Local Solutions

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LOCAL AND GLOBAL EXISTENCE OF SOLUTIONS TO SEMILINEAR PARABOLIC INITIAL VALUE PROBLEMS
Department of Mathematics,Lanzhou University, Lanzhou, Gansu 730000, People's Republic of China
Abstract: This paper is devoted to establishing local and global existence theorems for autonomous semilinear parabolic initial value problems. The local existence theorems do not require Lipchitz condition onnonlinear term. The global existence theorem is an extension of the well-known result of Fujita-Weissler for semilinear heat equations to general autonomous semilinear parabolic equations and systems. Key words and phrases: Semilinear parabolic system, initial value problem, solution, local existence, global existence. AMS 1991 subject classi cation number: 35K45.

S. B. Cui (Cui Shangbin)I. Introduction
In this paper we study local and global existence of solutions to the initial value problem
8 <

@t u = j j=m A @x u + F (u; @x u; :u(x; 0) = '(x); x 2 RN ;

P

m ; @x ?1 u); x 2 RN ; t > 0;

(1.1)

where m is an integer greater than 1, u = (u1 ; u2; ; un ); F = (F1 ; F2 ; ; Fn ) and ' = ('1 ; '2 ; ; 'n ) are n-dimensional real (resp. complex) vector functions, A 's areconstant N k real (resp. complex) n n matrices, 's represent indices in Z+ , and @x u represents the set of functions f@x u1 ; @x u2; ; @x un : j j = kg, which we regard as a vector function. We are only concerned with parabolic problems, i.e.,P we denote by ( ) ( 2 C N ) the maximum if of real parts of all eigenvalues of the matrix j j=m (i ) A , then it satis es the following assumption: (A1 )there exist positive constants a and b such that ( ) ?ajRe jm + bjIm jm ;

8 2 CN :

(1.2)

(Remark: As one can easily verify, the above assumption implies that m is an even number.)

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PARABOLIC INITIAL VALUE PROBLEMS

A typical example is the following reaction-di usion-convection system: @t u = A4u + F (u; @x u); x 2 RN ; t > 0; (1.3) where A is a diagonal matrix with positivediagonal elements. Another example is as follows: m = 2l; n = 1 and X A @x = (?1)l?1 4l : It is easy to verify that (1.2) is satis ed by both of the above examples. The rst major topic of this paper is concerned with local existence of solutions to (1.1). As is well-known, under the assumption that nonlinear lower-order terms satisfy Lipchitz condition, local existence of solutions to semilinearevolution equations can be obtained rather widespreadly. In fact, such results have been proved for almost all the semilinear evolution equations whose linear principal parts are well-posed (c.f. 1, 2] for instance). A natural as well as important problem is whether local existence of solutions can be obtained under weaker conditions such as the nonlinear lower-order terms are only continuous. Fornonlinear ordinary di erential equations and systems, such result, called Peano's theorem, is well-known (c.f. 1, Theorem 4, p.4] and 3, Theorem 1, Chapter 4]). For semilinear parabolic equations and systems, we can use the theory of semilinear evolution equations associated to compact C0 semigroup established by Pazy 4] and Lightbourne and Martin 5] to get similar results for their initial-boundaryvalue problems on bounded domains and periodic initial value problems (c.f. also 2, Section 6.2 and Section 8.2]. However, as far as general initial value problems are concerned, since the corresponding semigroups are not compact, this theory is no longer applicable. Our rst part of the main results lls partially in this gap. By following a di erent approach, we prove directly (without applying thesemigroup theory) that under the assumption (A1 ), if F is continuous, not required to satisfy Lipchitz m? condition, then for any ' 2 CBU 1 (RN ) (see the following section for this notation) (1.1) has a local weak solution; if in addition F satis es the assumption (A2 ) for any M > 0 there exists a corresponding consatant CM > 0 such that jF (w0 ; w1 ; ; wm?1 )j CM (jw0 j + jw1 j + + jwm?1 j)...