High order nonlinear parabolic equations

171. Wolf yon Wahl, "Regularity of weak solutions to elliptic equations of arbitrary order," J. Different. Equat., 29, No. 2, 235-240 (1978). 172. K.-O. Widman, "Local bounds for solutions of higher order nonlinear elliptic partial differential equations," Math. Z., 12!., No. I, 81-95 (1971). 173. S. A. Williams, "A sharp sufficient condition for solution of a nonlinear elliptic boundary value problem," J. Different. Equat., 8, No. 3, 580-586 (1970).

HIGH ORDER NONLINEAR PARABOLIC EQUATIONS

UDC 517.956.45

Yu. A. Dubinskii

The basic results and methods of the theory of high order nonlinear parabolic equations are described. In the first chapter boundary problems for quasilinear parabolic equations having divergent form are considered. In the second chapter nonlinear parabolic equations of general form are considered~ Attention is mainly paid to methods of study of nonlinear parabolic problems~ In particular, the methods of monotonicity and compactness, the method of a priori estimates, the functional-analytic method, etc. are described.

PREFACE The present survey is devoted to one of the actively developing directions of contemporary nonlinear analysis, the theory of high order nonlinear parabolic equations. Just as in the corresponding elliptic theory, the theory of high order nonlinear parabolic problems counts nearly a quarter of a century in its development (M. I. Vishik's first paper on nonlinear parabolic equations of order 2m appeared in 1962.) However in this comparatively short period it has been enriched with such important results, and what is no less important, methods of study, that it has rightfully become one of the fundamental directions of contemporary research. The object of the present paper is the description of the basic achievements of this theory. The paper consists of two chapters. In the first chapter a survey of various results in the theory of nonlinear parabolic equations of divergent form is given. In the second chapter boundary problems for nonlinear parabolic equations of general form are described. Since a brief scientific and historical annotation prefaces each section (cf. also the comments in the "Literature" section) we shall not dwell on the content of the paper by chapters here in the preface but characterize it on the whole. First of all we note that in the choice of material we have chosen papers in which the methods of studying nonlinear boundary problems have been developed. Among such methods are the method of compactness, the method of monotonicity, the method of a priori estimates, the function-analytic method, etc. Further, as is known there are two general approaches to the theory of nonlinear parabolic problems. The first approach is based on the consideration of the initial-boundary value problem for a nonlinear parabolic equation as a Cauchy problem for the corresponding nonlinear differential-operator equation in a Banach space.

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 37, pp. 89-166, 1990.

0090-4104/91/5604-2557512.50

9 1991 Plenum Publishing Corporation

2557

The second approach is based on the proper study of the boundary problem as an object of the theory of nonlinear partial differential equations. At the present time both approaches have a sufficient amount of both factual material and methods of study in their arsenals and thus deserve completely independent accounts. However, in the literature, attention is paid mainly to the first approach. Here we have a number of detailed monographs and surveys among which we note the books of Browder [103], Lions [120], Brezis [94], Barbu [91], Gajevski, GrSger, and Zacharias [iii], Vainberg [Ii], Levitan and Zhikov [63], and Pankov [71], and also the surveys of Dubinskii [31], Krylov and Rozovskii [52], Krein and Khazan [48], published by the National Institute of Scientific and Engineering Information (VINITI) of the Academy of Sciences of the USSR in the series "Itogi Nauki i Tekhniki." The survey of Krein and Khazan cited has the most complete bibliography, containing more than 700 citations. Thus, nonlinear differential-operator equations are quite completely represented in the scientific and reference literature. As to the second approach, here, on the contrary, there are no general surveys as far as we know. In view of what has been said, in the present survey we do not deal with differentialoperator equations and we consider nonlinear parabolic problems only from the point of view of partial differential equations. Of course this choice is dictated not only by the hope of filling the lacuna cited in the literature. The fact is that by no means all the results of the theory of nonlinear parabolic problems can be included in the framework of the differential-operator approach, so that actually the theory of nonlinear parabolic partial differential equations is richer than a simple consequence of the theory of differential-operator equations in Banach spaces (the reader will see this particularly in the second chapter). Moreover, on this route one is able to consider the material recounted from single scientific and methodological points of view. CHAPTER 1 QUASILINEAR PARABOLIC EQUATIONS OF DIVERGENT FORM i.

Methods of Solution of Quasilinear Parabolic Problems

i.i. Method of Monotonicity. The method of monotonicity or method of monotone operators is an active contemporary method of study of nonlinear equations. We apply the nonstationary version of this method, described below, to a large class of initial-boundary value problems for quasilinear parabolic equations having divergent form [cf. (i)]: the first boundary problem, the problem with periodic conditions on x, the Neumann type problem, etc. To be specific we dwell, for example, on the case of the first boundary problem since the main facets of the method of monotonicity are identical in all cases. Let G c R n be a bounded domain with piecewise-smooth boundary F; Q = [0, T] • G be a cylinder in the space R ~+I of variables t~[0,T], x60 with lateral surface S=[0, 7]XF (T being an arbitrary fixed number). In the cylinder Q we consider the first boundary problem for the quasilinear parabolic equation of divergent form

Ou ~ (--1)mO~A~(t, x, DVu)=h(t, x), O_uOtq- L (t, x, D) (tt) =---~i-q-r~lr

u(O,x) =~p(x), x~a, D~uls=/~(s), sES, [co[~<m--1.

( 1) (2) (3)

Here u=u(t, x), ((t, x)6Q)is the function sought, D = ( D I ..... D,), Dj=O/Oxj(l-.<j-./I is an integer. The functions As(t, x, $7) are (generally) nonlinear functions }71~ 1 is a number, r n > 0 is an integer), and Wp(G) be the subspace of the space W~(G) consisting of functions u(x)EW~(O),which on F satisfy the conditions

D~ulr=O, I~ I < : m - - 1 . 0

We denote the dual space (W~(G))* by Wp~(G), p' = p/(p - ]).

u(t, x):lluIti.---

L.(0, T.

Further, let

Ilu(t, x)ll 2co) a t
aoHu.-.vl[~,p,

Q ]al<m

where a 0 > 0 i s a c o n s t a n t . III. The i n i t i a l and boundary f u n c t i o n s ~(x) and f ~ ( s ) admit e x t e n s i o n t o Q from t h e space W(f), i . e . , t h e r e e x i s t s a f u n c t i o n f ( t , x) h a v i n g f i n i t e norms I[fllm, p and Jlc)r/at[t--m,r, such t h a t f(O, x)=tp(x), DOf[s=fo(s), I ~ l ~ m - - 1 . THEOREM i.i.

If conditions I-III hold, then the map

~ + L(t,x, D) (u):W(y)-+ Lp.(O, T; W-~.m(O)) ot i s a homeomorphism. To prove the proper solvability of the problem (1)-(3) condition II is excessive and can be relaxed. Namely, let us assume that the following conditions hold: II'. one has

[Coercivity of the Operato r L(t, x~ D)(u).]

For any function u{[, x)@Lp(O, T; W2(O))

2559

Re [L (t, x, D)(u), ~] =----Reff X

where

c(~):R~-~R~ is

As(t, x, D~u)D'adxdt>c([[u[[m.~),

q l=l~m a continuous function

such that

c(g)/g

§ +~ a s r + +~.

II" [Monotonicity of the Operator L(t~ X~ D)(u).]

For any functions u(t, x ) , v ( t , 0 from the space Lp(0, T; W~(G)] such that u(t, X)--9(t, x)6Lp(0, T; ~7~(O~), one has

Re[L(t, x, O )(u)--L(t, x, D)(v~, u~v]>O. THEOREM 1.2. If conditions I and III hold as any right side h(t, x)ELp,(0, T; W;9(O)) there exists problem (1)-(3). If in addition the operator L(t, (4) equality to zero is only possible when u(t, x)

x)

(4)

well as conditions II' and ll", then for at least one solution u(t,x)fiW(~) of the x, D)(u) is strictly monotone, i.e., in ~ v(t, x), then the solution is unique.

The proof of Theorems I.i and 1.2 goes by the method of monotone operators, which has now become classical. Applied to the problem (I)-(3) this method consists of the following: I.

Investigation of the solvability of the finite-dimensional nonstationary approximations of the problem (1)-(3) (Galerkin-Hopf-Faedo approximations).

2.

Derivation in Lp(0, T; W~(G)) of uniform a priori estimates for the approximate solutions uN(t, x).

3.

Passage to the limit as N § ~.

It is important to stress (and this is of importance in the method of monotonicity), that condition II' lets one pass to the limit from the finite-dimensional approximations to the original problem (I)-(3) starting from only the simplest estimates of the approximate solutions in Lp(0, T; W~(G)) and consequently using only weak convergence of derivatives of order m. In other words, equations with monotone operators are remarkable in that they are weakly closed with respect to Galerkin approximations. Example.

We consider the equation

o~

Ot F X

(5)

a=m=(Im=uJP-~D~u)=h(t;x) ,

[al<m w h e r e a=~O, p > l Then f o r

are numbers. p~2

(5)

satisfies

all

the hypotheses

of Theorem 1.1 b e c a u s e

and as calculation shows,

Re [L (t, x, D)(u)-- L (t, x, D)( v), u-- 91 > a o [Iu -- v l[ ~,p (% > 0). Consequently, the operator of the left side of (5) defines a homeomorphism of the spaces W(f) and Lp,(0, T;w?m(o)). If 1 < p < 2 then the operator L(t, x, D)(u) does not satisfy condition II, however it is coercive, which is obvious, and strictly monotone, which follows from the inequality

RelL(t, x, D)Cu)--L(t, x, D)(v), u - - v ] > X a~

]D=u--D%[Pdxdt+2--fi:-7_,

D~u+

x

D=u_ Dav l2dxdt), where

Q+.=={(t,x)6Q:JD=u--D%I~E(u)--K~ =----X [a(DVu) D~u, D=u]--l(, I~l=l~I=m [al=m

where the function a(~) (ITl--<m--1), satisfies a(~,)>ao(l+ K > 0 being a constant. II".

Z

I~,lq") ,

Here [ , ] is the integral

If the function

ao>O,

qv>O,

over the cylinder

u(t,x) :Q-+C I h a s f i n i t e e n e r g y i n t e g r a l A ~ (t; x, DTu) D~uELq(Q),

Q.

E(u),

then for all

e and B

w h e r e q > 1 i s a number. H e r e i f t h e s e t o f f u n c t i o n s u ( t , x) i s s u c h t h a t E(u)- 0 i s a c o n s t a n t , t h e n t h e c o r r e s p o n d i n g s e t A ~ 8 ( t , x , DXu)D~u i s bounded i n L q ( Q ) . III".

x) has t h e form h ( t , x) - h z ( t , x) + h 2 ( t , x) + h 3 ( t , x ) , w h e r e k~(t,x)6L2(Q), k2(t,x)6L~(O,r;w~"r and ka(t,x)GLq,o(O,T;W-,(m-1)((~)) , w h e r e q0 = m i n q v + 2, q~ is the conjugate exponent. % u(t;

The f u n c t i o n

Definition 1.2. x) such that

i) E(u)
0 and in t with exponent ~/2m on each compact subset. II (Strong Ellipticity Condition.)

,v

One has

oA~ (t, x, D

N k

I

s

k,t=Z ["[=lPl=m where

(t, x)fQ, ~vER, t~ 1 "q~eR~,

(N

k=l 1131<m

)

while a o > O , p~>2.

III (Subordination Condition.

)

For the same values

IA~ (t,;, ~) I~< K =p-I(D, where K > 0 is a constant. Analogous inequalities also hold for

0 A~(t, x, ~), ~--FA~(t, x, ~), e ~ ( t , x , t ) , Ox 02 02 e= A~( ~ t ,x,~). Ot'Oi'b A~ (t, x, D, ~O~'Ox~A~ (t, x, t), We introduce the space of generalized solutions of the system (17). ~m denote by ~ , p (Q) the space of functions having finite norm

Namely, first we

2567

vraimax]]

u (t, x) llL.w)+ ~ [[D~tt {t, X) IIr.{Q).

O~t 0),

1~I= m

where (', 9 ) i s t h e s c a l a r p r o d u c t and tl'll i s t h e E u c l i d e a n norm i n t h e f i n i t e - d i m e n s i o n a l space. Condition II. The vector-functions fa(t, x, ~ ) and fa(t, x, ST) satisfy a Caratheodory condition and the following conditions of growth of nonlinearity: a) for I~I = m

]lf~(t,x,~OH.. O , l~ O, m 2 > 0 are constants. We consider a generalized solution u(t, x) of (20) satisfying the conditions of the first boundary problem

u(O,x)=~(x), xOO; D'uls=0, I~l2~--2, p > n + l .

Then if ~(x)6C~(G), then o

uCt, x)~L,(O, T; W ~ (a)) ~L, (0, T; Wg'(o)), o~ CLp (Q). Ot Zhou Y u - l i n

and Fu Hong-yuan [138] s t u d y t h e Cauchy p r o b l e m f o r t h e s y s t e m o f e q u a t i o n s

c)2mu

0~ (--1)~A(t) 0~, =f(~), (t>0, xeR'), Ot u(O, x) =~(x), 2569

where u = (ul..... u~), A(O-----(ai~(t))~• is a nondegenerate nonnegative definite matrix with continuous elements aij(t) for all i90. The nonlinear function f(u) is assumed to be smooth of class C 2re+h, k~>l, while 8f/3u is semibounded above. The following result is established: if ~(x)GW~m+k(l~1), then there exists a generalized solution of the Cauchy problem while the derivatives aZu/axl(l~2ra--I and the additional condition of smoothness of aij(t) the solution is classical. In the class of bounded functions a classical solution is unique. In the case of constant matrix the Cauchy problem is studied for periodic initial data. Zhou Yu-lin [137] studies the solvability of the first boundary problem for some classes of pseudoparabolic equations with one spatial variable

(_l)~yiT+

D ~'~~--B(t,~--

x,u, . . . . D~"-'u) D~tz + g (t, x, u . ... .

D~"-lu),

where u = (Ul..... uN), g = (gt..... gN), A (t, x)-- (Ai~(t,X))NxN is a symmetric positive definite matrix with continuous coefficients. The basic restrictions on the nonlinear summands are: I) the matrix B(t,x, u ..... D2n-tu) is semibounded above; 2) the function g_(t,x, ~0.... ,~2n-i) satisfies

(M > 0 being constant), $0, .... $ 2 n - 1 "

i.e., has no higher than linear growth in the variables

Under the restrictions cited the unique solvability of the first boundary problem in the classes W~,~ n is established, i.e., the classical solvability is established. Additional results are obtained for the equation 0u ( - - I ) n 7Ou{ + A ( t . x)D ~"OF ----f (u, Du,

. . . .

D~-Itt)

with special restrictions on the nonlinear function f(~0 ..... ~2n-l). 3.

Nonlinear Parabolic Equations of Infinite Order

3.1. Sobolev Spaces of Infinite Order. Nontriviality Criteria. bolic problems for an equation or system of infinite order

In considering para-

oo

0s

Ot + ~ (-I~!D~A:(t" x, DVtt)=h(t, x), t h e r e a r i s e n a t u r a l l y (as i n t h e spaces, but of infinite order.

For example,

elliptic

case too)

energy

t~ J~ dt = l < tt~, "o > dr.

0

0

0

for any function ~ (g, x)ELa ((3, T; lt~~ {a=, p}(G)). a Banach s p a c e i f as norm one s e t s

O b v i o u s l y t h e s p a c e Lo,(0, T;

W-~'{a=, p'}(G)) becomes

T

I! 1 is a number, and the weight functions q~

q~

9~(X)=--Xl ...xn ~ X q~, where q~, I~[ 0 a r e c o n s t a n t s . IV.

(Lower S e m i b o u n d e d n e s s o f V a r i a t i o n . )

+ liB ( u , ) - - B 0

where K > 0 i s a c o n s t a n t ,

One h a s

(u~)ll2~ c~. r~ i s t h e d u a l i t y o f t h e s p a c e s W ; ~'W~'o' LatH'E) i s t h e s p a c e o f o p e r a t o r s r + E such that r are Hilbert-Schmidt o p e r a t o r s , where Q i s a s y m m e t r i c n o n n e g a t i v e k e r n e l o p e r a t o r i n H. The following theorem holds.

2574

THEOREM 4. i. Let conditions I-IV hold. Then the problem (28)-(30) has a unique strong solution (cf. the definition at the beginning of the section). This solution depends continuously on the initial data. One should note that in concrete situations the verification of conditions III and IV guaranteeing the strong parabolicity of the problem can be assured by the following sufficient conditions of algebraic character. Namely, let the functions A~(t, x, ~; $u be differentiable with respect to $7 everywhere except for a set which has a finite number of points of intersection with each line in the space of coordinates ~, IyI~<m, and the derivatives

A~(t, x, ~; ~T)-----OA~(t,x, o; ~)10~ be locally summable on each line in this space. Let us assume that there exist numbers s > 0 and N > 0 such that for all~T, N=, ~ (IYl, I~[, l~I~.m), and also t6~, T], xeG, ~ one has

I=l, I~l<m

I~l<m

I=[=m

I

~ ~ 0 is a constant. 5.

Numerical Methods of Solution of High Order Nonlinear Parabolic Equations

As is known the standard explicit difference schemes for parabolic equations of order 2 m (m>/]) require quite strong restrictions on the smallness of the ratio T/h 2m, where T is the step of the net in time and h is the step of the net in the spatial variable, to guarantee stability. As a rule implicit difference schemes are free of this deficiency but their numerical realization leads to excessive "machine" expenditure since on each time layer one has to solve rather complicated difference systems. By economical difference schemes we mean schemes combining the virtues of ordinary implicit and explicit schemes: they should be absolutely stable and let one make the numerical passage from one temporal layer to another upon performance of a number of arithmetic actions of order of the number of nodes of the spatial net. Questions of the construction and study of economical difference schemes for nonlinear parabolic equations are considered in D'yakonov [34, 35]~ Namely, let Qc~-Rn be a domain composed of parallelepipeds. We denote the boundary of by F. In the cylinder Q = [0, T] • ~ we study the first boundary problem

~-+(--I)

~

D=(A=~(t,x,DVu)D~)=~(t,.x, DVu),

I=I,I~T=~

(31)

l~l<m. u(O,x)=~(x), D~uii~rl•

i~l<m--1, x,fir.

(32)

Basic assumptions: I.

The functions A~$(t, x, Sy) and also their derivatives with respect to $~ are continuous and uniformly bounded. Analogous requirements are imposed on the right side

h(t, x, ~u II. For any sufficiently smooth function v(t, x) the operator

L,(u)------(--1)= ~

O=(A~(t, x, DVv)D~u)

is a strongly elliptic operator. 2575

The economical difference schemes constructed for this problem are generated by familiar methods of variable directions and are based on the use of difference splitting operators on the upper layer. In the simplest form these schemes have the form

where uiq = u(qT, ilh,...,inh) is the value of the net function u at time q~ at the corresponding node and L h and A s are finite-difference approximations of the differential operators

L(u)=(--1) ~ ~ and (-l)mD~ m, respectively

D=(A=~(x,DYu)Dgu)

(s = I ..... n).

In addition, E is the identity operator and o > 0 is a parameter on whose choice the stability of the difference scheme depends. The presence in the difference scheme of a splitting operator lets us reduce the solution of the multidimensional systems which arise to the successive solution of n one-dimensional systems. The indicated schemes themselves can be considered as ordinary difference schemes for the problem (31), (32) with "viscous" perturbation. For example in the twodimensional case (n = 2) the perturbed equation has the form 2

where ~i > 0, g2 > 0 are constants. Under conditions ! and II the convergence of the difference scheme in the energy metry corresponding to the problem (31), (32) was studied. It was proved that if a generalized solution of the problem (31), (32) exists, then one has convergence in the metric of the net space of type L2(0, T; W~(~)). Moreover, under the additional condition of uniqueness of a sufficiently smooth solution of the original problem one gets an estimate of the rate of convergence of order h 2 + ~. W e note that analogous results are obtained on the Cauchy problem for (31) for the case of periodic conditions with respect to the spatial variables and also for the problems indicated in the case of systems of quasilinear equations of high order. Under other conditions difference schemes for quasilinear parabolic equations were studied in Lyashko [64-67], Lapin and Lyashko [62, 61]; Karchevskii, Lapin, and Lyashko [42]. We describe some of the results of these papers. We consider the first boundary problem

O_tu Ot + X ( -1)r~tD~A~(t'x'D~u)=f(t' x), I~fl~<m,

(33)

[=l~m

u(0, x ) - - ~ ( x ) ,

D~215

=0,

[~

(34)

The problem ( 3 3 ) , (34) i s c o n s i d e r e d in t h e c y l i n d e r Q = [0, T] • ~ whose base i s a s q u a r e , i . e . , f~={0 0 being a c o n s t a n t and O ~ < p ~ < 1 . It follows from this that each integral (46) can be represented as a finite sum of in ~ tegrals of the type of

[rn~ (t, x, D Vu) ] D~u iar~D~tt, D~v], where the derivatives D~u obviously occur only in combination with factors makes sense if (by definition) we let

ID~u] p~ which now

p.

]D"tt I~'D~u= ID"u [p./2 p~+iDi(IDguj 2 2 D~u). We apply precisely this regularization as the values of the integrals Further, that

(46).

as already noted, from the fact that u(t,x)6W it does not generally follow for [al = m. However the "monotone" function

D=u~L2 ( Q)

cp~( D~u) ~ [D.u IvUIiD~uEW~(Q),

if u(I,x)~W, Consequently the functions %(D ~u) vanish on S in the sense of Li(Q). this in mind in saying that D~u[s=0, ]m]~rn--l,"in the mean~

We have

One has the following existence theorem for generalized solutions. THEOREM 6.2.

If the conditions formulated above hold then for any right side h(t,x)ELq,(0 , of the problem

'T;W~-,m(G)),q*=q/(q--1),there exists at least one generalized solution u(t,x)~W

(41)-(43).

The proof, of the theorem goes by the method of compactness analogous to the case of weakly nonlinear problems. The most essential point here is the application of the "nonlinear" version of the compactness theorem formulated in point 1.3 of Sec. i. 6.3. Implicit Degeneration of "Boundary Layer" TyEeo Now we turn to the case of implicit degeneration caused by the vanishing of derivatives DYu of order 2m - 2. Boundary problems for degenerate equations of the type indicated were studied by Soltanov in tlhe energy metric. In [84] he considers the first boundary problem for a quasilinear parabolic equation with one spatial variable ( (t,x) ~Q=[0, ~X[a, b])

ou

( - - 1 ) m ~ q-A([, .v, u . . . . . Dim-iu) Dimtt~h( [, x),

(47)

u(O, x)=O,

(48) (49)

DIg(t, a)=D]tt(t, b)~O, L e t us assume t h a t t h e n o n l i n e a r t h e o d o r y c o n d i t i o n and has t h e form

j = O , 1 . . . . . m--1.

f u n c t i o n A(t,x,~), ~= (~0, ~, . - . , ~=m-2), s a t i s f i e s

a Cara-

2581

A (t, x, ~)~A, (t, x, ~)I ~2~-21 ~, where 0 < a o ~ A , (t, x, ~)~.~M 0 are certain integers. Further, 9

are vectors with nonnegative

9 ..

integral components, I vlb=2b~'o+W+

O~/~

and by definition 9 9 9 +~,~

(analogously for [~k]b). The collections of multi-indices ~k are such that I~kI~g~--l, where k = 1 ..... N. The functions a~j(t,x,~=k),al{t,x,~=k) , and fi(t, x) are real functions while if t k = 0 then these functions are independent of the corresponding variables $~k. On the lateral surface S of the cylinder Q there are given boundary conditions N

(2) 7=1 1 3 % ~ l + t /

where ~ = 1 .... ,br; oj are negative integers; b~j(t, x) and ~ ( t , if o~ + tj < 0 then b~..(t, x) -- 0. x3 Moreover, for t = 0 let the initial conditions

o%ot~I ,=o=~ (j = 1 .... ,N) hold. tj = 0.

(3)

Here one does not impose initial conditions on the component uj(t, x) if

We introduce the spaces needed. N

i=o, 1,. .., t)--!

x) are real functions while

Let

0 t',t..

HI(Q)=HW~p'(Q) 1=I

be the space of solutions, 2583

N

]-]2(Q)~---]~Lp(Q)' be the space of right sides,

j=l

br

(s)= II

o

!-(

~- ~st - T~/i , - o ~-~1

(s)

f=1

be the space of boundary values. Norms are introduced in these spaces as the sums of the corresponding norms. Let us assume that the following conditions hold: I (Smoothness Condition.)

The functions a~j (t, x, ~=k) and a~(t,x,~ k) are defined and con-

tinuous for (t, x)6Q and all $~kcontinuous partial derivatives.

Moreover, let us assume that these functions have everywhere The coefficients b~j(t, x) belong to the class 1/

1

II

r,~-~l,-9-;,-~t-~-+s/e ~ 8 ~ 0 being arbitrary. t, x ~ol, II (Parabolicity Condition.) system

Let us assume that for any fixed function u(t,x)'~H1 the

N

~

mk

aTj (t, .x, m uk) DV~j= , j (t, x)

(] = 1 . . . . . N)

(4)

1=1 [ Vlb=t j

which i s l i n e a r [ w i t h respect to v ( t , Solonnikov (for s i = 0).

x)]

is parabolic

in the sense of D o u g l i s - N i r e n b e r g -

III (Complementability Condition. ) Suppose for any fixed vector-function u(t,x)~Hi(Q) the boundary conditions (2) relative to the linear system satisfy the familiar Lopatinskii condition (or complementability condition; in algebraic form applicable to the system considered these conditions are indicated, for example, in Ladyzhenskaya, Solonnikov, and Ural'tseva [60], Chap. VII, Sec. 9). IV (Existence of an A Priori Estimate.) problem (i)-(3) one has

Suppose for any solution u(t, x)EHz(Q) of the

IIu I! ,v cm < C ([l f IIH, CQ)+ it II where C(p) is a continuous function on [0, +~). Under the conditions formulated one has THEOREM i.i. if conditions 1-IV hold and p > n + 2b then for any vector-function f(t, x)~H2(Q) there exists at last one solution u(t,x)6Hl(Q)of the problem (1)-(3). If, in addition, one assumes that for any N

u(t,x)OHI(Q) and w(t,x)GHl(Q) the

system

1

!~lb~tj 0

which is linear with respect to v(t, x) is parabolic in the sense of D o u g l i s - N i r e n b e r g Solonnikov and the boundary conditions (2) satisfy a complementability condition [in v(t, x)] with respect to this system, then a solution u(t,x)~ff1{Q) of the original problem is unique. The proof is based on the application of the following operator theorem (Pokhozhaev, [73], Theorem 2). Let X and Y be Banach spaces while Y is uniformly convex. In addition, let A'X § Y be a nonlinear operator defined o n all of X and Frechet differentiable at each point x6X. THEOREM 1.2.

Let the following two conditions hold:

i) the domain of values A(X) is closed in Y; 2) for any

x~X

Ker(A'(x))*={0}, where A' (x) is the Frechet derivative.

Then A(X) = Y, i.e., for any y~Y the equation A(x) = y has at least one solution x@X. 1.2. Normal Solvability. Questions of normal solvability (in the sense of Fredholm) of general nonlinear parabolic boundary problems were considered by Babin. In [3] he establishes a nonlinear analog of the familiar "finiteness" theorems of Fredholm maps.

2584

Let D(S)u(t, t,x x), s = 0,...,lm mean the vector composed of all derivatives of the function u(t, x) of the form D~D~u such that lal + 2b$ < s, where b > 0 is a parabolic weig_h_t, i.e., ~(2m).

an integer which is a divisor of m.

We denote the dimension of the vector u~,x ~ by r.

Suppose in addition a(t, x, z), bj(t, x, z), j = i, .... m, where z ~ N , smooth functions of the variables (L x)OQT and z6R N.

are arbitrary

In the cylinder QT = [0, T] • G one considers the nonlinear parabolic problem

Lo(u)~a (t, x, O(lm)u)= f (r x),

(5)

B~ (~)-- Oi it, x, D ~ P ~ is =g] (t, x'),

( 6)

x'eOO, ]= 1, 2 . . . . . m

(mj~0

being integers).

We make correspond to the operators L0(u) and Bj(u) the polynomials

L~

Lx, z, ~, p)--

X

a ~ ( t , x , z)$~p~,

f~l+21~b=m

B~~(t, x, z, ~, p)----_-

2~

bm (t, x, z) ~p~,

l r [+ 2b[~=m ]

where ~ER n, zGR r, p E Q = { q @ C I : R e q ~ ; a=~ and bj~$ are the derivativs of the functions a(t, x, D(lm)u) andbj(t, x, D(mj)u) with respect to the variables D~D~u. Let us assume that the original problem is 2b-parabolic which means that the familiar conditions on the distribution of the roots p of the polynomial L~ x, z, $, p) and the complementability conditions (cf. point i.I) holdo 0 ~l,l

We shall seek a solution of the problem (5), (6) in the space Wl,2 (Qr), where the "zero" sign means that for t = 0 we consider null initial conditions. Further, let 0

0

fE 2,~ (Q~), gjew~,~ J(S), where c = l / 2 b , ~ = l - - 2 m , g j = l - - m l - - l / 2 ,

j = l . . . . . m.

In addition we shall assume that

where n = dimG.

Then from the familiar imbedding theorems for anisotropic Sobolev spaces it 0

follows that solutions tt(f,x)@l~z~,2 at.t{Qr) are classical solutions [we stress that no growth conditions with respect to the variable z are imposed on the functions a(t, x, z) and bj(t,

x, z)]. In order to formulate the result on the normal solvability of the problem (5), (6) precisely we introduce the Hilbert space (of right sides of the problem) 0

m

(QT)=WI,2 (QT)•

Wlff ](S). ]=1 0

We denote by oi, e 2 .... an orthonormal basis for the space

Wyffl,l

w2.2 (Qr) and let

B I,R,, RI, N) = {tt= t~+ v, y~L ~, II v II~, ~; R,, *~L -N, II ~ II~, ~R=}, where L N is the linear span of the first N vectors el, ...,eN; L -N is the orthogonal complement, R l > 0, R 2 > 0 are numbers; N = i, 2, .... One has the following theorem. THEOREM 1.3. Let the problem (5), (6) be parabolic. Then for any numbers R l > 0 and R > 0 and sufficiently large R 2 > 0 one can find a collection of functionals

F, (~, a) . . . . . ~ (n, a), defined on the set

{(,1, h):neR ~, t'1 (~ 0 one can find a number 6 > 0 such that for all u0(x) satisfying the inequality [u0[im+=,~6 and for which the compatibility conditions hold, the problem (10)-(12) has a unique solution U(t,x)~C~im+')/mn'=m+=(QT) for all T > 0 (QT = [0, T] x G) while

l u (t, x) 1~+.,~-~0 as t § +~. Let us assume that the following conditions hold: for any vl(t, x) and vi(t, x) such that

max{I v' 12a+~,Qr, I ~'212~+~,Qr} ~< ~ ~1. 2.

Nonlinear Parabolic Equations with Two Independent Variables

2.1. Boundary Problem for Equations of Partially Divergent Form. The method[ of monotonicity considered in Chap. 1 is also suited for the study of certain classes of boundary problems having generally nondivergent form. The largest class of such problems relates to the case of one spatial variable.

2589

Let (k, ~, r, s) be an aggregate on collections of nonnegative integers where k + ~ = 2m ( m ~ l ! being an integer). We denote the maximum of the numbers % in these collections by ~0 the corresponding value of k by k 0 (k 0 = 2 m - ~0). Let us assume that ko.~ I) and is monotonic, then for any right side h(t,x)ELr problem (20)-(22) has a unique solution u(t,

x~Wr 1 , 2 m

"

where

p !

= p/(p

--

1)

9

In particular the operator

-- 1..0~ ) ~/-+lA~ulm2A~u IF7 l'2ra/f]~

maps t h e space--u',p~-~J

2590

o n t o t h e s p a c e Lp,(Q) h o m e o m o r p h i c a l l y .

Interesting classes of divergent and partially divergent equations are studied in Kozhanov [44-46]. Namely, in the rectangle Q = [0, T] • [0, i] one considers the equation n~

Lu=-~-+ X ( - - I)~Dk%(u . . . . . Dkt0+~(u . . . . . Dktt)=h(t, x)

(23)

k=0

under initial-boundary conditions of two forms: u(0, x)=u0(x), Dku(t, 0)=Dhu(t, I)=0, 0~0,

~)I-.<M, O ~ < l < k ,

0 0 is sufficiently small. The following theorem holds. THEOREM 3.1. Let r > 1/2 + n/4m, where we recall m = dimG, 2m is the order of the operator A2m(X, D). If the conditions on F(x, DXu) indicated above hold, then the operator ( ~ ( u ) , ~ is analytic for sufficiently small p > 0. If in addition (30) holds, then the oper0

ator (~(u), ~00) has an analytic inverse R(f, u 0) defined in a ball ~p,~r'2mr-~Z2m(r+I/2)(O) with sufficiently small radius Pz > 0. The proof of the theorem is based on the Use of the familiar theorem on an inverse operator for an analytic map. Namely, let: i) ~(u) be an analytic map of a neighborhood of zero of the Banach space E l into a neighborhood of zero of the Banach space E2; 2) the Frechet derivative d~(0) of the operator ~(u) at the point 0 effect an isomorphism of the spaces E~ and E 2. Then locally ~(u) has an inverse operator ~(v) which is an analytic operator in a neighborhood of the point ~(0)~E=. The operator ~(v) having these properties is unique. 3.2. Solvability Theorem. The functional-analytic approach lets one get a number of results on the solvability of boundary problems for parabolic quasilinear equations. In the cylinder Q = [0, T] • G we consider the quasilinear 2b-parabolic equation

s

~ ] a~(D~b)tt)D~D~tt(t, [~,~1=M

(zo)tt)__f (t,x), (36)

(t, x)GQ, where [~, v] = I v l / 2 b + p; m, b, M a r e n a t u r a l numbers, m < M; D( 2 b ) d e n o t e s t h e row o f a l l 11 v p a r t i a l d e r i v a t i v e s o f t h e form DtDxtt, [ix,~]~<m. The f u n c t i o n s a ~ v ( - ) and a ( ' ) a r e a n a l y t i c functions of all their arguments, a(0) = 0. Let us assume that the following conditions hold: I.

2b-Parabolicity Condition.

For any z the roots of the polynomial

~(z;r,p)=_ ~

[p,v]=M

a~(z)p~(i~) ~,

satisfy the inequality

Re p ~ - - 6 1 ~l 2~, where ~ > 0 is a constant. Further, for Eq. (36) one poses the following boundary and initial conditions:

Oju-----

gj~DtDxuls=~j, 1 < j 0 arbitrary) with lateral surface S one studies the boundary problem 0u

Uls=O, 74/s ou =0'

(40)

u It=o= ~o(X), ~ ]~=o= ~(x)'

(41)

where 8u/3v is the derivative with respect to the inner normal to S. One defines the class of functions a(t, x, u, Du, 3u/3t) for which the problem (39)-(41) is solvable in the space 2,4 W2nOQ). Here a(t, x, u, p, q) is a real function of its arguments, uGR I, P Elan, q @RI" Let us assume that the following conditions hold: I. The function a(t, x, u, p, q) satisfies a Caratheodory condition, i.e., for almost all values (t,x)~Q it is continuous in the collection of variables (u, p, q) and measurable in (/,x)eQ for all (u,p,q)6R1XRn) 4 then n

n

n+2"]

la(...) ]~ 0 is an arbitrary fixed number. Under the conditions

indicated,

i = I, 2 are constants depending

the operator

~4 A (a) =--a (t, x, u, Du, 7/-~ W ~:2(Q)~-4(Q) is completely continuous and consequently defines a subordinate operator in relation to the principal part ~.

2,4

Let us assume in addition that the function a(t, x, u, p, q) satisfies the following functional condition: III. Let us assume that for any function v(t, x) which is sufficiently smooth in Q, and satisfies the conditions

V]s----O,

ls=o,

Ov

Ov

=

one has t

OQ

o~, ~: ~, ,

t

0 G

f o r any t6[O, T]. 2597

Here C I > 0, C 2 > 0 are constants depending on ~0(x) and ~2(x); ~ is an arbitrary constant from the interval (a, ~), where

(s.. + s.)I/21 s:',

[Vwledx:

s**=sup

~ s7~/2, =

(Aw)2dx----1, ~[oe-----~)- oo = 0 9

G

Under t h e c o n d i t i o n s

indicated

one h a s t h e f o l l o w i n g

theorem.

THEOREM 4 . 1 . I f c o n d i t i o n s I - I I I h o l d , t h e n f o r any r i g h t s i d e f ( t , x ~ L 2 ( Q ) and f o r any functions ~o(x)EVr and ~I(x)CW/~(G), satisfying the compatibility condition (~0[0o--~O~o/Ov== ~llao=0), the problem (39)-(41) has at least one solution E(t,x )~W/24 ~:~(Q). Under certain additional restrictions on 3a/au, 3a/~p, and 8a/Sq the solution is unique. For the proof one uses the principle of extension with respect to a parameter (the LeraySchauder principle), and in addition an essential role is played here by the a priori estimates of possible solutions obtained under conditions II and III. 4.2. Equations with Lowest Terms of Arbitrary Growth. First of all we give a result of Brezis and Browder in whose proof one uses both the methodology of monotone operators and that of compactness. In [97] these authors consider the first boundary problem for a nonlinear parabolic equation of the form

0u O-t-+ A (t) (u) q- g (t, ~, u) = f (t, x)

(42)

without restrictions on the growth of the nonlinear summand g(t, x, u) with respect to u. Suppose for each fixed t~[0,T]

A(t)(u)~

~

(--1)I~ID~A~(t,x, DVa) ([?l<m) 0m

i s a s t a n d a r d m o n o t o n e o p e r a t o r a c t i n g from Wp(G) (p>~2) t o t h e d u a l s p a c e w~m(G) ( c f . p o i n t 1.1). F u r t h e r , as u s u a l , l e t g ( t , x, u) s a t i s f y a C a r a t h e o d o r y c o n d i t i o n , i . e . , be m e a s u r a b l e in(t,x)EQ and c o n t i n u o u s i n u. I n a d d i t i o n , on t h e n o n l i n e a r p e r t u r b a t i o n g ( t , x , u) we impose the following basic condition: Condition a). There exists a nondecreasing function h(r):Rt-~l~$, h (0)----O, - such that for all( t,x)GQ and all r,

rg(t, x, r)~O, [g(t, x, r)[~h(r), while

h(r)~C(Ig(t,

x, r) 1+ lrl ~+1),

where C > 0 is a constant. [Obviously these inequalities do not impose any a priori restrictions on the growth of the function g(t, x, u) in u. ] 0

If condition a) holds one seeks a solution it(t,Jc),ELp(0, T; I~/~(O)), u(0, x)=0, of (42). Condition b) (restriction on the domain G ~ n ) . Let G6' ={xEG:dist(x,~G) 0. We shall say that the domain G satisfies condition b) if there exist constants C > 0 and 50 > 0 such that for all 5 < 50 and any ~(x)~'D(G) one has

I [~(x)IPdx