Existence and attractors of solutions for nonlinear parabolic systems

We prove existence and asymptotic behaviour results for weak solutions of a mixed problem (S). We also obtain the existence of the global at- tractor and the regularity for this attractor in H 2 () 2 and we derive estimates u1 = u2 = 0 in @ (0;T ) (b1(u1(x; 0);b2(u2(x; 0)) = (b1('0(x));b2( 0(x))) in where is a bounded open subset in R N , N 1, with a smooth boundary @ : (S) is an example of nonlinear parabolic systems modelling a reaction dif- fusion process for which many results on existence, uniqueness and regularity have been obtained in the case where bi(s) = s ( see, for instance (6; 7; 18)). The case of a single equation of the type (S) is studied in (1; 2; 3; 4; 5; 8; 9; 19): The purpose of this paper is the natural extension to system (S) of the results by (8), which concerns the single equation @ (u) @t u +f(x;t;u) = 0: Actually, our work generalizes the question of existence and regularity of the global attractor obtained therein. In the rst section of this paper, we give some assumptions and preliminaries and in section 2, we prove the existence of absorbing sets and the existence of the gobal attractor; while in section 3, we present the regularity of the attractor and show stabilization property. Finally, section 4 is devoted to estimates of the Haussdorf and fractal dimensions. 1. Preliminaries, Existence and Uniqueness 1.1 Notations and Assumptions Letbi, (i = 1; 2) be continuous functions withbi(0) = 0: We dene fort 2 R i(t) = R t 0 bi( )d : Then the Legendre transform of is dened by i ( ) = sup2R f s i(s)g: stands for a regular open bounded subset of