The Existence Of A Class Of Quasi Linear Cross-diffusion Equations

We investigate a biological model described by the following cross-diffusion system which was first proposed by N.Shigesada[5j:This paper is concerned with the local existence and global existence of the solutions for the system (1).l)The existence of local solutionsIn the part of the existence of local solutions,we investigate the intial boundary value problems for the more general quasilinear system:We consider the existence of local solutions of the system (2) when it may be a cross-diffusion system and u G(G is a open subset of Rn).In order to use the results of H.Amann [1] on local existence for quasilinear systems (2),we find that the results of H.Amann on the corresponding linear system were not proved completely.especially for the case of cross-diffusion systems (2). First.for linear systems in more general divergent form stated in [1],we give a complete proof on the generation of an analytic semigroup for the linear system by verifying the general sufficient condition on the analytic semigroup((.A,B,) is a-regular elliptic BVP),which make up for the insufficiency in the existence of local solutions which was established by H.Amann[2].Thus the H.Amann's theory on local existence and global existence of (1) is valid for cross-diffusion system (1).2)The existence of global solutionsIn the part of the existence of global solutions,we study the existence of local solutions of the system (1) which we got in the first part (n = l,n = 2,or n = 3). We use several different Banach space including V2(Q(T))([11]),Lp,q(QT),W(QT) and Lp,and the imbeding theorem between them,we combine the "energy" method with the classical estimates to obtain a series of a priori estimatessuch as estimation,until the priori estimates for global existence needed in H.Amann's theroy [2] hold.. When n=1,if a22 = 0 and there exists a unique global solution of (1);if a22 > 0 and there exists a unique global solution of (1).When n = 2,there exists a unique global solution of (1) when.When n = 3,a11 = 0,if a22 = 0 and ,there exists a unique global solution of (1);if a22>0 and ,there exists a unique global solution of (1).