FINITE ELEMENT METHODSFOR INTEGRODIFFERENTIAL EQUATIONS
SERIES ON APPLIED MATHEMATICS Editor-in-Chief: Frank Hwang Associate Editors-in-Chief: Zhong-ci Shi and U Rothblum
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Finite Element Methods for Integrodifferential Equations by C. M. Chen and T. M. Shih
Series on Applied Mathematics Volume9 A
FINITE ELEMENT METHODS FOR INTEGRODIFFERENTIAL EQUATIONS Chen Chuanmiao Hunan Normal University
Shih Tsimin
The Hong Kong Polytechncc University
World Scientific Singapore • NewJersey London •HongKong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Chen, Chuanmiao. Finite element methods for integrodifferential equations / by Chen Chuanmiao and Shih Tsimin. p. cm. - (Series on applied mathematics ; v. 9) Includes bibliographical references and index. ISBN 9810232632 (alk. paper) 1. Integro-differential equations - Numerical solutions. 2. Finite element method. I. Shih, Tsimin. II. Title. III. Series. QA431.C463 1997 515'.38-dc21 97-18449 CIP
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Preface It is known that many physical phenomena, such as heat conduction and dif fusion, can be reduced to a parabolic differential equation (PDE): ut - Au = f in Q = fi x J J = (0, T], u(x,t)=Q on 9 0 x 7 , u(x,0) — uo(x) in O,
(0.1)
where O is a bounded domain in d-dimensional space Rd and dQ is the bound ary of ft. In the past century, in many fields of science and engineering, such as heat conduction in materials with memory, viscoelasticity and reactor dy namics, there appeared a parabolic integrodifferential equation (PIDE) of the following form (
r* ut — Au — I Bu(s)ds + / I u(x, t ) = 0 on 3 0 x J , \ u(x,Q) = UQ(X) in Q,
in
Q = Q x J, ^' '
where B in the Volterra integral (memory term) is an arbitrary partial dif ferential operator of order f3 < 2. In general, this integral term reflects the memory, feedback or other mechanisms of a dynamical system. In order to explain the important role of the Volterra integral term we consider the following special case, ut==
- s)Au{s)ds f K(t K{t-
Jo
+ /.
If K(t) = 1, after differentiation with respect to £, it leads to the wave equa tion. If K(t) is the Delta function it becomes the heat equation. Therefore, the integrodifferential equation is an intermediate state of the heat and wave v
VI
Preface
equations, which possibly possesses the double features, i.e. propagating with finite speed and smoothing t h e discontinuities. It seems t h a t the importance of integrodifferential equations includes the following: 1. Many physical phenomena as mentioned above can be accurately de scribed by equations with memory; 2. There appear some new physical and mathematical properties when t h e kernel K(t) is singular; 3. For nonlinear integrodifferential equations, both parabolic and hyper bolic t y p e , when t h e kernel has a certain singularity, it is possible to obtain t h e global solution even for large data. Naturally, t h e presence of t h e Volterra integral term also causes some new difficulties in b o t h theoretical analysis and numerical computation. T h e first contribution to the numerical solution of P I D E was m a d e by J. Douglas and B. Jones in 1960s using t h e finite difference method. Many results in this direction were summarized by Brunner (1982). T h e early work on finite element analysis for P I D E was given by B. Neta (1982). In recent decade, many researchers, such as V. Thomee, I.H. Sloan, L.B. Wahlbin, G. Fairweather, E.G. Yanik, J.R. Cannon, Y.P. Lin, N.Y. Zhang, M.N. LeRoux, C. Lubich, W . Mclean, A.K. Pani, Y.Q. Huang , T. Zhang, and C M . Chen et al., have done valuable work in applications of finite element methods to integrodifferential equations and have attained remarkable achievements. T h e difficulties caused by the Volterra integral term mainly come from t h e following aspects: limitation on t h e integrability of solutions, especially in cases where the equations have nonsmooth initial values, weakly singular kernels or nonlinearities, and discretization of the integral term with respect to time t causes large amount of computation and huge computer storage etc. T h e purpose of this book is to summarize t h e results obtained in this di rection and give a clearer picture of researches in this field. Of t h e 13 chapters, the first two chapters introduce the practical background and the m a t h e m a t ical properties of solutions to parabolic integrodifferential equations, and the remaining chapters are devoted to the finite element approximations to inte grodifferential equations. T h e main focus of the book is on parabolic problems, except in the last two chapters where the hyperbolic problems and problems with positive memory are briefly discussed. For completeness, the theory of partial differential equation is briefly quoted in some sections, but t h e emphases are on investigating t h e inherent properties related to integrodifferential equa tions, t h e differences between t h e m and the difficulties involved in the finite element solution to integrodifferential equations. T h e contents of each chapter
Preface
vn
are summarized as follows. T h e first chapter introduces the practical background of integrodifferential equations. Many problems in various fields of science and engineering can be formulated as integrodifferential equations. In particular, heat flow in materi als with memory, viscoelasticity, biomechanics, nuclear reactor dynamics and pressure in porous medium. Besides, some new properties of relevant IDE, b o t h parabolic and hyperbolic, and their situation in recent research are briefly in troduced. We hope t h a t the chapter will be of interest to b o t h engineers and m a t h e m a t i c a l analysts. T h e second chapter discusses the mathematical solvability of linear P I D E , mainly t h e regularity estimates of solutions. First, the energy method is used to obtain the estimates in L2 space. Further, we use the known results of P D E to derive t h e corresponding regularity estimates of P I D E in Lp and C a spaces. In this chapter, the Volterra integral term is always treated as a perturbation to a purely parabolic problem. Various forms of the Gronwall inequality will play an i m p o r t a n t role throughout this book. Second, t h e semigroup m e t h o d is used to get some fine estimates of P I D E . There have been many publications concerning the solvabilities of P I D E . We select only some of the results which will be used in later chapters. In C h a p t e r 3 finite element methods of elliptic problem are summarized. We introduce briefly some useful results, in particular, results for curved bound ary domain, and some important operators such as the interpolation operator I hi t h e L 2 -project ion operator P^ and the Ritz-projection operator JR^,. T h e Ritz—Volterra projection operator Vh proposed by Cannon and Lin are dis cussed in detail because it is useful throughout the present book and is partic ularly i m p o r t a n t in the study of super convergence in Chapter 10. T h e remaining 10 chapters are entirely dedicated to the finite element anal ysis of integrodifferential equations. Three methods, i.e. energy method, semi group (or operator) m e t h o d and Green function (or weighted norm) method will be used in various chapters. In C h a p t e r 4 discretization of the spatial variables is first considered. T h e energy m e t h o d is used here to obtain the following error estimate in L2(Q) space: \\Uh(t) - U(t)\\ < Chr(\\u(0)\\r
+ / \\ut\\rds). Jo
(0.3)
T h e time variable t is also discretized. T h e time-stepsize is chosen to be k = T/N, tn — nk, n — 0 , 1 , . . . , N. T h e backward Euler scheme as well as the Crank—Nicolson scheme are discussed. T h e proofs in this chapter are basically
Preface
Vlll
the same as those for the purely parabolic case, but the Gronwall inequality will be applied repeatedly. Instead of RhU in parabolic case, V^u will be used as a comparison function. In C h a p t e r 5 the discretization of t h e integral term with respect to t is especially emphasized. In t h e calculation of U71 by the above schemes, all t h e values of C/-7 obtained in the previous n levels are used, which will occupy a large a m o u n t of computer storage. In order to reduce t h e storage needed, Sloan and T h o m e e proposed an economic scheme, i.e. the Volterra integral t e r m is evaluated by high accuracy quadrature formulae with a larger stepsize k\. T h e storage requirement decreases significantly from O(N) to 0(N1^2) 1//4 even t o 0(7V ). However, the solution should be more smooth and the prop erties of t h e coefficient matrix become worse. Moreover, an additive scheme proposed by Y.Q. Huang is introduced. Smooth kernels are approximated by piecewise polynomials with a larger time stepsize. T h e computer storage needed is further reduced to 0(lnN) and the smoothness requirement on the solution is also weakened. In C h a p t e r 6 the case with nonsmooth initial value u G L2 is discussed. It is different from the parabolic differential equation in b o t h the regularity of the solution for homogeneous parabolic integrodifferential equation and t h e optimal order of the finite element approximation are restricted by t h e Volterra integral term. For instance, when t h e initial value u0 G L 2 ( 0 ) , the following sharp estimate of regularity \\u(t)\\„ 0 is the constant of thermal conductivity, and the heat flux vector q is linearly dependent on \/u, the gradient of the temperature. It is also assumed t h a t t h e internal energy e depends linearly on the t e m p e r a t u r e e — eo 4- Cu, where C is the capacity and eo > 0 is a constant. Substituting the above two relations into (1.1), it leads to the classical linear heat equation ut = a2Au 4- / ,
where a2 = k/(Cp),
f = h/{Cp).
(1.4)
This theory has, however, two shortcomings. First, it is unable to account for memory effects which may be prevalent in some materials, particularly at low t e m p e r a t u r e . Secondly the heat equation (1.4) predicts an unrealistic result t h a t a thermal disturbance at one point propagated instantly to everywhere of the b o d y (though not equally). W h e n the domain is an infinite rod and / = 0, this can be seen from the following well-known Poisson formula
u{x t]
I
r°°
/
(£-x)2\
' = uTt Lexp r 4^r)««• °>*- < > °-
If t h e initial value is u(£, 0) = 1 in a small interval (xo, #o -+- e) and equals zero elsewhere, then u(x,t) > 0 for any x and t > 0. This implies t h a t by Fourier heat conductors, the thermal discontinuities must propagate with infinite speed (it also indicates t h a t the initial discontinuities are at once smoothed o u t ) . These observations lead one to believe t h a t Fourier's law may be a limiting approximation (perhaps for sufficiently steady t e m p e r a t u r e fields). Coleman [31] (1964), G u r t i n and Pipkin [67] (1968) proposed a nonlinear memory theory of heat conduction which is independent of the present value of \/u and has finite wave speed. W h e n this constitutive relation is linearized, it leads to t h e following heat flux relation q(t) = -
I K(s) V u(t - s)ds Jo
(1.5)
for isotropic materials. T h e special form has proved useful in describing the transmission of heat pulses observed in liquid helium and some dielectrics at low t e m p e r a t u r e .
4
Chapter 1. Some Practical Problems and Their Properties
Coleman and Gurtin [32] (1967), and Nunziato [125] (1971) further con sidered a memory theory which also depends upon the present value of \/u, i.e., the heat flux is given by q(t) = - a ( 0 ) V u{t) -
I
a'{s) v u(t - s)ds,
(1.6)
J — oo
where a(s) is the heat conduction relaxation function and a(0) > 0. Clearly, (1.6) reduces to (1.4) if a'(s) = 0, and to (1.5) if a(0) = 0. A similar relation for the internal energy e(t) = e0 + b(0)u(t) + /
b'(t)u(t - s)ds
(1.7)
J — CO
can also be assumed. Thus it leads to a new heat equation b(0)ut - a(0)Au - b'(0)u = [ [a'(s)Au(t - s) - b"(s)u(t - s)]ds + ft.
(1.8)
Jo
Miller [120] (1978) studied the existence and uniqueness of the solution and its dependence on parameters. Further, Nohel [124] (1981) systematically studied the nonlinear relations q(t) = -a(0)(p(ux) - / a!(t - s)<j>(ux(s))ds, Jo and, more extensive, q(t) = -a(0)i/j(ux)
- / a'{t - s)(ux(s))ds. Jo
It is important to note that the constants a(0) > 0 and 6(0) > 0 guarantee the parabolicity of (1.8). When the kernels a'(s) and b"(s) are regular, i.e., the principal part b(0)ut — a(0)Au + / is parabolic, then the memory term can always be treated as a perturbance. However, when a(t) and b(t) are more general functions, such as singular, integrable or distribution, then (1.8) can behave as different types of equations and the corresponding solutions will have new interesting properties. As a typical example, let us consider a linear equation with positive mem ory ut = j K(t - s)Au(s)ds.
(1.9)
1.1. Heat Conduction in Materials with Memory
5
This time, the memory term cannot be treated as a perturbed term. The equation must be studied as a whole and the properties of the solution may be essentially changed. If K(t) = a2 > 0 is a constant, after differentiating (1.9) with respect to £, it results in the wave equation uu = a2Au, where a is the propagation speed of the wave, which can be seen from the D'Alambert formula with initial values u(x,0) = 4>(x) and ut(x,0) = I/J(X) in an infinite rod, , ^ (x + at) + (x L- at) 1 *) = — - ^ + Ya\
^
fx+at',t^^ ^
It also indicates that the initial discontinuities of 0,
C 0 > 0,
(1.11)
where K is t h e Laplace transfrom of K, then the nonlinearity is quantitatively restricted by t h e positive constant Co, see Gripenberg [63]. However, when K is nonincreasing and convex, and y/tK{t) > 0 as t —> + 0 (i.e., K is more singular at 0 t h a n £ - 1 / 2 ) , and 0 < C\ < (f>'{p) < C3 < 00, then the global existence is also proved and even (1.11) need not hold, see Gripenberg [64] (1994) a n d literatures listed therein. In addition, another approach is to find the smooth solutions for suffi ciently small initial d a t a or for a finite time only. There is a wealth of results in this direction, see Gripenberg (1994) and references listed therein.
1.2
Viscoelast icity
Denote by u t h e displacement vector of continuous media, p the density, / the external force and o t h e 3 x 3 square matrix of stresses. T h e motion equation is of t h e form fnZtt = / + d i w .
(1.12)
It remains to supplement the constitutive relations between t h e stresses cr and strains e. It is well-known t h a t , for an ideal linear elastic body, o = A(div?2)/ + 2/xe,
and
1 (dui e = ei3; = - I
duj h —-
t
where e is t h e Cauchy strain tensor, and A and p are Lame constants. Sub stituting t h e m into (1.12) we obtain the second order hyperbolic system (also called L a m e equations): putt = (A + 0 or it does not gain any differentiability, depends on the boundedness of K(t)/lnt rather t h a n K(t). T h u s the models with singular kernels should have nicer existence prop erties t h a n those with regular kernels. However, one cannot expect t h a t the previous m e t h o d for proving the existence (i.e., the memory term is treated as perturbation) can also be applied to singular kernels. Note t h a t if singular kernels lead to smoothing process then t h e opposite sign of t h e integral must lead t o instantaneous blow-up, and a local exstence theorem cannot hold. Narain and Joseph [121] (1984) carefully analyzed the following linear stepwise shear displacement problem, making use of Laplace transform, uu = (M + a{fy)uxx
+ /
o!(t - s)uxx(s)ds,
0 < x < oo, = a u
u(x, 0) = ut(x, 0) = 0, where \i is t h e shear modulus. T h e relaxation function a'(t) is assumed to be continuously differentiable, monotonically decreasing, a ( + 0 ) ' = a(0), a(oo) = 0, and a(t) and a'(t) are 0(e~at) as t —► oo for some a > 0. T h e following conclusions are derived.
8
Chapter 1. Some Practical Problems and Their Properties
1). If 0 < a(0) < oo and —oo < a'(0) < 0, then the linear viscoelastic body subjected to a stepwise shear displacement fails and the discontinuity in displacement propagates into the interior of the body. If a(0) = oo or a'(0) = —oo, the discontinuity will not propagate. 2). If a(0) = oo, there is a diffusion-like smooth process of the discontin uous data. If a{i) = XS(t) 4- &(£), where 8 is the Delta function, A the small viscosity coefficient and b(t) a smooth kernel, then the smoothing process will take place in a propagating layer which has the same order as A. 3). If a'(0) = — oo, the solution is infinitely smooth, but the boundary of support of the solution propagates at a constant wave speed. 4). If 0 < a(0) < oo, — oo < a'(0) < 0, the material accommodates stress waves under step traction leading to an elastic steady state. The following nonlinear viscoelastic problem utt = {ux)x 4- / K(tJo
s)^{ux{s))xds
+ /
(1.17)
has been studied extensively during the past decade. For purely elastic case (i.e., K = 0), it is known that the smooth solution develops singularities in finite time even for data that are small and compactly supported. For one-dimensional Cauchy problem, the global existence of weak solutions follows from the recent famous works of Diperna [35-36] (1983-85) and are based on compensated compactness arguments. For the case that the spatial dimension d > 2, there are many results on existence but they only involve local time or small data. In spite of intense efforts by many outstanding mathematicians, most questions concerning global existence of both weak and classical solutions to mixed problem with large data for the nonlinear wave equation remain open. For the nonlinear viscoelastic problem (1.17) with regular kernels, there are many results on existence which also involve local solution and small data, the amplitude of accelerating waves should decay at a rate related to if (0). If if (0) = oo, it is reasonable to expect a regularizing effect from such a singularity. The case of singular kernel was first considered by Londen [104] (1978), in which the singularity of the kernel is stronger than logarithmic, and then further discussed by Hrusa and Renardy [80] (1986), where the kernel K (t) ~ t~a, 0 < a < 1, as £ ~ 0, but in essence, small data is demanded. The global existence of the weak solution with large data was studied by Engler [48] (1991), under various assumptions on K, f,uo,ui,ip and . For example, there are
1.2.
9
Viscoelasticity
0 < K(t) ~ t~a,
0 < a < l ,
and I^'(P)I + I^'(P)I < C j < o o .
(1.18)
T h e latter means t h a t the increment orders of 4> and I/J do not exceed 1. Problems involving propagation speed, smoothing effect and asymptotic stability are also analyzed. For three-dimensional nonlinear viscoelastic systems, a nicer argument to study t h e global existence for the weak solution u € L 1 ( 0 , T ; H1^)), with large d a t a , was recently proposed by Bellout, Bloom and Necas [7] (1993), in which it assumed t h a t 0 < K(t) ~ £~ a , 0 < a < 1/2, / 0 °° K{t)dt < oo and some restrictions similar to (1.18). It is conceivable t h a t for some singular kernels under more extensive as sumptions t h a n (1.18), t h e global smooth solution of (1.17) also exist for large d a t a . However, it has not been verified yet. A monograph written by M. Renardy, W. Hrusa and J. Nohel [139] (1987) is available for an overall review of this field. To conclude, several other examples simulating the viscoelastic problem are cited in the following. Example 1 is t h e dynamic rectilinear shear flow of incompressible nonNewtonian liquids, such as polymer melts and concentrated polymer solutions, see Laun [92] (1978), and Doi and Edwards [40] (1978). Under some suitable assumptions, the Cauchy stress a(x, t) of a particle at time t can be described by t h e gradient \/u of displacement in t h e form
a(x,t) = / K(t-s)(f)(\\ Jo
\/u{x,t)
- \7u(x,s)\\)s/(u(x,t)-u(x,s))ds.
(1.19)
Substituting into (1.12) it leads to a nonlinear hyperbolic integrodifferential equation. If t h e liquid adheres t o t h e boundary of t h e container, t h e n u = 0 on ^ x ( 0 , T ] . Example 2 involves biomechanics. In the history of elastic animal tissue, Fung [59] (1967) proposed the constitutive relation between the stress a and strain e = ux of t h e following form (see also Blatz, Chu and Waylan [8] (1969))
a(x,t) = J*K(t-s)^^.d..
(1.20)
Furthermore, Li, Chen and Chen [94] (1989) showed experimentally t h a t t h e h u m a n tendon in vitriol subjects the following nonlinear relation (within 60
Chapter
10
1. Some Practical Problems
and Their
Properties
seconds) a(t) = E I (1 + fjln{t - s))e(s)e'{s)ds,
(1.21)
Jo where E = 3 9 1 1 M N / m 2 and /i = - 0 . 0 3 4 3 . Note t h a t t h e kernel is of logarith mic singularity. E x a m p l e 3 is t h e compression of poro-viscoelastic media, see Habetler and Schiffman [68] (1970). Predictions of the compression process of a water s a t u r a t e d porous medium are based upon t h e theory of consolidation, where t h e porous mass is assumed to be a two-phase continuum. T h e liquid phase is incompressible, while e is the dilation of the porous matrix (soil skeleton) and o is t h e effective stress obeying a linear, time-dependent relationship e = m(t)cr,
(1.22)
where m is t h e compressibility of the matrix. A general, linear formulation of the dilation-effective stress relationship is modeled by a spring with M Kelvin units, all coupled in series
e = an 4- 2_. ^3 /
0 is the viscous coefficient. A corre sponding equation with viscoelastic forces in non-Newtonian fluid is ut 4- uux = (1 I (t - s)~auxx(s)ds, ./o
a = 1/2,
(1.26)
1.3. Reactor
11
Dynamics
or, a simplified form, ut = n
(t-
s)~°iuxx(s)ds1
(1-27)
Jo which is an equation with positive memory. J. Sanz and Serna [141], L o p e z Marcoz [106], and Hu [82] et al. discussed the numerical solutions of (1.27), b u t sharp results on b o t h regularities of the solution and the finite element approximation are obtained by Mclean and Thomee [118 ], see Chapter 13.
1.3
Reactor Dynamics
In analysis of the dynamics of a nuclear reactor, if t h e effect of t h e linear t e m p e r a t u r e feedback is taken into consideration, then the one group neutron flux u and the fuel t e m p e r a t u r e v in the reactor is governed by t h e coupled equations u
t ~ V(a(x) V u) = ( c i + c2v)u, vt = cv + C3U,
(x, t ) G f l x J ,
(1
2g.
where a(x) > 0 is the diffusion coefficient, and c l 5 C2,c 3 and c are physical quantities, see Kastenberg and Chambre [87]. Multiplying e~ct to the second equation, it leads to v(x, t) = ectv(xi
0) + c 3 / ec^~s^u(x,
s)ds.
Jo
Substituting into the first equation we obtain the following nonlinear integrodifferential equation ut - \/{a(x)
sj u) = (ci 4- c2ectv(x,
0))u + c2c3u
/ Jo
ec^~s^u(s)ds.
In particular, if c = 0 and v(x, 0) = const., then it becomes ut — \/(a(x)
v u) — \u + \iu \ u(s)ds, Jo
(1-29)
where A = c\ + C2v(0) and /x = c2c3 are positive constants. Consider the following initial boundary value conditions u(x,0)
= UQ(X) > 0 3u R(u) = a(x)—+(3{x)u
in Q, = 0
on 0 f i x ( O , r ] ,
(1.30)
12
Chapter 1. Some Practical Problems and Their Properties
where d/dn is the outward normal derivatives on dQ, and ct(x) > 0 and (3{x) > 0 are given functions satisfying a + /? > 0. It is known that if fi = 0 then the zero steady-state solution of the linear system (1.29) is asymptotically stable for A < Ai, and is unstable for A > Ai where Ai is the least eigenvalue of the eigenproblem v ( a V VO + W = 0 in f2, R(ip) — 0 on 0 (positive feedback) and as a sink when p < 0 (negative feedback), it is reasonable to expect that the nonlinear system remains asymptotically stable when A < Ai, /x < 0, and becomes more unstable when A > Ai, \i > 0. Pao [130] (1979) extensively discussed the existence of global solution, asymptotic behavior and blow-up properties in the finite time t'. Z.Q. Yan [166] (1994) further improved Pao's results as follows. With the help of the monotone method (see Gilbarg and Trudinger [60]), the "refined" upper and lower solutions for (1.29) are constructed by Yan. It is easy to check that u(x, t) — i(j(x)p(t) is an upper solution of (1.29) and (1.30), if p(t) satisfies p' > (A - Ai)p + [ip I p(s)ds, Jo
p(0) >
U0(X)/IIJ(X),
while u{x,t) — tp(x)p(t) is a lower solution if p(t) satisfies p' < (A - Ai)p + ftp / p(s)ds, p(0) < UQ(X)/II){X). Jo For this, Yan solved exactly the following Cauchy problem for an integrodifferential equation p' = aip + a2p / p(s)ds, Jo
p(0) = p0.
(1.31)
Setting q(t) = ai/a,2 + f0 p(s)ds, it leads to the following second order ordinary differential equation q" = a2qq',
with
q(0) = a i / a i ,
q'(0) = p 0 ,
or, after integration, q' = Yti2 + 6^
with
^(o) = «i/«2,
(i.32)
1.3. Reactor Dynamics
13
where the constant 6 = (2p0«2 - a\)/a22. Obviously, this equation can be exactly solved. Regarding the three cases of the sign of 0), no matter how large the value of A may be. This is really surprising. When fi > 0 and A < Ai, if u0(x) < ((Ai - X)2/2^m)^{x) then blow-up takes place. When /x > 0, A > Aii a(x) > 0 and u0(x) > M^(x) for some M > 0, then the solution blows-up in finite time t' < oo. The estimates to the asymptotic behavior and escape time are improved, and which are the best in the sense that when p{x) = 0, hence ijj{x) = 1, and the initial value u0{x) = constant, then the upper and all lower solutions constructed are the solutions of (1.31) and (1.30). In addition to this the following coupled system in man-environment epidemics, Ii ut = aAu --clUciu + + / / K(x,K(x,y)v(y,t)dy, y)v{y, t)dy,
(1.33.
^y vtt = -c22v + f(u),
was considered similarly by Capasso [14], and the existence and asymptotic stability of the solution was studied. H. Bellout [6] (1987) studied the blow-up of solution of the initial boundary value problem /1l
(
ut = Au+ K(t~s)g{u(s))ds K{t-s)g(u{s))ds I tiu = 0 on° 0 is greater than 1, so that F{u) = ^ g{s)ds satisfies the necessary condition /»oo /•OO
/ Jo
F(s)-^2ds
< oo.
Assume that, for s > 0, K(s)eC\ K(s) e C\ g{s)eC\ g{s)eC\
K(s) K{s)>m>0, > m > 0, s(0)>0, g{0)>0,
K'(s) K'{s)0,
g"(s)>0, g"(s)>0,
14
Chapter 1. Some Practical Problems and Their Properties 0
\\Daf\\PL1,{n))1/p,
ifl 1 -type, d > 2, 1 < p < oo and kp < d, then Wfc'p(ft)->L9(ft), where
2.1. Solvability in Spaces L 2 , Lp and Ca 1
1
k
19
dp
2. VgG [l,oo), if kp = d; 3. Wqe [l,oo], if fc = d, p = 1. Theorem 2.2 (imbedding to C z ' A (ft)). Let ft be C°' 1 -type, d > 2,1 < p < oo and kp > d, then W fc ' p (ft)->C*' A (ft), where 1. A = A; — — — Z,
ifO Z + l o r p = l , fc = d+"Z + l. Theorem 2.3 (imbedding to L*(ft m )). Let ft be C ^ - t y p e and ftm be an m-dimensional piecewise smooth manifold in ft where m < d, 1 < kp < i
i
i
m
mp
d
a, d — kp < m, — > q p
k (i.e. q
i/|H|?,
VveSo-
(2.3)
By Lax—Milgram theorem it is easy to show t h a t the weak solution u G S^ of (2.2) exists and is unique. If the boundary value ip G H 1 ( ^ ) , then there is an estimate
Nli 0 and F(t) are absolutely integrable, and the integrable function y(t) > 0 satisfies the integral inequality y(t)
+ /
Jo
u(Q) = u0.
And t h e n they determined t h e fundamental solution by means of a fixedpoint argument, and applied Laplace transform to the case t h a t A is timeindependent. W h e n A(t) is variable, see also Priiss [135]. T h e sharp regularity results were given by D a P r a t o and Iannelli [33]. Another approach is t o con sider t h e Volterra integral term in (2.4) as a perturbation of purely parabolic problem (without constructing the resolvent operator) and then use a fixedpoint argument. See Lunardi and Sinestrari [113-114], and Acquistapace and Terreni [2] (the latter contains a more detailed review of t h e references related to t h e linear case). In t h e case of singular kernel, see Lunardi [110].
2.2
Semigroup Method
In order to derive t h e fine regularity estimates for the solution of (2.4), in this section we shall t u r n to t h e semigroup method which provides a use ful tool in studying the regularities of the solution in spaces Lp(J;Hl(Q)) or Ca(J; Hl(Q)). We begin with a purely homogeneous parabolic problem ut + Au = 0 in Q x J, u = 0 onffixJ, u(0) = UQ in fi,
J = (0, T], (2.10)
where A is a second order symmetric positive definite elliptic operator with coefficients independent of t. T h e eigenvalue problem associated with A is Aifj = Xj(fj
in Q,
cpj = 0
on dft.
(2.11)
It is well-known t h a t this problem has a series of positive eigenvalue {Aj}f° and corresponding orthonormal eigenf unctions {j}i°. An arbitrary function
Chapter 2. Parabolic Integrodifferential
26
Equations
v G L2(Q) can then be expressed as oo
3= 1
and the Parseval equality oo
(v,w) =
'^T(v,(pj)(w, 0, define a Hilbert space HS(Q) = {ve Hs(ty,
Ajv = 0 on dfi, 0 < j < s/2}
with the norm oo
M- = \MH. = E A K ^ ^ ) ] 1 / 2 = H^/2»H3=1
It is proved that the norms \v\s and \\v\\s = \\V\\HS(Q) a r e equivalent in HS(Q), see Thomee [150]. In particular, H2 and H2 H-^o a r e equivalent. The solution u of the homogeneous problem (2.10) can be expressed as oo
u{x,t) = E(t)u0 = ^ e - ^ o , ^ ) ^ ) ,
(2.12)
3= 1
where E(t) is called the solution operator which is a semigroup generated by A and satisfies DtE(t) =-AE(t)
and
[ AE(s)ds = I - E{t),
(2.13)
Jo
or E(t) = -A-1DtE(t)
and
/ E(s)ds = A~X{I -
E{t)).
Jo
It is easy to see that for any v G L2(Q) and any s > 0, E(t)v G HS(Q) for t>0. If 0 < I < s and i; G #'( f t )> then |2^j>lJ"JS7(t)vU < C7t" i_:? "" (a -- |)/2 |v|i for z , j > 0 , t > 0. In particular, for any v G L2(S7),
ii£(*M < Nl
(2.14)
2.2.
Semigroup
27
Method
and \D\AjE(t)v\s
< Ct-t-'-'^WvW
for ij
> 0, t > 0.
2
If v <E # ( ^ ) , we have oo
= ^2e~~Xjt(Av,ipj)ipj
Au = AE(t)v
=
E(i)Avi
3= 1
and hence \\AE(t)v\\
< \\Av\\.
For nonhomogeneous parabolic problem of the following form
{
ut 4- Au = f in D x J, u = 0 onSfixJ,
(2.15)
it(0) = UQ in Q, one can formally express its solution, by Duhamel principle, as u(t, x) = E(t)u0
+ /
jE?(t - s)f(s)ds
= E(t)u0
+ E * /,
(2.16)
where E * f = f0 E{t — s)f(s)ds is Laplace convolution of £" and / , and there is the following result (see Pazy [132]). T h e o r e m 2 . 8 . Assume t h a t / e L 2 ( L 2 ( ^ ) ) or C a ( L 2 ( f t ) ) , 0 < a < 1, and ^o € L 2 (f£) or H2(Q), then t h e solution it(£, x) of (2.15) has t h e following estimates K*)||
< H
+ f \\f{s)\\ds Jo
for / e
L\L2(Q))
and \\ut(t)\\ + \\Au(t)\\ < C\u0\2 + C | | / | | C - ( L * ) . P r o o f . T h e first estimate follows immediately from ||i£(£)|| < 1. In order to estimate Au, one cannot directly differentiate AE(t — s) under t h e integral sign, because / * \\AE(t - s)f(s)\\ds Jo
< C [ (t Jo
ay^WMWda
is divergent. Rewrite (2.16) as u(t, x) = E(t)u0
+ f E(tJo
s)(f(s)
- f{t))ds
+ f E{tJo
8)dsf(t),
Chapter 2. Parabolic Integrodifferential
28
Equations
thus An = AE{t)uo + / AE(t - s){f{s) - f{t))ds + (I Jo
E{t))f{t),
and | | A u | | < C | u 0 | 2 + C j f t ( t - « r - 1 | | / | | c « > ( j ) d s + ll/(t)ll
0
are valid and will be useful in the analysis of the time discretization. Finally, turn to the Lp error estimates of Vhu(t). Following Rannacher and Scott [137], a new gradient type Green function G(t) G HQ can be defined as follows: A(G, v)+ [ B(G(s), v)ds = Dz8hcre(t) in QTi (3.34) Jo and its discrete analogue Gh{t) G S£ are studied by Lin and Zhang [100], where (T£(t) is a modifier function satisfying the following conditions: 0 2, Lin, Thomee and Wahlbin [97] (1991) presented a simpler duality argument (without using Green function) and got the following result. T h e o r e m 3 . 5 . Let ft be a smooth domain of any dimension, p = Vhu(t) — u(t), 2 < p < oo, then \\p(t)\\o,p + h\\p{t)\\liP
< Cphr(\\u(t)\\riP
+ f
\\u(s)\\r,pds).
(3.39)
Jo
P r o o f . For any
p ' < C*p IM|o,p' = Cp , 1 < p < oo, which is valid at least for the smooth domain ft. Taking Wh = RhW, then (Dtp, (f) = (p, -Dup)
= A(p, w)
— A(p,w-wh)
+
B(p(s),wh)ds Jo
= A(Rhu-u,w)
+
B(p(s),wh)ds Jo
= (Di(Rhu-u),(p)+
B(p(s),wh)ds Jo
< \\RhU-u\\liP\\ip\\oiP'+C
/
H/o^lli^ll^fclli^/ds.
Jo
By t h e boundedness of Rh in W1,p
, | | ^ ^ | | i , P ' < Cp||w||i j P / < CP, we have
\\p(t)\\i,P00ds), Jo
€>0
(3.40)
3.3. Ritz—Volterra
Projection
51
VhU
and t h e L p -estimates of t h e derivatives with respect to t can also be derived. Finally a super convergence estimate between V^u and Rh,u will be useful. T h e o r e m 3 . 6 . Assume t h a t the triangulation is quasiuniform. Let a(t, x) be a n arbitrary function and B\ be an arbitrary first-order operator. Assume t h a t B can be expressed as B — a A + B\, then the super convergence estimate \\Vhu(t)
- Rhu(t)\\x
< Chr
< C [\hU(s)h Jo f
\\u(s)\\rds,
+
te
U(*)\\o)ds
J.
(3.41)
= 0 + £, 6 = Vhu-Rhu
G S^ satisfies (3.33).
JO
holds, where £ = R^u — u. P r o o f . From (3.32), Vhu-u Rewriting t h e bilinear form as
B(u, v) = A(u, av) + (u, Bxv), Denoting w — av and w^ — Ih{&v) G S^ B(H, v) = A(£, w-wh)
v G S£.
and using
+ ( £ , B l V ) < CU\U\\w
-wh\\1+
C\\t\\o\Mi
and t h e super approximation estimate for v G 5 ^ ,
Ik - «*ll?.n = E I™ - «"ill?,r < Ch2r~2 £ II^Hlt r
r