear term B(·, ·) is defined by a bilinear form acting on u(x, t) with respect to x only. This nonlinearity will also be assumed to satisfy a scaling property. To set it up, first given f : Rn → R we define the rescaled function fλ (x) = f (λx) for each λ > 0. We extend this definition for all f ∈ §0 in the standard way. Definition 1. The bilinear form B(·, ·) is said to have the scaling order equal to b ∈ R if B(fλ , gλ ) = λb B(f, g) λ

0

for any λ > 0 and all f, g ∈ § (R ), for which the both sides make sense. n

Our main requirement is that the bilinear form B(·, ·) has the scaling order equal to b < 2. Now suppose we are able to construct local-in-time solutions in C([0, T ); E), where the Banach space E consists of tempered distributions. Assume, moreover, that the equation is invariant under some scaling transformations of the The paper is the extended abstract of the paper [1].

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independent and dependent variables. We show that these two assumptions combined with a scaling property of k · kE allow us to obtain global-in-time solutions for suitably small initial data. To get such results we introduce a new Banach space of distributions which, roughly speaking, is a homogeneous Besov type space modeled on E. This approach allows us to get solutions for initial data less regular than those from E. In this abstract setting, we also study large-time behavior of constructed solutions. We find a simple condition (in terms of decay properties of the heat semigroup) which guarantees that solutions have the same asymptotic behavior as t → ∞.

References [1] Karch, G., Scaling in nonlinear parabolic equations: locality versus globality, Report of the Mathematical Institute, University of Wroclaw 92 (1997) 1–29, submitted.



Almost Sharp Conditions for the Existence of Smooth Inertial Manifolds Norbert Koksch Departement of Mathematics, Technical University Dresden, 01062 Dresden, Germany Email: [email protected]

Abstract. We consider the nonlinear evolution equation u˙ + Au = f (u) in a separable, real Hilbert space H assuming that A is a linear, selfadjoint, positive operator on H with compact resolvent. The nonlinearity f is assumed to belong to Cbk (D(Aα ), D(Aβ )) with k ∈ N>0 ∪ {1−} and nonnegative α, β satisfying 0 ≤ α − β ≤ 12 . Let PN be the orthogonal projection of H onto the subspace generated by the eigenvectors corresponding to the first N eigenvalues λi of A. We state an existence theorem for an inertial C k manifold graph(ϕ) with ϕ ∈ Cbk (PN D(Aα ), (I − PN )D(Aα )) using an almost sharp spectral gap condition √ α−β λN+1 − kλN > 2 Lip (f ) λα−β . N+1 + kλN Assuming the existence of an absorbing ball BD(Aα ) (r) in dom(Aα ), and √ √ assuming only f |BD(Aα ) ( 2r) ∈ Cbk (BD(Aα ) ( 2r), D(Aβ )), we state the existence of a globally attracting, locally positively invariant C k manifold graph(ϕ) ∩ BD(Aα ) (r) using the spectral gap condition √ √ α−β λN+1 − kλN > 2 Lip f |BD(Aα ) ( 2r) λα−β N+1 + kλN where r > r. For it a special preparation of f is used. The proofs of the theorems base on comparison theorems for special twopoint boundary value problems and for inequalities in ordered Banach spaces. AMS Subject Classification. 34C30, 35K22, 34G20, 47H20 Keywords. smooth inertial manifolds, spectral gap condition, graph transformation, boundary value problems, comparison theorems

1

Introduction

Let H be a separable, real Hilbert space with inner product h·|·i and norm | · |. We consider the nonlinear evolution equation u˙ + Au = f (u) for u ∈ H where A satisfies This is the final form of the paper.

(1.kok )

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Assumption 1. A is a linear, self-adjoint, positive operator on H with compact resolvent. Thus, −A is the infinitesimal generator of an analytic semigroup on H. Let λ1 ≤ λ2 ≤ λ3 ≤ · · · denote the eigenvalues of A repeated with their multiplicities, and let ei denote corresponding orthonormal eigenvectors of A. By the properties of A the eigenvectors ei form an orthonormal basis in H. We can define the fractional powers Aα for α ∈ R, see [Hen81]. The domains Uα := D(Aα ) of Aα are Hilbert spaces with respect to the scalar product hu|viα := hAα u|Aα vi, and the corresponding norm | · |α is equivalent to the graph norm. With PN we denotes the orthogonal projection of H onto ∞ P span{e1 , . . . , eN }. Since Uα = {u ∈ H : hu, ej i2 λ2α j < ∞}, we have PN H ∩ j=1

Uα = PN Uα and (I − PN )H ∩ Uα = (I − PN )Uα . Further PN commutes with Aγ for γ ≥ 0. The nonlinear term f is assumed to satisfy at least Assumption 2. There are k ∈ N>0 ∪ {1−}, κ ∈ [0, 1[ with κ = 0 iff k = 1− and nonnegative constants α, β satisfying 0 ≤ α − β ≤ 12 such that f |Ω belongs k+κ to Cbu (Ω, Uβ ) for any bounded set Ω ⊂ Uα . 1− (E, F) denotes the Banach Here f |Ω denotes the restriction of f onto Ω. Cbu space of the bounded continuous functions from E into F being uniformly Lipk+κ schitz. For k ≥ 1, Cbu (E, F) denotes the Banach space of the k-times κ-Hölder continuously differentiable functions from E into F with bounded derivatives up to the order k. In the following, we calculate with 1− as with 1. We denote by Lip (g) the smallest Lipschitz constant of g on its domain dom(g). For a subspace U of Uα endowed with the induced topology let

BU (r) := {u ∈ U : |u|α < r} be the open ball in U centered at 0 with radius r ≤ ∞. Applying the results of [Hen81], equation (1.kok ) generates a (local) semigroup S in Uα , such that the (classical) solution at time t in the existence interval through an initial point u0 ∈ Uα is given by u(t) = S(t)u0 . For t > 0, u(t) is ˙ ∈ Uβ . more regular than the initial point, with u(t) ∈ U1+β ⊆ D(A) and u(t) These regularity results make it possible to work with the equation itself and take inner products rather than have to use the variation of constant formula. In particular expressions such as

1 d 2α−β |u(t)|2α = Aβ u(t)|A ˙ u(t) 2 dt make sense for t > 0 since U1+β ⊆ U2α−β because of α − β ≤ 12 and since u(t) ˙ ∈ Uβ and u(t) ∈ U1+β for t > 0. Recall that an inertial C k manifold M is a subset of H with the following properties (see [MPS88,FST88,Tem88] for k = 1−):

141

Smooth Inertial Manifolds

1. M is a finite dimensional C k manifold in Uα ⊆ H. 2. M is positively invariant; i.e., if u0 ∈ M then S(t)u0 ∈ M for all t ∈ [0, ∞[. 3. M is exponentially attracting; i.e., there is a γ > 0 such that for every η ∈ Uα there is a C such that dist(S(t)η, M) ≤ Ce−γt

(t ≥ 0).

In some papers the exponential attracting property is supplemented by the exponential tracking property ([FST89]) or asymptotical completeness property ([CFNT89,Rob96,Tem97]): There is γ > 0 such that for every η ∈ Uα there are ηˆ ∈ M and C ≥ 0 with |S(t)η − S(t)ˆ η |α ≤ Ce−γt dist(η, M) for all t > 0. Usually we are looking for an inertial C k manifold M which is constructed as the graph graph(ϕ) := {ξ + ϕ(ξ) : ξ ∈ PN Uα } of a C k function ϕ : PN Uα → (I − PN )Uα . Because of the attraction property of M, the asymptotical behavior of the solutions of (1.kok ) is governed by the asymptotical behavior of the solutions on the finite-dimensional manifold M. The dynamic on M is determined by the ordinary differential equation (inertial form) x˙ + Ax = PN f (x + ϕ(x)) in the N -dimensional Banach space PN Uα . Instead of Assumptions 2 usually one assumes f ∈ Cbk (Uα , Uβ )

(2.kok )

with suitable α ≥ β: For k = 1− we have for example α = 1, β = 12 in [FST88], β = α − 12 in [Tem88], α = β = 0 in [MPS88], 0 = β ≤ α < 1 in [Rom94], 0 ≤ α − β ≤ 12 in [Rob93], 0 ≤ α − β < 1 in [CLS92]. Thus our assumption 0 ≤ α − β ≤ 12 assumed for technical reason is not the weakest possible one. A spectral gap condition mostly of the form α−β ) λN +1 − λN > C1 Lip (f ) (λα−β N +1 + λN

(3.kok )

plays an important role where C1 is a number depending on α, β, and Lip (f ). Romanov [Rom94] found (3.kok ) with C1 = 1 ensuring the existence of a Lipschitz inertial manifold for (1.kok ) with 0 = β ≤ α < 1. He gave counter-examples satisfying a spectral gap condition (3.kok ) with C1 < 1 but not having an inertial manifold. That means, the spectral gap condition (3.kok ) with C1 = 1 is a sharp condition for Lipschitz inertial manifolds.√As corollary of our Theorem 8 we have a spectral gap condition (3.kok ) with C1 = 2, i.e. our spectral gap condition is a little stronger than Romanov’s one. The weakest known spectral gap condition in the form (3.kok ) for inertial C 1 manifolds was found by Ninomiya [Nin92] with C1 = 2 for 0 ≤ α − β < 1/2. Our

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√ Theorem 8 will allow C1 = 2, i.e. our spectral gap condition is a little weaker than Ninomiya’s one. For k > 1 and (2.kok ), Chow et al. [CLS92] have a spectral gap condition of the form α−β λN +1 − kλN > C1 λα−β , λN > C0 + λ N +1 N but with unknown C0 , C1 depending on α, β, k and Lip (f ). Additionally they get the a priori estimate Lip (ϕ) ≤ 1. Inserting (10.kok ) with Q = Uα in (14.kok ) we obtain the spectral gap condition √ α−β λN +1 − kλN > 2 Lip (f ) λα−β N +1 + kλN ) even for k ≥ 1. Moreover, we for the existence of an inertial C k manifold of (1.kok get the better a priori estimate Lip (ϕ) ≤ χ1 where the number χ1 < 1 is defined in Lemma 4. Let Q be an open set in Uα . In order to include also manifolds which are subsets of Q, we introduce the following notion: A set M is called inertial C k manifold in Q if 1. M is a finite dimensional C k manifold in Uα . 2. M ∩ Q is locally positively invariant; i.e., if u0 ∈ M ∩ Q then there is ε > 0 such that S(t)u0 ∈ M ∩ Q for all t ∈ [0, ε[. 3. M is exponentially attracting for all orbits in Q; i.e., there is a γ > 0 such that for any u0 with S(t)u0 ∈ Q for t > 0 there is a constant C such that dist(S(t)η, M) ≤ Ce−γt

(t ≥ 0).

If M is an inertial manifold in Q then the asymptotical behavior of the orbits of (1.kok ) in Q is determined by the orbits of (1.kok ) in M ∩ Q. If Q = Uα and dom(ϕ) = PN Uα then an inertial manifold M = graph(ϕ) in Q is an inertial manifold in the usual sense. If f does not satisfy (2.kok ), it is usually modified by a trunctation method to a new function f so that the asymptotic behavior of the solutions of (1.kok ) is not ) then f is modified changed but f satisfies (2.kok ): If B(r) is an absorbing set of (1.kok outside of B(r) in such a way that f (u) = 0 outside of B(2r), and such that Lip (f ) of the new function is not greater than Lip (f |B(2r)) of the old function. Then an inertial manifold M of the prepared equation is an inertial manifold in ). B(r) of the original equation (1.kok Let Q = BPN Uα (r) + B(I−PN )Uα (r) and Q = BPN Uα (r) + B(I−PN )Uα (r) with r > r > r arbitrary close to r. Theorem 8 allows to use Lip f |Q instead of Lip (f |B(2r)) even for k ≥ 1 such that weakening in the √ we get an additional spectral gap condition since Lip f |Q ≤ Lip f |B( 2r) ≤ Lip (f |B(2r)) for √ r < 2r. A crucial role in the proof of Theorem 8 plays comparison system (8.kok ). System (8.kok ) has a linear inertial manifold in R2≥0 if and only if the spectral gap condition (5.kok ) is satisfied.

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2 2.1

Main Results Existence of Smooth Inertial Manifolds in a Set Q

Let N ∈ N be suitable chosen. In order to simplify notation we shall use Λ1 := λN ,

Λ2 := λN +1 ,

π1 := PN , α Uα 1 := π1 U ,

π2 := I − PN , α Uα 2 := π2 U

α such that Uα = Uα 1 ⊕ U2 . To avoid repetition, we agree that i always ranges over the integers 1 and 2. We assume that the set Q has the special form

Q := BUα1 (r1 ) + BUα2 (r2 ), where ri ∈ R≥0 or r1 = r2 = ∞. In order to ensure the existence of an inertial manifold in Q, we introduce Assumption 3. There are numbers γi > 0 and r i with ri < r i < ∞ or ri = ri = ∞ such that the one-sided Lipschitz inequalities

2α−β A π1 u∆ |Aβ π1 [f (u1 ) − f (u2 )] ≥ −γ1 Λβ−α |π1 u∆ |2α−β |u∆ |α , 1

2α−β β−α β A π2 u∆ |A π2 [f (u1 ) − f (u2 )] ≤ γ2 Λ2 |π2 u∆ |2α−β |u∆ |α

(4.kok )

hold for any ui ∈ Q ∩ U1+β where u∆ = u1 − u2 and Q := BUα1 (r 1 ) + BUα2 (r 2 ). The following technical lemma gives a connection between the spectral gap condition (5.kok ) and a comparison problem (8.kok ) in the plane: Lemma 4. Let the spectral gap condition 3/2 2/3 2/3 Λ 2 − Λ 1 > γ 1 + γ2

(5.kok )

be satisfied. Then we have: p 1. There are χ2 > 3 γ2 /γ1 > χ1 > 0 and %2 < %1 which are uniquely determined by q q (6.kok ) %i = −Λ1 − γ1 1 + χ2i = −Λ2 + γ2 1 + χ−2 i . Moreover, %2 < 0.

(7.kok )

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Norbert Koksch

2. The sets Ψi := {w ∈ R2≥0 : w2 = χi w1 } are integral manifolds of the comparison system (8.kok ) w˙ 1 = −Λ1 w1 − γ1 |w|, w˙ 2 = −Λ2 w2 + γ2 |w|, p where |w| = (w1 )2 + (w2 )2 . The function ψi : R≥0 → R2≥0 defined by ψi (t) := e%i t (1, χi )

(t ≥ 0)

is the solution of (8.kok ) through (1, χi ) ∈ Ψi at t = 0. Proof. First we note that Ψ = {w ∈ R2≥0 : w2 = χw1 } is an integral manifold of (8.kok ) if χ ≥ 0 is a zero of the function p : R>0 → R defined by p p p(χ) = Λ1 − Λ2 + γ1 1 + χ2 + γ2 1 + χ−2 . = +∞. The function p is strongly convex with limχ→0 p(χ) = limχ→∞ p(χ) p 3 Hence p has at most two positive zeroes. p is minimized at χ0 := γ2 /γ1 and p(χ0 ) < 0 because of (5.kok ). Therefore, the existence of positive zeroes χ1 , χ2 of p with χ1 < χ0 < χ2 follow. By definition of p, these numbers χ1 , χ2 satisfy (6.kok ). Thus Ψi are integral manifolds of the comparison system (8.kok ) and the functions ψi are solutions on Ψi with the stated properties. ). t u Since Λ1 > 0 we have %2 < %1 < 0 and hence (7.kok ) in R2≥0 . Remark 5. Ψ1 is an inertial manifold of (8.kok Remark 6. Requiring p(1) < 0 one gets the little stronger gap condition √ Λ2 − Λ1 > 2(γ2 + γ1 ).

(9.kok )

Assuming (9.kok ) we have χ1 < 1 < χ2 and √ √ %1 > −Λ1 − 2γ1 , %2 < −Λ2 + 2γ2 . Remark 7. If ri < ∞ and Lip f |Q > 0 the existence of numbers γi satisfying Assumption 3 follows from Assumption 2: We can choose . (10.kok ) γi = Lip f |Q Λα−β i The spectral gap conditions (5.kok ), (9.kok ) read now

Λ2 − Λ1 > Lip f |Q and Λ2 − Λ1 > in the well-known form.

2(α−β)/3

Λ2

2(α−β)/3

3/2

+ Λ1

√ α−β 2 Lip f |Q Λα−β + Λ 2 1

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Theorem 8 (Inertial manifold in Q). Let the Assumption 1, 2, 3 be satisfied. If (5.kok ) then there is a ϕ ∈ Cb1− (BUα1 (r1 ), Uα 2 ) with Lip (ϕ) ≤ χ1 and being uniquely defined if ri = ∞ such that M := graph(ϕ) is an inertial C 1− manifold in Q with dist(S(t)u0 , M) ≤

χ 2 + χ1 |π2 u0 − ϕ(π1 u0 )|α e%2 t χ2 − χ1

(t ≥ 0)

(11.kok )

for any u0 with S(t)u0 ∈ Q for t ≥ 0. Moreover, for any Q ⊆ Q with positive distance to ∂Q if ri < ∞ and any u0 with S(t)u0 ∈ Q for t ≥ 0 there are u ˆ0 ∈ M ∩ Q and T ≥ 0 with S(t)ˆ u0 ∈ M ∩ Q for t ≥ 0 and |π2 u0 − ϕ(π1 u0 )|α i ψ2 (t + T ) χ2 − χ1

|πi [S(t + T )u0 − S(t)ˆ u0 ]|α ≤

(t ≥ 0)

(12.kok )

where T = 0 if r2 = ∞. If in addition k ≥ 1 and %2 > k%1

(13.kok )

then ϕ ∈ Cbk (BUα1 (r1 ), BUα2 (r2 )). Theorem 8 will be proved by means of Theorem 11 concerning the existence of special overflowing invariant manifolds. p Remark 9. Since χ1 < 3 γ2 /γ1 < χ2 we have q q 2/3 2/3 2/3 2/3 2/3 2/3 γ1 + γ2 , %2 < −Λ2 + γ2 γ 1 + γ2 %1 > −Λ1 − γ1 such that (13.kok ) can be replaced by Λ2 − kΛ1 >

2/3 (γ2

q

+

2/3 kγ1 )

2/3

γ1

2/3

+ γ2 .

Assuming (9.kok ), this inequality can be replaced by the stronger condition √ Λ2 − kΛ1 > 2(γ2 + kγ1 ). 2.2

(14.kok )

Overflowing Invariant Manifolds

Theorem 8 will be reduced to the following Theorem 11 concerning the existence of an overflowing invariant manifold for the prepared evolution equation u˙ + Au = f˜(u).

(15.kok )

∗

α A set M = graph(ϕ∗ ) with ϕ∗ : cl W0 → Uα 2 and W0 ⊆ U1 is called overflowing invariant with respect to (15.kok ) (compare [Wig94]) if:

– M∗ := graph(ϕ∗ |W0 ) is locally positively invariant with respect to (15.kok ).

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– The vector field of (15.kok ) is pointing strictly outward on the boundary ∂M∗ = ∗ ∗ M \M . – The vector field of (15.kok ) is nonzero on ∂M∗ . Besides Assumption 1 we need Assumption 10. There are rî with 0 < rˆ1 < r1 < ∞ or 0 < rˆ1 ≤ r1 ≤ ∞ and 0 < rˆ2 < r 2 ≤ ∞ such that f˜|Q belongs to Cbk (Q, Uβ ), and such that f˜ satisfies D E A2α−β π1 u∆ |Aβ π1 [f˜(u1 ) − f˜(u2 )] ≥ −γ1 Λβ−α |π1 u∆ |2α−β |u∆ |α , 1 E D (16.kok ) β−α 2α−β β π2 u∆ |A π2 [f˜(u1 ) − f˜(u2 )] ≤ γ2 Λ2 |π2 u∆ |2α−β |u∆ |α A for ui ∈ Q ∩ U1+β where u∆ = u1 − u2 , and E D A2α−β π1 u| − A1+β π1 u + Aβ π1 f˜(u) > 0 E D A2α−β π2 u| − A1+β π2 u + Aβ π2 f˜(u) < 0

if |π1 u|α = rˆ1 , if |π2 u|α = rˆ2

(17.kok )

for u ∈ Q ∩ U1+β . The inequalities (17.kok ) ensure some inflowing and outflowing properties of the vector field on the boundary of ˆ := BUα (ˆ Q r1 ) + BUα2 (ˆ r2 ). 1 Let S˜ denote the local semiflow of (15.kok ) in Q. Theorem 11 (Overflowing invariant manifold). Let Assumption 1 and 10 r1 ). Then there is a unique ϕ∗ ∈ as well as (5.kok ) be satisfied and let W0 := BUα1 (ˆ 1− ˆ2 for ξ ∈ cl W0 such that Cb (cl W0 , Uα 2 ) with Lip (ϕ) ≤ χ1 and |ϕ(ξ)|α ≤ r ∗ ∗ M := graph(ϕ ) is overflowing invariant with respect to the prepared evolution ˆ with S(t)u ˆ ˜ equation (15.kok ). Moreover, for any u0 ∈ Q 0 ∈ Q for t ≥ 0 there is ˜ u0 ∈ M∗ for t ≥ 0 and u ˆ0 ∈ M∗ with S(t)ˆ ∗ ˜ ˜ u0 ]|α ≤ |π2 u0 − ϕ (π1 u0 )|α ψ i (t) |πi [S(t)u 0 − S(t)ˆ 2 χ2 − χ1

(t ≥ 0).

If k ≥ 1 and (13.kok ) then ϕ∗ ∈ Cbk (cl W0 , Uα 2 ). In order to show the existence of a C 1− manifold with the properties stated in Theorem 11 we proceed as follows. For fixed γ ∈ [α, β + 1[ we introduce the Banach space G0 := Cb0 (cl W0 , Uγ2 ) equipped with the supremum norm ||ϕ0 ||0 := sup |ϕ0 (ξ)|γ . Let ξ∈cl W0

Φ0 := {ϕ0 ∈ G0 : ||ϕ0 ||0 ≤ rˆ2 , Lip (ϕ0 ) ≤ χ1 }. Note that Φ0 is a closed subset of G0 .

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We introduce the two-point boundary value problems u˙ + Au = f˜(u)

(18a.kok )

π1 u(ϑ) = ξ, π2 u(0) = ϕ0 (π1 u(0))

(18b.kok )

on [0, ϑ] with ξ ∈ cl W0 , ϑ > 0, and ϕ0 ∈ Φ0 . Showing that (18.kok ) has a unique solution U0 (·, ϑ, ξ, ϕ0 ) satisfying U0 (t, ϑ, ξ, ϕ0 ) ∈ Q

(t ∈ [0, ϑ])

for any ϑ > 0, ξ ∈ cl W0 , ϕ0 ∈ Φ0 , we can define the G0 (ϑ) : Φ0 → G0 by (G0 (ϑ)ϕ0 )(ξ) = π2 U0 (ϑ, ϑ, ξ, ϕ0 )

(ϑ ≥ 0, ξ ∈ cl W0 , ϕ0 ∈ Φ0 ).

Using some properties of U0 (t, ϑ, ξ, ϕ0 ) we can show that G0 (ϑ) maps Φ0 into itself and that G0 (ϑ) is uniformly contractive for ϑ ≥ T0 and sufficiently large T0 . Hence there is a unique fixed-point ϕ∗0 (ϑ) in Φ0 for ϑ ≥ T0 . Showing the existence of these fixed-points for all ϑ > 0 and showing their independence of ϑ we get the locally positive invariance of graph(ϕ∗0 |W0 ). The exponential tracking property can also be proved reducing it to the estimation of solutions of boundary value problems. In order to show higher smoothness of ϕ0 assuming the spectral gap condition (13.kok ), we shall use the fiber contraction principle [Van89,CLS92,Tem97]. Since C k smoothness for k ≥ 3 can be proved similarly to the C 2 -smoothness, we restrict us to k ≤ 2. First let k = 2. Let the spectral gap condition (13.kok ) be satisfied and let γ ∈]α, β + 1[ be fixed. Applying the implicit function theorem one can show that U0 (t, ϑ, ·, ϕ0 ) is twice continuously differentiable for t ∈ [0, ϑ], ϑ > 0, ϕ0 ∈ Φ0 . We introduce G1 := Cb0 (cl W0 , L(Uγ1 , Uγ2 )), G2 := Cb0 (cl W0 , L(Uγ1 × Uγ1 , Uγ2 )). G1 , G2 are complete with respect to the norms || · ||1 and || · ||2 defined by ||ϕ1 ||1 := sup

max

||ϕ2 ||2 := sup

max

(1) ξ∈cl W0 h∈cl BUγ 1

|ϕ1 (ξ)h|γ ,

(1) ξ∈cl W0 hi ∈cl BUγ 1

|ϕ2 (ξ)(h1 , h2 )|γ

for ϕ1 ∈ G1 , ϕ2 ∈ G2 . Further we introduce the closed sets Φ1 := {ϕ1 ∈ G1 : ||ϕ1 ||1 ≤ χ1 }, Φ2 := G2 . One can show that for any ϑ > 0, ξ ∈ cl W0 , (ϕ0 , ϕ1 , ϕ2 ) ∈ Φ0 × Φ1 × Φ2 , h1 , h2 ∈ Uγ1 there are a unique classical solution U1 (·, ϑ, ξ, ϕ0 , ϕ1 , h1 ) of u˙ + Au = Df˜(U (t))u, π1 u(ϑ) = h1 ,

π2 u(0) = ϕ1 (π1 U (0))π1 u(0)

(19.kok )

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Norbert Koksch

on [0, ϑ] and a unique classical solution U2 (·, ϑ, ξ, ϕ0 , ϕ1 , ϕ2 , h1 , h2 ) of u˙ + Au = Df˜(U (t))u + R1 (t), π1 u(ϑ) = 0,

(20.kok )

π2 u(0) = ϕ1 (π1 U (0))π1 u(0) + R2

on [0, ϑ] where U (t) = U0 (t, ϑ, ξ, ϕ0 ), R1 (t) = D2 f˜(U (t))(U1 (t, ϑ, ξ, ϕ0 , ϕ1 , h1 ), U1 (t, ϑ, ξ, ϕ0 , ϕ1 , h2 )), R2 = ϕ2 (π1 U (0))(U1 (0, ϑ, ξ, ϕ0 , ϕ1 , h1 ), U1 (0, ϑ, ξ, ϕ0 , ϕ1 , h2 )). We define G1 (ϑ) : Φ0 × Φ1 → G1 , G2 (ϑ) : Φ0 × Φ1 × Φ2 → G2 by (G1 (ϑ)(ϕ0 , ϕ1 ))(ξ, h1 ) = π2 U1 (ϑ, ϑ, ξ, ϕ0 , ϕ1 , h1 ),

(21.kok )

(G2 (ϑ)(ϕ0 , ϕ1 , ϕ2 ))(ξ, h1 , h2 ) = π2 U2 (ϑ, ϑ, ξ, ϕ0 , ϕ1 , ϕ2 , h1 , h2 ) for ϑ > 0, ξ ∈ cl W0 , (ϕ0 , ϕ1 , ϕ2 ) ∈ Φ0 × Φ1 × Φ2 , hi ∈ Uγ1 . There are T2 ≥ 0 and closed Φ˜j ⊂ Φj such that G0 (T2 ), G1 (T2 )(ϕ0 , ·), G2 (T2 )(ϕ0 , ϕ1 , ·) are uniformly ˜1 . contractive selfmappings on Φ˜0 , Φ˜1 , Φ˜2 respectively, for (ϕ0 , ϕ1 ) ∈ Φ˜0 × Φ ˜ ˜ ˜ Because of these contraction properties, the mapping G : Φ0 × Φ1 × Φ2 → Φ˜0 × Φ˜1 × Φ˜2 defined by G(ϕ0 , ϕ1 , ϕ2 ) := (G0 (T2 )(ϕ0 ), G1 (T2 )(ϕ0 , ϕ1 ), G2 (T2 )(ϕ0 , ϕ1 , ϕ2 )) ˜1 ×Φ˜2 has a unique fixed-point (ϕ∗0 , ϕ∗1 , ϕ∗2 ) ∈ Φ˜0 ×Φ˜1 ×Φ ˜2 . for (ϕ0 , ϕ1 , ϕ2 ) ∈ Φ˜0 ×Φ ˜2 , the fiber Showing the continuity of G1 (·, ϕ1 ), G2 (·, ·, ϕ2 ) for (ϕ1 , ϕ2 ) ∈ Φ˜1 × Φ contraction principle implies the attractivity of (ϕ∗0 , ϕ∗1 , ϕ∗2 ), i.e., the convergence of the iterates (n)

(n)

(n)

(ϕ0 , ϕ1 , ϕ2 ) := Gn (ϕ0 , ϕ1 , ϕ2 ) to (ϕ∗0 , ϕ∗1 , ϕ∗2 ) ∈ Φ˜0 × Φ˜1 × Φ˜2 for any (ϕ0 , ϕ1 , ϕ2 ) ∈ Φ˜0 × Φ˜1 × Φ˜2 . Choosing (ϕ0 , ϕ1 , ϕ2 ) = (0, 0, 0) we have (n)

Dϕ0 (n)

This and ϕ0

→ ϕ∗0 , ϕ1

(n)

= ϕ1 ,

(n)

(n)

D2 ϕ0

→ ϕ∗1 , ϕ1

(n)

(n)

= ϕ2

(n ∈ N).

→ ϕ∗1 imply

Dϕ∗0 = ϕ∗1 ,

D2 ϕ∗0 = ϕ∗2 ,

i.e., the C 2 -smoothness of graph(ϕ∗0 |W0 ). For k = 1 the proof proceeds similar to the case k = 2 where we use G : Φ˜0 × Φ˜1 → Φ˜0 × Φ˜1 defined by G(ϕ0 , ϕ1 ) := (G0 (T2 )(ϕ0 ), G1 (T2 )(ϕ0 , ϕ1 )). In order to study (18.kok ), (19.kok ), (20.kok ) we shall develop and use comparison theorems for such boundary value problems. The main difficulties are here that the comparison problem in R2≥0 will be a nonlinear one (in order to get an almost sharp gap condition) and that the differential inequality in general holds only in a part of R2≥0 (because of the nonequivalence of | · |α and | · |2α−β for α > β.)

149


2.3

Proof of Theorem 8

In order to apply Theorem 11, we have to determine numbers rî and a suitable modification f˜ of f satisfying Assumption 10. Let the assumptions of Theorem 8 be satisfied. First let ri = r i = ∞. In this case we can choose f˜ = f and rˆ1 = ∞. Remains the choice of rˆ2 < ∞ satisfying (17.kok ). Because of Assumption 2, there is a constant K0 with |f (u)|β ≤ K0 for K0 satisfies (17.kok ) since u ∈ Uα . One can show that a any rˆ2 > Λ−1+α−β 2 D E −A1+β π2 u + Aβ π2 f˜(u)|A2α−β π2 u ≤ (−Λ2 rˆ2 + K0 Λα−β )ˆ r2 < 0 2 for any u ∈ Q ∩ U1+β with |π2 u|α ≥ rˆ2 . Thus Assumption 10 is satisfied. Theorem 11 implies the existence of an inertial manifold M = graph(ϕ) with α ϕ ∈ Cb1− (Uα 1 , π2 U ) and Lip (ϕ) ≤ χ1 . 0 α Let graph(ϕ ) with ϕ0 ∈ Cb1− (Uα 1 , U2 ) be another inertial manifold with Lip (ϕ0 ) ≤ χ1 . Choosing rˆ2 > max{||ϕ||, ||ϕ0 ||, Λ−1+α−β K0 }, Theorem 11 implies 2 ϕ = ϕ0 . Thus Theorem 8 is proved in the case r i = ∞. Let now ri < r i < ∞. Let rî with ri < rî < r i be arbitrary. In order to construct the function f˜ let b ∈ C ∞ (−∞, ∞) be a bump function with the following properties: b(w) = 0 for w ≤ 0, b(w) = 1 for w ≥ 1, Db(w) ≥ 0. We introduce fî ∈ C ∞ (Uα , Uα ) defined by |πi u|2α − ri2 πi u (u ∈ Uα ). fî (u) := b rî2 − ri2 Then for any u ∈ Uα we have πi fˆj (u) = 0 if i 6= j, fî (u) = 0 if |πi u|α ≤ ri , fî (u) = πi u if |πi u|α ≥ rî . (22.kok ) Further

D

E Aα πi h|Aα πi Dfî (u)h ≥ 0

(u, h ∈ Uα ).

Applying the mean value theorem to the scalar-valued function D E τ 7→ Aα πi h|Aα πi fî (u + τ h) we obtain

D

E Aα πi h|Aα πi [fî (u + h) − fî (u)] ≥ 0

and hence D E A2α−β πi h|Aβ πi [fî (u + h) − fî (u)] ≥ 0

(u, h ∈ Uα )

(u ∈ Uα , h ∈ U2α−β ).

(23.kok )

150

Norbert Koksch

There is µ1 > 0 satisfying −Λ1 r2 − γ1 r + µ1 r2 ≥ 1 for r ∈ [ˆ r1 , r 1 ]. Further 2 γ + 1)/ˆ r . let µ2 := ( 14 − Λ−1 2 2 2 Now we are in position to introduce f˜ : Uα → Uβ defined by f˜(u) := f (u) + µ1 fˆ1 (u) − µ2 fˆ2 (u)

(u ∈ Uα )

satisfying Assumption 10: The inequalities (16.kok ) follows from (23.kok ) and (4.kok ). For any u ∈ Q ∩ U1+β with |π1 u|α ≥ rˆ1 , we have D E −A1+β π1 u + Aβ π1 f˜(u)|A2α−β π1 u ≥ 1 by choice of µ1 and hence the first inequality in (17.kok ). Further one can show D E −A1+β π2 u + Aβ π2 f˜(u)|A2α−β π2 u ≤ −1 ) is for any u ∈ Q ∩ U1+β with |π2 u|α ≥ rˆ2 . Thus the second inequality in (17.kok satisfied, too. Applying Theorem 11 to the prepared evolution equation (15.kok ) we get an ∗ overflowing invariant manifold M = graph(ϕ∗ ) with the properties stated in this theorem. Let ϕ := ϕ∗ |BUα1 (r1 ) and M := graph(ϕ). Then Lip (ϕ) ≤ χ1 . Because of (22.kok ), we have f˜(u) = f (u)

(u ∈ Q).

Therefore, the manifold M ∩ Q is locally positively invariant with respect to (1.kok ). ˜ Let u0 ∈ Q with S(t)u 0 = S(t)u0 ∈ Q for t ≥ 0. By means of Theorem 11 there ˆ and ˜ u0 ∈ Q is u ˜0 ∈ M∗ with S(t)˜ ˜ ˜ u0 ]|α ≤ |π2 u0 − ϕ∗ (π1 u0 )|α ψ i (t) |πi [S(t)u 0 − S(t)˜ 2 for t ≥ 0. Thus dist(S(t)u0 , M) ≤ |π2 S(t)u0 − ϕ(π1 S(t)u0 )|α ˜ u0 |α + |ϕ∗ (π1 S(t)˜ ˜ u0 ) − ϕ∗ (π1 S(t)u0 )|α ≤ |π2 S(t)u0 − π2 S(t)˜ ≤ |π2 u0 − ϕ∗ (π1 u0 )|α (χ1 ψ21 (t) + ψ22 (t)) for t ≥ 0 such that (11.kok ) follows. If Q = Q = Uα , the exponential attracting property follows directly from Theorem 11. ˜ Let Q ⊂ Q have positive distance to ∂Q and let u0 satisfy S(t)u 0 = S(t)u0 ∈ Q ˜ u0 ∈ M∗ for t ≥ 0 and for t ≥ 0. By means of Theorem 11 there is u ˜0 with S(t)˜ ˜ u0 ]|α ≤ |π2 u0 − ϕ∗ (π1 u0 )|α ψ2i (t) |πi [S(t)u0 − S(t)˜ ˜ u0 ∈ Q for t ≥ 0. Using these inequalities the existence of T ≥ 0 follows with S(t)˜ ˜ for t ≥ T . Let u ˆ0 := S(T )˜ u0 . Then S(t)ˆ u0 ∈ M ∩ Q for t ≥ 0 and the inequalities (12.kok ) follow. The smoothness properties of ϕ follow directly from Theorem 11. Thus Theorem 8 is proved. t u

151


3

Some Comparison Theorems for Two-point Boundary Value Differential Inequalities

Let the assumptions of Theorem 11 be satisfied. The following Lemmas 12, 13, 14 give a connection between solutions or the difference of solutions of the boundary value problems (18.kok ), (19.kok ), (20.kok ) and a solution v ∈ C([0, ϑ], R2≥0 of the boundary value differential inequality v˙ 1 (t) ≥ (−Λ1 − %)v 1 (t) − γ1 |v(t)| − A1 for a.e. t ∈ [0, ϑ], v˙ 2 (t) ≤ (−Λ2 − %)v 2 (t) + γ2 |v(t)| + A2 if v(t) ∈ V+ (%, A2 ), v (ϑ) ≤ B1 , 1

(24.kok )

v (0) ≤ χ1 v (0) + B2 2

1

where A1 , A2 , B1 , B2 are nonnegative numbers, % > −Λ2 , and V+ (%, A2 ) := {v ∈ R2≥0 : −2(−Λ2 − %)v 2 > γ2 |v| + A2 }. For a compact time interval T let min T (max T) denote the lower (upper) boundary point of T. The main goal of this section is to develop Theorem 16 and 17 for the comparison of solutions v ∈ C([0, ϑ], R2≥0 ) of (24.kok ) with solutions w ∈ C([0, ϑ], R2≥0 ) of the boundary value problem w˙ 1 (t) = (−Λ1 − %)w1 (t) − γ1 |w(t)| − a1 , w˙ 2 (t) = (−Λ2 − %)w2 (t) + γ2 |w(t)| + a2 , w1 (max T) = b1 ,

(t ∈ T),

w2 (min T) = χ1 w1 (min T) + b2

(25a.kok ) (25b.kok )

ˆ ∈ C([0, ϑ], R2≥0 ) of the where ai = Ai , bi = Bi , T = [0, ϑ], or with solutions w boundary value differential inequality w˙ 1 (t) ≤ (−Λ1 − %)w1 (t) − γ1 |w(t)| − a1 , w˙ 2 (t) ≥ (−Λ2 − %)w2 (t) + γ2 |w(t)| + a2 w (max T) ≥ b1 , 1

(t ∈ T),

(26.kok )

w (min T) ≥ χ1 w (min T) + b2 2

1

where ai = Ai , bi = Bi , % ∈ R, T = [0, ϑ]. In an intermediate step we shall compare solutions v ∈ C(T, R2≥0 ) of v˙ 1 (t) ≥ (−Λ1 − %)v 1 (t) − γ1 |v(t)| − a1 , v˙ 2 (t) ≤ (−Λ2 − %)v 2 (t) + γ2 |v(t)| + a2 v 1 (max T) ≤ b1 ,

(t ∈ int T),

v 2 (min T) ≤ χ1 v 1 (min T) + b2 .

with solutions w and w ˆ of (25.kok ) or (26.kok ), respectively.

(27.kok )

152

Norbert Koksch

Lemma 12. Let u1 : [0, ϑ] → Q be a solution of the boundary value problem ). Then v ∈ C([0, ϑ], R2≥0 ) (18.kok ) and let u2 : [0, ϑ] → Q be a solution of (18a.kok defined by v(t) = (|π1 [u1 (t) − u2 (t)]|α , |π2 [u1 (t) − u2 (t)]|α ) satisfies the boundary differential inequality (24.kok ) with % = 0, A1 = A2 = 0, B1 ≥ |π1 u2 (ϑ) − ξ|α , B2 ≥ |π2 u2 (0) − ϕ0 (π1 u2 (0))|α . Proof. 1. First we want to show E D −A1+β π1 [u1 − u2 ] + Aβ π1 [f˜(u1 ) − f˜(u2 )]|A2α−β π1 [u1 − u2 ] ≥ −Λ1 |π1 [u1 − u2 ]|2α − γ1 |u1 − u2 |α |π1 [u1 − u2 ]|α for any u1 , u2 ∈ Q ∩ U1+β . Moreover, we will show D E −A1+β π2 [u1 − u2 ] + Aβ π2 [f˜(u1 ) − f˜(u2 )]|A2α−β π2 [u1 − u2 ] ≤ −Λ2 |π2 [u1 − u2 ]|2α + γ2 |u1 − u2 |α |π2 [u1 − u2 ]|α

(28.kok )

(29.kok )

for any u1 , u2 ∈ Q ∩ U1+β with (|π1 [u1 − u2 ]|α , |π2 [u1 − u2 ]|α ) ∈ V+ (0, 0).

(30.kok )

Let u1 , u2 ∈ Q∩U1+β be arbitrary. For shortness let u∆ := u1 −u2 . Inequality (28.kok ) follows directly from (16.kok ) and |π1 u∆ |2α−β ≤ Λα−β |π1 u∆ |α . 1 Further (16.kok ) implies (−A1+β π2 u∆ + Aβ π2 [f˜(u1 ) − f˜(u2 )]|A2α−β π2 u∆ ) ≤ −Λ1−2α+2β |π2 u∆ |22α−β + γ2 Λβ−α |u∆ |α |π2 u∆ |2α−β . 2 2 Thus (29.kok ) is shown if α = β. Let now α > β. Since τ 2 + γ2 Λβ−α |u∆ |α τ τ 7→ −Λ1−2α+2β 2 2 γ2 |u∆ |α , we can use the estimate is monotonously decreasing for τ ≥ 12 Λ−1+α−β 2 |π2 u∆ |2α−β ≥ Λα−β |π2 u∆ |α 2 in order to get (29.kok ) if Λα−β |π2 u∆ |α ≥ 2

1 −1+α−β Λ γ2 |u∆ |α 2 2

i.e. if (30.kok ) holds. 2. Now let u1 , u2 be solutions of (18a.kok ) with the properties as required in the lemma and let v as defined in the lemma. Futher let u∆ = u1 − u2 . Then d v i (t)v˙ i (t) = 12 dt |πi u∆ |2α E D = −A1+β πi u∆ + Aβ πi [f˜(u1 (t)) − f˜(u2 (t))]|A2α−β πi u∆ .

153


Using (28.kok ), (29.kok ) we find v 1 (t)v˙ 1 (t) ≥ −Λ1 (v 1 (t))2 − γ1 |v(t)|v 1 (t) for a.e. t > 0 and v˙ 2 (t) ≤ −Λ2 v 2 (t) + γ2 |v(t)| for t > 0 with v(t) ∈ V+ (0, 0). 3. We have v 1 (ϑ) = |π1 u2 (ϑ) − ξ|α ≤ B1 and v 2 (0) = |[ϕ0 (π1 u1 (0)) − ϕ0 (π1 u2 (0))] + [ϕ0 (π1 u2 (0)) − π2 u2 (0)]|α ≤ χ1 |π1 [u1 (0) − u2 (0)]|α + |ϕ0 (π1 u2 (0)) − π2 u2 (0)|α ≤ χ1 v 1 (0) + B2 . t u

Thus Lemma 12 is proved. Similarly to Lemma 12 one can prove the following two lemmas.

Lemma 13. Let k ≥ 1 and let U ∈ C([0, ϑ], Q). Let u : [0, ϑ] → Uα be a solution 2 of (19.kok ) on [0, ϑ] with ϕ1 ∈ Φ1 , h1 ∈ Uα 1 . Then v ∈ C([0, ϑ], R≥0 ) defined by v(t) = (|π1 u(t)|α , |π2 u(t)|α ) satisfies (24.kok ) with % = 0, A1 = A2 = 0, B1 ≥ |h1 |α , B2 = 0. Lemma 14. Let k ≥ 1 and let U ∈ C([0, ϑ], Q). Let u : [0, ϑ] → Uα be a solution of (20.kok ) on [0, ϑ] with ϕ1 ∈ Φ1 . ) with Ai = % = 0, If R1 = 0 then v(t) = (|π1 u(t)|α , |π2 u(t)|α ) satisfies (24.kok B1 = 0, B2 ≥ |R2 |α . ˜ ˜ , |R2 | ≤ Ke−%ϑ , K > 0 then v ∈ C([0, ϑ], R2≥0 ) defined If |R1 (t)| ≤ Ke%(t−ϑ) −1 −%(t−ϑ) ˜ by v(t) = K e (|π1 u(t)|α , |π2 u(t)|α ) satisfies (24.kok ) with % = %˜, A1 = α−β , A = Λ , B = 0, B = 1. Λα−β 2 1 2 1 2 Now let ai , bi nonnegative numbers, % ∈ R and let T be a compact time interval. We introduce the cone KT := C(T, R2≥0 ) in the Banach space C(T, R2 ) equipped with the norm || · || defined by ||w|| := max |w(t)|. t∈T

For v1 and v2 belonging to KT we say v1 ≤ v2 if and only if v2 − v1 ∈ KT . We say v1 v2 if v2 − v1 belongs to the interior of KT . If v1 ≤ v2 and v1 6= v2 then we say v1 < v2 . Note that KT is a closed and normal cone. Here the normality of the cone means the semi-monotony of the norm, i.e. there is a number M such that ||v1 || ≤ M ||v2 || for any v1 , v2 ∈ KT with v1 ≤ v2 . We introduce the nonlinear but homogene, isotone and completely continuous integral operator LT,% : KT → KT defined by (L1T,% w)(t) := (L2T,% w)(t) :=

max R T t Rt min T

e(t−τ )(−Λ1 −%) γ1 |w(τ )| dτ

e(t−τ )(−Λ2 −%) γ2 |w(τ )| dτ + e(t−min T)(−Λ2 −%) χ1 w1 (min T)

154

Norbert Koksch

for w ∈ KT , and the function q(T, %, a1 , a2 , b1 , b2 ) ∈ KT defined by (q 1 (T, %, a1 , a2 , b1 , b2 ))(t) :=

max R T

e(t−τ )(−Λ1 −%) a1 dτ + e(t−max T)(−Λ1 −%) b1 ,

t

2

(q (T, %, a1 , a2 , b1 , b2 ))(t) :=

Rt

e(t−τ )(−Λ2 −%) a2 dτ + e(t−min T)(−Λ2 −%) b2

min T

for t ∈ T. Then the fixed-point problem LT,% w + q(T, %, a1 , a2 , b1 , b2 ) = w

(w ∈ KT )

(31.kok )

is equivalent to the two-point boundary value problem (25.kok ) in KT . A function v ∈ KT is called lower solution of (31.kok ) if v ≤ LT,% v + q(T, %, a1 , a2 , b1 , b2 ). Analogously, a function v ∈ KT is called upper solution of (31.kok ) if LT,% v + q(T, %, a1 , a2 , b1 , b2 ) ≤ v. One can show that a solution w ˆ ∈ KT of (26.kok ) is an ) is a lower solution of upper solution of (31.kok ) and that a solution v ∈ KT of (27.kok (31.kok ). Lemma 15. Let ai ≥ 0, bi ≥ 0, % ∈ R and let T be a compact time interval. If v ∈ KT is a solution of (27.kok ) then ˆ v ≤ w∗ ≤ w, ) in KT and w ˆ ∈ KT is a solution where w∗ ∈ KT is the unique solution of (25.kok of (26.kok ). Proof. Let v be a solution of (27.kok ), i.e. a lower solution of (31.kok ). 1. We show that there is a solution w∗ of (31.kok ) with v ≤ w∗ . For it we introduce w0 ∈ int KT defined by w0 (t) = e−%(t−min T) ψ2 (t − min T)

(t ∈ T).

) with ai = bi = 0 and graph(w0 ) ⊂ Ψ2 . Since Note that w0 is a solution of (25a.kok w01 (max T) > 0 and w02 (min T) = χ2 > χ1 = χ1 w01 (min T) we have q0 := LT,% w0 − w0 0. ¯ := ηw0 we have w ¯ ∈ KT There is η > 0 with v ≤ ηw0 and q ≤ ηq0 . Setting w and v ≤ w, ¯

¯ + q ≤ w. ¯ LT,% w

˜ := LT,% w + q for w ∈ KT . ˜ : KT → KT be defined by Lw For shortness let L Since v is a lower solution of (31.kok ), the isotony of LT,% implies ˜ ≤L ˜w v ≤ Lv ¯ ≤ w. ¯ ˜ k w) ¯ k∈N is monotone decreasing in the normal cone KT . Using The sequence (L ˜ k w) ¯ k∈N to a solution w∗ ∈ KT [EL75, Theorem 3.1] we get the convergence of (L ∗ of (31.kok ). Since v ≤ w ¯ we have v ≤ w , too.

155


2. Now we show the uniqueness of w∗ . Assume there are two different solutions ). Then w1 and w2 are lower solutions of (31.kok ). Proceeding as above w1 , w2 of (31.kok we get the existence of a third solution w3 of (31.kok ) with w1 ≤ w3 , w2 ≤ w3 . Therefore, without loss of generality, we can assume w1 < w2 . Let w∆ = w2 − w1 . Then w∆ > 0. Since w∆ = LT,% w2 − LT,% w1 ≤ LTm,% w∆ , w∆ is a lower solution of w = LT,% w.

(32.kok )

˜ Thus is a solution w ˜ of (32.kok ) such that w∆ ≤ w. Since (32.kok ) is equivalent to the boundary value problem (25.kok ) with ai = bi = ˜ T) belongs to the 0, w ˜ is a solution of (25.kok ) with ai = bi = 0. Since w(min invariant set Ψ1 , the point w(max ˜ T) belongs to Ψ1 , too. Since w ˜1 (max T) = 0, this inclusion implies w(max ˜ T) = 0. By uniqueness of the solutions of the corresponding initial value problem, we have w ˜ = 0 such that the contradiction w∆ = 0 follows. Thus w∗ is the unique solution of (31.kok ) and hence of (25.kok ) in KT . ), i.e. let w ˆ be an upper solution of (31.kok ). Since 3. Let w ˆ ∈ KT be a solution of (26.kok ˜ k w) (L ˆ k∈N is monotonously decreasing and converging to a solution of (31.kok ), the ˆ follows from the uniqueness of w∗ . t u inequality w∗ ≤ w ) with Ai = % = 0. Then Theorem 16. Let v ∈ K[0,ϑ] satisfy (24.kok v≤w≤w ˆ

(33.kok )

) and w ˆ ∈ K[0,ϑ] is a solution of (26.kok ) with where w ∈ K[0,ϑ] is the solution of (25.kok ai = Ai , bi = Bi , T = [0, ϑ]. Proof. First we note that the existence and uniqueness of w as well as w ≤ w ˆ ˆ is an upper follow from Lemma 15 with T = [0, ϑ], %0 , ai = 0, bi = Bi since w solution of (31.kok ). Studying the phase portrait of (8.kok ) we find −Λ2 w2 + γ2 |w| < 0 if w2 > χ1 w1 , w ∈ R2≥0 . ˜ − := ˜ + := {v ∈ R2 : v 2 > χ1 v 1 } ⊂ V+ (0, 0). Further we introduce V Hence V ≥0 2 2 1 {v ∈ R≥0 : v < χ1 v }. ˜ + for t ∈ [0, ϑ]. Then v is a solution of (27.kok ) with T = [0, ϑ], 1. Assume v(t) ∈ cl V % = 0, ai = 0, bi = Bi . The claim of the theorem follows directly from Lemma 15. ˜ − . We want show that there is 2. Assume now there is a t ∈ [0, ϑ] with v(t) ∈ V a ϑ1 ∈ [0, ϑ] with ˜ + for t ∈ [0, ϑ1 [, v(t) ∈ V

˜ − for t ∈ [ϑ1 , ϑ]. v(t) ∈ cl V

(34.kok )

Let ϑ1 be the first time point with v 2 (ϑ1 ) = χ1 v 1 (ϑ1 ). Assume there are ϑ2 ∈ [ϑ1 , ϑ[, ϑ3 ∈ ]ϑ2 , ϑ[ with v 2 (ϑ2 ) = χ1 v 1 (ϑ2 ) and v 2 (t) > χ1 v 1 (t) for t ∈ ]ϑ2 , ϑ3 ].

156

Norbert Koksch

We set T = [ϑ2 , ϑ3 ], b1 = v 1 (ϑ3 ), b2 = 0, ai = 0. Then (27.kok ) holds with % = 0. ¯ where w ¯ ∈ K[ϑ2 ,ϑ3 ] is the solution Applying Lemma 15 we obtain v|[ϑ2 , ϑ3 ] ≤ w of (25.kok ) on [ϑ2 , ϑ3 ]. Since w(ϑ ¯ 2 ) ∈ Ψ1 , we have graph(w) ¯ ⊂ Ψ1 . Hence v 2 (ϑ3 ) ≤ w ¯ 2 (ϑ3 ) = χ1 w ¯1 (ϑ3 ) = χ1 v 1 (ϑ3 ) ˜ − for t ≥ ϑ1 and in contrary to the choice of ϑ2 and ϑ3 . Therefore v(t) ∈ cl V (34.kok ) is shown. Applying Lemma 15 with T = [0, ϑ], % = 0, b1 = B1 + ε, b2 = B2 , ai = 0, ε > 0, the existence of the unique solution wε ∈ K[0,ϑ] of (25.kok ) follows. Moreover, Lemma 15 implies w ≤ wε . Let δ = wε1 − v 1 . Then p (v 1 (t))2 + (χ1 v 1 (t))2 , p w˙ε 1 (t) = −Λ1 wε 1 (t) − γ1 |wε (t)| ≤ −Λ1 wε 1 (t) − γ1 (wε 1 (t))2 + (χ1 wε 1 (t))2 v˙ 1 (t) ≥ −Λ1 v 1 (t) − γ1 |v(t)|

≥ −Λ1 v 1 (t) − γ1

for a.e. t ∈ [ϑ1 , ϑ] and hence ˙ ≤ −Λ1 δ(t) for a.e. t ∈ [ϑ1 , ϑ] with δ(t) ≥ 0. δ(t) Since ε > 0, we have δ(ϑ) > 0. Assume we do not have δ(t) > 0 for any t ∈ [ϑ1 , ϑ]. Then there is τ ∈ [ϑ1 , ϑ[ with δ(t) > 0 for t ∈ ]τ, ϑ] and δ(τ ) = 0. ˙ Thus we have δ(t) ≤ −Λ1 δ(t) for a.e. t ∈ [τ, ϑ] and δ(τ ) = 0. This differential inequality implies δ(t) ≤ 0 for t ∈ [τ, ϑ] in contrary to δ(ϑ) > 0. Therefore v 1 (t) ≤ wε 1 (t) for t ∈ [ϑ1 , ϑ]. Further v 2 (t) ≤ χ1 v 1 (t) ≤ χ1 wε 1 (t) ≤ wε 2 (t) for t ∈ [ϑ1 , ϑ]. Hence v|[ϑ1 , ϑ] ≤ wε |[ϑ1 , ϑ] in K[ϑ1 ,ϑ] . Since wε1 (ϑ1 ) ≥ v 1 (ϑ1 ) and wε2 (0) ≥ χ1 wε1 (0) + B2 , we have v|[0, ϑ1 ] ≤ wε |[0, ϑ1 ] in K[0,ϑ] such that v ≤ wε in K[0,ϑ] follows. ) with T = [0, ϑ], % = 0, ai = 1, b1 = 1, Let w ˜ ∈ KT be the solution of (25.kok b2 = 0. Then wε − w = εw ˜ such that wε → w as ε → 0. Since w ≤ w, ˆ the inequality (33.kok ) follows. t u , A2 = Λα−β . Then for any % ∈ ]%2 , %1 [ we have: Theorem 17. Let A1 = λα−β 2 N ) with ai = Ai . 1. There is a unique nonnegative stationary point w0 (%) of (25a.kok ˆ0 (%) ∈ K[0,ϑ] defined by w ˆ0 (%)(t) = 2. There are χ ˆ ∈ ]χ1 , χ2 [ and η(%) such that w (ˆ η , ηˆχ) ˆ is a constant solution of (26.kok ) with ai = Ai , b1 = 0, b2 = max{1, w02 (%)}, T = [0, ϑ]. 3. If v ∈ K[0,ϑ] satisfies (24.kok ) with B1 = 0, B2 = max{1, w02 (%)} then v ≤ w ˆ0 (%). Proof. Let % ∈ ]%2 , %1 [ be fixed and let W1 (w) = (−Λ1 − %)w1 − γ1 |w| − A1 , W2 (w) = (−Λ2 − %)w2 + γ2 |w| + A2 . 1. In order to show the existence of w0 (%) we note W1 (0, w2 ) < 0,

D1 W1 (w1 , w2 ) > −Λ1 − %1 − γ1 > 0,

D2 W1 (w1 , w2 ) < 0,

157


W2 (w1 , 0) > 0,

D1 W2 (w1 , w2 ) > 0,

D2 W2 (w1 , w2 ) < −Λ2 − %2 + γ2 < 0

for w > 0. Hence there are strongly increasing functions Ψî : R≥0 → R≥0 satisfying W2 (η, Ψˆ1 (η)) = 0 and W1 (Ψˆ2 (η), η) = 0 for η ≥ 0 and describing the isoclines w˙ 1 = 0 and w˙ 2 = 0 of (25a.kok ), respectively. Thus there is exactly one ). This stationary point has an unpositive stationary point w0 = w0 (%) of (25a.kok stable manifold graph(Ψ˜1 ) where Ψ1 : R≥0 → R≥0 is strongly increasing function η ) > Ψˆ1 (η) for η < w01 , and Ψ˜1 (η) < Ψˆ1 (η) for η > w01 . satisfying Ψ1 (˜ 2. Let B1 = 0, B2 = w02 (%). We define pi : R≥0 → R by p p p1 (χ) := −Λ1 − % − γ1 1 + χ2 , p2 (χ) := −Λ2 − % + γ2 1 + χ−2 . Then p1 is strongly concave and p2 is strongly convex with p1 (χ1 ) = p2 (χ1 ) = %1 − % > 0,

p1 (χ2 ) = p2 (χ2 ) = %2 − % < 0.

ˆ2 < χ ˆ1 and pi (χ î ) = 0. Let χ ˆ ∈ ]χ ˆ2 , χ ˆ1 [ be Thus there are χ î ∈ ]χ1 , χ2 [ with χ arbitrary. Then p1 (χ) ˆ > 0,

p2 (χ) ˆ < 0,

χ ˆ > χ1 .

Therefore there is ηˆ > 0 such that w = (η, η χ) ˆ satisfies 0 ≤ (−Λ1 − %)w1 − γ1 |w| − A1 , 0 ≥ (−Λ2 − %)w2 + γ2 |w| + A2 , w2 ≥ χ1 w1 + B2 . ) with ai = Ai , bi = Bi , Thus w ˆ0 (%) as defined in the theorem is a solution of (26.kok T = [0, ϑ]. 3. Let B1 = 0, B2 = w02 (%). By construction we have {w ∈ R2≥0 : w2 > Ψˆ1 (w1 )} ⊂ V+ (%, A2 ). Let ˜ := {w ∈ R2 : w1 ≤ max{Ψˆ2 (w2 ), w1 }}, V ≥0 0

ˆ + := {w ∈ V ˜ : w2 ≥ Ψ˜1 (w1 )}. V

˜ + . Further V ˆ + ⊂ V+ (%, A2 ). Then v˙ 1 (t) < 0 if t > 0 and v(t) 6= V ˜ Assume there is t1 ∈ [0, ϑ[ with v(t) 6∈ V. Because of v 1 (ϑ) = 0, there is a ˜ for t ∈ [t1 , t2 [ and t2 ∈ ]t1 , ϑ[ with v(t) 6∈ V v 1 (t2 ) = w01 ,

v 2 (t2 ) ≤ w02

(35.kok )

or v 1 (t2 ) = Ψˆ2 (v 2 (t2 )),

v 2 (t2 ) > w02 .

(36.kok )

˙ )(t2 − t1 ) ≤ 0 in If (35.kok ) then there is τ ∈ [t1 , t2 ] with v(t2 ) − v(t1 ) = v(τ contrary to v(t2 ) > v(t1 ).

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ˆ + implies the existence of t3 ∈ [t1 , t2 [ with v(t3 ) 6= V ˜ If (36.kok ) then v(t) 6= V 2 ˆ and v(t) ∈ V+ , i.e. with v˙ (t) ≤ 0 for t ∈ [t3 , t2 ]. Thus there are τ1 , τ2 with τi ∈ [t3 , t2 ] and v 1 (t2 ) − v 1 (t3 ) = v˙ 1 (τ1 )(t2 − t3 ) ≥ 0, v 2 (t2 ) − v 2 (t3 ) = v˙ 2 (τ2 )(t2 − t3 ) ≤ 0 ˜ in contradiction to the choice of t3 . which would imply v(t1 ) ∈ V ˜ for t ∈ [0, ϑ]. Therefore v(t) ∈ V Let ˆ + ∩ V, ˜ ˜ + := V V

˜ − := V ˆ − ∩ V. ˜ V

˜ + for ) satisfies w(t) ∈ V By the choice of B2 , the solution w ∈ K[0,ϑ] of (25.kok t ∈ [0, ϑ]. Thus we can proceed as in the proof of Theorem 16 in order to infer t u v≤w≤w ˆ0 (%).

4

Proof of Theorem 11

4.1 4.1.1

Existence and Properties of G0 (ϑ) Uniqueness and Estimates of U0 (·, ϑ, ξ, ϕ0 ). For ϕ0 ∈ Φ0 let ˜ + ϕ0 (ζ)) : ζ ∈ W0 } Wϑ (ϕ0 ) := {π1 S(ϑ)(ζ

(ϑ ≥ 0).

Then for any ϕ0 ∈ Φ0 , ϑ ≥ 0, ξ ∈ cl Wϑ (ϕ0 ) there is at least one solution U0 (·, ϑ, ξ, ϕ0 ) of the boundary value problem (18.kok ). Our goal is to prove that for any ϕ0 ∈ Φ0 , ϑ > 0, ξ ∈ cl Wϑ (ϕ0 ) there ) with is at most one solution U0 (·, ϑ, ξ, ϕ0 ) of the boundary value problem (18.kok maximal existence interval satisfying U0 (t, ϑ, ξ, ϕ) ∈ Q

(t ∈ [0, ϑ]).

(37.kok )

Further we show some estimates which we need for G0 (ϑ). Lemma 18. There hold: ) with ui (t) ∈ Q for t ∈ [0, T ] and with π1 ui (ϑi ) = ξi , 1. Let ui be solutions of (18a.kok π2 ui (0) = ϕ0 (π1 ui (0)) where ϑi ∈ [ϑ, ϑ] ⊂ [0, T ], ϕ0 ∈ Φ0 , ξi ∈ cl Wϑi (ϕ0 ). Then there is a constant K such that |πi [u1 (t) − u2 (t)]|α ≤ (K|ϑ1 − ϑ2 | + |ξ1 − ξ2 |α ) max ψ1i (t − Θ) Θ∈[ϑ,ϑ]

for t ∈ [0, max{ϑ1 , ϑ2 }].

(38.kok )

159


) with ui (t) ∈ Q for t ∈ [0, T ] and with π1 ui (ϑ) = ξ, 2. Let ui be solutions of (18a.kok π2 u1 (0) = ϕ0 (π1 u1 (0)), π2 u2 (0) = ϕ00 (π1 u2 (0)) where ϑ ∈ [0, T ], ϕ0 , ϕ00 ∈ Φ0 , ξi ∈ cl Wϑ (ϕ0 ) ∩ cl Wϑ (ϕ00 ). Then |πi [u1 (t) − u2 (t)]|α ≤

ψ2i (t) ||ϕ0 − ϕ00 || χ2 − χ1

(t ∈ [0, ϑ]).

(39.kok )

3. For any ϕ0 ∈ Φ0 , ϑ > 0, and ξ ∈ Wϑ (ϕ0 ) there is at most one solution U0 (·, ϑ, ξ, ϕ0 ) of (18.kok ) satisfying (37.kok ). Proof. 1. Let u1 , u2 have the properties as required in the first claim. Without loss of generality we can assume ϑ1 ≥ ϑ2 . Set ϑ := ϑ1 , ξ := ξ1 . We have |π1 [u2 (ϑ) − ξ]|α ≤ |π1 [u2 (ϑ1 ) − u2 (ϑ2 )]|α + |π1 u2 (ϑ2 ) − ξ|α . Since −Aπ1 is a linear, bounded operator and since f˜ maps bounded sets into bounded sets, there is a constant K with K ≥ | − Aπ1 u + π1 f (u)|α

(u ∈ Q).

Because of |π1 [u2 (ϑ1 ) − u2 (ϑ2 )]|α ≤ K|ϑ1 − ϑ2 |, π1 u2 (ϑ2 ) = ξ2 , the estimate |π1 [u2 (ϑ) − ξ]|α ≤ B1 follows where B1 := K|ϑ1 −ϑ2 |+|ξ1 −ξ2 |α . Moreover |π2 u2 (0)−ϕ0 (π1 u2 (0))|α = B2 := 0. Lemma 12 and Theorem 16 imply |πi [u1 (t)−u2 (t)]| ≤ w1i (t) for t ∈ [0, ϑ] where w1 ∈ K[0,ϑ] defined by w1 (t) = (M |ϑ1 − ϑ2 | + |ξ1 − ξ2 |α )ψ1 (t − ϑ1 ) is the ) follows. solution of (25.kok ) for these values of B1 and B2 . Thus (38.kok 2. Let u1 , u2 have the properties as required in the second claim. We have |π1 [u2 (ϑ) − ξ]|α = B1 := 0. Further |π2 u2 (0) − ϕ0 (π1 u2 (0))|α = |ϕ00 (π1 u2 (0)) − ϕ0 (π1 u2 (0))|α ≤ ||ϕ0 − ϕ00 ||0 =: B2 . The function w2 : [0, ϑ] → R2≥0 defined by w2 := ψ2 (t)(χ2 − χ1 )−1 ||ϕ0 − ϕ00 ||0

(t ∈ [0, ϑ])

is a solution of (26.kok ) for ai = 0, bi = Bi , T = [0, ϑ]. Lemma 12 and Theorem 16 ). imply |πi [u1 (t) − u2 (t)]| ≤ w2i (t) for t ∈ [0, ϑ], i.e. (39.kok 3. Let ϕ0 ∈ Φ0 , ϑ ∈ [0, T ], ξ ∈ Wϑ (ϕ0 ) be arbitrary. Assuming the existence of two different solutions of (18.kok ) satisfying (37.kok ) we obtain a contradiction to (38.kok ) t u with ϑ1 = ϑ2 = ϑ = ϑ = ϑ, ξ1 = ξ2 = ξ.

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Because of (17.kok ), there is a number T∗ > 0 with the following three properties ˜ S(t)u (t ∈ [0, T∗ ]), 0 ∈Q ˜ |π1 S(t)u0 |α > rˆ1 (|π1 u0 |α = rˆ1 , t ∈ [0, T∗ ]) if rˆ1 < ∞, ˜ |π2 S(t)u0 |α < rˆ2 (|π2 u0 |α = rˆ2 , t ∈ [0, T∗ ])

(40.kok )

ˆ for any u0 ∈ cl Q. ) and let ϕ0 ∈ Φ0 . Then Lemma 19. Let T∗ > 0 satisfy (40.kok cl W0 ⊂ Wϑ (ϕ0 )

(ϑ ∈ ]0, T∗ ])

Proof. Let ϕ0 ∈ Φ0 , ϑ ∈ ]0, T∗ ] be arbitrary. We define the continuous mapping H : [0, 1] × cl W0 → Uα 1 by ˜ ϑ)(ζ + ϕ0 (ζ)) H(τ ϑ, ζ) := π1 S(τ

(ζ ∈ cl W0 ).

By definition of T∗ and Wϑ (ϕ0 ) and by means of Lemma 18 there is a unique U0 (·, ϑ, ξ, ϕ0 ) for ξ ∈ Wϑ (ϕ0 ). Hence we can define the inverse H −1 (1, ·) of H(1, ·) by H −1 (1, ξ) := π1 U0 (0, ϑ, ξ, ϕ0 )

(ξ ∈ cl Wϑ (ϕ0 )).

Because of (38.kok ) with ϑ = ϑ1 = ϑ2 = ϑ = ϑ, t = 0, ϕ0 = ϕ00 , the func−1 tion H (1, ·) is continuous, too. Thus H(1, ·) is a homeomorphism from W0 onto Wϑ (ϕ0 ). If rˆ1 = ∞, i.e. W0 = Uα , the domain invariance theorem implies Wϑ (ϕ0 ) = W0 . If rˆ1 < ∞ then W0 is an open and bounded set. Because of (40.kok ) we have {H(τ, ξ) : τ ∈ [0, 1], ξ ∈ ∂W0 } ∩ W0 = ∅. Using an arbitrary base in the finite dimensional Banach space Uα 1 , the homotopy theorem implies deg(H(1, ·), W0 , ξ) = deg(H(0, ·), W0 , ξ) = deg(I, W0 , ξ) = 1 for any ξ ∈ W0 . Thus for any ξ ∈ W0 there exists ζ ∈ W0 with ξ = H(1, ζ). ), we have ∂Wϑ (ϕ0 ) ∩ cl W0 = ∅. Therefore, cl W0 ⊆ cl Wϑ (ϕ0 ). Because of (40.kok Since H(1, ·)|∂W0 is a bijection from ∂W0 onto ∂Wϑ (ϕ0 ) we have cl W0 ⊂ t u Wϑ (ϕ0 ). ). Let U0 (·, ϑ, ξ, ϕ0 ) be a solution of (18.kok ) Lemma 20. Let T∗ > 0 satisfy (40.kok satisfying U0 (t, ϑ, ξ, ϕ0 ) ∈ Q for t ∈ [0, ϑ] where ϑ ∈ [0, T∗ ], ξ ∈ Wϑ (ϕ0 ), ϕ0 ∈ Φ0 . Then |π2 U0 (t, ϑ, ξ, ϕ0 )|α ≤ rˆ2

(t ∈ [0, ϑ]).

Proof. The claim follows from (40.kok ) and |π2 U0 (0, ϑ, ξ, ϕ0 )|α ≤ rˆ2 .

t u

161


Lemma 21. Let T > 0 and let u be a solution of (18a.kok ) with u(t) ∈ Q for t ∈ [0, T ]. Further let ϑ = T , ξ = π1 u(T ) ∈ cl W0 , ϕ0 ∈ Φ0 and let U0 (·, ϑ, ξ, ϕ0 ) be a solution of (18.kok ) satisfying U0 (t, ϑ, ξ, ϕ0 ) ∈ Q for t ∈ [0, T ]. Then |πi [u(t) − U0 (t, ϑ, ξ, ϕ0 )]|α ≤ |π2 u(0) − ϕ0 (π1 u(0))|α ψ2i (t)

(t ∈ [0, T ]). t u

Proof. The proof proceeds similarly to the proof of Lemma 18.

4.1.2 The Graph Transformation G0 (ϑ). For simplicity let γ = α. For ϑ ≥ 0 let Φ0 (ϑ) be the set of all ϕ0 ∈ Φ0 for which U0 (·, ϑ, ξ, ϕ0 ) satisfies U0 (t, ϑ, ξ, ϕ0 ) ∈ Q for any t ∈ [0, ϑ] and any ξ ∈ cl W0 . We define G0 (ϑ) : Φ0 (ϑ) → G by (G0 (ϑ)ϕ0 )(ξ) := π2 U0 (ϑ, ϑ, ξ, ϕ0 )

(ϕ0 ∈ Φ0 (ϑ), ξ ∈ cl W0 , ϑ ≥ 0).

). Further let T∗ > 0 satisfy (40.kok Lemma 22. G0 possesses the following properties: 1. Φ0 (ϑ) = Φ0 , G0 (ϑ)Φ0 ⊆ Φ0 for ϑ ≥ 0. 2. (ϑ, ϕ0 ) 7→ G0 (ϑ)ϕ0 is continuous in (ϑ, ϕ0 ). 3. There are T0 > 0 and κ0 (T0 ) ∈ ]0, 1[ such that ||G0 (ϑ)ϕ0 − G0 (ϑ)ϕ00 ||0 ≤ κ0 (T0 )||ϕ0 − ϕ00 ||0

(ϑ ≥ T0 , ϕ0 , ϕ00 ∈ Φ0 ).

4. We have (ϑi ≥ 0).

G0 (ϑ2 )G0 (ϑ1 ) = G0 (ϑ1 + ϑ2 )

(41.kok )

Proof. 1. The first claim will be proved by induction. First we note that Φ0 (ϑ) = Φ0

(ϑ ∈ [0, T∗ ])

(42.kok )

follows from the definition of T∗ and from the Lemmata 18 and 19. Moreover, using Lemma 18 with t = ϑ1 = ϑ2 = ϑ = ϑ ∈ [0, T∗ ], the inequality |(G0 (ϑ)ϕ0 )(ξ1 ) − (G0 (ϑ)ϕ0 )(ξ2 )|α ≤ χ1 |ξ1 − ξ2 |α

(ξi ∈ cl W0 , ϕ0 ∈ Φ0 )

follows. By means of Lemma 20 we have |(G0 (ϑ)ϕ0 )(ξ)|α ≤ rˆ2 for any ξ ∈ cl W0 , ϕ0 ∈ Φ0 . Therefore G0 (ϑ)Φ0 ⊆ Φ0 for ϑ ∈ [0, T∗ ]. Let now Φ0 (ϑ) = Φ0 ,

G0 (ϑ)Φ0 ⊆ Φ0

(ϑ ∈ [0, mT∗ ])

) holds for m = m0 + 1, too. for m = m0 ∈ N. We want show that (43.kok Let ϕ0 ∈ Φ0 be arbitrary. Because of (43.kok ), we have (m0 )

ϕ0

:= G0 (m0 T∗ )ϕ0 ∈ Φ0 .

(43.kok )

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Norbert Koksch

Because of (42.kok ) for any (ϑ, ξ) ∈ [m0 T∗ , (m0 + 1)T∗ ] × cl W0 , there is a unique (m0 )

ξm0 (ϑ, ξ, ϕ0 ) := π1 U0 (0, ϑ − m0 T∗ , ξ, ϕ0

) ∈ cl W0 .

Moreover, there is a unique ξ0 (ϑ, ξ) := π1 U0 (0, m0 T∗ , ξm0 (ϑ, ξ), ϕ0 ) ∈ cl W0 . ˜ Thus S(·)(ξ ). Since 0 (ϑ, ξ) + ϕ0 (ξ0 (ϑ, ξ))) solves (18.kok ˜ S(t)(ξ (ˆ r1 ) + BUα2 (ˆ r2 )) 0 (ϑ, ξ) + ϕ0 (ξ0 (ϑ, ξ))) ∈ cl (BUα 1

(t ∈ [0, m0 T∗ ]),

we have ˜ S(t)(ξ 0 (ϑ, ξ) + ϕ0 (ξ0 (ϑ, ξ))) ∈ Q

(t ∈ [0, (m0 + 1)T∗ ]).

By means of Lemma 18 we have a unique U0 (·, ϑ, ξ, ϕ0 ) in Q and hence ˜ U0 (t, ϑ, ξ, ϕ0 ) = S(t)(ξ 0 (ϑ, ξ) + ϕ0 (ξ0 (ϑ, ξ))) ∈ Q for t ∈ [0, ϑ], (ϑ, ξ) ∈ [m0 T∗ , (m0 + 1)T∗ ] × cl W0 , too. Applying Lemma 18 once more (with T = ϑ = ϑi = ϑ = ϑ, ξi = ξ, ϕ0 = ϕ00 ) and by means of Lemma 19, the relation G0 (ϑ)ϕ0 ∈ Φ0 follows. Thus (43.kok ) is true for m = m0 + 1, too. By induction Φ0 (ϑ) = Φ0 , G0 (ϑ)Φ0 ⊆ Φ0 follow for any ϑ ≥ 0. 2. The continuity properties of G0 follows from Lemma 18 with ξ1 = ξ2 . 3. Let ϑ ≥ 0, ϕ0 , ϕ00 ∈ Φ0 . Lemma 18 implies ||G0 (ϑ)ϕ0 − G0 (ϑ)ϕ0 ||1 ≤ κ0 (ϑ)||ϕ0 − ϕ00 ||1 where κ0 (ϑ) :=

ψ22 (ϑ) χ2 = eϑ%2 . χ2 − χ1 χ2 − χ1

Because of (7.kok ), there is a number T0 > 0 such that κ0 (ϑ) ≤ κ0 (T0 ) < 1 for any ϑ ≥ T0 . 4. The solution u of (18a.kok ) with initial value ξ + (G0 (ϑ2 )G0 (ϑ1 )ϕ0 )(ξ) at ϑ1 + ϑ2 satisfies (18b.kok ) with ϑ = ϑ1 + ϑ2 . Lemma 18 implies (41.kok ). t u Lemma 23. For any ϑ > 0 there is ϕ∗0 ∈ Φ0 being the unique fixed-point of ϕ0 = G0 (ϑ)ϕ0

(ϕ0 ∈ Φ0 ).

Moreover, ϕ∗0 is independent of ϑ. Proof. Let ϑ ≥ T0 . By means of the first three claims of Lemma 22 the operator G0 (ϑ) is a continuous, κ0 (T0 )-contractive self-mapping of the closed set Φ0 in

163


the Banach space G0 . Thus for any ϑ ≥ T0 there is a unique fixed-point ϕ∗0 (ϑ) of G0 (ϑ) in Φ0 . Let m ∈ N \ {0}, T ≥ T0 be arbitrary. Because of G0 (T /m)ϕ∗0 (T ) = G0 (T /m)G0 (T )ϕ∗0 (T ) = G0 (T )(G0 (T /m)ϕ∗0 (T ) and the uniqueness of the fixed-point ϕ∗0 T of G0 (T ), the point ϕ∗0 (T ) is the unique fixed-point of G0 (T /m), too. Thus for any ϑ ≥ 0 there is a unique fixed-point ϕ∗0 (ϑ) of G0 (ϑ) in Φ0 . Moreover ϕ∗0 (ϑ) = ϕ∗0 (

k ϑ) m

(ϑ > 0, k, m ∈ N \ {0}).

Using this property and the continuity of G0 , i.e. the continuous dependence of ϕ∗0 (ϑ) on ϑ, the independence of ϕ∗0 (ϑ) of ϑ follows. Thus ϕ∗0 := ϕ∗0 (T0 ) is the t u unique fixed-point of G0 (ϑ) in Φ0 for any ϑ > 0. 4.2

∗

Invariance and Exponential Tracking Properties of M

Let M∗ = graph(ϕ∗0 |W0 ) where ϕ∗0 is the function as described in Lemma 23. ˜ Let u0 ∈ M∗ be arbitrary. There is τ > 0 such that π1 S(ϑ)u 0 ∈ W0 for ∗ ˜ ϑ ∈ [0, τ ]. Thus there exists U0 (·, ϑ, π1 S(ϑ)u 0 , ϕ0 ) and we have ∗ ∗ ˜ ˜ ˜ ˜ ϕ∗0 (π1 S(ϑ)u 0 ) = (G0 (ϑ)ϕ0 )(π1 S(ϑ)u0 ) = π2 U (ϑ, ϑ, π1 S(ϑ)u0 , ϕ0 ) = π2 S(ϑ)u0 , ∗ ∗ ˜ i.e. S(ϑ)u 0 ∈ M for ϑ ∈ [0, τ ]. Thus M is locally positively invariant. Because of (17.kok ), the vector field of (15.kok ) is pointing strictly outward and is ∗ ∗ ∗ nonzero on the boundary ∂M if ∂M 6= ∅. Thus M is overflowing invariant.

Now we shall prove the exponential tracking property. For it let u0 ∈ Q with ˜ ˜ π1 S(t)u 0 ∈ W0 for t ≥ 0. Further let τ > 0. Since S(mτ )u0 ∈ W0 , we may define ˜ by η = π U (0, mτ, S(mτ )u0 , ϕ∗0 ) for m ∈ N. Note that the sequence (ηm )∞ m 1 0 m=0 ηm ∈ W0 . ˜ ˜ Applying Lemma 21 with T = mτ , ξ = S(mτ )u0 , ϕ0 = ϕ∗0 , u(t) = S(t)u 0 for [0, T ] we get ∗ ∗ i ˜ ˜ |πi [S(t)u 0 − U0 (t, mτ, S(mτ )u0 , ϕ0 )]|α ≤ |π2 u0 − ϕ0 (π1 u0 )|α ψ2 (t)

(44.kok )

for t ∈ [0, ϑm ]. Especially we have |π1 u0 − ηm |α ≤ |π2 u0 − ϕ∗0 (π1 u0 )|α . Because of the compactness of cl W0 ∩ {η ∈ Uα 1 : |π1 u0 − η|α ≤ |π2 u0 − there is a subsequence (ηmj )∞ ˆ ∈ cl W0 . We j=0 converging to some η ˜ η + ϕ∗ (ˆ choose u ˆ(·, u0 ) = S(·)(ˆ η )). Let T > 0 and δ > 0 be arbitrary. Because of 0 the continuous dependence on the initial data, there is j0 = j0 (δ, T ) ∈ N such that mj τ ≥ T and ϕ∗0 (π1 u0 )|α },

˜ j τ )u0 , ϕ∗ )]|α ≤ δ|π2 u0 − ϕ∗ (π1 u0 )|α ψ i (t) |πi [ˆ u(t, u0 ) − U0 (t, mj τ, S(m 0 0 2

164

Norbert Koksch

for t ∈ [0, T ] and j ≥ j0 . Combining this inequality with (44.kok ) we find ˜ |πi [S(t)u ˆ(t, u0 )]|α ≤ (1 + δ)|π2 u0 − ϕ∗0 (π1 u0 )|α ψ2i (t) (t ∈ [0, T ]) 0−u for any T > 0 and any δ > 0 and hence, letting δ → 0, T → ∞ ˜ |πi [S(t)u ˆ(t, u0 )]|α ≤ |π2 u0 − ϕ∗0 (π1 u0 )|α ψ2i (t) (t ≥ 0). 0−u Moreover, we have u ˆ(t, u0 ) ∈ Q and π1 u ˆ(t, u0 ) ∈ cl W0 for any t ≥ 0. Thus ˆ(0, u0 ) ∈ W0 and u ˆ(t, u0 ) ∈ M∗ for t ≥ 0. π1 u Existence and Properties of G1 (ϑ), G2 (ϑ)

4.3

Let k ≥ 2 and let γ ∈]α, β + 1[ be fixed. For ϑ > 0 let Uϑ := C([0, ϑ], Uα ) be the Banach space equipped with the norm || · || defined by ||u|| := max |π1 u(t)|α + t∈[0,ϑ]

max |π2 u(t)|α . Let Fϑ be the open set of all continuous functions u ∈ Uϑ with

t∈[0,ϑ]

π1 u(0) ∈ W0 and u(t) ∈ Q. For ϑ > 0, (ϕ0 , ϕ1 ) ∈ Φ0 × Φ1 , U ∈ Uϑ we introduce the integral operators Fϑ,ϕ0 (·, ·) : Fϑ × Wϑ (ϕ0 ) → Uϑ and P (ϑ, ϕ1 , U ) : Uϑ → Uϑ defined by Fϑ,ϕ0 (u, ξ)(t) =

Rt

eπ1 A(t−τ ) π1 f˜(u(τ )) dτ +

ϑ

Rt

e(t−τ )π2 A π2 f˜(u(τ )) dτ

0

+e(t−ϑ)π1 A ξ + etπ2 A ϕ0 (π1 u(0)), (P (ϑ, ϕ1 , U )u)(t) =

Rt

eπ1 A(t−τ ) π1 Df˜(U (τ ))u(τ ) dτ +

ϑ

Rt + e(t−τ )π2 A π2 Df˜(U (τ ))u(τ ) dτ + etπ2 A ϕ1 (π1 U (0))π1 u(0) 0

for (u, ξ) ∈ Fϑ × Wϑ (ϕ0 ), t ∈ [0, ϑ]. Then the solution u = U0 (·, ϑ, ξ, ϕ0 ) ∈ Fϑ of (18.kok ) is a fixed-point of Fϑ,ϕ0 (·, ξ) in Fϑ and inversely. Moreover, a solution u = U1 (·, ϑ, ξ, ϕ0 , ϕ1 , h1 ) of (19.kok ) is a solution of the fixed-point problem u = P (ϑ, ϕ1 , U0 (·, ϑ, ξ, ϕ0 ))u + Q

(45.kok )

with Q = Q1 (ϑ, ξ, ϕ0 , ϕ1 , h1 ) defined by Q1 (ϑ, ξ, ϕ0 , ϕ1 , h1 ) = e(t−ϑ)π1 A h1 , and a solution u = U2 (·, ϑ, ξ, ϕ0 , ϕ1 , ϕ2 , h1 , h2 ) of (20.kok ) is a solution of (45.kok ) with Q = Q2 (ϑ, ξ, ϕ0 , ϕ1 , ϕ2 , h1 , h2 ) defined by Q2 (ϑ, ξ, ϕ0 , ϕ1 , ϕ2 , h1 , h2 ) = Rt π A(t−τ ) e 1 π1 D2 f˜(U0 (∗))(U1 (τ, ϑ, ξ, ϕ0 , ϕ1 , h1 ), U1 (τ, ϑ, ξ, ϕ0 , ϕ1 , h2 )) dτ ϑ

Rt + e(t−τ )π2 A π2 D2 f˜(U0 (∗))(U1 (τ, ϑ, ξ, ϕ0 , ϕ1 , h1 ), U1 (τ, ϑ, ξ, ϕ0 , ϕ1 , h2 )) dτ 0

+etπ2 A ϕ2 (π1 U0 (0, ϑ, ξ, ϕ0 ))(U1 (0, ϑ, ξ, ϕ0 , ϕ1 , h1 ), U1 (0, ϑ, ξ, ϕ0 , ϕ1 , h2 )) where U0 (∗) stands for U0 (τ, ϑ, ξ, ϕ0 ).

165


Lemma 24. The operator I − P (ϑ, ϕ1 , U ) : Uϑ → Uϑ is a linear homeomorphism from Uϑ onto itself for ϑ > 0, ϕ1 ∈ Φ1 , U ∈ Fϑ . Proof. Let ϑ > 0, ϕ1 ∈ Φ1 , U ∈ Fϑ be given. Obviously P (ϑ, ϕ1 , U ) is linear. Since γ > α one can show that P (ϑ, ϕ1 , U ) is completely continuous. Let u be a solution of u = P (ϑ, ϕ1 , U )u. Then u is a solution of (19.kok ) with h1 = 0, and the Lemma 13 and Theorem 16 imply u = 0 since A1 = A2 = B1 = B2 = 0 and w = 0 is the solution of (25.kok ). Therefore, I − P (ϑ, ϕ1 , U ) is injective. Since P (ϑ, ϕ1 , U ) is completely continuous, I − P (ϑ, ϕ1 , U ) is surjective, too. Thus I − P (ϑ, ϕ1 , U ) is a linear, continuous bijection from Banach space Uϑ onto itself which has a continuous inverse by means of Banach’s Theorem. t u Lemma 25. Let k = 2 and let ϑ > 0, ϕ0 ∈ Φ, ξ ∈ Wϑ (ϕ). If ϕ0 is C 2 then U0 (·, ϑ, ξ, ϕ0 ) is C 2 in ξ ∈ cl W0 . Moreover, D3 U0 (t, ϑ, ξ, ϕ0 )h1 = U1 (t, ϑ, ξ, ϕ0 , Dϕ0 , h1 ) D3,3 U0 (t, ϑ, ξ, ϕ0 )(h1 , h2 ) = U2 (t, ϑ, ξ, ϕ0 , Dϕ0 , D2 ϕ0 , h1 , h2 )

(46.kok )

for t ∈ [0, ϑ], ξ ∈ cl W0 , h1 , h2 ∈ Uα 1. Proof. Let ϑ > 0 and let ϕ0 ∈ Φ0 be twice continuously differentiable. We note that Fϑ,ϕ0 belongs to C 2 (Fϑ × Wϑ (ϕ0 ), Uϑ ). Since D1 Fϑ,ϕ0 (U, ξ) = P (ϑ, ϕ0 , U ), Lemma 24 implies that I − D1 Fϑ,ϕ0 (U, ξ) is a linear homeomorphism of Uϑ into itself. Since U0 (·, ϑ, ξ, ϕ0 ) solves u − Fϑ,ϕ0 (u, ξ) = 0 we can apply the implicit function theorem in order to conclude the C 2 -smoothness of ) follows from the implicit function U0 (·, ϑ, ξ, ϕ0 ) in ξ ∈ Wϑ (ϕ0 ). Moreover, (46.kok theorem. Since cl W0 ⊆ Wϑ (ϕ0 ), the lemma is proved. t u Similar to Lemma 22 but using some more technical estimates (since γ > α) one can show Lemma 26. There are T2 > 0 and closed sets Φ˜j ⊆ Φj with 0 ∈ Φ˜j for j = 0, 1, 2 such that: 1. G0 (T2 ), G1 (T2 )(ϕ0 , ·), G2 (T2 )(ϕ0 , ϕ1 , ·) are uniformly contractive on Φ0 , Φ1 , Φ2 , respectively, for (ϕ0 , ϕ1 ) ∈ Φ0 × Φ1 . ˜0 , G1 (T2 )(Φ˜0 × Φ˜1 ) ⊆ Φ˜1 , G2 (T2 )(Φ˜0 × Φ˜1 × Φ˜2 ) ⊆ Φ ˜2 . 2. G0 (T2 )Φ˜0 ⊆ Φ ˜ ˜ 3. G1 (T2 )(·, ϕ1 ), G2 (T2 )(·, ·, ϕ2 ) are continuous for (ϕ1 , ϕ2 ) ∈ Φ1 × Φ2 . Because of (46.kok ), we have DG0 (T2 )(ϕ0 ) = G1 (T2 )(ϕ0 , Dϕ0 ),

D2 G0 (T2 )(ϕ0 ) = G2 (T2 )(ϕ0 , Dϕ0 , D2 ϕ0 )

for twice continuously differentiable ϕ0 ∈ Φ˜0 . Choosing ϕ0 = 0 and applying the fiber contraction principle, the C 2 smoothness of the manifold follows. Thus Theorem 11 is proved. t u

166

Norbert Koksch

References [CLS92] [CFNT89]

[EL75]

[FST88] [FST89]

[Hen81] [MPS88]

[Nin92] [Rob93] [Rob96] [Rom94]

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[Wig94]

Chow, S.-N., Lu, K., Sell, G.R.: Smoothness of inertial manifolds. Journal of Mathematical Analysis and Applications 169 (1992) 283–312 Constantin, P., Foias, C., Nicolaenko, B., Témam, R.: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, volume 70 of Applied Mathematical Sciences. Springer 1989 Eisenfeld, J., Lakshmikantham, V.: Comparison principle and nonlinear contractions in abstract spaces. Journal of Mathematical Analysis and Applications 49 (1975) 504–511 Foias, C., Sell, G.R., Temam, R.: Inertial manifolds for nonlinear evolutionary equations. J. of Differential Equations 73 (1988) 309–353 Foias, C., Sell, G.R., Titi, E.S.: Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J. of Dynamics and Differential Equations 1 (1989) 199–244 Henry, D.: Geometric Theory of Semilinear Parabolic Equations, volume 850 of Lecture Notes in Mathematics. Springer, 1981 Mallet-Parret, J., Sell, G.R.: Inertial manifolds for reaction diffussion equations in higher space dimension. J. Amer. Math. Soc. 1 (1988) 805– 866 Ninomiya, H.: Some remarks on inertial manifolds. J. Math. Kyoto Univ. 32 (1992) 667–688 Robinson, J.C.: Inertial manifolds and the cone condition. Dyn. Syst. Appl. 2 (1993) 311–330 Robinson, J.C.: The asymptotic completeness of inertial manifolds. Nonlinearity 9 (1996) 1325–1340 Romanov, A.V.: Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations. Russ. Acad. Sci., Izv., Math. 43 (1994) 31–47 Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 1988 Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd ed. Springer, New York, 1988 Vanderbauwhede, A.: Centre Manifolds, Normal Forms and Elementary Bifurcations, pages 89 – 169. Dynamics Reported, Volume 2. John Wiley & Sons, 1989 Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Applied Mathematical Sciences 105. Springer, 1994



The Property (A) for a Certain Class of the Third Order ODE Monika Kov´ aˇcov´ a Department of Mathematics Fac. of Mechanical Engineering Slovak Technical University, n´ am. Slobody 17, 812 31 Bratislava, The Slovak Republic Email: kovacova [email protected] Abstract. We study oscillatory and non-oscillatory solutions of the third order ODE [g(t)(u00 (t) + p(t)u(t))]0 = f (t, u, u0 , u00 ),

(∗)

where g, p : [T, ∞) → [0, ∞) are bounded functions, g ≥ δ > 0. The function f is assumed to be continuous and f (x1 , x2 , x3 ) · x1 ≤ 0. Many authors have consider ODE’s of the form (∗), where the main part, i.e. the term u00 + pu is nonoscillatory. By contrast to these results we consider here the case of the oscillatory kernel function u00 + pu. The main goal is to show that any solution u of (∗) is either oscillatory or it is a solution of the second order ODE u00 (t) + p(t)u(t) = β(t) with vanishing right hand side β ≥ 0, β(t) → 0 as t → ∞. In the latter case all the derivatives u(n) (t) up to the second order tend to zero as t → ∞, i.e. eq. (∗) has the property (A). The results are generalizations of these obtained by I. T. Kiguradze [1]. AMS Subject Classification. 34C10, 34C15 Keywords. The Property (A), Oscillatory Solutions ODE

1

Introduction

In this paper we consider a nonlinear third order differential equation in the form (g(t) · [u00 (t) + p(t)u(t)])0 = f (t, u, u0 , u00 ).

(1.kov )

Let T, g1 , g2 , p2 be positive constants and let g : [T0 , ∞) → (0, ∞)

belong to the class C 1 [T0 , ∞),

0 < g1 ≤ g(t) ≤ g2 for all t ∈ [T0 , ∞), p : [T0 , ∞) → [0, ∞) belong to the class C 1 [T0 , ∞),

(2.kov )

0 ≤ p(t) ≤ p2 for all t ∈ [T0 , ∞), f : [T0 , ∞) × R3 → R is continuous function having the following

(3.kov )

sign property f (t, x1 , x2 , x3 ) · sign x1 ≤ 0 This is the final form of the paper.

for x1 6= 0.

(4.kov )

168

Monika Kov´ aˇcov´ a

The main goal of this paper is to describe oscillatory and nonoscillatory properties of solution of ordinary differential equation (1.kov ). This main result is a dichotomy property saying that any solution u of equation (1.kov ) is either oscillatory or u together with its derivatives up to the second order tend to zero. Several papers focused an aforementioned problem. An Oscillatory Criterion for a Class of Ordinary Differential Equations [1] becomes one of the main ones. The author assumes that a left-sided operator u(n) (t) + u(n−2) (t) is oscillatory at first. This assumption was considered as true, he searched necessary and sufficient conditions to fulfill that equation has a property A (B). Several authors studied differential equation (1.kov ), but they assumed that operator u00 (t) + p(t)u(t), which forms an equational kernel is nonoscillatorical. By contrast to these results we consider here the case of oscillatory kernel function u00 (t) + p(t)u(t).

2

Preliminaries

By a solution (proper solution) we mean a function u defined on an interval [T, ∞) ⊂ [T0 , ∞), having a continuous third derivative and such that sup{|u(t)| : t > T } > 0 for any t ∈ [T, ∞) and u satisfies equation. By an oscillatory solution we mean a solution of (1.kov ) having arbitrarily large zeroes. Otherwise, a solution is said to be nonoscillatory.

3

Auxiliary lemmata

We begin with several auxiliary lemmata which are needed in order to prove main results in the main section. Let us consider the equation y 00 (t) + p(t)y(t) = r(t),

(5.kov )

where p : [T, ∞) → [0, ∞) and r : [T, ∞) → (0, ∞) are continuous functions such that p(t) ≤ p2

and

0 < r1 ≤ r(t) ≤ r2

for all t ∈ [T, ∞),

(6.kov )

where r1 , r2 , p2 are positive constants, with coefficients p, r satisfying (6.kov ). ) Lemma 1. Let y ∈ C 2 [T, ∞) be a positive solution of differential equation (1.kov ) the conditions . and let r1 , r2 , p2 be positive constants which fulfill (6.kov r1 . Then for any δ > 0, small Let p0 > 0 be arbitrary large and put ε1 = 2p 2 ε1 ·r1 r1 δ 2 enough, and 0 < ε ≤ min 2(1+2p0 )2 ·r2 +r1 , 16 the solution y has the following property.

169

The Property (A) for a Certain Class of ODE

If we have

0 < y(t) < ε −

on some interval (t− , t+ )

y(t ) = y(t ) = ε, y(t) ≥ ε for any

then

and

+

(7.kov ) −

t ∈ [t , t + p0 (t − t )], +

+

+

−

t − t ≤ δ. +

(8.kov )

Proof. Suppose that 0 < y(t) < ε for any t ∈ [t− , t+ ] and y(t− ) = y(t+ ) = ε. Since y is a solution of (1.kov ) we have r1 2 0 < y(t) ≤ ε1

y 00 (t) = r(t) − p(t) · y(t) ≥ r1 − p2 ε1 = and Put t0 =

y 00 (t) ≤ r2

for each t such that

(9.kov )

min y(t).

t∈[t− ,t+ ]

Let us introduce the following auxiliary functions:

r1 (t − t0 )2 · t, 2 2 (t − t0 )2 , w(t) = y(t0 ) + y 0 (t0 )(t − t0 ) + r2 · 2 (t − t0 )2 , y(t) = y(t0 ) + y 0 (t0 )(t − t0 ) + y 00 (ξ) · 2 z(t) = y(t0 ) + y 0 (t0 )(t − t0 ) +

and

(10.kov ) ξ ∈ [t0 , t].

According to (9.kov ) and (10.kov ) we have the estimate z(t) ≤ y(t) ≤ w(t)

provided that

0 < y(t) ≤ ε1 .

(11.kov )

170


Then with regard to (10.kov ) there exist t1 , t2 , t3 ∈ [T, ∞) such that t0 > t3 and t1 , t2 > t0 roots w(t2 ) = ε1 ,

z(t1 ) = ε,

z(t3 ) = ε.

(12.kov )

Furthermore, there exists t++ > t+ such that y(t++ ) ≥ ε We conclude from the definitions of the functions w(t), z(t) and (11.kov ) that t++ − t+ ≥ t2 − t1 , t 1 − t3 ≥ t + − t− . In what follows, we will prove that t++ − t+ ≥ p0 · (t+ − t− ). As a consequence of this inequality we will obtain the statement (8.kov ). r1 Assume that 0 < ε ≤ r1 +2rε21(1+2p . Then 2 0) 2r2 ε1 ≥ ε (1 + 2p0 )2 · + 1 > 0. r1 As 0 < y(t0 ) =

(13.kov )

min y(t) < ε we obtain

t∈[t− ,t+ ]

ε1 − ε 2ε (ε − y(t0 )) ε1 − y(t0 ) ≥ ≥ (1 + 2p0 )2 · ≥ 2 · (1 + 2p0 )2 · . r2 r2 r1 r1

(14.kov )

It easily follows from (10.kov ) and (12.kov ) that 2 , r2 4 (t1 − t0 )2 = (ε − y(t0 )) · . r1

(t2 − t1 )2 = (ε1 − y(t0 )) ·

Therefore

w(t2 ) − y(t0 ) ε1 − y(t0 ) = = r2 r2

r2 2

· (t2 − t0 )2 . r2

With regard to (13.kov ), (14.kov ), (15.kov ), (16.kov ) we have z(t1 ) − y(t0 ) r1 ε1 − y(t0 ) ≥ 2(1 + 2p0 )2 · = 2(1 + 2p0 )2 · · (t1 − t0 )2 . r2 r1 4r1 Straightforward computations yield 1 1 (1 + 2p0 )2 · (t1 − t0 )2 ≤ (t2 − t0 )2 , 2 2 (t1 − t0 ) · (1 + 2p0 ) ≤ t2 − t0 , 2p0 · (t1 − t0 ) ≤ t2 − t1 , p0 (t − t ) ≤ (t1 − t3 ) · p0 ≤ t2 − t1 ≤ t++ − t+ . +

−

(15.kov ) (16.kov )

171


Hence

t++ − t+ ≥ p0 (t+ − t− ).

Thus y(t) ≥ ε on [t+ , t+ + p0 (t+ − t− )]. It remains to show the estimate t+ − t− ≤ δ. Due to (10.kov ), (12.kov ) ε = z(t1 ) = y(t0 ) + This is why t1 − t0 ≤

q

2ε r1 .

r1 r1 (t1 − t0 )2 ≥ (t1 − t0 )2 . 2 2

Since

√ t − t ≤ t1 − t3 = 2(t1 − t0 ) ≤ 2 2 +

and ε
0,

R∞

h(t)dt = +∞.

T

(ii) There exists δ > 0 and the sequence + − + − + T ≤ t− 1 < t1 < t2 < t2 < · · · < t k < tk → ∞

with the property t+ k − tk+1

− t− k ≤ δ, + − − t+ k ≥ t k − tk .

(iii) (a) Either there is h0 > 0 such that h(t) ≥ h0 > 0 on [T, ∞) (b) or h is a nonincreasing function on [T, ∞), such that h(t) → 0 as t → ∞ and there is k0 > 1 such that h(t) ≤ k0 · h(t + δ) for all t ≥ T .

172


Denote S=

∞ [

− [t+ k , tk+1 ],

∞ [

Sc =

k=1

+ (t− k , tk ).

k=1

Z

Then

h(t)dt = +∞. S

Proof.

S ∪ S c = (t− 1 , ∞) ⊂ [T, ∞).

With respect to (iii) the proof splits into two parts. The case (a) If µ(S c ) < ∞ ( µ is the Lebesgue measure) then µ(S) = ∞. If µ(S c ) = ∞ then according to (ii) weR again obtain µ(S) = ∞. Since h(t) ≥ h0 > 0 we may conclude h(t)dt = ∞. S R R The case (b) If h(t)dt < ∞ then clearly h(t)dt = ∞ Sc

S

(t− 1 , ∞),

h(t) ∈ C[T, ∞) and R On the other hand suppose that h(t)dt = ∞. because S ∪ S = c

R∞

h(t)dt = ∞.

T

Sc

Choose δ > 0 sufficiently small and p0 > 0 sufficiently large. Let ε > 0 satisfy the assumptions of Lemma 1. Obviously, there is a sequence + − + − + T ≤ t− 1 < t1 < t2 < t2 < · · · < t k < tk → ∞

such that + for t ∈ (t− k , tk ),

y(t) < ε

− for t ∈ [t+ k , tk+1 ],

y(t) ≥ ε

k = 1, 2, . . . .

− − + With regard to Lemma 1 we may conclude t+ k − tk ≤ δ. Then for any t ∈ [tk , tk ] + − we have h(t) ≤ k0 · h(t + δ) ≤ k0 · h(t + (tk − tk )) and thus +

+ − t+ k +(tk −tk )

+

Ztk

Ztk h(t)dt ≤

t− k

h(t +

(t+ k

−

Z

t− k ))dt

= k0 ·

t− k

Z

Z h(t)dt ≤

+∞ = Sc

It completes the proof of Lemma 2.

Z

h(u)du ≤ k0 · t+ k

Hence

t− k+1

h(t)dt. S

h(t)dt. t+ k

173


Lemma 3. Let (g(t) · [u00 (t) + p(t)u(t)])0 = f (t, u, u0 , u00 ), where p(.), g(.) and f (.) fulfill the following conditions: (i) p(.), g(.) fulfill conditions (2.kov ) a (3.kov ). (ii) f (t, x1 , x2 , x3 ) · sign x1 ≤ 0,

x1 6= 0,

f (t, x1 , x2 , x3 ) · sign x1 ≤ −h(t) · w(|x1 |),

(17.kov ) (18.kov )

where h(.) fulfill on [T, ∞) assumption (i), (ii), (iii) from Lemma 2. (iii) Let w : [0, ∞) → [0, ∞) be a nonincreasing function such that w(0) = 0,

w(s) > 0

for all s > 0.

(19.kov )

Then any proper solution of equation (1.kov ) on [T, ∞) is either oscillatory or there exists β(.) ≥ 0 such that lim β(t) = 0 and u(.) is a solution of equation t→∞

u00 (t) + p(t)u(t) = β(t) · sign u(t).

(20.kov )

Proof. Let u be a nonoscillatory solution. We will show the existence of a function β as stated in Lemma 3. According to (4.kov ), u solves (1.kov ) iff −u does. Therefore, without loss of generality we may assume that for all t ∈ [T, ∞)

(21.kov )

α(t) := g(t) · (u00 (t) + p(t)u(t)).

(22.kov )

u(t) > 0 Denote

Then from (17.kov ) and (21.kov ) we see α0 (t) = f (t, u(t), u0 (t), u00 (t)) ≤ 0 for all t ≥ T , is nonincreasing function on [T, ∞). We will consider three distinct cases: (i) lim α(t) < 0, t→∞

(ii) lim α(t) > 0, t→∞

(iii) lim α(t) = 0. t→∞

In the case (i) we have: Let there exist T1 > T such that α(t) ≤ −ε < 0 for all t ∈ [T1 , ∞). Then ) −ε (22.kov ) α(t) (2.kov

u00 (t) + p(t)u(t) =

g(t)

≤

g2

< 0.

According to (3.kov ), (21.kov ) we have p(.), u(.) > 0 on [T1 , ∞). Hence u00 (t) ≤ −ε g2 < 0 for all t ∈ [T1 , ∞) and so there is T2 , T2 ≥ T1 such that u(t) < 0 for t ≥ T2 . A contradiction.

174


In the case (ii) u(.) is the solution of equation (1.kov ). Let us mark: u00 (t) + p(t)u(t) =

α(t) =: r(t) g(t)

for all t ∈ [T, ∞).

We know that α(.) is a nonincreasing function on [T, ∞). Take α1 := lim α(t) > 0 and α2 := α(T ). Then according to the definition t→∞

of function α we have α2 ≥ α(t) ≥ α1 > 0 for any t ≥ T and therefore 0
0 sufficiently small and any p0 > 0 + − + sufficiently large there is ε > 0 and a sequence t− 1 < t1 < t2 < t2 . . . → ∞ such that − t+ k − tk ≤ δ,

+ for all t ∈ (t− k , tk ), − for all t ∈ [t+ k , tk+1 ],

y(t) < ε y(t) ≥ ε

k = 1, 2, . . . .

− Thus u(t) ≥ ε ⇒ α0 (t) ≤ −h(t)w(ε) on [t+ k , tk+1 ]. t R − And α(t) ≤ α(t+ h(t)w(ε)dt for any t ∈ [t+ k) − k , tk+1 ]. This yields the t+ k

following estimates. Zt α(t) ≤

α(t+ 1)

− w(ε)

h(t)dt, t+ 1 −

+ α(t− 2 ) ≤ α(t1 ) − w(ε)

Zt2

h(t)dt, t+ 1 −

+ α(t− 3 ) ≤ α(t2 ) − w(ε)

Zt3

−

h(t)dt ≤ α(t− 2 ) − w(ε)

t+ 2

Zt3

h(t)dt t+ 2

−

−

Zt2 Zt3 + h(t)dt + h(t)dt . ≤ α(t1 ) − w(ε) t+ 1

t+ 2

175


And in general: −

−

−

tk+1 Zt2 Zt3 Z − + h(t)dt + h(t)dt + · · · + h(t)dt . α(tk+1 ) ≤ α(t1 ) + w(ε) t+ 1

t+ 2

Hence lim

k→∞

α(t− k+1 )

≤

α(t+ 1)

t+ k

Z − w(ε) h(t)dt → −∞ S

R

because S h(t)dt = ∞ (see the Lemma 2), a contradiction. This way we have excluded the cases (i) and (ii). Thus the case (iii) must occur, i.e. lim α(t) = 0 . t→∞

α(t) g(t)

Finally, if we put β(t) = for all t ≥ T , then we have β(t) ≥ 0 (α is nonincreasing function ) and lim β(t) = 0 and the proof of Lemma 3 follows. t→∞

4

Main Theorems

Theorem 4. Let u be a solution of equation (1.kov ) (g(t)[u00 (t) + p(t)u(t)])0 = f (t, u, u0 , u00 ), where p(.), g(.) and f (.) fulfill conditions (2.kov ), (3.kov ), (17.kov ), (18.kov ) and (19.kov ). Let further, u00 (t) + p(t)u(t) = β(t)

on interval [T, ∞),

(23.kov )

where u ∈ C 2 [T, ∞),

u(t) > 0

for all t ≥ T,

β ∈ C [T, ∞),

β(t) ≥ 0

for all t ≥ T

2

(24.kov )

and lim β(t) = 0.

t→∞

If u is a nonoscillatory solution of (1.kov ), then lim u(t) = 0.

t→∞

Proof. It is sufficient to prove lim inf u(t) = 0 = lim sup u(t). t→∞

t→∞

(25.kov )

176


At first we show that lim inf u(t) = 0

(26.kov )

t→∞

We proceed by contradiction. If (26.kov ) is not valid, then according to (24.kov ) and (25.kov ) suppose that there is α > 0 and T1 ≥ T such that u(t) ≥ α for each t ≥ T1 . Thus u00 (t) = β(t) − p(t)u(t) ≤ β(t) − p1 α

for all t ∈ [T1 , ∞).

As lim β(t) = 0 , there exists T2 ≥ T1 , t→∞

u00 (t) ≤ −

p1 ·α 0, u00 (t∗∗ ) < 0.

Then according to (23.kov ) we have u00 (t)u0 (t) + p(t)u(t)u0 (t) = β(t)u0 (t)

in [T1 , ∞)

and after the integration Zt∗∗ Zt∗∗ Zt∗∗ u00 (t)u0 (t)dt + p(t)u(t)u0 (t)dt = β(t)u0 (t)dt. t∗

t∗

t∗

177


Because ∗∗

Zt

∗∗

p1 u(t)u0 (t)dt ≤

t∗

Zt

∗∗

p(t)u(t)u0 (t)dt ≤

t∗

Zt

p2 u(t)u0 (t)dt,

t∗

we get p1 1 0 ∗∗ 2 [u (t ) − u0 (t∗ )2 ] + · (u(t∗∗ ) − u(t∗ )) ≤ ε[u(t∗∗ ) − u(t∗ )]. 2 | {z } | {z } 2 =0

(28.kov )

=0

Due to the assumption lim β(t) = 0, therefore there exists T1 ; T1 ≥ T such t→∞

that β(t) < ε for all t ≥ T1 . According to inequality (28.kov ) we have p1 [u(t∗∗ )2 − u(t∗ )2 ] ≤ ε[u(t∗∗ ) − u(t∗ )], 2 u(t∗∗ ) ≤ u(t∗∗ ) + u(t∗ ) ≤

2ε p1

and this is a contradiction to (i) Now consider the case (ii). Suppose that there exists ε > 0 such that 0 < ε < lim sup u(t) < 2ε.

(29.kov )

t→∞

Let us choose β0 such that 0 < β0
0 thus according to (26.kov ) there exists T2 ≥ T1 such that u(T2 ) < min(u(T1 ), ε)

(32.kov )

and with aspect to (29.kov ) we have T4 ≥ T2 with the property u(T4 ) > ε. Thus (32.kov )

u(T2 ) < ε < u(T4 ) and we can find T3 such that u0 (T3 ) > 0.

(33.kov )

t0 = inf{t ≥ T1 , u0 (τ ) ≥ 0 for all τ ∈ [t, T3 ]}.

(34.kov )

T2 ≤ T3 ≤ T4 ,

u(T3 ) > ε,

Let

Since u(.) is continuous (33.kov ) implies the inequality t0 < T3 . ), u(.) is nondecreasing in [T1 , T3 ], what is If t0 < T1 , then according to (34.kov ). a contradiction to u(T2 ) ≤ u(T1 ), which follows from (32.kov We have T1 ≤ t0 < T3 . According to definition of t0 , u0 (t0 ) = 0. Then 0 0 (t0 ) u00 (t0 ) = lim u (t0 +δ)−u ≥ 0. Using (25.kov ), (31.kov ) and (21.kov ) we obtain δ δ→0+

0 ≤ u00 (t0 ) = β(t0 ) − p(t0 )u(t0 ) ≤ β0 − p1 u(t0 ), β0 ε ≤ . u(t0 ) ≤ p1 4

(35.kov )

According to definition (34.kov ) we have u0 (t) ≥ 0

for all t ∈ [t0 , T3 ].

(36.kov )

And by (35.kov ) we have u0 (t0 ) < 0, u(T3 ) > ε. Then there exists t1 , t1 ∈ (t0 , T3 ) such that u(t1 ) = ε2 . Hence we can obtain for u00 (t) the inequality on interval [t0 , t1 ]. (24.kov )

u00 (t) = β(t) − p(t)u(t)

(25.kov ),(ii)

≤

β0

on [t0 , t1 ].

By (36.kov ) it follows that u00 (t) · u(t) ≤ β0 u0 (t). And integrating we get Zt1

00

0

Zt1

u (t)u (t)dt ≤ t0

β0 u0 (t)dt,

t0

1 1 0 [u (t1 )]2 − [u0 (t0 )]2 ≤ β0 (u(t1 ) − u(t0 )), 2 2 1 0 ε [u (t1 )]2 ≤ β0 . clearly 2 2

(37.kov )

179


), (25.kov ), (30.kov ) If t ∈ [t1 , T3 ] then we can obtain from (24.kov u00 (t) = β(t) − p(t)u(t) ≤ β0 − p1 u(t1 ) ≤ −

p1 · ε . 4

Since u0 (t) ≥ 0 on [t1 , T3 ] we have ZT3

00

ZT3

0

u (t)u (t)dt ≤ t1

−

εp1 0 u (t)dt. 4

t1

Thus ε · p1 ε · p1 ε 1 0 [u (t1 )]2 ≥ [u(T3 ) − u(t1 )] ≥ . 2 4 4 2 And according to (37.kov ) 1 ε 2 p1 ε · β0 ≥ (u0 (t1 ))2 ≥ , 2 2 8 which implies β0 ≥ εp41 . The last inequality gives a contradiction to (ii). So lim inf u(t) = lim sup u(t) = lim u(t) = 0. t→∞

t→∞

t→∞

Theorem 5. Let u be a solution of equation (1.kov ) (g(t)[u00 (t) + p(t)u(t)])0 = f (t, u, u0 , u00 ), where p(.), g(.) and f (.) fulfill conditions (2.kov ), (3.kov ), (17.kov ), (18.kov ) and (19.kov ). Then equation (1.kov ) has the property A, so every proper solution of (1.kov ) is either oscillatory or it converges with its derivatives to zero as t → ∞. Proof. We proceed by contradiction. Let u(.) be a nonoscillatory solution. According to Lemma 3 and Theorem 4 we have lim u(t) = 0. Statement (iii) in Lemma 3 gives us that lim u00 (t) = 0. t→∞

t→∞

So we need only to prove that u0 (t) → 0 for t → ∞. Let lim u0 (t) 6→ 0, then lim sup |u0 (t)| ≥ A > 0. Thus in any neighbourhood t→∞

t→∞

of ∞ we can find t0 such that |u0 (t0 )| ≥ A 2 > 0. We have lim u(t) = 0, lim u00 (t) = 0. Then we take 0 < ε < 00

t→∞

t→∞

A 6

such that

|u (t)| ≤ ε, |u(t)| ≤ ε in [t0 − 1, t0 + 1]. Then on interval [t0 − 1, t0 + 1] we get the following inequalities (t0 is sufficiently great): u0 (t) − u0 (t0 ) = u00 (t0 + ξ(t − t0 ))(t − t0 ), |u0 (t)| = |u0 (t0 )| − |u00 (t0 + ξ(t − t0 ))| ·|t − t0 | , | {z } | {z } | {z } ≥A 2

≤ε

≤1

180


and therefore |u0 (t)| ≥

A −ε 2

for all t ∈ [t0 − 1, t0 + 1].

Farther |u(t0 )| ≤ ε and hence u(t) = u(t0 ) + u0 (t0 + ξ(t − t0 ))(t − t0 ), |u(t)| ≥ |u0 (t0 + ξ(t − t0 )| · |t − t0 | − |u(t0 )|, A for all t ∈ [t0 − 1, t0 + 1]. |u(t)| ≥ ( − ε)|t − t0 | − ε 2 A We put t = t0 + 1 . Then |u(t0 + 1)| ≥ ( A 2 − ε) − ε = 2 − 2ε. Since we A took 0 < ε < 6 , we get |u(t0 + 1)| > ε, what is a contradiction to assumption |u(t0 + 1)| ≤ ε. Thus lim u0 (t) = 0 and so equation (1.kov ) has the property A. t→∞

References [1] Kiguradze, I. T. , An oscillation criterion for a class of ordinary differential equations , J. Diff. Equations, 28(1992), 207–219



On Factorization of Fefferman’s Inequality Miroslav Krbec1 and Thomas Schott2 1

2

Institute of Mathematics, Academy of Sciences of the Czech Republic, ˇ a 25, 115 67 Prague 1, Czech Republic Zitn´ Email: [email protected] Fakult¨ at f¨ ur Mathematik und Informatik, Friedrich-Schiller-Universit¨ at, Ernst-Abbe-Platz 1–4, 07740 Jena, Germany Email: [email protected]

Abstract. This paper is concerned with conditions for a weight function V in order that Z 1/2 Z 1/2 u2 (x)V (x) dx ≤c (∇u(x))2 dx , u ∈ W01,2 (B), B

B

where B is a ball in R . This inequality has found wide applications in many areas of analysis and this has been the reason for an effort to obtain various conditions, either sufficient or necessary and sufficient. Here we survey some of them and we also present a method, using decomposition of imbeddings between Sobolev and Lorentz-Orlicz spaces (and/or their weak counterpart). We state sufficient conditions in terms of a membership of the weight function V in Lorentz-Orlicz spaces and pay an attention to the so called ‘size condition’ in order to discuss applications to the strong unique continuation property for |∆u| ≤ V |u| in dimensions 2 and 3. N

AMS Subject Classification. 46E35, 46E30 35J10 Keywords. Limiting imbeddings, Orlicz spaces, Orlicz-Lorentz spaces, strong unique continuation property

1

Introduction

Fefferman’s inequality [F] 1/2 Z 1/2 Z 2 2 u (x)V (x) dx ≤c (∇u(x)) dx , RN

u ∈ W 1,2 (RN ),

(1.krb )

RN

has turned out to be a very powerful tool to handle many topical problems in the PDEs including the strong unique continuation property (the SUCP in the sequel), distribution of eigenvalues and so on. This is the preliminary version of the paper.

182

Miroslav Krbec and Thomas Schott

After making a short trip into the history, when we recall some of the most important results, our concern will be to establish efficient and manageable conditions for the function V , guaranteeing validity of a local version of (1.krb ), that is, Z

1/2 Z 1/2 2 u (x)V (x) dx ≤c (∇u(x)) dx , 2

B

u ∈ W01,2 ,

(2.krb )

B

where B is a bounded domain in RN , say, a ball, |B| = 1. We shall use a natural idea of a decomposition of the imbedding in (2.krb ) into an imbedding of W01,2 into a suitable target space and an imbedding from this target into L2 (V ); we invoke imbedding theorems for the Sobolev space W01,2 — the classical Sobolev theorem and a refinement in terms of Lorentz spaces in the role of target spaces in the dimension N ≥ 3, and the limiting imbedding theorem due to BrézisWainger [BW] (see also [Zi], Lemma 2.10.5) in the dimension N = 2, which can be viewed as an analogous refinement of Trudinger’s celebrated limiting imbedding [T]. The method suggested for proving (2.krb ) is a kind of a generator of ndimensional Hardy inequalities or, alternatively, of weighted imbeddings W01,2 ,→ L2 (V ): general results of this nature will appear elsewhere. It is rather surprising that working with superpositions of imbeddings we do not lose much and that combining our conditions for validity of (2.krb ) with the conditions for the SUCP in Chanillo and Sawyer [CS] we recover or generalize some of known results about the strong unique continuation property for |∆u| ≤ V |u| in dimensions 2 and 3. In fact all the above imbeddings of the Sobolev spaces are sharp in the scale of spaces considered and the same is true for the weighted imbeddings. In the latter case we shall use only H¨ older’s inequality, nevertheless, we actually use conditions which are necessary as well.

2

Recent history — a partial survey

Let us start with an observation that the theory of weighted imbeddings is by no means complete; only special problems have been fully solved. For instance the particular type of power weights has been considered in [OK] — powers of distance to the boundary of the domain in question (that is, power type weights after flattening the boundary using local coordinates). Passing to more general weights, a natural idea is to apply what is known for the behaviour of Riesz potentials in weighted spaces since (1.krb ) follows for a weight function V provided the boundedness of the Riesz potential of order 1 from the Lebesgue space L2 into the weighted Lebesgue space L2 (V ) has been established. Let us observe that one of the peculiarities of the inequality (2.krb ) is that the powers at both sides are the same. Necessary and sufficient conditions have been found for the case of imbeddings of W 1,p into Lq (V ), see Adams’ inequality in [A] and Maz’ya [Ma], when p < q. If p = q = 2 and N ≥ 3, then a necessary and sufficient condition

183

On Factorization of Fefferman’s Inequality

is due to Kerman and Sawyer [KeSa]; it reads 2 Z Z Z V (y) dy dx ≤ K V (x) dx |x − y|N −1 RN

Q

(3.krb )

Q

for all dyadic cubes Q ⊂ RN , with a constant K independent of Q. This condition uses local potentials in an intrinsic way since it hangs on Sawyer’s theorem on two weight inequalities for the maximal function from [Sa] and on the goodλ-inequality due to Muckenhoupt and Wheeden [MW]; the latter giving a link between an inequality for the corresponding Riesz potential and for the associated fractional maximal function. The condition (3.krb ) can sometimes be hard to verify since it involves the local potential of V , or, alternatively, the fractional integral of V . Hence various sufficient conditions, including those preceding [KeSa] are of importance. The celebrated Fefferman’s paper [F] gave a sufficient condition, which we describe in the following. Let us recall the definition of the Fefferman-Phong class Fp , 1 ≤ p ≤ N/2. A function V belongs to Fp if kV kFp = sup r

2

x∈RN r>0

1/p

Z

1 |B(x, r)|

|V (y)| dy p

< ∞.

B(x,r)

Let us first formulate the basic result in the framework of the classes Fp . Theorem 1 (Fefferman [F]). Let N ≥ 3, 1 < p ≤ N/2, and V ∈ Fp . Then (1.krb ) holds. A particularly fine and elegant proof of (1.krb ) was given by Chiarenza and Frasca [CF]. It is worth observing that Fp2 ⊂ Fp1 for 1 ≤ p1 ≤ p2 ≤ N/2, and plainly FN/2 = LN/2 . Provided that we restrict ourselves to balls B(x, r) with radius smaller than some ε0 > 0 in the above definition the result can be identified with the Morrey space Lp,N −2p . We recall that, for 0 < λ ≤ N and 1 ≤ p < ∞, the Morrey space Lp,λ is the collection of all V ∈ Lploc such that kV kLp,λ = sup r x∈RN 0 0}. Restriction of these spaces to a domain in RN , say, Ω can be done in an obvious way, namely, by considering χΩ V instead of V in the above definitions. It will be useful to give relations between the spaces considered up to now. They are discussed e.g. in Zamboni [Za], Di Fazio [DiF] (the first inclusion in (i)), Piccinini [Pi] (the statement in (iii) below) and Kurata [K]; the last quoted author considers also other variants of the Stummel-Kato class to get a background tailored for more general elliptic operators. Proposition 2. The following statements are true: (i) L1,λ ⊂ S ⊂ Se ⊂ L1,N −2 , λ > N − 2. (ii) LN/2,∞ ⊂ Fp for every 1 ≤ p < N/2, where the former space denotes the weak LN/2 space (the Marcinkiewicz space). (iii) For each p ≥ 2 and each 0 < λ < n, there exists a function f ∈ Lp,λ \ Lq for every q > p. (iv) For every sufficiently small p > 1 there exists a function f ∈ Fp \ LN/2,∞ . (v) S(Ω) ⊂ F1 (Ω), and LN/2 (Ω) is incomparable with S(Ω). Let us observe that (ii) gives a sufficient condition for the validity of (1.krb ) in terms of another scale of function spaces, namely, of the weak Lebesgue spaces. We shall come to use of more general Lorentz spaces later in this paper. e it is possible to prove (see [Za]): Employing the class S, e Then for every r > 0 there is Cr depending only on Theorem 3. Let V ∈ S. η(V, r) and N such that Z Z 2 u (x)V (x) dx ≤ Cr |∇u(x)|2 dx RN

RN

holds for every u ∈ C0∞ supported in B(0, r).


185

A reader can find further results in Chang, Wilson and Wolff [CWW], who consider a certain Orlicz variant of Morrey spaces. An interesting Orlicz spaces type refinement of the well-known Adams’ inequality [A], has recently appeared in Ragusa and Zamboni [RZ]. The inequalities (1.krb ), (2.krb ) and further weighted imbeddings certainly deserve further study aimed at obtaining necessary and sufficient conditions or to get as close as possible to them; at the same time it is desirable that these conditions are described in a manageable way.

3

The size condition and some applications

For the sake of applications we shall pay a special attention the so called ‘smallness condition’ or the ‘size condition’ (see (5.krb ) below), playing a important role in the study of the strong unique continuation property. We shall restrict ourselves to a differential inequality arising from the Schrödinger operator, namely, |∆u| ≤ V |u|. Let us recall that a locally integrable function u is said to have a zero of infinite order at x0 if Z lim r−k |u(x)|2 dx = 0 r→0+

|x−x0 | (N − 1)/2 and proved the SUCP for potentials V which have locally small Fp -norm in the sense that lim sup kV χB(y,r) kFp ≤ ε(p, N ) r→0

for all y ∈ RN ,

(4.krb )

where ε(p, N ) is a sufficiently small constant. Since LN/2,∞ ⊂ Fp for all p < N/2 (see Proposition 2) this gives a result for V in a larger class than in [JK], [St], however, with the size constraint, this time in the Fp class; again the value of the constant appearing in the size condition is not specified. If N ≤ 3, then a condition for the SUCP in terms of the local smallness of the constant C in (1.krb ) appears; more specifically: Theorem 5 (Chanillo, Sawyer [CS]). Let us assume that N = 2 or N = 3 and that Ω is a bounded open and connected subset of RN . Let T (V ) denote the imbedding in (1.krb ). If lim sup kT (V χB(x,r) )k ≤ ε

(5.krb )

r→0+

2,2 with a sufficiently small ε > 0 for all x ∈ Ω, then any solution u ∈ Wloc of the inequality |∆u| ≤ V |u| in Ω has the SUCP.

It turns out that the size condition can be effectively verified in some cases. We shall consider the scale of Lorentz spaces in the dimension 3, and for N = 2 we present a general theorem, including [GL] as a special case. Proofs will appear elsewhere (see [KrSc]). We shall need some basic facts from the Orlicz, Lorentz-Zygmund and OrliczLorentz spaces theory. Let us agree that all the spaces in the sequel will be considered on a ball B ⊂ RN with the unit measure, N ≥ 2, or on the interval (0, 1); we shall usually omit the appropriate symbol for the domain since it will be clear from the context. We shall also need a finer scale of spaces, which includes Orlicz spaces in a rather same manner as Lorentz spaces include Lebesgue spaces. We refer to Montgomery-Smith [M-S]. Let us recall that an even and convex function Φ : R → [0, ∞) such that lim Φ(t) = lim 1/Φ(t) = 0 is called a Young function. A general reference for t→0

t→∞

the (non-weighted) theory of Orlicz spaces is [KR], more general modular spaces are subject of [Mu]. Let Φ and Ψ be Young functions. For a function g even on R1 and positive on (0, ∞) let us put   1/g(1/t), t > 0, ge(t) = e g (−t), t < 0,   g(0), t = 0.

187


Let V be a weight in B and let fV∗ denote the non-increasing rearrangement of f with respect to the measure V (x) dx. An Orlicz-Lorentz space LΦ,Ψ (V ) is the set of all measurable f on B for which the Orlicz-Lorentz functional e ◦ Ψe−1 kΨ kf kΦ,Ψ ;V = kfV∗ ◦ Φ ! Z∞ e Ψe−1 (t))) fV∗ (Φ( dt ≤ 1} = inf{λ > 0; Ψ λ

(6.krb )

0

is finite. A measurable function f defined on B belongs to a weak Orlicz (or Orlicz-Marcinkiewicz) space LΦ,∞ (V ) if its Orlicz-Marcinkiewicz functional e−1 (ξ)f ∗ (ξ) kf kΦ,∞;V = sup Φ V

(7.krb )

ξ>0

is finite. If V ≡ 1, we shall simply write LΦ,Ψ and LΦ,∞ instead of LΦ,Ψ (1) and LΦ,∞ (1), resp. For brevity and in accordance with a general usage we shall often use only the major part of a Young function (that is, functions equivalent to the Young function in question in a neighbourhood of infinity) in symbols for spaces. Let us observe that LΦ,Φ = LΦ , the Orlicz space. If Φ(t) = |t|p and Ψ (t) = |t|q , then LΦ,Ψ = Lp,q , the Lorentz space, LΦ,∞ = Lp,∞ , the Marcinkiewicz space; analogously for the weighted variants. Special cases of the Orlicz-Lorentz spaces are the Lorentz-Zygmund spaces, that is, logarithmic Lorentz spaces, investigated by Bennett and Rudnick [BR]. For 0 < p, q ≤ ∞ and α ∈ R1 , the Lorentz-Zygmund space Lp,q (log L)α consists of functions f with the finite functional Z1 kf kLp,q (log L)α =

[t1/p (log(e/t))α f ∗ (t)]q

dt t

1/q ,

for q < ∞,

0

kf kLp,∞(log L)α = sup t1/p (log(e/t))α f ∗ (t), 0 q2 , and α1 + 1/q1 > α2 + 1/q2 ; (iii) p1 = p2 < ∞, q1 ≤ q2 , and α1 ≥ α2 ; (iv) p1 = p2 = ∞, q1 ≤ q2 , and α1 + 1/q1 ≥ α2 + 1/q2 (see [BR], Theorems 9.1 and 9.3 and 9.5). Remark 7. According to the limiting imbedding theorem due to Brézis and Wainger [BW] we have, for N = 2, W01,2 ,→ L∞,2 (log L)−1 .

(8.krb ) 2

2

The latter space, as was observed above, is the Orlicz-Zygmund space Lexp t ,t , 2 2 2 a space smaller than Lexp t = Lexp t ,exp t , and this interpretation of the target space in (8.krb ) gives a natural analogue to the (sublimiting) imbeddings of Sobolev spaces into Lebesgue spaces and their Lorentz refinements.

4

Decomposition of imbeddings

Let us recall our agreement that for the sake of simplicity we shall suppose that the domain B is a ball, |B| = 1. We shall usually omit the symbol of the domain. We are seeking for sufficient conditions for (2.krb ) and (5.krb ); we shall even find a condition stronger than (5.krb ), namely, lim sup kT (V χA )k = 0.

δ→0 A⊂B |A| 2.

(12.krb )

Then (2.krb ) and (9.krb ) hold. Remark 10. The proofs of Theorems 8 and 9 can be carried out making use of the refined Sobolev imbedding W 1,2 ,→ L2N/(N −2),s for N ≥ 3 and of the refined limiting imbedding in (8.krb ) for N = 2 together with conditions (necessary and sufficient) for the imbeddings of weighted Orlicz-Lorentz spaces, taking, moreover, care about the quantitative behaviour of norms of the imbeddings. The details can be found in [KrSc]. Remark 11. The space L1,∞ (log L)2 can be identified with the Orlicz-Marcinkie2 β s wicz space Lt log t,∞ and L1,s (log L)β , 0 < s < ∞, with Lt log t,t . This can be 1,∞ 2 checked easily. Indeed, considering for instance V ∈ L (log L) , that is, if we have sup t(log(e/t))2 V ∗ (t) < ∞, then Fe−1 (t) = t(log(e/t))2 near the origin, 0 0. The domain Ω corresponds to a channel filled up by a fluid. Γ1 is a fixed wall of the channel and Γ2 involves the input and the output of the channel.

2

Classical Formulation of the Problem

Let T > 0 be a positive number. (0, T ) denotes the time interval, Q = Ω × (0, T ), eij (u) (for 1 ≤ i, j ≤ n) denotes ∂ui /∂xj + ∂uj /∂xi . The problem we will deal with can be classically formulated as follows: ∂ ∂ui ∂P ∂u − (ν · eij (u)) + uj · + = gi ∂t ∂xj ∂xj ∂xi divu = 0

in Q,

i = 1, . . . , n,

in Q,

u = 0 in Γ1 × (0, T ), −P · ni + ν · eij (u) · nj = σi in Γ2 × (0, T ), This is the final form of the paper.

(1.kuc ) (2.kuc ) (3.kuc ) (4.kuc )

194

Petr Kuˇcera

u(x, 0) = u(x, T ) in Ω,

(5.kuc )

u(., 0) = 0 on Γ1 .

(6.kuc )

Here u is the velocity, P is the pressure, ν denotes the viscosity, g is a body force, σ is a prescribed vector function on Γ2 and n = (n1 , . . . , nn ) is the outer normal vector. The problem (1.kuc )–(6.kuc ) will be called the time-periodic NavierStokes problem with the mixed boundary conditions. We suppose that ν is a positive constant in the whole paper. The used Dirichlet boundary condition expresses a non-slip behaviour of the fluid on the fixed walls of the channel. The condition (4.kuc ) means that we prescribe a normal component of the stress tensor on Γ2 . The Navier-Stokes equations with condition (4.kuc ) were already treated in the works [1]–[6].

3

Some Function Spaces and Their Properties

To formulate the problem (1.kuc )–(6.kuc ) weakly, we shall need some function spaces. Let us denote E(Ω) = {ϕ ∈ [C ∞ (Ω)]n ; div ϕ ≡ 0, supp ϕ ∩ Γ1 ≡ ∅}. The Banach spaces V k,p , resp. V 0,q , is defined as the closure of E(Ω) in the norm of the space [W k,p (Ω)]n , resp. [Lp (Ω)]n , where k > 0 (it need not be an integer) and 1 ≤ q ≤ ∞. For simplicity, we denote the space V 0,2 by the symbol H. Both the spaces V 1,2 and H are Hilbert spaces with the scalar products Z ((ψ, φ))1,2 = eij (ψ) · eij (φ) d(Ω) Ω

resp.

Z ψi · φi d(Ω).

((ψ, φ))0,2 = Ω

The symbol h., .i denotes the duality between elements from (V 1,2 )∗ and V 1,2 . It is obvious that V 1,2 , H and (V 1,2 )∗ are three Hilbert spaces, which satisfy the following conditions V 1,2 ,→,→ H ,→,→ (V 1,2 )∗ and H coincides with the interpolation [V 1,2 , (V 1,2 )∗ ]1/2 . Moreover, if u ∈ L2 (0, T, V 1,2 ), u0 ∈ L2 (0, T, (V 1,2 )∗ ), then u ∈ C([0, T ]; H) and ||u||L∞ (0,T ;H) ≤ c · (||u||L2 (0,T ;V 1,2 ) + ||u0 ||L2 (0,T ;(V 1,2 )∗ ) ), where c = c(Ω). If X is a Banach space then (X )∗ will denote its dual and Lp (0, T ; X ), 1 < p < ∞, will be the linear space of all measurable functions from the interval (0, T ) into X such that Z T ||u(t)||pX dt < ∞. 0

195

A Time Periodic Solution of Navier-Stokes Equations

Let X and Y be the following Banach spaces: X = {u; u0 ∈ L2 (0, T, V 1,2 ), u00 ∈ L2 (0, T, (V 1,2 )∗ ), u(0) = u(T ) ∈ V 1,2 , ||u||X

u0 (0) = u0 (T ) ∈ H}, = ||u||L2 (0,T ;V 1,2 ) + ||u ||L2 (0,T ;V 1,2 ) + ||u ||L2 (0,T ;(V 1,2 )∗ ) , 0

00

Y = {f ; f ∈ C([0, T ], (V 1,2 )∗ ), f 0 ∈ L2 (0, T, (V 1,2 )∗ ), f (0) = f (T ) ∈ (V 1,2 )∗ }, ||f ||Y = ||f ||L2 (0,T ;(V 1,2 )∗ ) + ||f 0 ||L2 (0,T ;(V 1,2 )∗ ) .

4

Weak Formulation of the Problem

The weak formulation of the problem (1.kuc )–(6.kuc ) will be based on an operator equation. Therefore we define operators S, B and N at first. The operator S from X to Y is defined by the equation hS(u), vi = ((u0 , v))0,2 + ν · ((u, v))1,2 for every v ∈ V 1,2 and almost every t ∈ (0, T ). b(ϕ, ψ, φ) will denote trilinear form on V 1,2 × V 1,2 × V 1,2 such that Z ∂ψi b(ϕ, ψ, φ) = ϕj · · φi d(Ω). ∂xj Ω It can be easily verified that b(ϕ, ψ, φ) satisfies the following estimate |b(ϕ, ψ, φ)| ≤ c · ||ϕ||V 1,2 · ||φ||V 1,2 · ||ψ||V 1,2 ,

(7.kuc )

where c = c(Ω). Integrating by parts and using the theorems about imbedding the space [W kp (Ω)]n into the space Lq (∂Ω) the following estimates are verified: |b(ϕ, ψ, φ)| ≤ c · ||ϕ||V 1,2 · ||ψ|| |b(ϕ, ψ, φ)| ≤ c · ||ϕ||

V

7 ,2 8

· ||v||V 1,2 ,

(8.kuc )

· ||ψ||V 1,2 · ||v||V 1,2 .

(9.kuc )

V

7 ,2 8

The symbols ϕ and ψ will sometimes also denote functions of the variable t with values in V 1,2 . B will be operator from X into Y defined by the equation hB(u), vi = b(u, u, v) for every v ∈ V 1,2 and almost every t ∈ (0, T ).

196

Petr Kuˇcera

Finally, operator N from X into Y is defined by the equation N (u) = S(u) + B(u). A function u ∈ X will be called a weak solution to the time-periodic NavierStokes problem with the right hand side f if N (u) = f.

Notice that

gi · vi d(Ω) +

hf, vi = Ω

5

Z

Z

σi · vi d(∂Ω). Γ2

The Local Diffeomorphism Theorem

Suppose that u0 and f0 are such elements of X and Y that N (u0 ) = f0 . (This means that u0 is a weak solution to the time-periodic Navier-Stokes problem with the right hand side f0 .) Our further aim is to investigate the solvability of the equation N (u) = f with f from some neighbourhood of point f0 in Y . To solve this problem, we will use the following very important theorem (the Local Diffeomorphism Theorem). Theorem 1. Let X , Y be Banach spaces, F be a mapping from X into Y belonging to C 1 in some neighbourhood V of point u0 . If F 0 (u0 ) is one-to-one from X onto Y and continuous, then there exists a neighbourhood U of point u0 , U ⊂ V and a neighbourhood W of point f (u0 ), W ⊂ Y, so that F is one-to-one from U to W .

6

The Fr´ echet Derivative of Operator N

It is obvious that if there exists a point u ∈ X in which the operator N satisfies the assumption of the Local Diffeomorphism Theorem then the equation N (u) = f is “locally solvable” (i.e. solvable in some neighbourhood of u). It is clean that N ∈ C 1 (X). Further, need to express the Fréchet derivative of operator N at point u and to find out, whether it is one-to-one. We will express the Fréchet derivative of N by means of operators K and G. K is the bilinear operator from X × X into Y defined by the equation hK(u), vi = b(u, w, v) + b(w, u, v) for every v ∈ V 1,2 and for almost every t ∈ (0, T ).


197

The operator G from X × X into Y is given by the equation G(u, w) = S(w) + K(u, w). It is possible to prove there exists a constant c = c(Ω) so that ||b(u, w, .)||Y ≤ c · ||u||X · ||w||X . Theorem 2. Let u ∈ X. Then the operator G(u, .) is the Fréchet derivative of N at point u and G ∈ C 1 (X × X, Y ). Proof. It is possible to prove for arbitrary u, w ∈ X following estimate ||b(u, w, .)||Y ≤ c · ||u||X · ||w||X , where c = c(Ω). Therefore and from the estimate ||N (u + w) − N (u) − G(u, w)||Y = ||b(w, w, .)||Y ≤ c · ||w||2X , we get

||N (u + w) − N (u) − G(u, w)||Y = 0. ||w||X →0 ||w||X lim

So G(u, .) is the Fréchet derivative of N at point u. The smoothness of G follows immediately from its definition. The proof is complete.

7

Local Properties of Operator N

We have proved that the operator G(u, .) has the form G(u, .) = S(.) + K(u, .) in the previous section. Further we will prove that operator S is a one-to-one linear operator from X onto X and K(u, .) is a compact linear operator from X into Y . So the operator G(u, .) is the sum of a one-to-one operator and a compact operator. Operators of this form have properties which will be used later. Lemma 3. S is a linear continuous one-to-one operator from X onto Y . Proof. The linearity and continuity of S are obvious. Next we prove that S is an operator from X onto Y . The form ((., .))1,2 is V 1,2 -elliptic. Then there exists w ∈ L2 (0, T, V 1,2 ) ∩ C([0, T ]; H), so that w0 ∈ L2 (0, T, (V 1,2 )∗ ), the equation d ((w(t), v))0,2 + ν · ((w(t), v))1,2 = hf 0 , vi dt holds for every v ∈ V 1,2 and w(0) = w(T ).

198

Petr Kuˇcera

Then there exists ω0 ∈ V 1,2 so that for every v ∈ V 1,2 holds ν · ((ω0 , v))1,2 = hf, vi − ((w(0), v))0,2 . Let

Z

t

w(s) ds, t ∈ (0, T ).

u(t) = ω0 + 0

Then u ∈ X and S(u) = f . Thus we have proved that S is from X onto Y . Let us suppose that S(u) = 0. Then u = 0. The proof is complete. Lemma 4. Let u ∈ X. Then K(u, .) is a linear compact operator from X into Y . Prior to the proof we recall a result from [7, Lemma 4.5]. Denote Z = {u; u ∈ L2 (0, T, V 1,2 ), u0 ∈ L2 (0, T, (V 1,2 )∗ )} with the norm

||u||Z = ||u||L2 (0,T ;V 1,2 ) + ||u0 ||L2 (0,T ;(V 1,2 )∗ )

(u0 is the Schwartz derivative in the sence of imbedding V 1,2 ,→ H ,→ (V 1,2 )∗ ). Then Z ,→,→ L2 (0, T ; V

7 8 ,2

)

(10.kuc )

Proof. Let wk ⊂ X be a bounded set in X. Using (10.kuc ) we get w ∈ X such that wk0 → w0 in L2 (0, T ; V

7 8 ,2

)

(11.kuc )

).

(12.kuc )

and wk (0) → w(0) in V

7 8 ,2

Combining it with (11.kuc ) we get wk → w in L∞ (0, T ; V

7 8 ,2

Note that ||K(u, wk ) − K(u, w)||Y = =||b(u, wk − w, .)||L2 (0,T ;(V 1,2 )∗ ) + ||b(wk − w, u, .)||L2 (0,T ;(V 1,2 )∗ ) + +||b(u, wk0 − w0 , .)||L2 (0,T ;(V 1,2 )∗ ) + ||b(u0 , wk − w, .)||L2 (0,T ;(V 1,2 )∗ ) + +||b(wk0 − w0 , u, .)||L2 (0,T ;(V 1,2 )∗ ) + ||b(wk − w, u0 , .)||L2 (0,T ;(V 1,2 )∗ )

(13.kuc )

We estimate the third and fourth additive terms. Let v ∈ V 1,2 . Use (8.kuc ) to get the estimate |b(u(t), wk0 (t) − w0 (t), v)| ≤ c · ||u(t)||V 1,2 · ||wk0 (t) − w0 (t)||

V

7 ,2 8

· ||v||V 1,2 .

199


Therefore ||b(u(t), wk0 (t) − w0 (t), .)||(V 1,2 )∗ ≤ c · ||u(t)||V 1,2 · ||wk (t) − w(t)||

V

7 ,2 8

for almost all t ∈ (0, T ), c = c(Ω). It follows that ||b(u, wk0 − w0 , .)||L2 (0,T ;(V 1,2 )∗ ) ≤ c · ||u||L∞ (0,T ;V 1,2 ) · ||wk0 − w0 ||

L2 (0,T ;V

7 ,2 8 )

. (14.kuc )

Similarly, we get |b(u0 (t), wk (t) − w(t), v)| ≤ c · ||u0 (t)||V 1,2 · ||wk (t) − w(t)||

V

7 ,2 8

· ||v||V 1,2

and therefore ||b(u0 (t), wk (t) − w(t), .)||(V 1,2 )∗ ≤ c · ||u0 (t)||V 1,2 · ||wk (t) − w(t)||

V

7 ,2 8

for almost all t ∈ (0, T ), c = c(Ω). It follows ||b(u0 , wk − w, .)||L2 (0,T ;(V 1,2 )∗ ) ≤ c · ||u0 ||L2 (0,T ;V 1,2 ) · ||wk − w||

L∞ (0,T ;V

7 ,2 8 )

. (15.kuc )

The same way we prove ||b(u, wk − w, .)||L2 (0,T ;(V 1,2 )∗ ) ≤ c · ||u||L2 (0,T ;(V 1,2 )∗ ) · ||wk − w||

L∞ (0,T ;V

7 ,2 8 )

,

(16.kuc ) ||b(wk − w, u, .)||L2 (0,T ;(V 1,2 )∗ ) ≤ c · ||u||L2 (0,T ;(V 1,2 )∗ ) · ||wk − w||

L∞ (0,T ;V

7 ,2 8 )

,

(17.kuc ) ||b(wk0 − w0 , u, .)||L2 (0,T ;(V 1,2 )∗ ) ≤ c · ||u||L∞ (0,T ;V 1,2 ) · ||wk0 − w0 ||

L2 (0,T ;V

7 ,2 8 )

(18.kuc )

and ||b(wk − w, u0 , .)||L2 (0,T ;(V 1,2 )∗ ) ≤ c · ||u0 ||L2 (0,T ;(V 1,2 )∗ ) · ||wk − w||

L∞ (0,T ;V

7 ,2 8 )

.

(19.kuc ) From (11.kuc )–(19.kuc ) we get ||K(u, wk ) − K(u, w)||Y → 0. The proof is complete. The operator G(u, .) is the sum of a one-to-one operator and a compact operator. The operators of this form are widely treated in mathematical literature and we can apply their known properties to prove the following theorem. Theorem 5. Let u ∈ V 1,2 . Then the following statements are equivalent: (a) G(u, .) is an injective operator . (b) G(u, .) is an operator onto (V 1,2 )∗ . Moreover, if the statements (a)–(b) are satisfied at point u then there exists an open neighbourhood U of point u in X and an open neighbourhood W of point N (u) in Y such that N is a one-to-one operator from U onto W .

200

Petr Kuˇcera

References [1] R. Glowinski, Numerical methods for nonlinear variational problems, Springer Verlag, Berlin-Heidelberg-Tokio-New York, 1984. [2] S. Kraˇcmar, J. Neustupa, Global existence of weak solutions of a nonsteady variational inequalities of the Navier-Stokes type with mixed boundary conditions. Proc. of the conference ISNA’92, (1992), Part III, 156–157. [3] S. Kraˇcmar, J. Neustupa, Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities. ZAMM 74, 6 (1994), 637– 639. [4] S. Kraˇcmar, J. Neustupa, Some Initial Boundary Value Problems of the NavierStokes Type with Mixed Boundary Conditions. Proc. of the seminar Numerical Mathematics in Theory and Practice, Pilsen, (1993), 114–120. [5] S. Kraˇcmar, J. Neustupa, Simulation of Steady Flows through Channels by Variational Inequalities. Proc. of the conference Numerical Modelling in Continuum Mechanics, Prague 1994, (1995), 171–174. [6] R. Rannacher, Numerical analysis of the Navier-Stokes equations. Proceedings of conference ISNA’92, part I. (1992), 361–380. [7] P. Kuˇcera, Z. Skal´ ak, Local solutions to the Navier-Stokes equations with mixed boundary conditions. Submitted to Acta Applicandae Mathematicae.



Numerical Solution of Compressible Flow M´ aria Lukáˇcov´ a-Medvid’ov´ a Department of Mathematics, Faculty of Mechanical Engineering, Technical University Brno Technick´ a 2, 616 00 Brno, Czech Republic Institute of Analysis and Numerics Otto-von-Guericke University Magdeburg PSF 4120, 39 106 Magdeburg, Germany Email: [email protected] [email protected]

Abstract. The main feature of the equations describing the motion of the viscous compressible flows, i.e. the Navier-Stokes equations, is the combination of dominating convective parts with the diffusive effects. These equations will be numerically solved by the combined finite volume — finite element method via operator inviscid-viscous splitting. The main idea of the method is to discretize nonlinear convective terms with the aid of the finite volume scheme, whereas the diffusion terms are discretized by piecewise linear conforming triangular finite elements. The nonlinear convective terms can also be solved by the method of characteristics. Numerical solution obtained by latter method is truly multidimensional and independent of the mesh character. We will present results of numerical experiments for some well-known test problems. AMS Subject Classification. 65M12, 65M60, 35K60, 76M10, 76M25 Keywords. Compressible Navier-Stokes equations, nonlinear convection-diffusion equation, finite volume schemes, finite element method, numerical integration, truly multidimensional schemes, evolution-Galerkin schemes

1

Formulation of the problem

We consider gas flow in a space-time cylinder QT = Ω × (0, T ), where Ω ⊂ R2 is a bounded domain representing the region occupied by the fluid and T > 0. By Ω and ∂Ω we denote the closure and boundary of Ω, respectively. The complete system of viscous compressible flow consisting of the continuity equation, Navier-Stokes equations and energy equation can be written in the form ∂w X ∂fi (w) X ∂Ri (w, ∇w) + = ∂t ∂xi ∂xi i=1 i=1 2


2

in QT .

(1.luk )

202

M´ aria Luk´ aˇcov´ a-Medvid’ov´ a

Here w = (w1 , w2 , w3 , w4 )T = (ρ, ρv1 , ρv2 , e)T , w = w(x, t),

(2.luk )

x ∈ Ω, t ∈ (0, T ),

fi (w) = (ρvi , ρvi v1 + δi1 p, ρvi v2 + δi2 p, (e + p) vi )T , Ri (w, ∇w) = (0, τi1 , τi2 , τi1 v1 + τi2 v2 + k∂θ/∂xi )T , ∂vj ∂vi , i, j = 1, 2. + τij = λ div v δij + µ ∂xj ∂xi From thermodynamics we have p = (γ − 1) (e − ρ|v|2 /2),

e = ρ(cv θ + |v|2 /2).

(3.luk )

We use the standard notation: t — time, x1 , x2 — Cartesian coordinates in R2 , ρ — density, v = (v1 , v2 ) — velocity vector with components vi in the directions xi , i = 1, 2, p — pressure, θ — absolute temperature, e — total energy, τij — components of the viscous part of the stress tensor, δij — Kronecker delta, γ > 1 — Poisson adiabatic constant, cv — specific heat at constant volume, k — heat conductivity, λ, µ — viscosity coefficients. We assume that cv , k, µ are positive constants and λ = − 32 µ. We neglect outer volume force. The functions fi are called inviscid (Euler) fluxes and are defined in the set D = {(w1 , . . . , w4 ) ∈ R4 ; w1 > 0}. The viscous terms Ri are obviously defined in D × R8 . (Due to physical reasons it is also suitable to require p > 0.) System (1.1), (1.3) is equipped with the initial conditions x∈Ω

w(x, 0) = w0 (x),

(4.luk )

(which means that at time t = 0 we prescribe, e. g., ρ, v1 , v2 and θ) and boundary conditions: The boundary ∂Ω is divided into several disjoint parts. By ΓI , ΓO and ΓW we denote inlet, outlet and impermeable walls, respectively, and assume that (i) ρ = ρ∗ ,

vi = vi∗ ,

(ii) vi = 0,

i = 1, 2,

(iii)

2 X i=1

τij ni = 0,

i = 1, 2,

j = 1, 2,

θ = θ∗ on ΓI , ∂θ = 0 on ΓW , ∂n

∂θ =0 ∂n

(5.luk )

on ΓO .

Here ∂/∂n denotes the derivative in the direction of unit outer normal n = (n1 , n2 ) to ∂Ω; w0 , ρ∗ , vi∗ and θ∗ are given functions. Let us note that nothing is known about the existence and uniqueness of the solution of problem (1.1), (1.3)–(1.5). Some solvability results for system (1.1) & (1.3) were obtained either for small data or on a very small time interval under simple Dirichlet boundary conditions (for reference, see e. g., [3, Par. 8.10]). We do not take care of the lack of theoretical results and deal with the numerical solution of the above problem. Since the viscosity µ and heat conductivity

203

Numerical Solution of Compressible Flow

k are small, we treat the diffusion terms on the right hand side of (1.1) as a perturbation of the inviscid Euler system and conclude that a good method for the solution of viscous flow should be based on a sufficiently robust scheme for inviscid flow simulation. Therefore, we will split the complete system (1.1) into inviscid and viscous part: ∂w X ∂fi (w) + = 0, ∂t ∂xi i=1

(6.luk )

∂w X ∂Ri (w, ∇w) = ∂t ∂xi i=1

(7.luk )

2

2

and discretize them separately. First we will pay attention to the inviscid flow problem.

2

Numerical solution of the Euler equations

In what follows we will describe some numerical methods for solving the Euler equations system (6.luk ). The first part of this section will be devoted to the finite volume methods, in the second part we will briefly describe truly multidimensional methods based on the method of characteristics, the so-called evolution Galerkin schemes. In the third part we present some numerical experiments for the Euler equations system. It is easy to realize that fj ∈ C 1 D; R4 for j = 1, 2. Thus, we can apply the chain rule to the function fj (w) and obtain a first order quasilinear system of PDE’s 2 ∂w X ∂w + Aj (w) = 0, (8.luk ) ∂t ∂xj j=1 where Aj (w) =

Dfj (w) Dw

are Jacobi matrices of fj (w) , j = 1, 2.

Definition 1. Let us consider general first order system of type (8.luk ). The system is said to be hyperbolic, if for arbitrary vectors w ∈ D and n = (n1 , n2 ) ∈ R2 the matrix 2 X P (w, n) = nj Aj (w) j=1

has four real eigenvalues λi = λi (w, n) , i = 1, . . . , 4, and is diagonizable, i.e. there exists a nonsingular matrix T = T (w, n) , s.t. T−1 · P · T = D (w, n) = diag(λ1 , λ2 , λ3 , λ4 ) Theorem 2. The system of Euler equations (8.luk ) is hyperbolic. The eigenvalues of the matrix P (w, n) are λ1 = λ2 = n1 v1 + n2 v2 , λ3 = λ1 + a|n|, λ4 = λ1 − a|n|. q Here a is a local speed of sound, i.e. a = kp ρ .

204 2.1


Finite volume schemes

The above properties of the Euler equations allow us to construct efficient numerical schemes for the solution of inviscid flow. We will carry out the discretization of system (6.luk ) with the use of the finite volume method (FVM) which is very popular because of its flexibility and applicability. Let Th be a triangulation of the domain Ωh which is a polygonal approximation of the domain Ω . The so-called dual finite volume partition of Ωh will be denoted by Dh = {Di }i∈J , J is a suitable index set. Moreover, it holds Ωh =

[

Di .

(9.luk )

i∈J

The dual finite volumes will be constructed in the following way: Join the centre of gravity of every triangle T ∈ Th , containing the vertex Pi , with the centre of every side of T containing Pi . If Pi ∈ ∂Ωh , then we complete the obtained contour by the straight segments joining Pi with the centres of boundary sides that contain Pi . In this way we get the boundary ∂Di of the finite volume Di . (See Figure 1.) Dual finite volume meshes were successfully used in a number of works. See, e.g., [1], [8]. r Pk a

C aa aa D

DkC H H C Pi @ HH AA H` r ` C Pj A C ` A ` C `r C ( ( Q J J DC Q iC D Q j J Q J C JJ HH C BB HH C HHC C Figure 1 If for two different finite volumes Di and Dj their boundaries contain a common straight segment, we call them neighbours. Then we write Γij = ∂Di ∩ ∂Dj = Γji .

(10.luk )

The index set of all neighbours for the dual volume Di will be denoted by S(i). Furthermore, we introduce the following notation: |Di | = area of Di , nij = (n1ij , n2ij ) = unit outer normal to ∂Di on Γij , ìj = length of Γij , and consider a partition 0 = t0 < t1 < . . . of the time interval (0, T ) and set τk = tt+1 − tk for k = 0, 1, . . . .

205


The finite volume method reads τk X g(wik , wjk , nij ) ìj , wik+1 = wik − |Di |

(11.luk )

j∈S(i)

Di ∈ Dh (i. e., i ∈ J), k = 0, 1, . . . . To derive (11.luk ) we integrate (6.luk ) over every set Di ×(tk , tk+1 ), use Green’s theorem, the approximation of the exact solution by a piecewise constant function with values wik on Di × {tk } and the approximation of the flux Z

2 X

fr (w) nr dS

Γij r=1

of the quantity w through the segment Γij in the direction nij with the aid of the so-called numerical flux g(wik , wjk , nij ) calculated from wik , wjk and nij . In order to ensure the stability of the scheme (11.luk ) the so-called CFL stability condition has to be fulfilled τk max max λs (wk , nij ) ≤ CFL |Di | j∈S(j) s=1,...,4

∀j ∈ J,

(12.luk )

where CFL ∈ (0, 1]. In literature one can find a lot of numerical flux functions, e.g. Steger-Warming, Osher-Solomon, Van-Leer, Vijayasundaram numerical fluxes. For references, see e.g., [3]. We do not discuss now the question of implementation of the boundary conditions. They have to be prescribed in such a way that the hyperbolic character of the equations is taking into account. For more details the reader is referred to [4]. In the approach described about only the piecewise constant approximation is considered. Nevertheless, also the higher order schemes, using for example discontinuous piecewise linear approximate functions, can be constructed. The details can be found, e.g., in [4]. 2.2

Evolution Galerkin methods

Although in the recent years the most commonly used methods for hyperbolic problems are the finite volume methods, it turns out that in special cases this approach leads to structural deficiencies in the solution (see, e.g., [7], [13]). This is due to the fact that the finite volume methods are based on a quasi dimensional splitting using one-dimensional Riemann solvers. The evolution Galerkin method, first considered by Morton et al. in [2] for scalar hyperbolic equation, combines the theory of characteristics for hyperbolic problems with the finite element ideas. The initial function is transported along the characteristic cone and then projected onto a finite element space. Let E(t) be the exact evolution operator for our hyperbolic problem (8.luk ), i.e. w(·, tk+1 ) = E(∆t)w(·, tk ),

(13.luk )

206


then the evolution Galerkin scheme reads: wk+1 = Ph E∆ wk ,

(14.luk )

where E∆ is an approximate evolution operator and Ph is a projection onto a finite element space. It can be shown that the method is unconditionally stable and the accuracy can be increased by increasing the order of the approximate space and the accuracy of the approximate evolution operator. Using different approximate evolution operators E∆ and projections Ph one obtains a class of the evolution Galerkin methods. The approach described can be fully exploited for simple problems, e.g. the linear hyperbolic system of wave equation (see Luk´ aˇcov´ a, Morton and Warnecke [9], [10], [11]). More details about the application of this method to the Euler equations can be found in the works of Fey [7] and Ostkamp [13]. 2.3

Numerical experiments

(1) Flow through the GAMM channel (10 % circular arc in the channel of width 1 m) for air, i. e. γ = 1.4, and inlet Mach number M := |v| a = 0.67 was solved by the Vijayasundaram higher order scheme applied on the dual mesh over a triangular grid. In Figure 2 the basic grid and dual mesh, respectively, are shown. Our aim was to obtain a steady state solution with the aid of the time marching process for tk → ∞. After 10000 time iterations the stability of the solution up to 10−5 was achieved. Figure 3 shows Mach number isolines and entropy isolines. We can see a sharp shock wave which is resolved very well.

Figure 2: Triangular mesh in the GAMM channel and the dual mesh

Figure 3: Mach number isolines and entropy isolines

207


(2) Two-dimensional Sod’s problem. Now we will show a comparison of the solution obtained by the finite volume method and by the evolution Galerkin scheme. It will be showed that some symmetry structures are better preserved by the truly multidimensional evolution Galerkin method than by the finite volume scheme. The computational domain is the square [-1,1] × [-1,1]. To ensure the CFL stability condition, the CFL number is taken 0.8. We choose periodical boundary conditions and the following initial data

ρ = 1, u = 0, v = 0, p = 1 ρ = 0.125, u = 0, v = 0, p = 0.1

if |x| ≤ 0.4 otherwise.

(15.luk )

In Figure 4 the first picture on the left hand side shows the isolines of pressure for the solution computed by the evolution Galerkin scheme at time T = 0.2 for quadrilateral grid with 200 × 200 grid cells. The symmetry of the data can be observed very well. The resolution of the flow phenomena is the same in all directions and information is moving in infinitely many directions in a circular manner. However this is not the case for the finite volume method. In the next two pictures of Figure 4 the isolines of pressure for the solution computed by the Osher-Solemn finite volume scheme on the quadrilateral mesh (middle) and on the dual mesh (right hand side) are plotted.

Figure 4: Evolution Galerkin scheme and the Osher-Solomon FVM on the quadrilateral mesh and the dual mesh

3

Discretization of the complete system of the Navier-Stokes equations

In this section we will describe the combined finite volume – finite element method which is used for the discretization of the Navier-Stokes equations (1.1). Let us note that we now use only the finite volume method in order to discretize the Euler equations, however also other possibilities are open (cf. Section 2.2 Evolution Galerkin methods).

208


First of all we will describe the finite element discretization of the purely viscous system (7.luk ) equipped with initial conditions (1.4) and boundary conditions (1.5). We use conforming piecewise linear finite elements. This means that the components of the state vector are approximated by functions from the finite dimensional space Xh = ϕh ∈ C(Ω h ); ϕh |T is linear for each T ∈ Th . Further, we set X h = [Xh ]4 and a) V h = {ϕh = (ϕ1 , ϕ2 , ϕ3 , ϕ4 ) ∈ X h , ϕi = 0 on the part of ∂Ωh approximating the part of ∂Ω where wi satisfies the Dirichlet condition} b) W h = {wh ∈ X h ; its components satisfy the Dirichlet boundary conditions following from (1.5)}. Multiplying (7.luk ) considered on time level tk by any ϕh ∈ V h , integrating over Ωh , using Green’s theorem, taking into account the boundary conditions (1.5) and approximating the time derivative by a forward finite difference, we obtain the following explicit scheme for the calculation of an approximate solution whk+1 on the (k + 1)-th time level a) whk+1 ∈ W h , Z Z whk+1 ϕh dx = b) Ωh

(16.luk ) whk ϕh dx −

Ωh

Z

2 X

−τk

Rs (whk , ∇whk )

Ωh s=1

∂ϕh dx ∀ ϕh ∈ V h . ∂xs

The integrals are approximated by a numerical quadrature, called mass lumping, using the vertices of triangles as integration points: Z F dx ≈ T

3 X 1 |T | F (PTi ) 3 i=1

(17.luk )

for F ∈ C(T ) and a triangle T = T (PT1 , PT2 , PT3 ) ∈ Th with vertices PTi , i = 1, 2, 3. The numerical integration yields

wik+1 = wik −

2 ∂ϕm τk X X i |T |Rsk , |Di | s=1 T ∂xs T

(18.luk )

T ∈Th

where i ∈ J, k = 0, 1, . . . , m = 1, . . . , 4, and ϕm i is a basis function from V h having the only non-zero component on the m-th position; namely ϕi ∈ Xh , which corresponds to the vertex Pi . Now we combine the finite volume scheme (11.luk ) with the finite element scheme (18.luk ). The resulting finite volume – finite element operator splitting scheme has the following form:

209


wi0 =

1 |Di |

k+1/2

wi

Z w(x, y, 0), Di

= wik −

τk X g(wik , wjk , nij ) ìj , |Di | j∈S(i)

wik+1 =

k+1/2 wi

2 ∂ϕm τk X X i − |T |Rsk+1/2 , |Di | s=1 T ∂xs T

(19.luk )

T ∈Th

i ∈ J, m = 1, . . . , 4, k = 0, 1, . . . . The above scheme can be applied only under some stability conditions. In the case of explicit discretization of the viscous terms we have to consider not only (12.luk ) but also the additional stability condition in the form 3 h τk max(µ, k) ≤ CFL, 4 ρ |T |

T ∈ Th ,

(20.luk )

where h is the length of the maximal side in Th and ρ = minT ∈Th ρT , ρT = radius of the largest circle inscribed into T . Concerning the theoretical results we are able to prove the convergence and the error estimates for the combined finite volume – finite element method. These results are obtained for one scalar nonlinear convection – diffusion equation. The convergence was proved by Feistauer, Felcman and Luk´ aˇcov´ a in [5] and by Luk´ aˇcov´ a in [12]. Using the piecewise constant approximate functions in the finite volume step and the piecewise linear approximation in the finite element step it is possible to show that the method is of first order, see Feistauer, Felcman, Luk´ aˇcov´ a and Warnecke [6]. 3.1

Computational Results

Viscous flow through the GAMM channel for γ = 1.4, µ = 1.72·10−5 kg m−1 s−1 , λ = −1.15·10−5 kg m−1 s−1 , k = 2.4·10−2 kg m s−3 K −1 , cv = 721.428 J ·kg·K −1 and the inlet Mach number M = 0.67 was computed by scheme (19.luk ). In Figure 5 Mach number isolines are drawn. Here we can see boundary layer at the walls, shock wave, wake and interaction of the shock with boundary layer.

Figure 5: Mach number isolines of viscous flow

210


Acknowledgements. This research was supported under the Grant No. 201/97/ 0153 of the Czech Grant Agency and the DFG Grant No. Wa 633/6-1 of Deutsche Forschungsgemeinschaft. The numerical experiments for the EG schemes were computed with the code based on the original work of S. Ostkamp. The author gratefully acknowledges these supports.

References [1] P. Arminjou, A. Dervieux, L. Fezoui, H. Steve and B. Stoufflet, Non-oscillatory schemes for multidimensional Euler calculations with unstructured grids, in: J. Ballmann, R. Jeltsch, Eds., Nonlinear Hyperbolic Equations-Theory, Computation Methods, and Applications, Notes on Numerical Fluid Mechanics 24 (Vieweg, Braunschweig, 1989) 1–10. [2] P. N. Childs and K. W. Morton, Characteristic Galerkin methods for scalar conservation laws in one dimension. SIAM J. Numer. Anal., 27 (1990), 553–594. [3] M. Feistauer, Mathematical Methods in Fluid Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 67 (Longman Scientific & Technical, Harlow, 1993). [4] M. Feistauer, J. Felcman and M. Luk´ aˇcov´ a-Medvid’ov´ a, Combined finite element – finite volume solution of compressible flow, Journal of Comput. and Appl. Math., 63 (1995), 179–199. [5] M. Feistauer, J. Felcman, and M. Luk´ aˇcov´ a-Medvid’ov´ a, On the convergence of a combined finite volume – finite element method for nonlinear convection - diffusion problems, Num. Methods for Part. Diff. Eqs., 13 (1997), 1–28. [6] M. Feistauer, J. Felcman, M. Luk´ aˇcov´ a-Medvid’ov´ a and G. Warnecke, Error estimates of a combined finite volume - finite element method for nonlinear convection - diffusion problems, Preprint 27, (1996), University of Magdeburg, submitted to SIAM J. Numer. Anal. [7] M. Fey, Ein echt mehrdimensionales Verfahren zur L¨ osung der Eulergleichungen, Dissertation, ETH Z¨ urich,1993. [8] L. Fezoui and B. Stoufflet, A class of implicit schemes for Euler simulations with unstructured meshes, J. Comp. Phys, 84 (1989), 174–206. [9] M. Luk´ aˇcov´ a-Medvid’ov´ a, K. W. Morton, and G. Warnecke, The evolution Galerkin schemes for hyperbolic systems in two space dimensions, in preparation. [10] M. Luk´ aˇcov´ a-Medvid’ov´ a, K. W. Morton, and G. Warnecke, The second order evolution Galerkin schemes for hyperbolic systems, in preparation. [11] M. Luk´ aˇcov´ a-Medvid’ov´ a, K. W. Morton, and G. Warnecke, On the evolution Galerkin method for solving multidimensional hyperbolic systems, to appear in Proceedings of the Second European Conference on Numerical Mathematics and Advanced Applications, ENUMATH’97, 1998. [12] M. Luk´ aˇcov´ a-Medvid’ov´ a, Combined finite element-finite volume method (convergence analysis), to appear in Comm. Math. Univ. Carolinae 1997. [13] S. Ostkamp, Multidimensional characteristic Galerkin schemes and evolution operators for hyperbolic systems, Math. Meth. Appl. Sci., 20 (1997), 1111–1125.



Structure of Distribution Null-Solutions to Fuchsian Partial Differential Equations Takeshi Mandai Faculty of Engineering, Gifu University, Yanagido 1-1, Gifu 501-11, Japan Email: [email protected] WWW: http://www.gifu-u.ac.jp/~mandai

Abstract. We give a structure theorem for distribution null-solutions to Fuchsian partial differential equations in the sense of M. S. Baouendi and C. Goulaouic. We assume neither that the characteristic exponents are real-analytic nor that the characteristic exponents do not differ by integer. AMS Subject Classification. 35D05, 35A07, 35C20 Keywords. Fuchsian partial differential operator, null-solutions, regular singularity, characteristic exponent (index)

1

Introduction

Consider a Fuchsian partial differential operator with weight ω := m − k in the sense of M. S. Baouendi and C. Goulaouic ([1]) : P = tk ∂tm +

k X j=1

0 ≤ k ≤ m,

aj (x)tk−j ∂tm−j +

X

X

bl,α (t, x)td(l) ∂tl ∂xα ,

l<m |α|≤m−l

d(l) := max{ 0, l − m + k + 1 },

(1.man )

(t, x) ∈ R × Rn .

When m = k (ω = 0), M. Kashiwara and T. Oshima ([5], Definition 4.2) called such an operator “an operator which has regular singularity in a weak sense along Σ0 := { t = 0 }.” In the following two categories (coefficients, data, solutions) : (a) functions real-analytic in (t, x), (b) functions real-analytic in x and of class C ∞ in t, M. S. Baouendi and C. Goulaouic showed the following results: A. The unique solvability of the characteristic Cauchy problems (Cauchy-Kovalevsky type theorem, Nagumo type theorem). This is the preliminary version of the paper.

212

Takeshi Mandai

B. The uniqueness in a wider class of solutions (Holmgren type theorem). H. Tahara ([8], and so on) also showed similar results to A and B in the category of (c) functions of class C ∞ in (t, x), for “Fuchsian hyperbolic operators”, which are, roughly speaking, operators being weakly hyperbolic in t > 0 and satisfying “Levi conditions”. In all cases, it easily follows that there exist no sufficiently smooth nullsolutions. Here, a distribution u near (t, x) = (0, 0) is called a null-solution for P , if P u = 0 near (0, 0) and if (0, 0) ∈ supp u ⊂ Σ+ := { t ≥ 0 }, where supp u denotes the support of u. K. Igari ([4]) showed the existence of a distribution null-solution under a weak additional condition in Case (a). This solution is real-analytic in x. The author ([6]) showed the existence of a distribution null-solution under no additional conditions in Case (a),(b),(c). This solution is also real-analytic (Case (a),(b)) or of class C ∞ (Case (c)) in x. The aim of this study is to make the structure of all solutions belonging to these classes as clear as possible. We consider the case (b) for simplicity. Namely, the coefficients of P are of class C ∞ in t and real-analytic in x. Many problems about Fuchsian partial differential equations have been considered by many authors. Almost all of them, however, have some assumptions on the characteristic exponents (indices), especially the one that the characteristic exponents do not differ by integer. In this study, we assume neither that the characteristic exponents (indices) are real-analytic in x nor that the characteristic exponents do not differ by integer. Notation: (i) The set of all integers (resp. nonnegative integers) is denoted by Z (resp. N). (ii) The real part of a complex number z is denoted by Rez. Ql−1 (iii) Put ϑ := t∂t and (λ)l := j=0 (λ − j) for l ∈ N. (iv) For a domain Ω in Cn , we denote by O(Ω) the set of all holomorphic functions on Ω. For a complete locally convex topological vector space X, we put O(Ω; X) := { f ∈ C 0 (Ω; X) | hφ, f iX ∈ O(Ω) for every φ ∈ X 0 }, where X 0 is the dual space of X and h·, ·iX denotes the duality between X 0 and X. Note that if Ω is a domain in Cl and D is a domain in Rn , then O(Ω; C ∞ (D)) = C ∞ (D; O(Ω)). (v) The space of test functions on an open interval I of R is denoted by D(I) and the space of distributions by D0 (I). The space of rapidly decreasing C ∞ functions is denoted by S(R) and the space of tempered distributions by S 0 (R). The duality between each pair of these spaces is denoted by h·, ·i. More generally, for a complete locally convex topological vector space X,

213

Structure of Distribution Null-Solutions

the space of all X-valued distributions is denoted by D0 (I; X), which is defines as L(D(I), X), where L(X, Y ) denotes the space of all continuous linear mappings from X to Y (See [7]). Note that D0 (I; O(Ω)) = O(Ω; D0 (I)). 0 Put D+ (I; X) := { f ∈ D0 (I; X) | f (t) = 0 in X for t < 0 }. Also, for N ∈ N put N C+ (I; X) : = { f ∈ C N (I; X) | f (t) = 0 in X for t < 0 }, −N 0 0 C+ (I; X) : = { ∂tN (f ) ∈ D+ (I; X) | f ∈ C+ (I; X) } .

(vi) For z ∈ C with Rez > −1, we put z t (t > 0) z , t+ := 0 (t ≤ 0) which is a locally integrable function of t with holomorphic parameter z, 0 (R; O({ z ∈ C | Rez > −1 })). By ∂t (tz+ ) = ztz−1 and hence belongs to D+ + , z this distribution t+ is extended to z ∈ C \ { −1, −2, . . . } meromorphically with simple poles at z = −1, −2, . . . ([3]). (vii) For a commutative ring R, the ring of polynomials of λ with the coefficients belonging to R is denoted by R[λ]. The degree of F (λ) ∈ R[λ] is denoted by degλ F .

2

Review of some Results for Ordinary Differential Equations

In this section, we review some results for ordinary differential equations with a regular singularity at t = 0, which will help us to understand our result. Consider P = tk ∂tm +

k X

aj tk−j ∂tm−j +

j=1

m X

bl (t)td(l) ∂tl ,

l=0

where 0 ≤ k ≤ m, aj ∈ C, and bl ∈ C ∞ (−T0 , T0 ). Namely, P is an operator with a regular singularity at t = 0 having C ∞ coefficients. Put C[P ](λ) = C(λ) := {t−λ+ω P (tλ )}|t=0 = (λ)m +

k X

aj (λ)m−j ∈ C[λ] ,

j=1

which is called the indicial polynomial of P . A root of C(λ) = 0 is called a characteristic exponent (index) of P . We can decompose C as e − ω) , C(λ) = (λ)ω C(λ e ](λ) = C(λ) e where C[P := (λ)k +

k X j=1

aj (λ)k−j ∈ C[λ].

Q e Let C(λ) = dl=1 (λ − λl )rl , where d ∈ N, rl ≥ 1, and (λ1 , . . . , λd ) are distinct.

214 2.1

Takeshi Mandai

Formal Solutions

First, we consider the solutions in the space F of formal series of the form u(t) = tρ

∞ X j=0

tj

qj X

aj,ν (log t)ν ,

ν=0

where ρ ∈ C, qj ∈ N, and aj,ν ∈ C. Note that we have only to consider tω P instead of P in this space, since P u = 0 if and only if tω P u = 0. Thus, we assume ω = 0 (k = m) without loss of generality. Theorem 1. Assume k = m and put KerF P := { u ∈ F | P u = 0 }. For every l with 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl , there exists vl,p = qj ∞ X X λl p−1 λl +j + t aj,ν (log t)ν ∈ KerF P , where aj,ν ∈ C. Further, these t (log t) j=1

ν=0

m(= r1 + · · · + rd ) solutions make a base of KerF P . Especially, there holds dim KerF P = m. Remark 2. (1) If the coefficients of P are holomorphic in a neighborhood of 0, then this formal solution converges in O(R(B \ {0})) for some domain B including 0, where R(V ) denotes the universal covering of V . (2) If rl = 1 for every l and if { λl } do not differ by integer, then we can take qj = 0 for every j ∈ N, that is, the solutions never include the terms with log t. 2.2

0 Solutions in D+

0 Next, we consider the solutions of P u = 0 in D+ (−T0 , T0 ). In this case, we can not reduce to the case where ω = 0, since P u = 0 is not equivalent to tω P u = 0. Put

G(z) = G(z; t) :=

tz+ 0 ∈ D+ (R; O(C)) , Γ (z + 1)

0 (R; O(C)) . G(j) (z) := ∂zj (G(z)) ∈ D+

(2.man ) (3.man )

Note that ∂th G(z) = G(z − h) (h ∈ N) and that G(−d) = ∂td (G(0)) = δ (d−1) (t) for d = 1, 2, . . . . 0 Theorem 3. Put KerD+0 P := { u ∈ D+ (−T0 , T0 ) | P u = 0 }. For every l with 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl , there exists ul,p ∈ KerD+0 P qj ∞ X X aj,ν G(ν) (λl + ω + j). Here, ∼ means satisfying ul,p ∼ G(p−1) (λl + ω) + j=1 ν=0

that for every N ∈ N, there holds ul,p − G(p−1) (λl + ω)−

qj N X X

aj,ν G(ν) (λl + ω +

j=1 ν=0 dReλ e+ω+N (−T0 , T0 ), C+ l

j) ∈ M ≥ a ∈ R.

where dae denotes the smallest integer M satisfying

215


Further, these k (= r1 +· · ·+rd ) solutions make a base of KerD+0 P . Especially, there holds dim KerD+0 P = k. (Cf. Similarly, we have dim KerD0 P = m + k.) Example 4. (1) Consider P = (ϑ − d + 1)∂t = ∂t (ϑ − d), where d ∈ N and d ≥ 1. We have m = 2, k = 1, ω = 1, and C(λ) = λ(λ − d). We have KerF P = KerF (tP ) = Span{1, td}, KerD+0 P = Span{td+ }, KerD0 P = Span{td+ , td , 1}. (2) Consider P = (ϑ + d + 1)∂t = ∂t (ϑ + d), where d ∈ N and d ≥ 1. We have m = 2, k = 1, ω = 1, and C(λ) = λ(λ + d). We have KerF P = KerF (tP ) = Span{1, t−d }, KerD+0 P = Span{δ (d−1) }, KerD0 P = Span{δ (d−1) , 1, (t + i0)−d }. (3) Consider P = (ϑ − d + 1)2 ∂t = ∂t (ϑ − d)2 , where d ∈ N and d ≥ 1. We have m = 3, k = 2, ω = 1, and C(λ) = λ(λ − d)2 . We have KerF P = KerF (tP ) = Span{1, td, td log t}, KerD+0 P = Span{td+ , td+ log t+ }, KerD0 P = Span{td+ , td+ log t+ , 1, td , td log(t + i0)}. (4) Consider P = (ϑ + 1)∂t = ∂t ϑ. We have m = 2, k = 1, ω = 1, and C(λ) = λ2 . KerF P = KerF (tP ) = Span{1, log t}, KerD+0 P = Span{t0+ }, KerD0 P = Span{t0+ , 1, log(t + i0)}. Remark 5. As in §2.1, if rl = 1 for every l and if { λl } do not differ by integer, then we can take qj = 0 for every j. 2.3

Rough Statement of our Result

We want to prove a similar fact for Fuchsian partial differential equations. We shall show that (O0 )k ∼ = (KerD+0 P )(0,0) ,

(4.man )

constructing the isomorphism (invertible linear map) rather concretely, where (. . . )0 and (. . . )(0,0) denote the spaces of all germs. Namely, O0 := indlim0∈Ω⊂Cn O(Ω) and (KerD+0 P )(0,0) := indlimT >0;0∈Ω⊂Cn KerD+0 (−T,T ;O(Ω)) P . We shall state in a little more detail. Consider an operator (1.man ) with aj ∈ O(Ω0 ) and bl,α ∈ C ∞ (−T0 , T0 ; O(Ω0 )), where T0 > 0 and Ω0 is a domain in Cn including 0. Define C(x; λ) := (λ)m +

k X

aj (x)(λ)m−j = {t−λ+ω P (tλ )}|t=0 ,

j=1

which is also called the indicial polynomial of P , and a root λ of C(x; λ) = 0 is called a characteristic exponent of P . The indicial polynomial can be decomposed as e λ − ω) , C(x; λ) = (λ)ω C(x; where e λ) := (λ)k + C(x;

k X j=1

aj (x)(λ)k−j .

216

Takeshi Mandai

e λ) = Qd (λ − λl )rl , where d ∈ N, rl ≥ 1, and (λ1 , . . . , λd ) are distinct. Let C(0; l=1 Rough Statement of our result is the following. Theorem 6. There exist T > 0 and a subdomain Ω of Ω0 including 0 such that for every l with 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl , there exists 0 (−T, T ; O(Ω)) satisfying the a continuous linear map ul,p from O(Ω0 ) to D+ following. For every a ∈ O(Ω0 ), there holds P (ul,p [a]) = 0 and ul,p [a]|x=0 ∼ a(0)G(p−1) (λl + ω) +

qh ∞ X X

ah,ν G(ν) (λl + ω + h)

h=1 ν=0

for some qh ∈ N and ah,ν ∈ C. 0 (−T0 , T0 ; O(Ω0 )) of P u = 0 is represented Conversely, every solution u ∈ D+ P by l,p ul,p [al,p ] for some al,p ∈ O(Ω) (1 ≤ l ≤ d, 1 ≤ p ≤ rl ) in a neighborhood of (0, 0). Further, if there exists (l, p) such that al,p 6≡ 0, then (0, 0) ∈ supp u, that is, u is a null-solution. In the result by T.Mandai ([6]) stated in Introduction, he constructed a e λl + solution corresponding to the solution ul,0 for a root λl satisfying that C(0; j) 6= 0 for j = 1, 2, . . . . e λ) = 0 is not The major difficulty of the proof is the fact that a root of C(x; necessarily holomorphic in x, and this becomes a bigger difficulty when there exist two roots with integer difference, as suggested from the case of ordinary differential equations.

3

Preliminaries

In this section, we give some lemmas and propositions needed for the proof of our result. For each 1 ≤ l ≤ d, we take a domain Dl in C enclosed by a simple closed curve Γl such that the following three conditions hold. (a) λl ∈ Dl (1 ≤ l ≤ d). (b) Dl ∩ Dl0 = ∅ if l 6= l0 . (c) if j ∈ N and if λl0 − j ∈ Dl , then λl0 − j = λl . (This is equivalent to “{ λl0 − j ∈ C | 1 ≤ l0 ≤ d, j ∈ N } ∩ Dl = { λl } for every e λ + j) 6= 0 for every λ ∈ Sd (Dl \ { λl }) l”. Also these are equivalent to “C(0; l=1 and for every j ∈ N”. ) Note that { λl0 − j ∈ C | 1 ≤ l0 ≤ d, j ∈ N } is a discrete set and hence we can take such Γl . There exist a domain Ω in Cn including 0, and monic polynomials El (x; λ) ∈ O(Ω)[λ] (1 ≤ l ≤ d) such that Q e λ) = d El (x; λ), (d) C(x; l=1 (e) El (0; λ) = (λ − λl )rl (1 ≤ l ≤ d),

217


(f) for 1 ≤ l ≤ d, if El (x; λ) = 0 and x ∈ Ω, then λ ∈ Dl , S e λ + j) 6= 0 for every x ∈ Ω, every λ ∈ d Γl , and every j ∈ N. (g) C(x; l=1 Further, by reducing Dl and Ω if necessary, we can take ε ≥ 0 and Ll ∈ Z (1 ≤ l ≤ d) such that e λ) = 0 and if x ∈ Ω, then Reλ − ε 6∈ Z. Further, Dl ⊂ { λ ∈ C | (h) If C(x; Ll + ε < Reλ < Ll + ε + 1 }. Definition 7.

For 1 ≤ l ≤ d, j ∈ N, and for φ ∈ O(Ω × Γl ), put

Hl,j [φ](t, x) :=

1 2πi

Z Γl

φ(x; ζ) G(ζ + j; t) dζ El (x; ζ)

0 ∈ D+ (R; O(Ω)) .

Also for 1 ≤ p ≤ rl , put wl,p (t, x) := Note that wl,p (t, 0) = Remark 8. where cj,k

(rl − p)! Hl,ω [∂ζp El ](t, x) . rl !

(5.man )

1 G(p−1) (λl + ω). (p − 1)!

P (k) (µj + ω), If we fix x = x0 , then wl,p (t, x0 ) = j,k:finite cj,k G e ∈ C, and {µj } are the roots of C(x0 ; λ) = 0.

e ϑ)∂tω for every fixed Proposition 9. (1) {wl,p (·, x)}l,p is a base of KerD+0 C(x; x ∈ Ω. 0 ω e (2) P If u ∈ D+ (R; O(Ω)) satisfies C(x; ϑ)∂t u = 0 for every x ∈ Ω, then u = l,p al,p (x)wl,p (t, x) for some al,p ∈ O(Ω). The point is the holomorphy of al,p . Example 10. Consider P = ϑ2 − x = E1 (x; ϑ), where d = 1(= l), r1 = 2, and ω = 0. We have Z √ √ 1 1 1 1 2ζ G(ζ; t) dζ = {G( x; t) + G(− x; t)}, w1,1 = H1,0 [2ζ] = 2 2 2 2πi Γ1 ζ − x 2 √ √ Z G( x; t) − G(− x; t) 1 1 1 2 √ G(ζ; t) dζ = . w1,2 = H1,0 [2] = 2 2 2πi Γ1 ζ 2 − x 2 x Proposition 11. (1) ∂th Hl,j [φ] = Hl,j−h [φ]. (2) th Hl,j [φ] = Hl,j+h [(ζ + j + h)h φ]. (3) For F (x; λ) ∈ O(Ω)[λ], there holds F (x; ϑ)Hl,j [φ] = Hl,j [F (x; ζ + j)φ]. (4) ∂xν Hl,j [φ] = Hl,j [Lν (φ)], where Lν (φ)(x; ζ) := (∂xν φ)(x; ζ) − Proposition 12.

(∂xν El )(x; ζ) φ(x; ζ) . El (x; ζ)

j+Ll Hl,j [φ] ∈ C+ (R; O(Ω)).

218

Takeshi Mandai

For 1 ≤ l ≤ d and 1 ≤ p ≤ rl , we can construct an asymptotic solution of P u = 0 in the form of u = a(x)wl,p (t, x) +

∞ X

Hl,ω+h [Sh (a)](t, x) ,

h=1

where Sh = Sl,p,h is a continuous linear map from O(Ω) to O(Ω × Γl ) of the form X Sh (a)(x; ζ) = sh,α (x; ζ)∂xα a(x) , |α|≤mh

where

1 × O(Ω × Dl ) , mh e j=0 C(x; ζ + j)

sh,α = sl,p,h,α ∈ Qh

for some mh ∈ N. Further, there exists qh ∈ N and ah,ν ∈ C (h ≥ 1; 0 ≤ ν ≤ qh ) such that ∞

u(t, 0) ∼ a(0)

XX 1 G(p−1) (λl + ω) + ah,ν G(ν) (λl + ω + h) . (p − 1)! ν=0 qh

h=1

4

Detailed Statement of our Result

Now, we can state our result in a full detail. Let Ω be a subdomain of Ω0 including 0 and T ∈ (0, T0 ). Theorem 13. There exist T 0 ∈ (0, T ) and a subdomain Ω 0 of Ω including 0 such that for every l with 1 ≤ l ≤ d and for every p with 1 ≤ p ≤ rl , the following holds: There exists a continuous linear map ul,p from O(Ω) to Ll +ω (−T 0 , T 0 ; O(Ω 0 )) such that for every a ∈ O(Ω), there holds C+ (i) P (ul,p [a]) = 0. P∞ (ii) ul,p [a](t, x) ∼ a(x)wl,p (t, x) + h=1 Hl,ω+h [Sh (a)](t, x), 0 Theorem 14. If u ∈ D+ (−T, T ; O(Ω)) satisfies P u = 0, then there exists a 0 of Ω including 0 and there exists a unique al,p ∈ O(Ω 0 ) such that subdomain Ω P u = l,p ul,p [al,p ] in a neighborhood of (0, 0). Further, if there exists (l, p) such that al,p 6≡ 0, then (0, 0) ∈ supp u, that is, u is a distribution null-solution for P.

These two theorems imply that (O0 )k ∼ = (KerD+0 P )(0,0) , as we already stated. m+k ∼ We can also show that (O0 ) = (KerD0 P )(0,0) , similarly. We can prove Theorem 13 by realizing the asymptotic solution constructed in the previous section. We shall give a sketch of a proof of Theorem 14 in the next section.

219


5

Sketch of the Proof of Theorem 14

First, we give a sketch of the uniqueness of al,p . e λ) = 0 (x ∈ Ω), then Reλ − ε 6∈ Z. We We have taken ε ≥ 0 such that if C(x; have also taken Ll ∈ Z such that if x ∈ Ω and if El (x; λ) = 0, then Ll + ε < Reλ < Ll + ε + 1. For L ∈ Z, we put U 0  N ϑs ∂t|L| (tε × C+ (−T, T ; X)) (L ≤ 0) s=0 (N ) . WL (−T, T ; X) := U N  s L+ε 0 × C+ (−T, T ; X)) (L ≥ 0) s=0 ϑ (t

Definition 15.

Note that (N )

(N )

WL (−T, T ; O(Ω)) ⊂ WL−1 (−T, T ; O(Ω)) and (N )

(N +1)

WL (−T, T ; O(Ω)) ⊂ WL

(−T, T ; O(Ω)).

Take χ(t) ∈ D(−T, T ) with χ(t) = 1 near t = 0. Then, we have the following. (0)

Hl,j [φ] ∈ WLl +j , if v ∈

(N ) WL ,

then hv, χ(t)e

−t/ρ

it = o(ρ

(6.man ) L+ε+1

) (ρ → +0) .

If we fix an arbitrary x, then there exists aj,k ∈ C such that P hwl,p , χ(t)e−t/ρ it = j,k:finite aj,k ρµj +ω+1 (log ρ)k + o(ρ∞ ) ( = o(ρLl +ω+ε+1 )) (ρ → +0) ,

(7.man )

(8.man )

e λ) = 0, since hG(ν) (λ), e−t/ρ it = ρλ+1 (log ρ)ν . where { µj } are the roots of C(x; P From these, we can show that if l,p ul,p [al,p ] = 0, then al,p = 0 for every (l, p). Next, we give a sketch of the existence of al,p for a given solution u. Proposition 16. if Te > T . (ii) t×

0 (−Te, Te; O(Ω)) ⊂ (i) D+

(N ) WL (−T, T ; O(Ω))

( ⊂ (

(N ) ∂t (WL (−T, T ; O(Ω))) (N )

⊂

S

(0) L∈Z WL (−T, T ; O(Ω)),

(N +1)

WL+1 (−T, T ; O(Ω)) (L ≤ −1) , (N ) WL+1 (−T, T ; O(Ω)) (L ≥ 0) (N )

WL−1 (−T, T ; O(Ω)) (L ≤ 0) , (N +1) WL−1 (−T, T ; O(Ω)) (L ≥ 1) (N +1)

ϑ(WL (−T, T ; O(Ω))) ⊂ WL

(−T, T ; O(Ω)).

220

Takeshi Mandai

(iii) For sufficiently large L, there holds (N )

KerD+0 (−T,T ;O(Ω)) P ∩ WL (−T, T ; O(Ω)) = {0} for every N ∈ N. (N ) (N ) (iv) For every g ∈ WL (−T, T ; O(Ω)), there exists v ∈ WL+ω (−T, T ; O(Ω)) e ϑ)∂tω v = g. such that C(x; (0) (v) For 1 ≤ l ≤ d and 1 ≤ p ≤ rl , there holds wl,p ∈ WLl +ω (−T, T ; O(Ω)). Further, for every L ∈ Z and every N ∈ N, there holds e ϑ)∂ ω ∩ W KerD+0 (−T,T ;O(Ω)) C(x; t L (−T, T ; O(Ω)) (N )

= Span{wl,p | 1 ≤ l ≤ d, Ll + ω ≥ L, 1 ≤ p ≤ rl } . From these, we can show the existence of al,p as follows. 0 (−T, T ; O(Ω)) and P u = 0. By (i) and by reducing T , there Let u ∈ D+ (0) e ϑ)∂tω + R, we exists L ∈ Z such that u ∈ WL (−T, T ; O(Ω)). Putting P = C(x; (m) e ϑ)∂ ω u = −Ru ∈ W have C(x; t L−ω+1 (−T, T ; O(Ω)) by (ii). By (iv), there exists (m) e ϑ)∂tω . Since (−T, T ; O(Ω)) such that u − v ∈ KerD0 (−T,T ;O(Ω)) C(x; v ∈ W L+1

+

(m)

u − v ∈ WL (−T, T ; O(Ω)), there exists al,p [0] ∈ O(Ω) (1 ≤ l ≤ d, Ll + ω ≥ L, 1 ≤ p ≤ rl ) such that X u−v = al,p [0]wl,p , 1≤l≤d, Ll +ω≥L, 1≤p≤rl

by (v). Put u[1] := u −

P

l,p ul,p [al,p [0]], (m) WL+1 (−T, T ; O(Ω)). P

show u[1] ∈ Similarly, we get u[2] := u[1] − (2m) WL+2 (−T, T ; O(Ω)),

then we have P (u[1]) = 0 and we can

l,p

ul,p [al,p [1]] ∈ KerD+0 (−T,T ;O(Ω)) P ∩

for some al,p [1] ∈ O(Ω) by reducing Ω and T . By repeat(N m)

ing this argument, we get u[N ] ∈ KerD+0 (−T,T ;O(Ω)) P ∩ WL+N (−T, T ; O(Ω)) P that can be written as u[N ] = u − l,p ul,p [al,p ] for somePal,p ∈ O(Ω). By (iii), we get u[N ] = 0, and hence u can be written as u = l,p ul,p [al,p ] for some al,p ∈ O(Ω). The research was supported in part by Grant-in-Aid for Scientific Research (Nos.07640191, 08640189), Ministry of Education, Science and Culture (Japan).

References [1] M. S. Baouendi and C. Goulaouic, Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math., 26 (1973), 455–475. [2] A. Bove, J. E. Lewis, and C. Parenti, Structure properties of solutions of some Fuchsian hyperbolic equations, Math. Ann., 273 (1986), 553–571.


221

[3] I. M. Gel’fand and G. E. Shilov, Generalized functions, Volume 1 : Properties and operations, Academic Press, 1964, Transl. by E. Saletan. [4] K. Igari, Non-unicité dans le problème de Cauchy caractéristique — cas de type de Fuchs, J. Math. Kyoto Univ., 25 (1985), 341–355. [5] M. Kashiwara and T. Oshima, Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math. (2), 106 (1977), 145–200. [6] T. Mandai, Existence of distribution null-solutions for every Fuchsian partial differential operator, J. Math. Sci., Univ. Tokyo, 5 (1998), 1–18. [7] L. Schwartz, Théorie des distributions a ` valeurs vectorielles, Ann. Inst. Fourier (Grenoble), 7 (1957), 1–141. [8] H. Tahara, Singular hyperbolic systems, III. On the Cauchy problem for Fuchsian hyperbolic partial differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 465–507.



A Functional Differential Equation in Banach Spaces Nasr Mostafa Faculty of Mathematics and Computer Science A. Mickiewicz University, Matejki 48/49 60–769 Pozna´ n, Poland Email: [email protected] Permanent address: Department of Mathematics Faculty of Science, Suez Canal University Ismailia, Egypt

Abstract. In this paper we prove the existence of pseudo-solution and weak solution for the Cauchy problem x0 = F x, x(0) = x0 , t ∈ [0, a]. AMS Subject Classification. 34G20

Keywords. Functional differential equation, existence theorem, weak solution, pseudo-solution

The study of the Cauchy problem for differential and functional differential equations in a Banach space relative to the strong topology has attracted much attention in recent years. However a similar study relative to the weak topology was studied by many authors, for example, Szep [11], Mitchell and Smith [9], Szufla [12], Kubiaczyk [6,7], Kubiaczyk and Szufla [8], Cicho´ n [1], Cicho´ n and Kubiaczyk [2], and others. Let E be a Banach space, E ∗ the dual space. We set Bb (x0 ) = {x ∈ E : kx−xo k ≤ b}, (b > 0). We denote by C(I, E) the space of all continuous function from I to E, and by (C(I, E), w) the space C(I, E) with the weak topology. Put e = {x ∈ C(J, E) : x(J) ⊂ Bb (xo ), kx(t) − x(s)k ≤ M |t − s|, for t, s ∈ J} , B e is nonempty, closed, bounded, convex and equicontinuous, where note that B b J = [0, h], h = min a, M and M > 0 is a constant. We deal with the Cauchy problem: x0 = F x,

x(0) = x0 ,

t ∈ I = [0, a],

(1.mos )

e into P (I, E) in the case of F being an bounded operator of Volterra type from B (the space of all Pettis integrable functions on I). Let us introduce the following definitions. This is the final form of the paper.

224

Nasr Mostafa

e and for any Definition 1. F is said to be of Volterra type if for x1 , x2 ∈ B so > 0 the equality x1 (t) = x2 (t) for t < so implies (F x1 )(t) = (F x2 )(t) for t ≤ so . Now fix x∗ ∈ E ∗ , and consider (x∗ x)0 (t) = x∗ ((F x)(t)),

t ∈ I.

(10 .mos )

Definition 2. A function x : I −→ E is said to be a pseudo-solution of the Cauchy problem (1.mos ) if it satisfies the following conditions: (i) x(·) is absolutely continuous, (ii) x(0) = xo , (iii) for each x∗ ∈ E ∗ there exists a negligible set A(x∗ ) (i.e., mes (A(x∗ )) = 0), such that for each t 6∈ A(x∗ ), x∗ (x0 (t)) = x∗ ((F x)(t)) . Here 0 denotes a pseudoderivative (see Pettis [10]). In other words, by a pseudo-solution of (1.mos ) we will mean an absolutely continuous ) a.e. for each x∗ ∈ E ∗ . function x(·), with x(0) = xo , satisfying (10 .mos Definition 3. A function r : [0, ∞) −→ [0, ∞) is said to be a Kamke function if it satisfies the following conditions: (i) r(0) = 0, (ii) u(t) ≡ 0 is the unique solution of the integral equation Z t r(z(s))ds , t ∈ I . z(t) = 0

Lemma 4 ([9]). Let H ⊂ C(I, E) be a family of strongly equicontinuous functions. Then βc (H) = sup β(H(t)) = β(H(I)) , t∈I

where βc (H) denote the measure of weak noncompactness in C(I, E) and the function t → β(H(t)) is continuous. Now suppose that: (∗) For each strongly absolutely continuous function x : J :−→ E, (F x)(·) is Pettis integrable, F (·) is weakly-weakly sequentially continuous, then the existence of a pseudo-solution of (1.mos ) is equivalent to the existence of a solution for Z t x(t) = xo + (F x)(s)ds , (2.mos ) 0

where the integral is in the sense of Pettis (see [10]).

225

A Functional Differential Equation in Banach Spaces

e Theorem 5. Let F be a bounded continuous operator of Volterra type from B into P (I, E) and under the assumption (∗) and β

[ e ≤ r(β(X)) e , {(F x)[J] : x ∈ X}

(3.mos )

e of B, e where r is a non-decreasing Kamke function and holds for every subset X β is the measure of weak noncompactness. Then the set S of all pseudo-solutions of the Cauchy problem (1.mos ) on J is non-empty and compact in (C(J, E), w). Proof. Put

Z

t

T u(t) = xo +

F u(s)ds

,

t ∈ I,

e, u∈B

0

where the integral is in the sense of Pettis. e into B. e By our assumptions the operator T is well defined and maps B Using Lebesgue’s dominated convergence theorem for the Pettis integral (see [4]), we deduce that T is weakly sequentially continuous. e We will prove that V Suppose that V = Conv({x} ∪ T (V )) for some V ⊂ B. is relatively weakly compact, thus Theorem 1 in [7] is satisfied. e and Lemma 4 it follows that the function v : t → From the definition of B β(V (t)) is continuous on J. For fixed t ∈ J, divide the interval [0, t) into m parts: 0 = to < t1 < · · · < tm = t, where

ti = it/m ,

i = 0, 1, 2, . . . , m .

Put V ([ti−1 , ti ]) = {u(s) = u ∈ V,

ti−1 ≤ s ≤ ti } .

By Lemma 4 and the continuity of v there is si ∈ [ti−1 , ti ] such that β(V ([ti−1 , ti ])) = sup{β(V (s)) : ti−1 ≤ s ≤ ti } = v(si ) . On the other hand, by the mean value theorem we obtain T u(t) = xo +

m−1 X Z ti+1 i=0

F u(s)ds ∈ xo +

ti

m−1 X

(ti+1 − ti )ConvF u([ti , ti+1 ])

i=0

for each u ∈ V . Therefore T V (t) ⊂ xo +

m−1 X

(ti+1 − ti )ConvF ([V ])([ti , ti+1 ]) .

i=0

(4.mos )

226

Nasr Mostafa

By (4.mos ) and the corresponding properties of β it follows that β(T (V )(t)) ≤ β(xo +

m−1 X

(ti+1 − ti )ConvF ([V ])([ti , ti+1 ])) ≤

i=0

≤ ≤ ≤

m−1 X

(ti+1 − ti )β(F (V )([ti , ti+1 ])) ≤

i=0 m−1 X

(ti+1 − ti )r(β(V [ti , ti+1 ])) ≤

i=0 m−1 X

(ti+1 − ti )r(β(V (si )) , for some si ∈ [ti , ti+1 ]

i+0

=

m−1 X

(ti+1 − ti )r(v(si )) .

i=0

By letting m → ∞, we have Z

t

β(T (V (t)) ≤

r(v(s))ds .

(5.mos )

0

), Since V = Conv({x} ∪ T (V )) we have β(V (t)) ≤ β(T (V (t))) and in view of (5.mos Rt it follows that v(t) ≤ 0 r(v(s))ds for t ∈ J. Hence applying now a theorem on differential inequalities (cf. [5]) we get v(t) = β(v(t)) = 0. By Lemma 4, V is relatively weakly compact. e which is actually a pseudoSo, by Theorem 1 in [7], T has a fixed point in B solution of (1.mos ). As S = T (S), by repeating the above argument with V = S we can show that S is relatively compact in (C(J, E), w). ω Since T is weakly sequentially continuous on S(J) , S is weakly sequentially closed. By Eberlein-Smulian Theorem [3], S is weakly compact. Remark 6. One can easily prove that the integral of a weakly continuous function is weakly differentiable with respect to the right endpoint of the integration interval and its derivative equals the integral at the same point (see [6], Lemma 2.3). In this case a pseudo-solution is, actually, a weak solution. Moreover, in some classes of spaces our pseudo-solutions are also strong C-solutions (in separable Banach spaces, for instance).

References [1] Cicho´ n, M., Weak solutions of differential equations in Banach spaces. Discuss. Math. — Diff. Inclus. 15 (1995), 5–14.

A Functional Differential Equation in Banach Spaces

227

[2] Cicho´ n, M. and Kubiaczyk, I., On the set of solutions of the Cauchy problem in Banach spaces. Arch. Math. 63 (1994), 251–257. [3] Edwards, R. E., Functional Analysis. Holt Rinehart and Winston, New York 1965. [4] Geitz, R. F., Pettis integration . Proc. Amer. Math. Soc. 82 (1981), 81–86. [5] Hartman, P., Ordinary Differential Equations. New York 1964. [6] Kubiaczyk, I., A functional differential equation in Banach spaces. Demonstratio Math. 15 (1982), 113–129. [7] Kubiaczyk, I., On a fixed point theorem for weakly sequentially continuous mappings. Discuss. Math. — Diff. Inclus. 15 (1995), 15–20. [8] Kubiaczyk, I. and Szufla, S., Kneser’s theorem for weak solutions of ordinary differential equations in Banach spaces. Publ. Inst. Math. 32 (1982), 99–103. [9] Mitchell, A. R. and Smith, C., An existence theorem for weak solutions of differential equations in Banach spaces. pp. 387–404 in, Nonlinear Equations in Abstract Spaces, ed. by V. Lakshmikantham 1978. [10] Pettis, B. J., On integration in vector spaces. Trans. Amer. Math. Soc. 44 (1938), 277–304. [11] Szep, A., Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Studia Sci. Math. Hungar. 6 (1971), 197–203. [12] Szufla, S., Kneser’s theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Ibid. 26 (1978), 407–413.



On the Limit Cycle of the van der Pol Equation Kenzi Odani Department of Mathematics, Aichi University of Education, Igaya-cho, Kariya-shi, Aichi 448-8542, Japan. Email: [email protected] WWW: http://www.auemath.aichi-edu.ac.jp/

Abstract. In the paper, we estimate the amplitude (maximal x-value) of the limit cycle of the van der Pol equation x˙ = y − µ(x3 /3 − x),

y˙ = −x

from above by ρ(µ) < 2.3439 for every µ = 6 0. The result is an improvement of the author’s previous estimation ρ(µ) < 2.5425. AMS Subject Classification. 34C05, 58F21 Keywords. Van der Pol equation, limit cycle, amplitude

1

Introduction

We are interested in the limit cycle (isolated periodic orbit) of the Liénard equation: x˙ = y − F (x),

y˙ = −g(x).

(L.oda )

The following is our result. Theorem A. Suppose that Liénard equation satisfies the following conditions: (1) F, g are of class C 1 and odd; (2) g(x) has the same sign as x; (3) F has a positive zero β such that F (x) < 0 on (0, β) and > 0 on (β, ∞); (4) there are two piecewise differentiable, continuous mappings φ, ψ : [0, β] → [β, ∞) such that (ii) −φ0 (x)f (φ(x)) ≥ −f (x), (i) −φ0 (x)g(φ(x))F (φ(x)) ≥ −g(x)F (x), 0 (iv) ψ 0 (x)f (ψ(x)) ≥ f (x), (iii) ψ (x)g(ψ(x))F (ψ(x)) ≥ −g(x)F (x), 0 (vi) φ(0) ≤ ψ(β), (v) ψ (x)g(ψ(x)) ≤ g(x), where f = F 0 . Then it has a periodic orbit in the strip |x| < ψ(β). The above theorem is effective to estimate the amplitude (maximal x-value) of the limit cycle of the van der Pol equation: x˙ = y − µ(x3 /3 − x),

y˙ = −x.

(vdP.oda )

We know that the van der Pol equation has a unique limit cycle for every µ 6= 0; see [O] for example. The following is an application of Theorem A. This is the preliminary version of the paper.

230

Kenzi Odani

Theorem B. The amplitude ρ(µ) of the limit cycle of the van der Pol equation is estimated by ρ(µ) < 2.3439 for every µ 6= 0. The upper bound 2.3439 is better than previous results, namely, 2.8025 of Alsholm [A] and 2.5425 of the author [O]. Due to a computer experiment, we expect that the amplitude ρ(µ) < 2.0235 for every µ 6= 0. So Theorem B is not a sharp result in comparison with it. We give the result of the experiment in Section 4.

2

Proof of Theorem A

We consider an orbit γ which starts from a point on the left half of the curve y = F (x) and reaches to the right half of it. Then we can regard the y-coordinate of γ as a function of x, that is, y = y(x). In the proof of Theorem A, we use the following notation: v1 (x) = y(x) − F (x),

v2 (x) = y(−x) + F (x).

(1.oda )

Then the functions v1 , v2 must satisfy the following differential equations: g(x) dv1 =− − f (x), dx v1

g(x) dv2 =− + f (x). dx v2

(2.oda )

By the definition of γ, we know that v1 (x), v2 (x) ≥ 0 on [0, ψ(β)]. Proof (of Theorem A). We assume that the orbit γ starts from the curve y = F (x) at x = −ψ(β), that is, v2 (ψ(β)) = 0. We want to prove that the orbit γ gets across the curve at the left-hand side of x = ψ(β). To prove it by a contradiction, we assume that v1 (x) is defined on [0, ψ(β)]. By using (i), we know that φ0 (x) < 0 on (0, β). So by using (ii), we calculate as follows: d v1 (x) − v1 (φ(x)) dx φ0 (x)g(φ(x)) g(x) + − f (x) + φ0 (x)f (φ(x)) ≤ 0. (3.oda ) =− v1 (x) v1 (φ(x)) By integrating it on [x, β], we obtain that v1 (x) − v1 (φ(x)) ≥ v1 (β) − v1 (φ(β)) = y(β) − y(φ(β)) + F (φ(β)) > 0

(4.oda )

because y(x) is strictly decreasing on [−φ(β), φ(β)]. On the other hand, by using (iv), (v), we calculate as follows: d v2 (x) − v2 (ψ(x)) dx ψ 0 (x)g(ψ(x)) g(x) + + f (x) − ψ 0 (x)f (ψ(x)) =− v2 (x) v2 (ψ(x)) g(x) v2 (x) − v2 (ψ(x)) . ≤ v2 (x)v2 (ψ(x))

(5.oda )

231

On the Limit Cycle of the van der Pol Equation

By integrating it on [x, β], we obtain that Z v2 (x) − v2 (ψ(x)) ≥ v2 (β) − v2 (ψ(β)) exp −

β

x

g(u)du > 0. v2 (u)v2 (ψ(u))

We can easily confirm the following equality: Z x g(x)F (x) d 1 2 y(x) + . g(u)du = − dx 2 y(x) − F (x) 0

(6.oda )

(7.oda )

By integrating it on [0, ψ(β)], we obtain that Z ψ(β) Z ψ(β) 1 g(x)F (x) g(x)F (x) 2 2 y(ψ(β)) − y(−ψ(β)) = − dx − dx. (8.oda ) 2 v (x) v2 (x) 1 0 0 By using (i), (4.oda ), we calculate the first term of (8.oda ) as follows: Z ≤− 0

β

Z φ(0) g(x)F (x) g(x)F (x) dx − dx v1 (x) v1 (x) φ(β) Z β 0 Z β g(x)F (x) φ (x)g(φ(x))F (φ(x)) dx + dx < 0. (9.oda ) =− v (x) v1 (φ(x)) 1 0 0

On the other hand, by using (iii), (6.oda ), we calculate the second term of (8.oda ) as follows: Z β Z ψ(β) g(x)F (x) g(x)F (x) ≤− dx − dx v (x) v2 (x) 2 0 ψ(0) Z β 0 Z β g(x)F (x) ψ (x)g(ψ(x))F (ψ(x)) dx − dx < 0. (10.oda ) =− v (x) v2 (ψ(x)) 2 0 0 By combining (8.oda ), (9.oda ), (10.oda ), we obtain that y(ψ(β))2 < y(−ψ(β))2 = F (ψ(β))2 .

(11.oda )

It is in contradiction with v1 (ψ(β)) ≥ 0. So the function v1 (x) does not defined on [0, ψ(β)], that is, the orbit γ gets across the curve y = F (x) at the left-hand side of x = ψ(β). Thus the orbit γ winds toward inside. On the other hand, every orbit near the origin winds toward outside. Hence the equation has a periodic orbit in the strip |x| < ψ(β). t u

3

Proof of Theorem B

In the proof of Theorem B, we use the following functions: 1 f (x) 3(x2 − 1) f (x) =µ x− , Q(x) := = 4 P (x) := . g(x) x g(x)F (x) x − 3x2

(12.oda )

By checking the derivatives, we know that the function P is √ strictly increasing on √ (0, ∞) and that the function Q is strictly decreasing on (0, 3) and on ( 3, ∞).

232

Kenzi Odani

Proof (of Theorem B). We can assume without loss of generality that µ > 0 because the transformation (x, y, t, µ) → (x, −y, −t, −µ) preserves the form of the equation. We first define φ(x) by the following algebraic equation: Z

φ

uF (u)du = x

µ 5 (φ − 5φ3 − x5 + 5x3 ) = 0. 15

(13.oda )

√ √ Of course, φ( 3) = 3. By differentiating it, we obtain that (14.oda ) − φ0 (x)φ(x)F (φ(x)) + xF (x) = 0. √ Since φ0 (x) < 0 on [0, 3], the mapping φ is strictly decreasing (orientation reversing) on it. Since the function √ −Q(φ(x)) + Q(x) is strictly decreasing on √ (0, 3), it has a unique zero ξ1 in (0, 3). A computer experiment indicates that ξ1 ≈ 0.6941, ξ2 := φ(ξ1 ) ≈ 2.2043. By substituting φ0 (x) from (14.oda ) and by the definition of ξ1 , we obtain that (15.oda ) φ0 (x)f (φ(x)) − f (x) = −xF (x) −Q(φ(x)) + Q(x) ≤ 0 √ ) does not hold on [0, ξ1 ), the definition (13.oda ) is valid only on [ξ1 , √ 3]. Since (15.oda on [ξ1 , 3]. On the interval [0, ξ1 ), we define φ(x) by the following algebraic equation: Z

Z

φ

ξ1

f (u)du +

f (u)du x

ξ2

=

µ 3 (φ − 3φ − x3 + 3x − ξ23 + 3ξ2 + ξ13 − 3ξ1 ) = 0. 3

(16.oda )

By differentiating it, we obtain that φ0 (x)f (φ(x)) − f (x) = 0

(17.oda )

) and by the definition of ξ1 , we obtain on [0, ξ1 ). By substituting φ0 (x) from (17.oda that xF (x) −Q(φ(x)) + Q(x) ≥ 0 (18.oda ) −φ0 (x)φ(x)F (φ(x)) + xF (x) = − Q(φ(x)) on [0, ξ1 ). Hence the mapping φ satisfies (i), (ii) of Theorem A. We first define ψ(x) by the following algebraic equation: Z

Z

ψ

x

uF (u)du + θ2

uF (u)du = θ1

√ µ 5 (ψ − 5ψ 3 + x5 − 5x3 + 4 6) = 0, 15

(19.oda )

p √ √ √ where θ1 , θ2 := 2 ∓ 3 = ( 3 ∓ 1)/ 2. Of course, ψ(θ1 ) = θ2 . By differentiating it, we obtain that ψ 0 (x)ψ(x)F (ψ(x)) + xF (x) = 0.

(20.oda )


233

√ Since ψ 0 (x) > 0 on [0, 3], the mapping ψ is strictly increasing (orientation preserving) on it. Since the function Q(ψ(x)) + Q(x) is strictly decreasing on √ √ (0, 3), it has a unique zero η1 in (0, 3). A computer experiment indicates that η1 ≈ 1.3784, η2 := ψ(η1 ) ≈ 2.2006. By substituting ψ 0 (x) from (20.oda ) and by the definition of η1 , we obtain that (21.oda ) ψ 0 (x)f (ψ(x)) − f (x) = −xF (x) Q(ψ(x)) + Q(x) ≥ 0 √ on [0, η1 ]. Since (21.oda ) does not hold on (η1 , 3], the definition (19.oda ) is valid only on [0, η1 ]. √ On the interval (η1 , 3], we define ψ(x) by the following algebraic equation: Z

Z

ψ

x

f (u)du − η2

f (u)du η1

=

µ 3 (ψ − 3ψ − x3 + 3x − η23 + 3η2 + η13 − 3η1 ) = 0. 3

(22.oda )

By differentiating it, we obtain that √

ψ 0 (x)f (ψ(x)) − f (x) = 0

(23.oda )

) and by the definition of η1 , we on (η1 , 3]. By substituting ψ 0 (x) from (23.oda obtain that xF (x) Q(ψ(x)) + Q(x) ≥ 0 (24.oda ) ψ 0 (x)ψ(x)F (ψ(x)) + xF (x) = Q(ψ(x)) √ A. on (η1 , 3]. Hence the mapping ψ satisfies (iii), (iv) pof Theorem √ To prove (v), we prepare the mapping χ(x) := x2 + 2 3 . By the proof of Example 2 of [O], we obtain that F (χ(x)) ≥ −F (x)

(25.oda )

χ0 (x)χ(x)F (χ(x)) ≥ −xF (x) = ψ 0 (x)ψ(x)F (ψ(x)).

(26.oda )

√ ) and (25.oda ), we obtain that on [0, 3]. By combining (20.oda

By integrating it on [x, θ1 ], we obtain that Z

ψ(x)

uF (u)du ≥ 0

(27.oda )

χ(x)

√ on [0, θ1 ]. Since uF (u) > 0 on ( 3, ∞), we obtain that ψ(x) ≥ χ(x) on [0, θ1 ]. So we obtain that F (ψ(x)) ≥ F (χ(x)) ≥ −F (x) on [0, θ1 ].

(28.oda )

To prove the same inequality as (28.oda ) on (θ1 , η1 ], we consider the minimum of the function F (ψ) + F (x) under the restriction (19.oda ). We denote by ψ0 , x0 the

234

Kenzi Odani

variables which attain the minimum. To find the minimum, we consider the following function: Z Λ(ψ, x) = F (ψ) + F (x) − λ

Z

ψ

x

uF (u)du +

θ2

uF (u)du .

(29.oda )

θ1

By the Lagrange’s method of indeterminate coefficients, we obtain that Λψ.(ψ0 , x0 ) = f (ψ0 ) − λψ0 F (ψ0 ) = 0, Λx (ψ0 , x0 ) = f (x0 ) − λx0 F (x0 ) = 0.

(30.oda ) (31.oda )

By the first equality, we obtain that λ > 0. So we obtain that F (ψ(x)) + F (x) ≥ F (ψ0 ) + F (x0 ) = (1/λ) P (ψ0 ) + P (x0 ) ≥ (1/λ) P (θ2 ) + P (θ1 ) = 0

(32.oda )

on (θ1 , η1 ]. By substituting ψ 0 (x) from (20.oda ) and by using (28.oda ) and (32.oda ), we obtain that x F (ψ(x)) + F (x) ≥ 0 (33.oda ) x − ψ 0 (x)ψ(x) = F (ψ(x)) on [0, η1 ]. On the other hand, by substituting ψ 0 (x) from (23.oda ), we obtain that x − ψ 0 (x)ψ(x) =

x P (ψ(x)) − P (x) ≥ 0 P (ψ(x))

(34.oda )

√ on (η1 , 3]. Hence the mappings φ, ψ satisfy all the conditions of Theorem A except (vi). √ A computer experiment indicates that φ(0) ≈ 2.3439, ψ( 3) ≈ 2.3233. So we must replace ψ by the following mapping: p ˆ (35.oda ) ψ(x) := ψ(x)2 − ψ(β)2 + φ(0)2 . ˆ Of course, ψ(β) = φ(0). Moreover, we can calculate as follows: ˆ ψˆ0 (x)ψ(x) = ψ 0 (x)ψ(x) ≤ x, ˆ ˆ ˆ (ψ(x)) = ψ 0 (x)ψ(x)F (ψ(x)) ψˆ0 (x)ψ(x)F ≥ ψ 0 (x)ψ(x)F (ψ(x)) ≥ −xF (x), ˆ ˆ = ψ 0 (x)ψ(x)P (ψ(x)) ≥ ψ 0 (x)ψ(x)P (ψ(x)) ψˆ0 (x)f (ψ(x)) = ψ 0 (x)f (ψ(x)) ≥ f (x). Hence the mappings φ, ψˆ satisfy all the conditions of Theorem A.

(36.oda ) (37.oda ) (38.oda ) t u

235


4

A Conjecture

Since the limit cycle of the van der Pol equation is unique, its amplitude ρ(µ) is a continuous function of the parameter µ 6= 0. In [L], the following facts are proved: ρ(µ) → 2 as µ → 0,

ρ(µ) → 2 as µ → ∞.

(39.oda )

More precisely, it is proved in [H] that ρ(µ) = 2+(7/96)µ2 +O(µ3 ) for sufficiently small µ > 0 and in [C] that ρ(µ) = 2+(0.7793 · · · )µ−4/3 +o(µ−4/3 ) for sufficiently large µ > 0. By a computer experiment, we have the following table. µ ↓0 ρ ↓2 µ 3.3 ρ 2.02342

0.1 2.00010 3.4 2.02341

1.0 2.00862 4.0 2.02296

2.0 2.01989 5.0 2.02151

3.0 2.02330 10 2.01429

3.2 2.02341 ↑∞ ↓2

We calculate the amplitude ρ of the above table by using the Runge-Kutta method with a step size 2−20 . In comparison with the above table, we realize that Theorem B is not a sharp result. So we want to pose the following conjecture. Conjecture. The amplitude ρ(µ) of the limit cycle of the van der Pol equation is estimated by 2 < ρ(µ) < 2.0235 for every µ 6= 0. However, to estimate the amplitude is a very difficult problem. An attempt to estimate the amplitude is done by Giacomini and Neukirch [GN]. Acknowledgement. The author wishes to thank Professors K. Shiraiwa and K. Yamato for reading the manuscript.

References [A] [C]

[GN] [H] [L] [O] [Y] [Z]

P. Alsholm, Existence of limit cycles for generalized Liénard equation, J. Math. Anal. Appl. 171 (1992), 242–255. M. L. Cartwright, Van der Pol’s equation for relaxation oscillation, in “Contributions to the Theory of Non-linear Oscillations II”, S. Lefschetz, ed., Ann. of Math. Studies, vol. 29, Princeton Univ. Press, 1952, pp. 3–18. H. Giacomini and S. Neukirch, On the number of limit cycles of Liénard equation, Physical Review E, to appear. W. T. van Horssen, A perturbation method based on integrating factors, SIAM J. Appl. Math., to appear. S. Lefschetz, “Differential Equations: Geometric Theory”, 2nd Ed., Interscience, 1963; reprint, Dover, New York, 1977. K. Odani, Existence of exactly N periodic solutions for Liénard systems, Funkcialaj Ekvacioj 39 (1996), 217–234. Y.-Q. Ye et al., “Theory of Limit Cycles”, Transl. of Math. Monographs, vol. 66, Amer. Math. Soc., 1986. (Eng. transl.) Z.-F. Zhang et al., “Qualitative Theory of Differential Equations”, Transl. of Math. Monographs, vol. 102, Amer. Math. Soc., 1992. (Eng. transl.)



Lp Solutions of Non-linear Integral Equations Alejandro Om´ on Arancibia1 and Manuel Pinto Jimenez2 1

2

Departamento de Ingenier´ıa Matem´ atica, Universidad de Chile, Casilla 170, correo 3, Santiago, Chile Email: [email protected] Departamento de Matem´ aticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile Email: [email protected]

Abstract. We study nonlinear Volterra integral operators of first and second kind on unbounded domains. We get bounded and Lp solutions on all [0, ∞) as domain with Schauder’s fixed point theorem over unbounded sets. AMS Subject Classification. 45B05, 45D05, 47H15 Keywords. Volterra integral equation of first and second kind, Schauder’s fixed point theorem, Arzela-Ascoli and Fréchet-Kolmogorov compactness theorems

1

Introduction

We wish to find solutions x(t) for the following nonlinear problems of Volterra’s kind: Zt F (t, s, x(s)) ds,

g(t) = 0

t ≥ 0,

(1.omo )

Zt

x(t) = x0 (t) +

k(t, s, x(s)) ds,

t ≥ 0.

(2.omo )

0

General existence results can be found in [2], [3] essentially using standard techniques of functional analysis. On first kind equation (1.omo ), which is very difficult for its implicit character, very few methods have been implemented on its study. In this article for both equations we look for bounded and Lp solutions with 1 ≤ p < +∞, when easily checkable conditions are imposed to functions g, F, x0 and k. The main technique is based in compactness method which do not ensure uniqueness. The uniqueness problem in nonlinear integral equations is very interesting but also not too touched; in [2] and [3] there are some interesting results. For the Lipschitz situation where the Banach contraction theorem is used, there are important results in [5] and [6]. This is the final form of the paper.

238

Alejandro Om´ on and Manuel Pinto

In equation (2.omo ) (of second kind) the results are not hard to extend to the Fredholm integral equation of second kind, namely Z∞ x(t) = x0 (t) +

k(t, s, x(s)) ds

(2’.omo )

0

For our purpose we need a compactness criterion over not bounded subsets of the whole real axis which are given in the first and second lemmas, and are a well known generalization of the Arzela-Ascoli theorem and Fréchet-Kolmogorov theorem [1].

2

Bounded Solutions

Initially, consider the equation (1.omo ). Assuming that g is differentiable and F has partial derivative with respect to the first variable (that will be denoted Ft = ∂F ) we obtain: ∂t ), then differentiating (1.omo Zt

0

g (t) = F (t, t, x(t)) +

Ft (t, s, x(s)) ds.

(3.omo )

0

Let us denote C the Banach space of continuous and bounded functions over [0, ∞), normed by the supremum over all [0, ∞). Now let us define the operator T: C → C, such that given any x in C b T x(t) = e g (t) + Fbx(t) + Kx(t),

t ≥ 0,

(4.omo )

where Zt

0

g (t) = g (t) − F (t, t, 0) − e

Ft (t, s, 0) ds, 0

Fb x(t) = x(t) + F (t, t, 0) − F (t, t, x(t)), Zt b Kx(t) = − (Ft (t, s, x(s)) − Ft (t, s, 0)) ds. 0

With this definition any solution of equation (1.omo ) satisfies the problem T x = x. For this approach we will need the following definition and lemma: Definition 1. Let f : [0, ∞) × [0, ∞) × Cn → Cn . We say that f (t, s, u) is t-locally equicontinuous with respect to s and u if ∀ε > 0 ∃δ > 0 s.t. |t1 − t2 | < δ ⇒ |f (t1 , s, u) − f (t2 , s, u)| < ε, uniformly on s over compact sets and u over bounded sets.

Lp Solutions of Non-linear Integral Equations

239

Lemma 2. Given A ⊂ C bounded, locally equicontinuous and equiconvergent, then A is relatively compact on C. Theorem 3. Let F = F (t, s, u) : [0, ∞) × [0, ∞) × Cn → Cn be continuous on each variable, derivable with respect to the first one ; Ft continuous on each variable and t-locally equicontinuous with respect to s and u. Assume I. a) The functional Fb is equicontinuous over any M ⊂ C bounded. b) sup |F (t, t, 0)| < +∞. t∈[0,∞)

c) There exists a : [0, ∞) → R+ ∪ {0} bounded, continuous, such that a(t) → 0 as t → ∞ and verifying |x + F (t, t, 0) − F (t, t, x)| ≤ a(t)|x|

∀x ∈ Cn , ∀t ∈ [0, ∞).

II. There exists a function L : [0, ∞) × [0, ∞) → R+ ∪ {0} such that Zt L(t, s)ds is continuous, and it goes to zero as t → ∞, a) 0

b) |Ft (t, s, u) − Ft (t, s, 0)| ≤ L(t, s)|u| ∀t, s ∈ [0, ∞), ∀u ∈ Cn . Zt III. a)

|Ft (t, s, 0)| ds < ∞,

sup t∈[0,∞)

0

Zt b)

sup [a(t) + t∈[0,∞)

L(t, s)ds] < 1. 0

IV. g 0 ∈ C and g(0) = 0. Then, there exists a solution x ¯ ∈ C of first kind equation (1.omo ). Proof. To prove the theorem we use a fixed point approach, showing first that the operator T , given by (4.omo ) is well defined in C; let us take any x ∈ C, then by I.c) and II.b): b |T x(t)| ≤ |e g (t)| + |Fbx(t)| + |Kx(t)| Zt ≤ |e g (t)| + [a(t) + L(t, s) ds]kxk∞ . 0

Then T x(·) is bounded. Moreover, T x(·) is continuous for all x ∈ C, indeed: ge(·) Zt 0 |Ft (t, s, 0)|ds are continuous) ; Fbx(·) is continuous is continuous (g , F , and 0

240


b (x, F (·, ·, 0), F (·, ·, ·) are continuous) ; Kx(·) is continuous, because b b 0 )| ≤ |Kx(t) − Kx(t

Zt

|Ft (t, s, x(s)) − Ft (t0 , s, x(s))| ds ≤

0

Zt

Zt

0

|Ft (t, s, 0) − Ft (t , s, 0)| ds +

L(t0 , s)dskxk∞

t0

0

Zt and Ft is t-locally equicontinuous and

L(t0 , s)ds −→0 0. Then, T x(·) is well t→t

t0

defined and continuous. Secondly, using Lemma 2 we prove that T is a compact operator. Let M ⊆ C bounded, i.e., ∀x ∈ M , kxk∞ ≤ R < ∞, we will show that: i) T (M ) is bounded in C ; ii) T (M ) is locally equicontinuous in C ; iii) T (M ) is equiconvergent. i) T (M ) is bounded. As the functions ge(.), a(.) and L(.,.) are bounded, given x ∈ M , we use I.b), I.c), II.b) and III.b), and we get that: b |T x(t)| ≤ |e g (t)| + |Fb x(t)| + |Kx(t)| ≤ Zt L(t, s) ds]R ≤ |e g (t)| + R.

|e g (t)| + [a(t) + 0

ii) T (M ) is equicontinuous. First e g is continuous over [0, ∞) because g’, F (., ., 0) and Ft (., s, 0) are continuous on [0, ∞). Moreover, Fb x is equicontinuous on bounded subsets of C. With these considerations, given [a, b] compact in [0, ∞) and t1 ≤ t2 on [a, b], then the equicontinuity of T (M ) follows easily from Zt2 g (t1 ) − ge(t2 )| + |Fˆ x(t1 ) − Fˆ x(t2 )| + |T x(t1 ) − T x(t2 )| ≤ |e

|Ft (t2 , s, 0)| ds t1

Zt1

|Ft (t1 , s, x(s)) − Ft (t2 , s, x(s))| ds

+ 0 Zt2

|Ft (t2 , s, x(s)) − Ft (t2 , s, 0)| ds

+ t1

Zt2

≤ |e g (t1 ) − ge(t2 )| + |Fˆ x(t1 ) − Fˆ x(t2 )| +

|Ft (t2 , s, 0)| ds t1


241

Zt1

Zt2 |Ft (t1 , s, x(s)) − Ft (t2 , s, x(s))| ds + R

+

L(t2 , s) ds. t1

0

iii) T (M ) is equiconvergent, because Zt L(t, s) ds} −→ 0.

ˆ |T x(t) − e g (t)| ≤ |Fˆ x(t)| + |Kx(t)| ≤ R{a(t) +

t→∞

0

So, by Lemma 2, T (M ) is relatively compact in C. From Schauder’s fixed point theorem ∃¯ x ∈ C such that x¯ = T x ¯. Integrating this last fixed point equation and as g(0) = 0, implies that x ¯ is a solution of equation (1.omo ). t u Now, consider the equation of second kind: Zt k(t, s, x(s)) ds, t ≥ 0,

x(t) = x0 (t) +

(2.omod )

0

where k : [0, ∞) × [0, ∞) × C → C is continuous in s and x. We can formulate now the following n

n

Theorem 4. Assume that the function k : [0, ∞)×[0, ∞)×Cn → Cn is t-locally equicontinuous and satisfies Zt I.

|k(t, s, 0)|ds < ∞.

sup t∈[0,∞)

0

II. There exists L : [0, ∞) × [0, ∞) → R+ ∪ {0} such that: a) ∀(t, s, u) ∈ [0, ∞) × [0, ∞) × Cn with t ≥ s, |k(t, s, u) − k(t, s, 0)| ≤ L(t, s)|u|. Zt L(t, s)ds is continuous and converges to 0 as t → ∞.

b) 0

Zt c)

sup t∈[0,∞)

L(t, s)ds < 1. 0

) has a solution x ¯ ∈ C. Then for all x0 ∈ C, equation (2.omod Proof. Consider the operator T : C → C defined as Zt T x(t) = x0 (t) +

Zt (k(t, s, x(s)) − k(t, s, 0)) ds.

k(t, s, 0)ds + 0

0

Proceeding as in the previous Theorem, from Schauder’s fixed point theorem we obtain that there exists x ¯ ∈ C such that x ¯ = Tx ¯, and then a solution of equation (2.omod ). t u

242


Lp Solutions

3

Now, we will find Lp [0, ∞) solutions to equation (2.omod ) with 1 ≤ p < ∞. For the whole section, q will be the Holder conjugate of p, i.e., p1 + 1q = 1. We will need the next definition and lemma for our result over Lp spaces. Definition 5. The function k(t, s, u) is t-locally Lp equicontinuous if given a, b, c, d ∈ R+ , such that a ≤ b and c ≤ d, then Zb Zd |k(t + h, s, x(s)) − k(t, s, x(s))|q dsdt −→ 0, h→0

a

c

with x on bounded subsets of Lp [0, ∞). Lemma 6. A ⊂ Lp [0, ∞) bounded will be relatively compact if: a) The restriction A|[a,b] where [a, b] ⊂ [0, ∞) is a compact interval, satisfy the Lp -equicontinuity of the translations (Fréchet-Kolmogorov criterion). Z∞ p b) Equiconvergence: there exists u in L [0, ∞) such that |x(s) − u(s)|p ds → 0 as t → ∞ uniformly for x ∈ A.

t

Now, our next result is Theorem 7. Assuming k is t-locally Lp equicontinuous, and I. There exist a function L : [0, ∞) × [0, ∞) → R+ ∪ {0} such that: pq Z∞ Z∞ q a) L (t, s) ds dt < ∞, 0

0

Z∞ Zt b)

q

sup t∈[0,∞)

pq

L (t, s)ds 0

dt < 1,

0

c) ∀(t, s, u) ∈ [0, ∞) × [0, ∞) × Cn with t ≥ s, |k(t, s, u) − k(t, s, 0)| ≤ L(t, s)|u|. Zt |k(t, s, 0)| ds ∈ Lp [0, ∞).

II. 0

) has a solution on Lp [0, ∞). Then, given x0 ∈ Lp [0, ∞), equation (2.omod


243

Proof. Let T : Lp [0, ∞) → Lp [0, ∞) defined by Zt (k(t, s, x(s)) − k(t, s, 0)) ds,

T x(t) = x e0 (t) + 0

where Zt x e0 (t) = x0 (t) +

k(t, s, 0) ds. 0

Clearly x e0 is in L , and there exists a constant c ≥ 0, such that p

n Z t p o p x0 (t)| + |k(t, s, x(s)) − k(t, s, 0)|ds |T x(t)| ≤ c |e p

0

and using hypothesis I.b) and Holder’s inequality we get that o n Z t pq p |T x(t)| ≤ c |e x0 (t)| + Lq (t, s) ds kxkpLp . p

0

Then by hypothesis I.a) and II.), we have that T x ∈ Lp [0, ∞). Now we want to see that T is a compact operator from Lp [0, ∞) to Lp [0, ∞). To this end, we use Lemma 6. Let us take M ⊆ Lp [0, ∞) bounded, i.e., ∀x ∈ M , kxkLp ≤ R < ∞, then we must prove that T (M ) is relatively compact. First, by I.a), I.b), II. and the last inequality T (M ) is bounded. Moreover, we have a) Equicontinuity in the translations. Given [a, b] ⊂ [0, ∞) compact, there exists a constant c ≥ 0 such that n |T x(t + h) − T x(t)|p ≤ c |e x0 (t + h) − x e0 (t)|p + t+h ip hZ |k(t + h, s, x(s)) − k(t + h, s, 0)| ds t

hZt +

|k(t + h, s, x(s)) − k(t, s, x(s))| ds

ip o

0 t+h Z pq n p e0 (t)| + Lq (t + h, s) ds kxkpLp ≤ c |e x0 (t + h) − x t

hZt + 0

|k(t + h, s, x(s)) − k(t, s, x(s))| ds

ip o .

244


Then, from Holder’s inequality, we have Zb |T x(t + h) − T x(t)| dt ≤ c p

nZb

a

|e x0 (t + h) − x e0 (t)|p dt +

a t+h Zb Z pq p R Lq (t + h, s)ds dt a

t

Zb Zb +b

p a

o |k(t + h, s, x(s)) − k(t, s, x(s))|q dsdt .

0

and then, due to x e0 ∈ L , the t-equicontinuity of k, and the integrability of L(., .), we obtain the equicontinuity in the translations. b) Finally, the Lp -equiconvergence follows from I.a) and I.b) because: p

|T x(t) − x e0 (t)| ≤ p

Z t

q

L (t, s)ds

pq Zt

0

≤ Rp

|x(t)|p dt

0

Z t

Lq (t, s) ds

pq

0

and

Z∞ |T x(t) − x ˜0 (t)| dt ≤ R p

e t

p

Z∞Zt e t

Lq (t, s) ds

pq

dt.

0

Thus, T is a compact operator from Lp [0, ∞) to Lp [0, ∞) and Schauder’s ¯ = x ¯, and fixed point theorem implies there exist x ¯ ∈ Lp [0, ∞) satisfying T x p then x¯ is an L solution of equation (2.omod ). t u As an example of the first theorem consider a function F (., ., .) as follows: F (t, s, u) = 1 + (t − s) /4e−t u + f (t, s), such that sup |f (t, t)| < ∞, t∈[0,∞)

Zt ∂f sup (t, s) ds < ∞. ∂t t∈[0,∞) 0

The function a = 0 satisfies I.c). Moreover, conditions II.a) and II.b) are fulfilled since |Ft (t, s, x) − Ft (t, s, 0)| ≤ |1 + (t − s)|/4e−t |x|, and

Zt 0

|1 + (t − s)|/4e−t ds → 0


245

as t → ∞. Then for any function g that satisfies g(0) and g 0 in C, theorem 3 implies that the equation Zt g(t) =

(1 + (t − s))/4e−t x(s) + f (t, s) ds

0

has a continuous and bounded solution. Acknowledgement. Fondecyt 1980835 Catedra Presidencial , res. 031 , 21-09-1996

References [1] Brezis, H., Analyse Fonctionnelle, Masson, 1993 (5e tirage) [2] Corduneanu, C., Integral Equations and Applications, Cambridge Univ. Press, 1991 [3] Krasnoselskii, M. A., Topological Methods in the Theory of Nonlinear Equations, Pergamon Press, 1964 [4] Rejto, P., Taboada, M., Unique solvability of nonlinear Volterra equations in weighted spaces, J. Math. Anal. Appl. 167 (1992), 368–381 [5] Rejto, P., Taboada, M., Weighted resolvent estimates for Volterra operators on unbounded intervals, J. Math. Anal. Appl. 160 (1991), 223–235 [6] Willet, D., Nonlinear vector integral equations as contraction mappings, Arch. Rational Mech. Anal. 15, No 1 (1964), 79–86



Rothe’s Method for Degenerate Quasilinear Parabolic Equations Volker Pluschke Department of Mathematics and Computer Science, Martin-Luther-University Halle-Wittenberg, P.O.Box, 06 099 Halle, Germany Email: [email protected] WWW: http://www.mathematik.uni-halle.de/~analysis/pluschke

Abstract. In the contribution we state local existence of a weak solution u to a degenerate quasilinear parabolic Dirichlet problem. Degeneration occurs in the coefficient g(x, t, u) ≥ 0 in front of the time derivative, which is not assumed to be bounded below and above, resp., by positive constants. The nonlinear coefficients and the right-hand side are defined with respect to u only in a neighbourhood of the initial function. The quasilinear parabolic problem is approximated by linear elliptic problems by means of semidiscretization in time (Rothe’s method). We obtain L∞ -estimates for the approximations and uniform convergence to a H¨ older continuous weak solution. An essential tool for this are estimates of the first order derivatives uniformly for all subdivisions in the space L∞ ([0, T ], Lν (G)) with certain ν > 2.

AMS Subject Classification. 35K65, 65M20, 35K20

Keywords. Degenerate equations, Rothe’s method, L∞ -estimates

1

Introduction

In this contribution we formulate a local existence result for the parabolic initial boundary value problem g(x, t, u) ut + A(t, u)u = f (x, t, u) u(x, t) = 0 u(x, 0) = U0 (x)

in QT , on Γ,

(1.plu ) (2.plu )

x ∈ G,

(3.plu )

where A(t, v)u = −

N X i,k=1

∂ ∂xk

X N ∂u ∂u aik (x, t, v) + ai (x, t, v) , ∂xi ∂xi i=1

The paper is an overview article summarizing the results of [8] and [9].

(4.plu )

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Volker Pluschke

which we obtain by means of semidiscretization in time (Rothe’s method). Here we denote by G ⊂ RN , N ≥ 2, a simply connected, bounded domain with boundary ∂G ∈ C 1 , I = [0, T ], QT = G × I, Γ = ∂G × I. In the following we give an overview on the results of the author’s papers [8] and [9]. In [8] the non-degenerated quasilinear problem (1.plu )–(3.plu ) is investigated where g ≡ 1. The aim of the paper [9] is to deal with the case where the coefficient g(·, t, u) may degenerate (i.e. g = 0 or g = ∞) on some sets St,u ⊂ G with meas(St,u ) = 0 (there A = A(t) is linear). In both papers it is supposed that the nonlinear coefficients and the righthand side f are defined only in a neighbourhood MR (U0 ) = {(x, t, u) ∈ RN +2 : x ∈ G, t ∈ I, |u − U0 (x)| ≤ R} of the initial function for some given R > 0. In order to have convergence of the Rothe approximations to a solution we have to ensure that the approximations belong to the ball ¯ : ku − U0 kC(G) BR (U0 ) = {u ∈ C(G) ¯ ≤ R}. This holds for small time t ∈ Iˆ = [0, Tˆ ] due to L∞ -estimates which are derived by means of the technique of Moser [7] combined with recursive estimates due to Alikakos [1]. Because of the nonlinear coefficient g we only can apply this technique to the semidiscrete problem if we have uniform boundedness of the discrete time derivative δuj in L∞ (I, Lν (G)) with sufficiently large ν > 2. In standard literature on Rothe’s method (cf. Kaˇcur [2], Chapter 2) such an estimate of δuj is derived under the assumption of monotonicity of the nonlinear operator A. Moreover, one obtains this estimate for ν = 2 only. Without assumption of full monotonicity and with nonlinear coefficients in general one only obtains an estimate in L2 (I, L2 (G)) (cf. e.g. Kaˇcur [4], Lemma 2.7, where a similar problem with nonlinear coefficient of ut is treated). We use Lp -theory with p > 2, powertype test functions, interpolation arguments, and nonlinear Gronwall lemma to derive the desired a priori estimate. Moreover, degeneration forces to work in weighted Lebesgue spaces. These strong a priori estimates also yield strong convergence results for the approximates despite of weak regularity of the data (Lebesgue data). We obtain uniform convergence of the Rothe functions in Hölder space with respect to space and time variables.

2

Preliminaries and result

We use standard notations of the function and evolution spaces, resp. (cf. [5]). By k · kp , k · k1,p , and k · k0,λ we denote the norms in Lp (G), W01,p (G), and ¯ respectively. Lp,g (G) denotes the weighted Lebesgue space with norm C λ (G), R kukp,g = ( G g|u|p dx)1/p for nonnegative g ∈ L1 (G). h·, ·i is the duality between ¯ the operator A(t, v) Lp (G) and Lp0 (G) (1/p + 1/p0 = 1). For t ∈ I and v ∈ C(G) 1,p 1,p0 from (4.plu ) generates a bilinear form on W0 (G) × W0 (G) denoted by A(t,v) (·, ·).

249

Degenerate Parabolic Equations

First we formulate the complete assumptions which we fix throughout the paper. Assumptions. For given R > 0 let g, aik ai , and f be Carathéodory functions defined on MR (U0 ). Let further r1 , r2 , r3 , µ1 , µ2 , µ3 , µ4 , ν1 , ν, σ, κ be real (κ−1) numbers fulfilling the relations 2 ≤ κ < ∞, r1 > N , r2 > N2κ−N , r3 > N2 , N σ Nκ Nκ Nκ 2 < µ1 ≤ ν1 = σ+1 κ, µi ≤ ν < N −2 (i = 2, 3, 4), κ−2 < µ2 , 2κ−2 < µ3 , Nκ 2κ+N −2 < µ4 , σ > 1. Then we suppose for arbitrary t, t0 ∈ I and u, u0 ∈ BR (U0 ) o

(i) U0 ∈ W 1r1 (G) , A(0, U0 )U0 ∈ L1 (G); (ii) g(·, t, u) : I × BR (U0 ) → Lr2 (G) is bounded in Lr2 (G) and fulfils the Lipschitz condition kg(·, t, u) − g(·, t0 , u0 )kµ1 ≤ l1 (|t − t0 | + ku − u0 kν1 ). Furthermore, g(x, t, u) ≥ 0 for all (x, t, u) ∈ MR (U0 ) and 1/g(·, t, u) : I × BR (U0 ) → Lσ (G) is bounded in Lσ (G). ¯ and ai (·, t, u) : I × BR (U0 ) → L∞ (G) are (iii) aik (·, t, u) : I × BR (U0 ) → C(G) bounded mappings which fulfil the Lipschitz conditions kaik (·, t, u) − aik (·, t0 , u0 )kµ2 ≤ l2 (|t − t0 | + ku − u0 kν ) kai (·, t, u) − ai (·, t0 , u0 )kµ3 ≤ l3 (|t − t0 | + ku − u0 kν ) as well as ellipticity condition (a > 0) P 2 for all (x, t, v) ∈ MR (U0 ) and ξ ∈ RN . i,k aik (x, t, v) ξi ξk ≥ a ξ (iv) f (·, t, u) : I × BR (U0 ) → Lr3 (G) is bounded in Lr3 (G) and fulfils the Lipschitz condition kf (·, t, u) − f (·, t0 , u0 )kµ4 ≤ l4 (|t − t0 | + ku − u0 kν ). (v) It holds the compatibility condition f (·, 0, U0 ) − A(0, U0 )U0 /g(·, 0, U0 ) ∈ Lκ,g(·,0,U0 ) (G) . We remark that the coefficient g may not only decay to zero on some sets (this decay is governed by the assumption 1/g ∈ L∞ (I, Lσ (G)) but it also may have singularities because it belongs to the Lebegue space L∞ (I, Lr2 (G)). This is equivalent to some degeneration of ellipticity of the operator A. In order to solve the problem by semidiscretization in time (Rothe’s method) we subdivide the time interval I by points tj = jh (h > 0, j = 0, . . . , n) and replace (1.plu )–(3.plu ) by the time discretized problem (in weak formulation) hgj δuj , vi + Aj (uj , v) = hfj , vi uj = 0 u 0 = U0 ,

o

∀v ∈ W 1p0 (G) ,

(1j .plu )

on ∂G ,

(2j .plu ) (30 .plu )

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Volker Pluschke

j = 1, . . . , n, where δuj := (uj − uj−1 )/h, gj := g(x, tj , uj−1 ), fj := f (x, tj , uj−1 ), and Aj (·, ·) := A(tj ,uj−1 ) (·, ·). This is a set of linear elliptic boundary value problems to determine the approximate uj if uj−1 is already known. However, we do not know if uj ∈ BR (U0 ), i.e. whether the data of (1j+1 ), (2j+1 ) are well-defined because of the local assumptions. Therefore we define global extensions ( ψ(x, t, u) for (x, t, u) ∈ MR (U0 ) R ψ (x, t, u) = ψ(x, t, U0 (x) + R sign (u − U0 (x)) otherwise R R and replace g, aik , ai , and f by g R , aR ik , ai , and f , respectively. By Lemma 3 we state that uj ∈ BR (U0 ) for tj ≤ Tˆ, hence we omit the superscript R. For sufficiently small fixed h now we can solve these elliptic boundary value problems applying the Lax-Milgram theorem (after an interpolation procedure to deal with the weighted norm with weight gj ) and a regularity theorem (cf. [6, Theorem 5.5.4’]).

Lemma 1. Let assumptions (i)–(iv) be fulfilled. Then there are h0 > 0, r > N ), (2j ), .plu (30 .plu ) has a unique solution such that for 0 < h ≤ h0 the problem (1j .plu o

uj ∈ W 1r (G), j = 1, . . . , n. Especially, the embedding theorem implies continuity of uj . By interpolation with respect to time we obtain the Rothe functions u ñ (x, t) =

tj − t t − tj−1 uj−1 (x) + uj (x) h h

t ∈ [tj−1 , tj ]

,

and ( u ¯n (x, t) =

uj (x) U0 (x)

if t ∈ (tj−1 , tj ], if t ≤ 0.

Our result is the following Theorem 2. Suppose assumptions (i)–(v). Then the following assertions hold: a) There is an interval Iˆ = [0, Tˆ] such that problem (1.plu )–(3.plu ) has a unique weak soo ˆ ˆ W 10 (G)∩ lution u with u(·, t) ∈ BR (U0 ) for any t ∈ I fulfilling for all v ∈ L1 (I, r L%0 (G)) (%0 = r20 κ0 ) the relation Z Z Z hg(·, t, u)ut , vi dt + A(t,u) (u , v) dt = hf (·, t, u) , vi dt Iˆ

Iˆ

Iˆ

and initial condition (3.plu ). b) The solution u belongs to the spaces o

1 ˆ ¯ ˆ ) ∩ L∞ (I, ˆ W 1r (G)) ∩ W∞ (I, Lν1 (G)) u ∈ C α (Q T

ˆ Ls (G)) for s < for some r > N and α > 0. Moreover, ut ∈ Lκ (I, ˆ L% (G)) with % = r2 κ . g(·, ·, u)ut ∈ L∞ (I, r2 +κ−1

Nκ N −2

and

251


c) The solution u is pointwise uniformly approximated by the Rothe functions u ñ with further convergence properties u ñ −→ u u ñ , u ¯n −→ u ∗ u ñ , u ¯n −* u ∗ u ñt −* ut

¯ ˆ) in C α (Q T ˆ C λ (G)) ¯ in L∞ (I,

(λ < 1 − N/r)

o

ˆ W 1 (G)) in L∞ (I, r ˆ Lν1 (G)) in L∞ (I,

as n tends to infinity. d) It holds an error estimate sup k˜ un (·, t) − u(·, t)kν1 ≤ c hn1/2 . t∈Iˆ

The r > N may be explicitly given in terms of the Lebesgue exponents from the assumptions. Furthermore, because of uniform boundedness of the approxio ˆ W 1r (G)) an interpolation inequality yields an error estimate in mations in L∞ (I, Hölder space, too, un (·, t) − u(·, t)k0,λ ≤ c hn(1−λ−N/r)/2 , sup k˜

0 < λ < 1 − N/r .

t∈Iˆ

3

A priori estimates for the approximations

In this final section we sketch some steps of the proof of Theorem 2. For the details compare [8] and [9]. In order to prove uj ∈ BR (U0 ) we have to estimate zj := uj − U0 in L∞ (G). ) into Therefore, we rewrite (1j .plu hgj δzj , vi + Aj (zj , v) = hfj , vi − Aj (U0 , v)

o

∀v ∈ W 1r0 (G).

(5.plu )

We use the Moser iteration technique [7]. The idea of this technique is to estimate kzkp for arbitrary p ≥ p0 and then pass with p to infinity. Since kzkp → kzk∞ (cf. [2, Theorem 2.11.4]) one obtains an estimate in L∞ (G). In our case, because of degeneration, we have to work with the weighted norm kzkp,g . But we have the same property kzkp,g → kzk∞ as p → ∞, i.e. the influence of the degeneration vanishes in the limit. In order to obtain an estimate of the weighted ) with v = |zj |p−2 zj and obtain after some Lp,g -norm for arbitrary p we test (5.plu manipulations p p 2 − kzj−1 kp,g + ch kwj k1,2 kzj kp,g j+1 j (p−2)/2 ≤ ch kzj kpp + cph kfj kr3 kzj krp−1 0 (p−1) + cph kU0 k1,r0 kwj k1,2 kzj ks 3

+ cph (1 + kδuj kν1 ) kzj kµp0 p , 1

(6.plu )

252

Volker Pluschke

where wj := |zj |(p−2)/2 zj . The last item on the right-hand side appears since p arising from (5.plu ) on the left-hand side must be replaced by the item kzj kp,g j p kzj kp,g . Hence, for our L -estimate we need uniform boundedness of kδuj kν1 ∞ j+1 (cf. Lemma 3). Then we have to estimate the unweighted norms of zj on the right-hand side by weighted norms occuring on the left-hand side. For this we use the Nirenberg-Gagliardo interpolation inequality θ kwks ≤ C kwk1,2 kwk11−θ

for some θ ∈ (0, 1), s < 2N/(N − 2). This enables us to insert the weight by means of Cauchy-Schwarz’ inequality p/2

1/2

kwj k1 = kzj kp/2 ≤ k1/gj+1 k1

p/2 kzj kp,g . j+1

(7.plu )

Summing up the inequalities (6.plu ) for j = 1, . . . , i we would then come to an estimate of the weighted norm kzi kp,gi+1 . However, in our case it is not possible to hold the bounds depending on p uniformly bounded as p → ∞. Therefore, for the limit process we use a recursive approach due to Alikakos [1]. Since 1/g ∈ Lσ (G) with σ > 1 is supposed we may even obtain the weighted norm kzj kλp,gj+1 with λ < 1 on the right-hand side of (7.plu ). Then we derive an recursive estimate of the form β(p)p p p c ≤ cp t max kz k + max kz k max kzj kp,g j j λp,g λp,g j+1 j+1 j+1 tj ≤t

tj ≤t

tj ≤t

which is investigated for the special sequence pk = λ−k p0 . Passing to the limit k → ∞ this yields an estimate of kzj k∞ in terms of kzj kp0 ,gj+1 for fixed p0 . After estimation of this norm for fixed p0 we obtain Lemma 3. Let be kδuj kν1 ≤ C for j = 1, . . . , n independent of the subdivision. Then there are constants c, γ > 0 such that max kuj − u0 k∞ ≤ c tγ .

0≤tj ≤t

¯ due to Lemma 1 we have uj ∈ BR (U0 ) for all Obviously, since uj ∈ C(G) tj ∈ Iˆ := [0, Tˆ ] if we fix Tˆ for given R > 0 by cTˆ γ = R. Remark 4. In [3, Theorem 4.17] J. Kaˇcur proves a L∞ -estimate for quasilinear equations without the assumption kδuj k ≤ C of our Lemma 3. The reason is that the degeneration in [3] corresponds to the case g(x, t, s) = b0 (s), i.e. the item concerning the time derivative is written in the form b(u)t . This is not possible in our case. Moreover, we have weaker regularity of the data with respect to x. On the other hand, the technique in [3] allows stronger degeneration with respect to u. The next task is to check the assumption of Lemma 3. One obtains an es)–(1j−1 .plu ) and testing the resulting timate of δuj by forming the difference (1j .plu relation with an appropriate test function (cf. [2, Chapters 2.1, 2.2]). However,

253


we run into problems since we have no full monotonicity of the nonlinear op)–(1j−1 .plu ) with erator Au := A(t, u)u unlike in [2, Example 2.2.17]. Testing (1j .plu v = |δuj |κ−2 δuj in order to estimate the weighted norm kδuj kκ,gj we are forced to deal with an item ch (1 + kδuj−1 kν ) kuj k1,r kωj k1,2 kδuj ks(κ−2)/2 (8.plu ) (ωj = |δuj |(κ−2)/2 δuj ) arising from Aj − Aj−1 (uj , v) on the right-hand side. Hence, we have to estimate the space-like derivative kuj k1,r in order to obtain an estimate of the discrete time derivative. This is possible by means of a priori ) is. However, we are not able to split estimates for elliptic equations like (1j .plu u u gj δuj into gj hj and gj j−1 ,resp., and then to use estimates of the solution uj h u of the elliptic equation with right-hand side fj + gj j−1 since we need a priori h ) in the form bounds uniformly with respect to h > 0. Hence we write (1j .plu Aj uj = fj − gj δuj =: Fj where we obtain from Lp -theory for elliptic equations (cf. [6, Theorem 5.5.5’] the estimate j X kδui kν1 . kuj k1,r ≤ c (kFj kρ + kuj k1 ) ≤ c 1 +

(9.plu )

i=1

The constant c now is independent of the subdivision since the coefficients of the elliptic operator Aj are uniformly bounded. Inserting this estimate into (8.plu ) we notice that the total power of kδuj k on the right-hand side of the resulting estimate is κ + 1 while we have the power κ on the left-hand side, only. This seems to contradict the intention to obtain boundedness of kδuj k by these estimations. However, after some very technical manipulations, we are able to apply a nonlinear discrete version of the Gronwall lemma (cf. Willett, Wong [10, Theorem 4]) to obtain at least a local bound for small tj . Since the unweighted norm kδuj kν1 may be estimated by the weighted norm kδuj kκ,gj we obtain Lemma 5. Suppose assumptions (i)–(v). Then for h ≤ h0 there is a time interval [0, T ∗] such that the estimate kδuj kν1 ≤ C1

∀tj ∈ [0, T ∗ ]

holds independent of the subdivision. By means of (9.plu ) this lemma also yields boundedness of the space-like derivatives. Lemma 6. For all h ≤ h0 the estimate kuj k1,r ≤ C2

∀tj ∈ [0, T ∗]

holds independent of the subdivision. ˆ If Tˆ > T ∗ The time T ∗ is a bound for the length of our local existence interval I. ∗ ˆ ˆ for the T choosen after Lemma 3 we have to fix I := [0, T ]. The a priori estimates from Lemma 3, 5, and 6 now provide the tools to prove the convergence results of Theorem 2.

254

Volker Pluschke

References [1] N. D. Alikakos, Lp -bounds of solutions of reaction-diffusion equations, Comm. Part. Diff. Equ., 4 (8) (1979), 827–868. [2] J. Kaˇcur, Method of Rothe in Evolution Equations, B. G. Teubner Verlagsges., Leipzig, 1985. [3] J. Kaˇcur, On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev spaces II, Math. Z., 203 (1990), 569–579. [4] J. Kaˇcur, Solution to strongly nonlinear parabolic problems by a linear approximation scheme, Comenius University, Faculty of Mathematics and Physics, Preprint No. M1-94 (1994). [5] A. Kufner, O. John, S. Fuˇc´ık, Function Spaces, Noordhoff Intern. Publ., Leyden/ Academia, Prag, 1977. [6] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer, Berlin–Heidelberg–New York, 1966. [7] J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure and Appl. Math., 13 (3) (1960), 457–468. [8] V. Pluschke, Local Solutions to quasilinear parabolic equations without growth restrictions, Z. Anal. Anwend., 15 (1996), 375–396. [9] V. Pluschke, Rothe’s method for parabolic problems with nonlinear degenerating coefficient, Martin-Luther-University Halle, Dept. of Math., Report No. 14 (1996). [10] D. Willett, J. Wong, On the discrete analogues of some generalizations of Gronwall’s inequality, Monatshefte f¨ ur Math., 69 (1965), 362–367.



A Posteriori Error Estimates for a Nonlinear Parabolic Equation? Karel Segeth ˇ a 25, Mathematical Institute, Academy of Sciences, Zitn´ CZ-115 67 Praha 1, Czech Republic Email: [email protected]

Abstract. A posteriori error estimates form a reliable basis for adaptive approximation techniques for modeling various physical phenomena. The estimates developed recently in the finite element method of lines for solving a parabolic differential equation are simple, accurate, and cheap enough to be easily computed along with the approximate solution and applied to provide the optimum number and optimum distribution of space grid nodes. The contribution is concerned with a posteriori error estimates needed for the adaptive construction of a space grid in solving an initial-boundary value problem for a nonlinear parabolic partial differential equation by the method of lines. Under some conditions, it adds some more statements to the results of [2] in the semidiscrete case. Full text of the contribution will appear as a paper [4].

AMS Subject Classification. 65M15, 65M20

Keywords. A posteriori error estimate, nonlinear parabolic equation, finite element method, method of lines

1

A Nonlinear Model Problem

The principal ideas of semidiscrete a posteriori error estimation for nonlinear parabolic partial differential equations can be demonstrated with the help of a simple initial-boundary value one-dimensional model problem. We consider the nonlinear equation ∂ ∂u ∂u (x, t) − a(u) (x, t) + f (u) = 0, ∂t ∂x ∂x

0 < x < 1,

0 < t ≤ T,

for an unknown function u(x, t) with the homogeneous Dirichlet boundary conditions u(0, t) = u(1, t) = 0, 0 ≤ t ≤ T, ?

This work was supported by Grant No. 201/97/0217 of the Grant Agency of the Czech Republic. This is the preliminary version of the paper.

256

Karel Segeth

and the initial condition u(x, 0) = u0 (x),

0 < x < 1.

In the above formulae, T > 0 is a fixed number and a, f , and u0 are smooth functions. Let 0 < µ ≤ a(s) ≤ M, s ∈ R, and let a and f satisfy the global Lipschitz conditions |a(r) − a(s)| ≤ L|r − s|, |f (r) − f (s)| ≤ L|r − s|,

r, s ∈ R.

We employ the usual L2 (0, 1) inner product to introduce the weak solution u(x, t) ∈ H 1 ([0, T ], H01 (0, 1)) of the model problem by the identity ∂u ∂v , v + a(u) , + (f (u), v) = 0 ∂t ∂x ∂x

∂u

holding for almost every t ∈ (0, T ] and all functions v ∈ H01 , and the identity ∂u ∂v ∂u0 ∂v = a(u0 ) , a(u0 ) , ∂x ∂x ∂x ∂x holding for t = 0 and all functions v ∈ H01 .

2

Discretization

Finite element solutions of the model problem are constructed from this weak formulation, too. We first introduce a partition 0 = x0 < x1 < · · · < xN −1 < xN = 1 of the interval (0, 1) into N subintervals (xj−1 , xj ), j = 1, . . . , N , and then put hj = xj − xj−1 ,

j = 1, . . . , N,

We further use the notation

Z

and h = max hj . j=1,...,N

xj

(v, w)j =

v(x)w(x) dx xj−1

for the L2 (xj−1 , xj ) inner product. We construct the finite dimensional subspace S0N,p ⊂ H01 with a piecewise polynomial hierarchical basis of degree p ≥ 1 in the following way. We put p N −1 N X n o X X Vj1 ϕj1 (x) + Vjk ϕjk (x) , S0N,p = V | V ∈ H01 , V (x) = j=1

j=1 k=2

A Posteriori Error Estimates for a Nonlinear Parabolic Equation

257

where ϕj1 are the usual piecewise linear shape functions of the finite element method, ϕj1 (x) = (x − xj−1 )/hj , = (xj+1 − x)/hj+1 , = 0 otherwise, while for k > 1,

p

ϕjk (x) =

2(2k − 1) hj

=0

Z

xj−1 ≤ x < xj , xj ≤ x ≤ xj+1 ,

xj

xj−1 ≤ x ≤ xj ,

Pk−1 (y) dy, xj−1

otherwise

are bubble functions with Pk being the kth degree Legendre polynomial scaled to the subinterval [xj−1 , xj ] (see, e.g., [5]). ¯ (x, t) is the semidiscrete finite element approximate We say that a function U solution of the model problem if it belongs, as a function of the variable t, into H 1 ([0, T ], S0N,p), if the identity ∂U ¯ ¯ ¯ ) ∂ U , ∂V + (f (U ¯ ), V ) = 0 , V + a(U ∂t ∂x ∂x holds for each t ∈ (0, T ] and all functions V ∈ S0N,p , and if the identity ¯ ∂V ∂U ∂u0 ∂V , = a(u0 ) , a(u0 ) ∂x ∂x ∂x ∂x holds for t = 0 and all functions V ∈ S0N,p . The procedure for constructing the approximate solution ¯ (x, t) = U

N −1 X j=1

¯j1 (t)ϕj1 (x) + U

p N X X

¯jk (t)ϕjk (x) U

j=1 k=2

described above is the method of lines. It transforms the solution of the original initial-boundary value problem for a parabolic partial differential equation into an initial value problem for a system of ordinary differential equations for ¯jk (t) that, in practice, is solved by proper numerical the unknown functions U software.

3

A Posteriori Semidiscrete Error Indicators

Let ¯ (x, t) e(x, t) = u(x, t) − U be the error of the semidiscrete approximate solution. We employ the finite dimensional subspace N n o X Sˆ0N,p+1 = Vˆ | Vˆ ∈ H01 , Vˆ (x) = Vˆj ϕj,p+1 (x) j=1

258

Karel Segeth

of piecewise polynomial bubble functions of degree p + 1 equal to zero at the ¯ t) = grid points xj to construct error indicators. We say that a function E(x, N,p+1 1 ˆ ¯ ) is a parabolic nonlinear a posteriori semidiscrete error EPN ∈ H ([0, T ], S0 indicator if the identities ∂E ¯ ˆ ¯ ¯ + E) ¯ ∂E , ∂V , Vˆ + a(U ∂t ∂x ∂x j j ¯ ˆ ¯ ¯ + E), ¯ Vˆ )j − ∂ U , Vˆ − a(U ¯ + E) ¯ ∂U , ∂V = −(f (U ∂t ∂x ∂x j j hold for j = 1, . . . , N , all t ∈ (0, T ] and all functions Vˆ ∈ Sˆ0N,p+1 , and if the identities ¯ ∂ Vˆ ¯ ) ∂ Vˆ ∂E ∂(u0 − U a(u0 ) , , = a(u0 ) ∂x ∂x j ∂x ∂x j N,p+1 hold for j = 1, . . . , N , t = 0 and all functions Vˆ ∈ Sˆ0 . Note that the special N,p+1 ˆ results in an uncoupled system of choice of the bubble function space S0 ordinary differential equations. On each interval (xj−1 , xj ), the error indicator ¯ t) = E(x,

N X

¯j (t)ϕj,p+1 (x) E

j=1

is computed independently of the other intervals. The indicator thus has a local character and its computation is rather cheap. ¯ is neglected in the argument of the functions a and f we say that When E ¯ t) = E ¯PL ∈ H 1 ([0, T ], SˆN,p+1) is a parabolic linear a posteriori a function E(x, 0 semidiscrete error indicator if the identities ¯ ∂E ˆ ¯ ˆ ¯ ¯ ¯ ) ∂E , ∂V ¯ ), Vˆ )j − ∂ U , Vˆ − a(U ¯ ) ∂U , ∂V , Vˆ + a(U = −(f (U ∂t ∂x ∂x j ∂t ∂x ∂x j j j hold for j = 1, . . . , N , all t ∈ (0, T ] and all functions Vˆ ∈ Sˆ0N,p+1 , and if the identities ¯ ∂ Vˆ ¯ ) ∂ Vˆ ∂E ∂(u0 − U a(u0 ) , , = a(u0 ) ∂x ∂x j ∂x ∂x j N,p+1 hold for j = 1, . . . , N , t = 0 and all functions Vˆ ∈ Sˆ0 . The practical computation of the linear error indicator is thus easier. ¯ The task to compute error indicators can be simplified if the derivative ∂ E/∂t is neglected. The corresponding a posteriori semidiscrete error indicator is then called (linear or nonlinear) elliptic indicator since the resulting uncoupled algebraic system does not depend on t. Moreover, the practical computation of such an elliptic indicator need not be carried out for each t but only when required. ¯ t) = E ¯EN that maps [0, T ] into SˆN,p+1 is an We thus say that the function E(x, 0 elliptic nonlinear a posteriori semidiscrete error indicator if the identities ¯ ˆ ¯ ¯ ˆ ¯ + E), ¯ Vˆ )j − ∂ U , Vˆ − a(U ¯ + E) ¯ ∂U , ∂V ¯ + E) ¯ ∂E , ∂V = −(f (U a(U ∂x ∂x j ∂t ∂x ∂x j j


259

hold for j = 1, . . . , N , all t ∈ (0, T ] and all functions Vˆ ∈ Sˆ0N,p+1 , and if the identities ¯ ∂ Vˆ ¯ ) ∂ Vˆ ∂E ∂(u0 − U , , a(u0 ) = a(u0 ) ∂x ∂x j ∂x ∂x j hold for j = 1, . . . , N , t = 0 and all functions Vˆ ∈ Sˆ0N,p+1 . ¯ t) = E ¯EL that maps [0, T ] into SˆN,p+1 Finally, we say that the function E(x, 0 is an elliptic linear a posteriori semidiscrete error indicator if the identities

¯) a(U

¯ ˆ ¯ ¯ ∂ Vˆ ∂E ¯ ), Vˆ )j − ∂ U , Vˆ − a(U ¯ ) ∂U , ∂V , = −(f (U ∂x ∂x j ∂t ∂x ∂x j j

hold for j = 1, . . . , N , all t ∈ (0, T ] and all functions Vˆ ∈ Sˆ0N,p+1 , and if the identities ¯ ∂ Vˆ ¯ ) ∂ Vˆ ∂E ∂(u0 − U a(u0 ) , , = a(u0 ) ∂x ∂x j ∂x ∂x j hold for j = 1, . . . , N , t = 0 and all functions Vˆ ∈ Sˆ0N,p+1 . To assess properties of the above semidiscrete a posteriori error indicators, we introduce the quantity ¯ 1 kEk Θ= kek1 called the effectivity index of the respective error indicator. The norm used is the H 1 (0, 1) norm. Then we can prove the following statement. ¯ (x, t) ∈ S N,p and E ¯ ∈ SˆN,p+1 . Theorem 1. Let u(x, t) ∈ H01 be smooth, let U 0 0 ¯ and its elliptic Let the norm of the difference between the semidiscrete solution U projection is a nondecreasing function of t. Let the same hold for the norm of ¯ and its elliptic projection. the difference between the error indicator E Moreover, let kek1 ≥ Chp . Then lim Θ = 1

h→0

holds for ΘPN , ΘPL , and ΘEL . The exact assumptions as well as a complete proof will be published in [4].

4

A Numerical Example

We present numerical results obtained by the finite element method of lines for a nonlinear parabolic initial-boundary value problem (a reaction-diffusion model) with a simple grid adjustment procedures described in [1] (Fig. 1) and [3] (Fig. 2). Both the procedures are based on the equidistribution of error. The example fully confirms the above statement.

260

Karel Segeth

The differential equation solved is ∂u ∂ 2 u − 2 − D(1 + α − u) exp(−δ/u) = 0, ∂t ∂x D=R

exp δ , 0 < x < 1, 0 < t ≤ 0.6, αδ α = 1, δ = 20, R = 5,

with the boundary conditions ∂u (0, t) = 0, ∂x

u(1, t) = 1,

0 < t ≤ 0.6,

and the initial condition u(x, 0) = 1,

0 < x < 1.

We used piecewise linear shape functions, i.e. p = 1, for the computation of ¯ and required a very small error bound in the integration of the the solution U corresponding system of ordinary differential equations by a standard differential system solver. The model describes a single step reaction of a reacting mixture of temperature u in a region 0 < x < 1. Further, α is the heat release, δ is the activation energy, D is called the Damkohler number, and R is the reaction rate. For small times, the temperature gradually increases from unity with a “hot spot” forming at x = 0. At a finite time, ignition occurs and the temperature at x = 0 jumps rapidly from near 1 to near 1+α. A sharp flame front then forms and propagates towards x = 1 with velocity proportional to 12 exp(αδ)/(1 + α). In real problems, α is about unity and δ is large. The flame front thus moves exponentially fast after ignition. The problem reaches a steady state once the flame propagates to x = 1. The trajectories of nodes of the partition of interval (0, 1) as constructed by the two procedures mentioned are shown in Figs. 1 and 2. The grid is rather slow and is unable to follow the dynamics of the problem properly. The integration with respect to t requires small time steps and is expensive during this rapid transience as the solution changes rapidly along the grid node trajectories. The problem is very difficult and yet adaptive grid methods are capable of finding a solution with relative ease.

References [1] Adjerid, S., Flaherty, J. E. : A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations. SIAM J. Numer. Anal. 23 (1986) 778–796 [2] Moore, P. K. : A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension. SIAM J. Numer. Anal. 31 (1994) 149–169


261

Fig. 1. Trajectories of nodes constructed by the procedure of [1]

[3] Segeth, K. : A grading function algorithm for space grid adjustment in the method of lines. Software and Algorithms of Numerical Mathematics 10. (Proc. of Summer School, Cheb 1993.) Plzeˇ n, University of West Bohemia (1993) 139–152 [4] Segeth, K. : A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension (to appear) [5] Szab´ o, B., Babuˇska, I. : Finite Element Analysis. New York, J. Wiley & Sons 1991

262

Karel Segeth

Fig. 2. Trajectories of nodes constructed by the procedure of [3]



The Solvability Conditions of the Infinite Trigonometric Moment Problem with Gaps G. M. Sklyar1 and I. L. Velkovsky2 1

Dept. of Math. Analysis, Kharkov State University Svoboda sqr. 4, 310077, Kharkov, Ukraine Email: [email protected] 2 Reiterstrasse 11, 93 053, Regensburg, Germany Email: [email protected]

Abstract. The infinite Markov trigonometric moment problem with periodic gaps is considered. The precise analytical description of the solvability set of the problem is given. The introduced approach is based on investigation of the special subclass of the Carathéodory function class corresponding to given periodic law. AMS Subject Classification. 42A70 Keywords. Markov trigonometric moment problem, periodic gaps, periodic law, Carathéodory function class

Let p be a natural number and M = {m0 , . . . , mν } , where 0 ≤ m0 < m1 < m2 < · · · < mν < p, is the subset of the set {0, 1, . . . p − 1}. ∞

Definition 1. We will refer to the sequence M = {mk }k=0 , which is the p-periodic extension of M to the set N ∪ {0}, i.e. 0 ≤ m0 < m1 < · · · < mν < p ≤ mν+1 = m0 + p < · · · · · · < m2ν+1 = mν + p < · · · , as a p-periodic law generated by M. If from l ∈ M implies p − l ∈ M , we will say, that the p-periodic law is a symmetric one. ∞

Let M = {mk }k=0 be a p-symmetric periodic law. Consider the infinite Markov trigonometric moment problem of the form: Zθ eimk t f (t) dt = smk ,

|f (t)| ≤ 1,

0

This is the preliminary version of the paper.

t ∈ (0, θ), k = 0, 1, 2, . . . ,

(1.skl )

264

G. M. Sklyar and I. L. Velkovsky

where 0 < θ < 2π p . Our first goal is to give conditions for the complex sequence {smk }∞ k=0 = {amk + ibmk }∞ (if 0 ∈ M then b = 0) to be a moment one. That means that 0 k=0 there exists at least one measurable function f satisfying moment equalities (1.skl ). As it is known, the classical trigonometric moment problem (mk = k) is closely connected with Carathéodory coefficient problem [1] and based on the technique of the Carathéodory functions [2]. Remind that for the class C of Carathéodory functions one can write: C := {F : F is holomorphic, Re F (z) > 0 for |z| < 1} . Further we need the following theorem describing properties of certain functions from this class: N S Theorem 2. Let T = Tj be a collection of nonintersecting intervals Tj = j=1 τj , τj0 ⊂ [0, 2π] . Then the following statements are equivalent to each other:

i) A function F (z) ∈ C is holomorphic for z = eiτ and ImF (eiτ ) = 0, τ ∈ T. ii) The following representation holds: ) ( Z eit + z i ϕ (t) dt , F (z) = |F (0)| exp 4 eit − z [0,2π]\T

−1 ≤ ϕ(t) ≤ 1, t ∈ [0, 2π]\T. iii) Two functions N P

±

F (z) = F (z) ·

0 N 0 i Y eiτj − z − 2 j=1(τj −τj ) · e eiτj − z j=1

!± 12

belong to the class C . Introduce the subclass of the Carathéodory function class associated with a p-periodic law. Definition 3. For the subclass C(M ) corresponding to the periodic law M we call the set of functions F(z), satisfying the following conditions: i) F ∈ C. ii) F is holomorphic and real on the arc z = eiτ , where τ ∈ F (z) iii) Power series for the function ln |F is of the form: (0)| ln

F (z) |F (0)|

=

∞ X k=0

ρk z m k ,

|z| < 1.

2π p

(p − ν) , 2π .

265

The Solvability Conditions

Further we give a multiplicative description of the class C(M ) . Introduce the polynomial rM (z) of the form: rM (z) =

q Y

p−1 X k 1 − e−γ z = rl z l , p

k=1

l=0

where ep is a primitive root of unity, of order p, Γ = {γ1 , . . . , γq } = {0, 1, . . . , p − 1} \M. Note that r0 = 1, rl = 0, l > q = p − ν − 1. Besides, if M is a symmetric law then rk are real, k = 0, 1, . . . , p − 1. Theorem 4. A function F (z) ∈ C(M ), where M is a p-periodic symmetric law, iff ) ( Z2π p 2π q X ei(t+ p l) + z i ϕ (t) dt , rl F (z) = |F (0)| exp 2π 4 ei(t+ p l) − z 0 l=0 where −µ ≤ ϕ (t) ≤ µ, µ−1 = max |rl | , l = 0, q . ∞

problem ). Complete Let a sequence {smk }k=0 be a moment one for the (1.skl 2π then the definition of the function f (t) by f (t) = 0, t ∈ 0, p . Next consider the function ϕ(t), t ∈ (0, 2π) , of the form: 2π 2π 2π l , t∈ l, (l + 1) = ∆l , ϕ(t) = µrl f t − p p p

l = 0, 1, . . . p − 1.

Note that |ϕ(t)| ≤ 1,

t ∈ (0, 2π) ,

|ϕ(t)| ≤ µ,

t ∈ ∆0 ;

Next consider the complex function ) ( Z2π eit + z i ϕ (t) dt . F (z) = exp 4 eit − z

(2.skl )

0

Note that |F (0)|= 1. Hence due to Theorem 4 obtain that F (z) ∈ C(M ). Besides 2π ϕ(τ ) ≡ 0, τ ∈ θ, p , then the function F (z) is holomorphic and real on the arc z = eiτ , τ ∈ θ, 2π (Theorem 2). Applying Theorem 2 once more and p ± considering T = θ, 2π ∪ 2π p p (q + 1), 2π , we obtain that F (z) ∈ C, where ±

F (z) = F (z) ·

± 12 ep − z 1−z i 2π · (p − q) − θ . · exp − eiθ − z eq+1 2 p −z p

(3.skl )

266

G. M. Sklyar and I. L. Velkovsky

Let F ± (z) = α± +

∞ X

j α± j z ,

|z| < 1.

(4.skl )

j=1

Express now coefficients α± , α± j , j = 1, 2, . . . of the expansion via elements of the moment sequence smk , k = 0, 1, . . . Let ∞ X j ln F ± (z) = s± (5.skl ) j z , j=0

then s± j

iµ =

2

n0 =

i 4

where

nj =

(2 − δ0j )rM (ejp )sj ± nj , j ∈ M, j∈ / M, ±nj ,

2π (p − q) − θ , p

e−iθ + e−

2π p (q+1)i

− e−

2π p i

j

−1

(6.skl )

(7.skl ) ,

One can see from (4.skl ) and (6.skl ) that: ± α± j = exp s0 ,

± ± α1 2α± 2 · · · jαj ± ± α α± 1 · · · αj−1 = j α± n s± . A± j = j . . . . . . . . . . . . . . . .±. . 0 0 · · · α1

(8.skl )

Thus, taking account of (6.skl ) we can regard (8.skl ) as recurrent expressions of the coefficients α± , α± j , j = 1, 2, . . . via smk , k = 0, 1, 2 . . . Now we are able to formulate the main result of the paper. Theorem 5. A sequence {smk }∞ k=0 is a moment one for the Markov trigonometric moment problem (1.skl ) associated with the p-symmetric periodic law M iff A± n ≥ 0, n = 0, 1, 2, . . . where ± ∞ — the coefficients α± , α± j ,j = 1, 2, . . . are expressed via sequence sj j=0 by means of (8.skl ), — s± ), j , j = 0, 1, . . . are of the form (6.skl ), — nj , j = 0, 1, 2, . . . are defined from (5.skl n ± ± , — An , n = 0, 1, . . . are symmetric matrices such that A± n = αk−j k,j=1

where

α± 0

±

±

±

= α + α = 2Re α ,

α± −j

=

α± j ,

j = 1, 2, . . .

The Solvability Conditions

267

References ¨ [1] Carathéodory, C., Uber Variabilit¨ atsbereich der Koefficienten von Potenzreichen, die gegebene Werte nicht annehmen, Math. Annalen, 64 (1907) ¨ [2] Akhiezer, N. I., Krein, M. G., Uber Fourierische Reichen beschr¨ ankter summierbarer Functionen und ein neues Extremumproblem, Common. Soc. Math., Kharkov, 9 (1934)



Elliptic Equations with Decreasing Nonlinearity I : Barrier method for Decreasing Solutions Tadie Matematisk Institut, Universitetsparken 5 2100 Copenhagen, Denmark Email: [email protected]

Abstract. In this note, we establish existence theorems for positive and classical solutions of the problem (Ea.tad ) below using a barrier method. Moreover we show that the existence of such solutions can be obtained from the sole existence of a supersolution or of a subsolution of the equation.

AMS Subject Classification. 35J70, 35J65 34C10

Keywords. Quasilinear elliptic, integral operators, fixed points theory

1

Introduction

Let f ∈ C 1 ([0, ∞) × (0, ∞)) be such that f1) ∀r ≥ 0,

f (r, .)+ := max{0, f (r, .)} ∈ C 1 ((0, ∞)) and non increasing;

f2) ∀S, T > θ > 0, if f (r, S), f (r, T ) > 0 then ∃k1 (θ), k2 (θ) > 0 such that |f (r, T ) − f (r, S)| ≤ k1 (θ)f2 (r, k2 (θ) )|T − S|; f2 (., S) := |∂f (., S)/∂S|. For a > 1, p ∈ (1, 2] and Dap u := (ra |u0 |p−2 u0 )0 , consider in R+ the problem Ea(u) := Dap u + ra f (r, u)+ = 0;

u(0) > 0; u0 (0) = 0.

(Ea.tad )

Definition 1. Let M be a positive number, finite or not. Let IM := [0, M ) and w, v ∈ C 1 (IM ) be piecewise C 2 be non increasing. 1) v will be said to be a supersolution (subsolution) of the problem (Ea.tad ) in IM if Ea(v) ≥ 0 (Ea(v) ≤ 0) almost everywhere in IM ; 2) w and v will be said to be Ea-compatible in IM if i) Ea(w) ≤ 0 ≤ Ea(v) a.e. in IM , ii) 0 < w ≤ v and w0 ≤ v 0 ≤ 0 in IM , iii) ∀r ∈ IM , f (r, .) > 0 and decreasing in [w(r), v(r)]. This is the final form of the paper.

270

Tadie

For a non-increasing positive φ ∈ C 1 (IM ) define 1/(p−1) Z r Z t Φ(r) = T φ(r) := φ(0) − dt (s/t)a f (s, φ)ds . 0

(T)

0

Definition 2. A non increasing (respectively decreasing) positive supersolution v (resp. subsolution w) of (Ea.tad ) in IM will be said to be Ea-compatible if T v and v (resp. w and T w) are Ea-compatible in IM . In the sequel super- and subsolutions are supposed to be C 1 and piecewise C 2 in the corresponding domains. Also for ease writing, under the integral signs we will write f (., .) for f (., .)+ . The main results are the following: ) Theorem 3. If there are w and v which are Ea-compatible in IM , then (Ea.tad has a solution u ∈ C 2 (IM ) such that w ≤ u ≤ v in IM . Theorem 4. Assume that there is a non increasing (resp. decreasing) positive supersolution v (resp. subsolution w) which is Ea-compatible in IM . Then (Ea.tad ) has a decreasing solution u ∈ C 2 (IM ) such that T v ≤ u ≤ v (resp. w ≤ u ≤ T w) in IM . Theorem 5. 1) Assume that there are w and v which are Ea-compatible in [0, ∞) such that Z ∞ {1 + sp−1 }f (s, w)ds < ∞. (1.tad ) 0

Then (Ea.tad ) has a solution u ∈ C 2 ([0, ∞)) such that w ≤ u ≤ v in [0, ∞). 2) Assume that there is a non increasing (resp. decreasing) positive supersolution v (resp. subsolution w) Ea-compatible in R+ . Then (Ea.tad ) has a positive decreasing solution u ∈ C 2 ([0, ∞)) such that it holds T v ≤ u ≤ v (resp. w ≤ u ≤ T w) in [0, ∞). Theorem 6. 1) Assume that there are w and v which are Ea-compatible in [0, ∞) with Z ∞ {sf (s, w)}1/(p−1) < ∞. (2.tad ) 0

Then (Ea.tad ) has a solution u ∈ C 2 ([0, ∞)) such that w ≤ u ≤ v. 2) Assume that there is a non increasing positive supersolution v of (Ea.tad ) in [0, ∞) such that Rt R∞ ); i) V (r) = Iv(r) := r dt{ 0 (s/t)a f (s, v)ds}1/(p−1) satisfies (2.tad ii) V and v are Ea-compatible in [0, ∞). Then (Ea.tad ) has such a solution u with V ≤ u ≤ v. Similarily if there is a decreasing positive subsolution w such that w and Iw are Ea-compatible in [0, ∞) and which satisfies (2.tad ), then (Ea.tad ) has such a solution u with w ≤ u ≤ W := Iw.

271

Barrier Method

2

Proof of the theorems

2.1

Preliminaries

Let Cf (M ) := {φ ∈ C(IM ) | f (r, φ) > 0 ∀r ∈ IM } and b := 1/(p − 1). For some A = φ(0), define on Cf (M ) the operator T by b Z r Z t a dt (s/t) f (s, φ)ds . (3.tad ) Φ(r) := T φ(r) := A − 0

0

+ r f (r, φ) = 0 in IM , Φ(0) = A, Φ0 (0) = 0 and Φ0 ≤ 0. Then From [5], as b ≥ 1, ∀t ≤ M , with s∗ := max{1, s}, Z t b p−1 sp−1 f (s, φ)ds ; |Φ(t)| ≤ a+1−p 0 ∗ Z t b 1 0 p−1 |Φ (t)| ≤ s f (s, φ)ds . t∗ 0 ∗ Rt Rt As Φ00 (t) = −b{ 0 (s/t)a f (s, φ)ds}b−1 {f (t, φ) − at 0 (s/t)a f (s, φ)ds}, Dap Φ

a

|Φ00 (t)| ≤ b

Z

t

(s/t)a f (s, φ)ds 0

(4.tad ) (5.tad )

b−1 Z a t f (t, φ) + f (s, φ)ds . t 0

(6.tad )

Thus T Cf (M ) ⊂ C 2 (IM ) and for φ ∈ Cf (M ), |T φ|C 2 ([0,M] ≤

2 CM (φ)

a := A + a+1−p

Z 0

b

M

sp−1 f (s, φ)ds ∗ Z

+ b−1

M

+ b(a + 1)|f (., φ)|C(IM )

f (s, φ)ds

.

(7.tad )

0

Lemma 7. Let w, v be those in Theorem 3 and define EM (w, v) := {φ ∈ C 1 (IM ) | w ≤ φ ≤ v; w0 ≤ φ0 ≤ v 0 inIM }. Then with A ∈ [w(0), v(0)],

T EM (w, v) ⊂ EM (w, v) ∩ C 2 (IM ) .

Proof. Let V := T v and W := T w; then in IM w≤W ≤V ≤v

and

w0 ≤ W 0 ≤ V 0 ≤ v 0 .

In fact, as V 0 , v 0 ≤ 0, Dap V −Dap v = (ra {|v 0 |p−1 −|V 0 |p−1 })0 ≤ 0 whence |v 0 |p−1 ≤ |V 0 |p−1 or V 0 ≤ v 0 ≤ 0. Because V (0) ≤ v(0) we then have V ≤ v in IM . Similarily we have w0 ≤ W 0 and w ≤ W in IM . Also in the same way, w ≤ v and W (0) = V (0) imply that W 0 ≤ V 0 and W ≤ V . If φ ∈ EM (w, v) then f (r, v) ≤ f (r, φ) ≤ f (r, w) in IM , hence Φ := T φ satisfies W ≤Φ≤V

and

W 0 ≤ Φ0 ≤ V 0 .

272

Tadie

Corollary 8. Let v (w) be a non increasing (decreasing) positive supersolution (subsolution) which is Ea-compatible in IM . Then T EM (v) ⊂ EM (v) ∩ C 2 (IM ), where EM (v) ≡ EM (T v, v) ( T EM (w) ⊂ EM (w) ∩ C 2 (IM ), where EM (w) ≡ EM (w, T W )). Proof. In the light of Lemma 7, it is enough to notice that V := T v (W := T w) is a subsolution (supersolution) of (Ea.tad ) in IM . Lemma 9. Let w and v be as in Theorem 3. Then, T : EM (w, v) −→ C 1 (IM ) is continuous and T EM (w, v) is equicontinuous in C 1 (IM ). Proof. The continuity follows from the fact that for φ, ψ ∈ EM (w, v) and | |r denoting the norm in C([0, r]), Z r |(|(T φ)0 |p−1 − |(T ψ)0 |p−1 )(t)| ≤ k1 (θ)|φ − ψ|r (s/r)a f2 (s, k2 (θ)) , 0

where φ, ψ > θ > 0 in IM is assumed (see f2) ) and a similar bound for |T φ− T ψ| is obtained easily. The equicontinuity in C 1 follows from (7.tad ). 2.2

Proof of Theorems 3 and 4

Lemma 7 and Lemma 9 imply that T has a fixed point in EM (w, v) by the Schauder-Tychonoff’s fixed point theorem [2]; (6.tad )–(7.tad ) imply that the fixed point is in C 2 (IM ). In the same way Corollary 8 and Lemma 9 imply that T has such a fixed point in EM (v) (EM (w)). 2.3

Proof of Theorem 5

We prove 1) only as 2) and 3) would be simple readaptations. If (1.tad ) holds, then V := T v and W := T w are in E(w, v) ∩ C 2 ([0, ∞)). With (1.tad ), (4.tad )–(7.tad ) imply that ∀φ ∈ E(w, v) := E∞ (w, v), 2 |T φ|C 2 (IM ) ≤ C∞ (w)

∀M > 0.

(8.tad )

Let (Mk )k∈N be an increasing sequence such that Mk % ∞ and (uk := uMk ) the corresponding solutions in Ik := IMk . uk is extended by uk := T uk ∈ C 2 (R+ ), say, which satisfies (8.tad ) and Ea(uk ) = 0 in Ik , uk (0) = A. By means of the Schauder-Tychonoff’s fixed point theorem, such a required solution is an inductive limit of the (uk ) ([3]). 2.4

Proof of Theorem 6

Define this time the inverse operator of (Ea.tad ) in IM , K := KM on Cf (M ) by Z Φ(r) = Kφ(r) :=

Z

M

a

dt r

b

t

(s/t) f (s, φ)ds 0

.

273

Barrier Method

Rt Rt From Jensen’s inequality {(1/t) o sa f (s, φ)ds}b ≤ (1/t) 0 {sa f (s, φ)}b ds and simple integrations by parts, as in (4.tad )–(7.tad ), ∀t ∈ IM , Z b(a − 1) Φ(t) ≤

M b sb f (s, φ)b ds := IM (φ);

|Φ0 (t)| ≤ (1/t)Itb (φ)

0

and (6.tad ) holds for this case. If necessary, we replace f by f1 := λf such that Z ∞ [(p − 1)/(a − 1)] {sf1 (s, w)}1/(p−1) ds < v(0)

in (2.tad );

0

the required solution will be u(r) := u1 (µr) for some suitable µ = µ(λ), u1 being obtained with f1 . So, without major difficulties the proof of this Theorem follows the same steps as that of Theorem 5.

References [1] Istratescu, V. I., Fixed point theory. Math. and its Appli., Reidel Publ. 1981. [2] Kufner, A. et al., Function Spaces. Noordhoff 1977. [3] Tadie, Weak and classical positive solutions of some elliptic equations in Rn , n ≥ 3: radially symmetric cases. Quart. J. Oxford 45 (1994), 397–406. [4] Tadie, Semilinear ODE (part 2): Positive solutions via super-sub-solutions method. Proc. Prague Math. Conf. 1996, 320–323. [5] Yasuhiro, F., Kusano, T. and Akio, O., Symmetric positive entire solutions of second order quasilinear degenerate elliptic equations. Arch. Rat. Mech. Anal. 127 (1994), 231–254.



Elliptic Equations with Decreasing Nonlinearity II : Radial Solutions for Singular Equations Tadie Matematisk Institut, Universitetsparken 5 2100 Copenhagen, Denmark Email: [email protected]

Abstract. By means of the super-sub-solutions method from [3], the existence of decreasing solutions of some singular elliptic equations will be established.

AMS Subject Classification. 35J70, 35J65, 34C10

Keywords. p-Laplacian, integral equations

1

Introduction

Let f ∈ C 1 ([0, ∞); R+ ) with f (r) > 0 ∀r ≥ 0 and f (r) ' r−θ at ∞ for some θ > 0. For some a > 1 and p ∈ (1, 2], assume that f ) ∃b ∈ (0, a + 1 − p]; for w(t) := (1 + t)−b/(p−1) , some γ > 0 and Z ∞ −γ ψ(r) := f (r)w(r) , sb+p−1 ψ(s)ds < ∞. 0

In this note, we investigate the existence of positive and decreasing solutions u ∈ C 2 := C 2 ([0, ∞)) of Qu ≡(ra |u0 |p−2 u0 )0 + ra Fqν (r, u)+ = 0, where q > 0, or

Fνq (r, u)

u0 (0) = 0,

Fνq (r, u) := f(r)u−γ − νuq ,

:= νf(r)u

−γ

q

+u ,

ν ≥ 0,

ν > 0.

   (Q.tadd )

 

For a = n − 1, n ∈ N, such u is a radial solution in Rn of the p-Laplacian equations div(|∇u|p−2 ∇u) + Fqν (|x|, u)+ = 0. For a positive and decreasing function φ, define Z Φ(r) = T φ(r) := φ(0) −

1/(p−1)

t a

dt 0


Z

r

(s/t) 0

Fqν (s, φ)+ ds

.

276

Tadie

Given such a function φ, the following result from [3] will be used: Z ∞ assume that (1 + sp−1 )Fqν (s, φ)+ ds < ∞;

(φ.tadd )

0

if ∀r ≥ 0 Qφ ≥ 0 (≤ 0 respectively) and Fqν (r, .) is positive and decreasing in [Φ(r), φ(r)] ( [φ(r), Φ(r)] respect.), then (Q.tadd ) has a decreasing solution u ∈ C 2 ([0, ∞)) such that Φ ≤ u ≤ φ (φ ≤ u ≤ Φ respect.) in [0, ∞). The main results are the following: Theorem 1 (Uniqueness). Assume that ∀r ≥ 0 t 7→ Fqν (r, t)+ is decreasing in t > 0. Then a) ∀b ≥ 0, if it exists the decreasing solution ub ∈ C 1 of (Q.tadd ) such that lim∞ ub = b is unique; ) such that b) ∀R > 0, if it exists the decreasing solution u ∈ C 1 ([0, R)) of (Q.tadd u(R) = 0 is unique. Theorem 2 (Existence). Suppose that for some γ > 0 and b ∈ (0, a + 1 − p] Z ∞ sb+p−1 f (s)(1 + s)bγ/(p−1) < ∞. (1.tadd ) 0

1) Then, the equation (ra |u0 |p−2 u0 )0 + ra f (r)u(r)−γ = 0

(2.tadd )

has a unique positive and decreasing solution u ∈ C 2 := C 2 ([0, ∞)) such that u ≤ C r−b/(p−1)

(u ' r−b/(p−1) if b = a + 1 − p) at ∞;

2) if also q > max{p(p − 1)/b, −γ + θ(p − 1)/b}, i) there is ν0 > 0 depending only on f such that for ν ∈ (0, ν0 ] (ra |v 0 |p−2 v 0 )0 + ra {f (r)v(r)−γ − νv(r)q }+ = 0

(3.tadd )

has a unique decreasing and positive solution v ∈ C 2 ; if in addition q > (p − 1)(b + p)/b, then v(r) ≤ C r−b/(p−1) at ∞; ii) there is ν1 > 0 depending only on f such that ∀ν > ν1 (ra |U 0 |p−2 U 0 )0 + ra {νf (r)U −γ + U q } = 0

(4.tadd )

has a positive and decreasing solution U such that U ≤ C r−b/(p−1) at ∞.

2

Preliminaries

Definitions and notations: R b ∈ (0, a + 1 − p]; w(r) := (1 + r)−m ; v(s) := Rµ := 1/(p − 1); m := µb, v(s)ds; ψ(r) := f (r)w(r)−γ ; t∗ := max{1, t} and Dap u := (ra |u0 |p−2 u0 )0 .

277

Elliptic Equations

2.1

Properties of some integrals

Define for t ≥ 0

∞ Z r

Z J(t) := t

We normalized f so that Z 1 Z Ψ1 := 0

0

µ

r

1 + m

ψ

0

Lemma 3. If Z ∞ sb+p−1 ψ(s) < ∞

µ s a ψ(s) . r

Z

(5.tadd )

µ

∞

s

b+p−1

≤ 1.

ψ

0 < γ < (p − 1)

or

0

(θ − b − p) , b

where b ∈ (0, a + 1 − p], then ∀t ≥ 0 Z 1 µ (p − 1) sa ψ ≤ J(t) ≤ Ψ1 t−m ∗ ; a+1−p 0 Z 1 µ −m a s ψ(s)ds b = a + 1 − p =⇒ mJ(t) ≥ t |J(t)0 | ≤

Z

µ

1

0

Z

ψ

∞

+

0

(6.tadd )

0

µ

sb+p−1 ψ

0

(7.tadd )

(8.tadd ) ∀t > 1;

t−m−1 ; ∗

(9.tadd ) (10.tadd )

|J(t)00 | ≤ (a + 1)µ|J(t)0 |(µ−1)/µ |ψ|∞ ,

(11.tadd )

where (7.tadd ) is not necessary for the lower bound in (8.tadd ). Proof. We have Z Z ∞ r−m−1 r−a+b+p−1 J(t) = t

µ

r

Z ≤

sa ψ

0

on one hand and

Z 1 Z

J(t) ≤

ψ 0

Z

∞

+

0

r−m−1

t

µ

r

∞

r−m−1

1

Z

∞

µ sb+p−1 ψ

0

Z

∞

µ sb+p−1 ψ

0

on the other hand; the RHS of (8.tadd ) then follows from integrations by parts . For t ≤ 1, µ Z 1 µ Z ∞ Z r Z ∞ r−a sa ψ ≥ sa ψ r−aµ dr J(t) ≥ 1

0

0

0

and for t > 1 , Z

µ Z

1

J(t) ≥

a

s ψ 0

t

∞

r−aµ dr.

278

Tadie

We thus get the LHS of (8.tadd ). R 1 a µ R ∞ −m−1 If b = a + 1 − p, J(t) ≥ ( 0 s ψ) t r dr and (9.tadd ) follows. For t > 1, as a > b + p − 1, Z ∞ µ µ Z t 0 −b+1−p b+p−1 −m−1 b+p−1 0 ≤ −J(t) ≤ t s ψ ≤t s ψ . 0

0

R1 For t ≤ 1 |J(t)0 | ≤ ( 0 ψ)µ and (10.tadd ) is obtained. For (11.tadd ), Z −a J(r) = −µ r 00

µ−1

r a

s ψ

−ar

−a−1

Z

0

r

s ψ(s) + ψ(r) a

0

hence from Z |J(r)00 | ≤ µ(a + 1)|ψ|∞ r−a

∞

µ−1 sa ψ

0

(11.tadd ) follows. Lemma 4. Under the assumptions (6.tadd )–(7.tadd ) (ra |U 0 |p−2 U 0 )0 + ra ψ(r) = 0;

r≥0

(12.tadd )

has a decreasing and positive solution U ∈ C 2 ([0, ∞)) such that U (r) ≤ (1 + r)−b/(p−1)

∀r ≥ 0.

(13.tadd )

Proof. It is easy to verify that U = J where J is defined in (5.tadd ) satisfies (12.tadd ). Then (8.tadd )–(11.tadd ) complete the proof. 2.2

Proof of Theorem 1

Let u and v be two such solutions with u > v > 0 in some [0, R). As they are decreasing, from the equations, in [0, R) {ra (|v 0 |p−1 − |u0 |p−1 )}0 = ra {Fqν (r, v) − Fqν (r, u)} > 0 with ra (|v 0 |p−1 − |u0 |p−1 )|r=0 = 0, whence |v 0 | > |u0 | or v 0 < u0 ≤ 0 in (0, R). This implies that u(r) − v(r) > u(0) − v(0) whenever v(r) > 0. 2.3

Proof of Theorem 2

In the lights of the super-sub-solutions methods established in [3], it suffices for each case to find an appropriate sub- or supersolution of the problem. 1) The function U in Lemma 4 is a supersolution of (2.tadd ) as ψ(r) = f (r)(1 + r)bγ/(p−1) ≤ f (r)U (r)−γ .

Elliptic Equations

279

The estimate for the case b = a + 1 − p follows from (9.tadd ). 2) i) The solution v, say, obtained in 1) satisfies v(r) ≤ (1 + r)−b/(p−1) . F (r, v) = v q {f (r)v −(γ+q) − ν} ≥ v q {f (r)(1 + r)b(γ+q)/(p−1) − ν}. So, as f (r) > 0 everywhere, there is ν0 := inf r>0 [f (r)(1 + r)b(q+γ)/(p−1) ] such that if ν ≤ ν0 , then F (r, v) := f (r)v −γ − νv q ≥ 0 and ∂v F (r, v) ≤ 0. Then v is a suitable subsolution of (3.tadd ) as the condition (1.tadd ) of Theorem 5 of [3] is guaranteed by q > max{p(p − 1)/b, −γ + θ(p − 1)/b (see (φ.tadd ) ). R∞ Rt a If in addition q > (b + p)(p − 1)/b, then V (r) := r ( 0 (s/t) F (s, v)ds)µ dt is a supersolution of the equation with rb/(p−1) V (r) bounded. ii) For G(r, φ) := νf (r)φ−γ + φq , ∂φ G(r, φ) = qφ−1−γ {φq+γ − γνf (r)/q} := qΦ−1−γ Ψν (r), where Ψν (r) := (1 + r)−b(γ+q)/(p−1) − νγf (r)/q. If q > θ(p − 1)/b − γ and φ < (1 + r)−b/(p−1) , then for some large R > 0 there is Ψν (r) < 0; in this case there is ν1 := sup[0,R] q{γ(1 + r)b(γ+q)/(p−1) f (r)} such that ν > ν1 implies that G is decreasing in such positive φ. The solution v obtained in 1) is then a suitable supersolution of (4.tadd ). This work is dedicated to my late uncle Toam Chatue J.B., ( † on 14/08/1997).

References [1] Furusho, Y. On decaying entire positive solutions of semilinear elliptic equations. Japan J. Math. 14 #1 (1988), 97–118. [2] Kusano T. and Swanson C. A. Radial entire solutions of a class of quasilinear elliptic equations. J. Differential Equations 83 (1990), 379–399. [3] Tadie Elliptic equations with decreasing nonlinearity I : Barrier method for decreasing radial solutions. [4] Tadie Subhomogeneous and singular quasilinear Emden-type ODE. Preprint # 11, Series 1996, Copenhagen University.



Mathematical Models of Suspension Bridges: Existence of Unique Solutions Gabriela Tajˇcov´ a Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306 14 Plzeˇ n, Czech Republic Email: [email protected] WWW: http://home.zcu.cz/~gabriela/

Abstract. In this paper, we try to explain two mathematical models describing a dynamical behaviour of suspension bridges such as Tacoma Narrows Bridge. Our attention is concentrated on their analysis concerning especially the existence of a unique solution. Finally, we include an interpretation of particular parameters and a discussion of known and obtained results. This paper is based on our diploma thesis which deals with a qualitative study of dynamical structures of this type.

AMS Subject Classification. 35B10, 70K30, 73K05

Keywords. Nonlinear beam equation, jumping nonlinearities, periodic oscillations

1

Introduction and historical review

One of the most problematic and not fully explained areas of mathematical modelling involves nonlinear dynamical systems, especially systems with so called jumping nonlinearity. It can be seen that its presence brings into the whole problem unexpected difficulties and very often it is a cause of multiple solutions. An example of such a dynamical system can be a suspension bridge. The nonlinear aspect is caused by the presence of supporting cable stays which restrain the movement of the center span of the bridge in a downward direction, but have no influence on its behaviour in the opposite direction. Our paper sets a goal to develop a simple model describing the behaviour of the suspension bridge, to make its analysis which means to determine under what conditions the existence of a unique stable solution is guaranteed, and to find out safe parameters of the bridge constructions. We do not try to model the bridge in its full complexity, but on the other hand, we would like to avoid some over-simplifications. That is why we consider only one dimensional model and neglect the torsional motion, but we do not simplify the problem even more — e.g. by eliminating the space variable at all. This is the final form of the paper.

282

Gabriela Tajˇcov´ a

As a result of this effort, we describe the behaviour of the suspension bridge by one beam equation, or by a system of two coupled equations of “string-beam” type, respectively. As a motivation of our interest, we can mention the event which changed radically the common view of these nonlinear dynamical systems. On July 1, 1940, the Tacoma Narrows bridge in the state of Washington was completed and opened to traffic. From the day of its opening the bridge began to undergo vertical oscillations, and it was soon nicknamed “Galloping Gertie”. As a result of its novel behaviour, traffic on the bridge increased tremendously. People came from hundreds of miles to enjoy riding over a galloping, rolling bridge. For four months, everything was all right, and the authorities in charge became more and more confident of the safety of the bridge. They were even planning to cancel the insurance policy on the bridge. At about 7:00 a.m. of November 7, 1940, the bridge began to undulate persistently for three hours. Segments of the span were heaving periodically up and down as much as three feet. At about 10:00 a.m., the bridge started suddenly oscillating more wildly. At one moment, one edge of the roadway was twentyeight feet higher than the other; the next moment it was twenty-eight feet lower than the other edge. At 10:30 a.m. the bridge began cracking, and finally, at 11:00 a.m. the entire structure fell down into the river. The federal report on the failure of the Tacoma Narrows suspension bridge points out that the essentially new feature of this bridge was its extreme flexibility. Already, the Golden Gate bridge exhibited travelling waves, or in the Bronx Whitestone Bridge, large amplitude oscillations were observed of such a magnitude to make a traveller seasick. But due to a combination of damping and readjusted stays, they were not considered threatening to the structure. As soon as the more flexible Tacoma Narrows bridge was built, it began to exhibit complex oscillatory motion with an order of magnitude higher than that of earlier mentioned bridges. This resulted in a pronounced tendency to oscillate vertically, under widely differing wind conditions. The bridge might be quiet in winds of forty miles per hour, and might oscillate with large amplitude in winds as low as three or four miles per hour. These vertical oscillations were standing waves of different nodal types. They were not considered to be dangerous, and it was expected that the bridge would be stabilized by a combination of the same devices as in case of the Bronx Whitestone bridge. The second type of oscillation was observed just before the collapse of the bridge. It was a pronounced torsional mode with some of the cables alternately loosening and tightening. Sometimes the oscillations even preferred one end of the bridge to the other. These phenomenons caused that a large portion of the center span fell into the river. Subsequently, the entire structure was destroyed, and a new, much more expensive bridge of more conventional and less flexible design was built in its place.

Mathematical Models of Suspension Bridges

283

The first standard explanation (see e.g. M. Braun [4]) claims that the frequency of a periodic force caused by alternating trailing vortices just happened to be very close to the natural frequency of the bridge, and caused the linear resonance. Thus, even though the magnitude of the forcing term was small, this could explain the large oscillations and eventual collapse of the bridge. However, the federal report includes the following paragraph: “It is very improbable that resonance with alternating vortices plays an important role in the oscillations of suspension bridges. First, it was found that there is no sharp correlation between wind velocity and oscillation frequency, as is required in the case of resonance with vortices whose frequency depends on the wind velocity. Second there is no evidence for the formation of alternating vortices at a cross section similar to that used in the Tacoma bridge . . . It seems that it is more correct to say that the vortex formation and frequency is determined by the oscillation of the structure than that the oscillatory motion is induced by the vortex formation.” But the precise cause of the large-scale oscillations of suspension bridges has not been satisfactorily explained yet. The aspect which distinguishes the suspension bridges is their fundamental nonlinearity. As we have mentioned above, it is caused by the presence of supporting cable stays which restrain the movement of the center span in a downward direction, but have no influence on its behaviour in the opposite direction. This type of nonlinearity, often called jumping or asymmetric, has given rise to the following principle: Systems with asymmetry and large uni-directional loading tend to have multiple oscillatory solutions: the greater the asymmetry, the larger the number of oscillatory solutions, the greater the loading, the larger the amplitude of the oscillations. As we mentioned above, our paper tries to analyze such nonlinear dynamical systems and to bring some new pieces of information into this area. First of all, we present two possibilities how to model suspension bridges — by a single beam and by a beam coupled with a vibrating string by nonlinear cables — and give a brief survey of known facts in this field. Then we introduce our own results concerning existence and uniqueness of time-periodic solutions of two chosen models. We use two different attitudes. The first one is based on the Banach contraction theorem which needs some restrictions on the bridge parameters. The second one works in relatively greater generality but with an additional assumption of sufficiently small external forces. In the end, we summarize our intention and results and make a short discussion where we compare our foundations with known facts. We would like to emphasis that this paper is a short abstract of our diploma thesis [21] and that is why it does not contain proofs of the assertions stated here.

284

2


Mathematical models and known results

One of the easiest ways how to model a behaviour of a suspension bridge is to consider only one dimension. We do not have to take into account the other two dimensions because proportions of the bridge in these dimensions are very small in comparison with its length and so can be omitted (see Fig. 1). If we also neglect the influence of the towers and side parts, we can use a model of a simply supported one-dimensional beam. towers

@ @

side-cable

? 6

main cable

side-cable @ @ @ @ @ cable stays @ @ Q Q @ R @ Q R @ Q ? + s Q

Q k Q

Z } Z

Z

Q

center-span

Z

Z

side-span

3 6

Z

6

>

road-bed

side-span

Fig. 1. The main ingredients in a model of a one-dimensional suspension bridge.

2.1

Single beam equation

In the first idealization, the construction holding the cable stays can be taken as a solid and immovable object. Then we can describe the behaviour of the suspension bridge by a vibrating beam with simply supported ends. It is subjected to the gravitation force, to the external periodic force (e.g. due to the wind) and in an opposite direction to the restoring force of the cable stays hanging on the solid construction. Our system is illustrated on Fig. 2. The displacement u(x, t) of this beam is described by nonlinear partial differential equation: ∂ 4 u(x, t) ∂u(x, t) ∂ 2 u(x, t) = −κu+ (x, t) + W (x) + εf (x, t), + EI +b 2 ∂t ∂x4 ∂t with the boundary conditions m

u(0, t) = u(L, t) = uxx (0, t) = uxx (L, t) = 0, u(x, t + 2π) = u(x, t), − ∞ < t < ∞, x ∈ (0, L).

(1.taj )

(2.taj )

285

Mathematical Models of Suspension Bridges An immovable object

Nonlinear springs under tension

@ I @

C

6

C CW

HH

HH

H j H

A bending beam with supported ends

Fig. 2. The simplest model of a suspension bridge — the bending beam with simply supported ends, held by nonlinear cables, which are fixed on an immovable construction.

The meaning of particular parameters used in the equation is the following: m E I b κ W εf L

mass per unit length of the bridge, Young’s modulus, moment of inertia of the cross section, damping coefficient, stiffness of the cables (spring constant), weight per unit length of the bridge, external time-periodic forcing term (due to the wind), length of the center-span of the bridge.

As we can see from the equation (1.taj ) and the boundary conditions (2.taj ), we are describing vibrations of a beam of length L, with simply supported ends. Its deflection u(x, t) at the point x and at time t is measured in the downward direction. The first term in the equation represents an inertial force, the second term is an elastic force and the last term on the left hand side describes a viscous damping. On the right hand side, we have the influence of the cable stays, the gravitation force and the external force due to the wind (we assume it to be timeperiodic). The cable stays can be taken as one-sided springs, obeying Hooke’s law, with a restoring force proportional to the displacement if they are stretched, and with no restoring force if they are compressed. This fact is described by the expression κu+ , where u+ = max{0, u} and κ is a coefficient, which characterizes the stiffness of the cable stays. We have not considered the inertial effects of the rotation motion (in a plane xu) in the equation since they are usually omitted. This model was introduced e.g. in a paper [16] by P. J. McKenna and A. C. Lazer and is used as a starting point for study of suspension bridges in the most of cited works by the other authors. It does not describe exactly the behaviour of a suspension bridge but on the other hand it is reasonably simple and applicable.

286


For further considerations, it would be useful to transform the equation (by making a change of the scale of the variable x and dividing by the mass m) to the following form: utt + α2 uxxxx + βut + ku+ = W (x) + εf (x, t), u(0, t) = u(π, t) = uxx (0, t) = uxx (π, t) = 0,

(3.taj )

u(x, t + 2π) = u(x, t), − ∞ < t < ∞, x ∈ (0, π), where α2 = EI m W , ε and f .)

π 4 L

6= 0 and β =

b m

> 0. (We use the same symbols for rescaled

As for as the results, which are known for this model, we can mention the theorem proved (by the degree theory) in [5] by P. Drábek. It says that the problem (3.taj ) has at least one solution for an arbitrary right hand side. Further, there is proved that in case that there is no external force (it means no wind), the bridge achieves a unique position (called the equilibrium) determined only by its weight W (x). Under some special assumptions on W (x), the paper [5] shows that in case of small external disturbances, there is always a solution “near” to the equilibrium. If we assume that W (x) = W0 sin x and a periodic function f (x, t) is of a special form then there is another solution which is in a certain sense “far” from this position. Another known result concerns the case when the damping term is equal to zero. This was studied by W. Walter and P. J. McKenna in paper [19]. Under an additional assumption α = 1 they proved the theorem which says that if W (x) ≡ W0 (positive constant) and f (x, t) is even and π-periodic in the time variable t and symmetric in the space variable x about π2 , then, if 0 < k < 3, the equation (3.taj ) has a unique periodic solution of the period π, which corresponds to small oscillations about the equilibrium. If 3 < k < 15, the equation has in addition another periodic solution with a large amplitude. In other words, this theorem says that strengthening the stays, which means increasing the coefficient k, can paradoxically lead to the destruction of the bridge. The similar result can be proved for the system of ordinary differential equations which we obtain from the equation (3.taj ) using the spatial discretization. The theorem proved in [1] by J. M. Alonso and R. Ortega says that if the condition k < β 2 + 2αβ holds then there exists N0 ∈ N such that if N ≥ N0 then the discretization of a suspension bridge equation has a unique bounded solution that is exponentially asymptotically stable in the large. This result has a similar sense as the previous one — the more flexible the cable stays are, then the better the situation is and oscillations of the bridge cannot be too high.


2.2

287

“String-beam” system

Another possible but a little more complicated process is not to consider the construction holding the cable stays as an immovable object, but to treat it as a vibrating string, coupled with the beam of the roadbed by nonlinear cable stays (see Fig. 3). The cable represented by a vibrating string

A v(x, t) AA 6 U ?

AA K A

OC S o S C S C S C 6 C u(x, t) S S C ? S C Nonlinear springs

The vibrating beam with supported ends

Fig. 3. A more complicated model of a one-dimensional suspension bridge — the coupling of the main cable (a vibrating string) and the roadbed (a vibrating beam) by the stays, treated as nonlinear springs. Instead of one equation, we have now a system of two connected equations in the following form: m1 vtt − T vxx + b1 vt − κ(u − v)+ = W1 + εf1 (x, t), m2 utt + EIuxxxx + b2 ut + κ(u − v)+ = W2 + εf2 (x, t),

(4.taj )

with boundary conditions u(0, t) = u(L, t) = uxx (0, t) = uxx (L, t) = v(0, t) = v(L, t) = 0, where v(x, t) measures the displacement of the vibrating string representing the main cable and u(x, t) means — as in the previous section — the displacement of the bending beam standing for the roadbed of the bridge. Both functions are considered to be periodic in the time variable. The nonlinear stays connecting the beam and the string pull the cable down, hence we have the minus sign in front of k(u − v)+ in the first equation, and hold the roadbed up, therefore we consider the plus sign in front of the same term in the second equation. We can transform both equations into a simpler form in the same way as in the previous section. It means that we divide by the mass m1 , and m2 respectively, and change the scale of the space variable x. Then we obtain

288


vtt − α21 vxx + β1 vt − k1 (u − v)+ = W1 + εf1 (x, t), utt + α22 uxxxx + β2 ut + k2 (u − v)+ = W2 + εf2 (x, t), u(0, t) = u(π, t) = uxx (0, t) = uxx (π, t) = v(0, t) = v(π, t) = 0,

(5.taj )

− ∞ < t < ∞, x ∈ (0, π), b1 b2 π 2 EI π 4 κ κ , α22 = m where α21 = mT1 L L , k1 = m1 , k2 = m2 , β1 = m1 and β2 = m2 . 2 We use the same symbols as in the previous equations for the other transformed parameters. We can find a description of this model again in A. C. Lazer and P. J. McKenna [16], but these authors consider the right hand sides in a rather purer form. In the first equation, they neglect the weight of the string W1 , and on the other hand, in the second equation, they ignore the external force εf2 (x, t). However, nobody (as far as we know) has treated this model in detail yet.

3

Application of Banach contraction principle

As we can see from the previous survey of known results, one of the problems is to prove the existence of the solutions of particular models and find out the conditions, under which the solution is unique and stable. In particular, it means that we are looking for conditions which guarantee that the bridge cannot exhibit large-scale oscillations and cannot be destructed by any wind of an arbitrary power. We have tried to clear up these problems with use of Banach contraction principle for both one-dimensional models — the first one considers the bridge as a single beam supported by nonlinear springs, and the second one describes the bridge as a beam coupled with a string by nonlinear cables. 3.1

The first case — a single beam

As we stated above, we model the suspension bridge as a one-dimensional beam with simply supported ends, which is held by nonlinear springs hanging on an immovable construction. This situation is described by the boundary value problem (3.taj ). Let us denote Ω = (0, π) × (0, 2π) the considered domain, H = L2 (Ω) the usual Hilbert space with the corresponding L2 -norm Z 12 2 ku(x, t)k = |u(x, t)| dxdt Ω

and D the set of all smooth functions satisfying the boundary conditions from equation (3.taj ). Now we can generalize the notion of a classical solution by which we mean a continuous function with continuous derivatives up to the fourth order with respect to x and up to the second order with respect to t in the set [0, π] × [0, 2π], satisfying the boundary value problem (3.taj ), and define a so called generalized solution of (3.taj ).

289


Definition 1. A function u(x, t) ∈ H is called a generalized solution of the boundary value problem (3.taj ) if and only if the integral identity Z Z 2 u(vtt + α vxxxx − βvt ) dxdt = (W + εf − ku+ )v dxdt Ω

Ω

holds for all v ∈ D. Remark 2. We can extend the generalized solution u = u(x, t) by 2π-periodicity in t to (0, π) × R. So, any generalized solution can be regarded as a function defined on (0, π) × R. ˜ = H + iH. As the set Let us consider a complex Sobolev space H {eint sin mx; n ∈ Z, m ∈ N} forms a complete orthogonal system in this space, each function u(x, t) can be represented by Fourier series u(x, t) =

∞ ∞ X X

unm eint sin mx.

(6.taj )

n=−∞ m=1

Moreover, we have XX n

|unm |2 < ∞,

and u−nm = u ¯nm

m

(see J. Berkovits and V. Mustonen [3]). Let p, r ∈ Z+ . If we use this Fourier interpretation, we can define the following spaces H p,r = {h ∈ H;

∞ ∞ X X

(n2r + m2p )|hnm |2 < ∞}

(7.taj )

n=−∞ m=1

and the corresponding norm khkH p,r =

∞ ∞ X X

! 12 (n2r + m2p )|hnm |2

.

(8.taj )

n=−∞ m=1

Then H p,r equipped with the norm k · kH p,r is the Sobolev space. In particular, H 0,0 = H. First of all, we will treat the solvability of the linear equation utt + α2 uxxxx + βut − λu = h.

(9.taj )

If we define a generalized solution of this equation in an analogous way as in Definition 1, then the following lemma is a consequence of the expansion (6.taj ) (cf. [2]).

290


Lemma 3. If unm and hnm are the corresponding Fourier coefficients of the functions u and h, then the equation (9.taj ) has a generalized solution if and only if (−n2 + α2 m4 + iβn − λ)unm = hnm

(10.taj )

holds for all n ∈ Z, m ∈ N. If we denote L(u) = utt + α2 uxxxx + βut the linear operator, and put Nλ = {(m, n) ∈ N × Z; α2 m4 − n2 − λ = 0}, S = {λ ∈ R; Nλ 6= ∅}, σ = {λ ∈ R; λ = α2 q 4 , q ∈ N}, then σ is a set of eigenvalues of the operator L, and σ ⊂ S holds. Further, we can rewrite the equation (9.taj ) into a new form L(u) − λu = h and formulate the following theorem (for the proof see G. Tajˇcov´ a [20]). Theorem 4. Let λ ∈ R. Then for an arbitrary h ∈ H the equation (9.taj ) has a unique generalized solution u ∈ H if and only if λ 6∈ σ. If λ 6∈ σ, then there exists a mapping Tλ : H → H,

Tλ : h 7→ u

with the following properties: ¯ (i) Tλ is linear and R(Tλ ) ⊂ C(Ω); (ii) Tλ : H p,r → H p+2,r+1 and there exists a constant c > 0 such that for any h ∈ H p,r , p, r ∈ N ∪ {0}, we have kukH p+2,r+1 ≤ ckhkH p,r , whenever u = Tλ h; ¯ (and thus from H into H) and for its (iii) Tλ is compact from H into C(Ω) norm we have kTλ k ≤

1 = max{dist(λ, S), min{β, dist(λ, σ)}} =

1 . min{dist(λ, σ), max{β, dist(λ, S)}}

291


Now we turn our attention to the equation (3.taj ) and deal with its solvability. As zero is not an eigenvalue of the operator L, we can rewrite this equation — in accordance with the previous paragraph — into an equivalent form u = T0 (−ku+ + W + εf ).

(11.taj )

Moreover, we have for the norm of the operator T0 the following estimate kT0 k ≤

max

m∈N, n∈Z

1 1 p = K0 . ≤ min{α2 , β} β 2 n2 + (α2 m4 − n2 )2

If we want to find out conditions for the existence of a unique solution, it is suitable to use the Banach contraction principle which reads as follows: Let the operator G : H → H be a contraction, i.e. there exists c ∈ (0, 1) such that kG(u) − G(v)k ≤ cku − vk ∀u, v ∈ H. Then there exists a unique u0 such that G(u0 ) = u0 . In our case G(u) = T0 (−ku+ + W + εf ) and kG(u) − G(v)k = kT0 (W + εf − ku+ ) − T0 (W + εf − kv + )k = = kT0 (kv + − ku+ )k ≤ ≤ kkT0 kkv + − u+ k ≤ ≤ kK0 kv − uk. If we require the operator G to be a contraction, the condition 0 < kK0 < 1 must be satisfied, and thus 0
0 (ii) Tλ : H × H → H such that for any h ∈ H p,r × H p,r , p, r ∈ N ∪ {0}, we have kwkH p+1,r+1 ×H p+2,r+1 ≤ ckhkH p,r ×H p,r , whenever w = Tλ h. ¯ × C(Ω) ¯ (and thus from H into H), and (iii) Tλ is compact from H into C(Ω) for its norm we have an estimate 1 1 , kTλ k ≤ max max λ ; max λ m,n |Anm | m,n |Bnm | where Aλnm = −n2 + α21 m2 + iβ1 n − λ, λ = −n2 + α22 m4 + iβ2 n − λ. Bnm As zero is not the eigenvalue of the operator L, we can define the operator T0 and to estimate its norm as follows 1 1 . kT0 k ≤ max max 0 ; max 0 m,n |Anm | m,n |Bnm | Further, 1 1 1 = max = max p ≤ 2 2 m,n | − n2 + α2 m2 + iβ1 n| m,n |A0nm | β1 n + (α21 m2 − n2 )2 1 1 , ≤ min{α21 , β1 } 1 1 1 = max p 2 max 0 = max ≤ 4 + iβ n| 2 m,n |Bnm | m,n | − n2 + α2 m,n m 2 β2 n + (α22 m4 − n2 )2 2 1 . ≤ min{α22 , β2 } max m,n

294


Hence we finally obtain kT0 k ≤ max

1 1 ; 2 min{α1 , β1 } min{α22 , β2 }

= =

1 ¯ 0. =K min {α21 , α22 , β1 , β2 }

(17.taj )

If we use this operator T0 , we can rewrite our equation (16.taj ) in the equivalent form w = T0 (h − F(w)).

(18.taj )

Since we want to prove its unique solvability, it is again suitable to apply the Banach contraction principle. In our case G(w) = T0 (h − F(w)). We have to verify, whether this operator is a contraction: kG(w1 ) − G(w2 )k = kT0 (h − F(w1 )) − T0 (h − F(w2 ))k = = kT0 kkF(w2 ) − F(w1 )k ≤ ≤ kT0 k(k1 + k2 )k(u2 − v2 )+ − (u1 − v1 )+ k ≤ ≤ kT0 k(k1 + k2 )k(u2 − v2 ) − (u1 − v1 )k = = kT0 k(k1 + k2 )k(u2 − u1 ) − (v2 − v1 )k ≤ ≤ kT0 k(k1 + k2 ) [ku2 − u1 k + kv2 − v1 k] ≤ ¯ 0 kw2 − w1 k. ≤ (k1 + k2 )K Hence it follows that the operator G is a contraction if the condition ¯0 < 1 0 < (k1 + k2 )K holds. Equivalently, k1 + k2 < min α21 , α22 , β1 , β2 . As we have k1 = mκ1 , k2 = mκ2 , we obtain a condition of the existence of a unique solution of the operator equation (16.taj ) in the following form

κ
0,

(u0 )x (π, t) < 0

for every t ∈ R. Remark 9. In particular, this means that the equation utt + α2 uxxxx + βut + ku+ = 0 has due to uniqueness only a trivial generalized solution for any k ∈ R.

(22.taj )

296


Our main result is the following. Theorem 10. Let ε ∈ R, k > 0, W (x) ≡ W0 > 0, f ∈ H 1,1 . Then there exists ε0 > 0 such that for |ε| < ε0 the problem (3.taj ) has a unique generalized solution u ∈ H 3,2 . Moreover, this generalized solution is strictly positive in (0, π) × R. The proof of this main result would be carried out in several steps. We know that there exists at least one generalized solution of the equation (3.taj ) for any right hand side (see P. Drábek [5]). Moreover, by Proposition 8, there exists a positive, time-independent solution u0 (x, t) = u ˜0 (x) of the equation utt + α2 uxxxx + βut + ku+ = W0 , ˜00 (π) < 0. with u ˜00 (0) > 0 and u Step 1. We prove that there exists a positive generalized solution u ∈ H 3,2 of (3.taj ) which is “close” to u0 from Proposition 8 with respect to the norm in H 3,2 . Step 2. There is no other positive generalized solution of (3.taj ) than u = u0 + uε . Step 3. There is no other generalized solution of (3.taj ) (changing signs) than u ˜ε = u0 + uε if |ε| < ε0 and ε0 is small enough. (For the complete proof see G. Tajˇcov´ a [21] or J. Berkovits, P. Drábek, H. Leinfelder, V. Mustonen and G. Tajˇcov´ a [2].) 4.2

The second case — the coupling of a beam and a string

Now, we can try to apply the previous ideas on the system of two coupled equations which model the suspension bridge as a simply supported beam and a string connected by nonlinear cable stays. We work again with a periodic-boundary value problem (5.taj ). We would like to formulate a similar assertion as in the previous section, it means to prove under some additional assumptions that if the weight of the bridge W1 and the weight of the main cable W2 are constant and the external forces εf1 (x, t) and εf2 (x, t) are sufficiently small, then our problem (5.taj ) has a unique solution which is symmetric and strictly positive in its both components and close to the stationary solution. However, as it can be seen later, we are not able to overcome some problems with regularity of the solution and thus we formulate statements which are more general and — in some sense — weaker. Similar argument as that used in [2] enables us to prove the following assertion.

297


Proposition 11. Let u, v ∈ H and h1 , h2 ∈ H, h1 , h2 are independent of t. Then [v, u]T is a generalize solution of vtt − α21 vxx + β1 vt − k1 (u − v)+ =h1 (x), utt + α22 uxxxx + β2 ut + k2 (u − v)+ =h2 (x)

(23.taj )

if and only if the functions v, u are independent of the variable t and [˜ v (x), u ˜(x)]T T = [v(x, t), u(x, t)] is a solution of the boundary value problem u − v˜)+ = h1 (x), −α1 v˜00 − k1 (˜ ˜(4) + k2 (˜ u − v˜)+ = h2 (x) in (0, π), α2 u ˜00 (π) = 0. v˜(0) = v˜(π) = u ˜(0) = u ˜(π) = u˜00 (0) = u

(24.taj )

As for as the uniqueness of the solution, the following statement holds. Proposition 12. Let k1 , k2 > 0 and h1 , h2 ∈ H, h1 and h2 are independent of t. Then (23.taj ) has at most one generalized solution w0 = [v0 , u0 ]T ∈ H which is independent of t. Remark 13. As a consequence of Propositions 11 and 12, we can state that for ε = 0 and W1 = W2 = 0 (it means no loading), the nonlinear system (5.taj ) vtt − α21 vxx + β1 vt − k1 (u − v)+ = 0, utt + α22 uxxxx + β2 ut + k2 (u − v)+ = 0, with standard string-beam boundary conditions, has only a trivial solution. Now, we have all auxiliary assertions to formulate the following theorem concerning the general existence of a solution of the system (5.taj ) for an arbitrary right hand side. The proof is based on the degree theory and is a direct analogy to the proof by P. Drábek in [5]. Theorem 14. Let ε ∈ R, k1 , k2 > 0, W1 (x), W2 (x) ∈ L2 (0, π), and f1 (x, t), f2 (x, t) ∈ H. Then the system (5.taj ) has at least one generalized solution w = [v, u]T ∈ H. Now, we can have a look at the case when the right hand sides are constant functions. It means that the corresponding solution is (according to Proposition 11) a stationary solution and should express the equilibrium of the suspension bridge. By a detailed analysis of a linear system −γ1 v 00 − u + v = h1 , x ∈ (0, π), γ2 u(4) + u − v = h2 , 00

(25.taj ) 00

v(0) = v(π) = u(0) = u(π) = u (0) = u (π) = 0 we can prove the following assertion.

298


Proposition 15. Assume in a boundary value problem (5.taj ) that W1 (x) ≡ W1 and W2 (x) ≡ W2 are nonzero constants and ε = 0. Moreover, let the weight W2 is “large enough”. Then (5.taj ) has a unique generalized solution w0 which is positive, time-independent, symmetric with respect to the line x = π2 in its both components and satisfies u0 (x, t) > v0 (x, t)

∀(x, t) ∈ (0, π) × R

and (u0 − v0 )x (0, t) > 0,

(u0 − v0 )x (π, t) < 0

for every t ∈ R. Now, we can have a closer look at the solution of the system (5.taj ) and its properties. We would like — on the basis of the previous statements — to formulate the analogy of Theorem 10. However, the only thing we know is that there exists at least one generalized solution of the boundary value problem (5.taj ) and, moreover, (see Proposition 15), that under some additional assumptions, there exists a symmetric, strictly positive, time-independent solution w0 = [v0 , u0 ]T of the system vtt − α21 vxx + β1 vt − k1 (u − v)+ = W1 , utt + α22 uxxxx + β2 ut + k2 (u − v)+ = W2 , where W1 and W2 are positive constants, and the conditions (u0 − v0 )x (0, t) > 0,

(u0 − v0 )x (π, t) < 0,

hold. But we are not able to prove the existence and uniqueness of the solution of the system (5.taj ) which would be “close” to this w0 . The obstacle is the fact that we have not manage to prove a better regularity that w ∈ H 2,2 × H 3,2 . It means (due to embedding theorems) that w ∈ C 0,0 × C 1,0 . And this is not enough to guarantee the existence of the positive solution w = w 0 + wε neither for ε sufficiently small.

5

Final remarks and discussion

In this chapter, we would like to clear up our results and compare them with known facts mentioned in Chapter 2. Our main effort was to determine sufficient conditions for the existence and uniqueness of the solution. Let us have a closer critical look at them.


5.1

299

The application of Banach contraction principle

First of all, we dealt with the uniqueness using the Banach contraction principle. The price we had to pay was a certain restriction on the magnitude of the stiffness κ of the cable stays. For the single beam model, it was in the roughest form (cf. (12.taj )) κ < m min{α2 , β}.

(26.taj )

The corresponding result for the string-beam model was (cf. (19.taj )) κ

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Equadiff 9: Conference on Differential Equations and their Applications : Proceedings

introduction to differential equations and their applications

Nonlinear partial differential equations and their applications

Equadiff 10: Czechoslovak international conference on differential equations and their applications, Prague, August 27-31, 2001 : papers

An elementary treatise on differential equations and their applications

Differential equations and applications

Differential equations and applications

Lectures on functional equations and their applications

Proceedings of the conference on differential and difference equations and applications

Proceedings of the Conference on Differential & Difference Equations and Applications

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Differential Equations: Theory and Applications

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