Strong Attractors for the Benjamin-bona-mahony Equation

|In this paper, we study the asymptotic behaviour of the solutions for the Benjamin-Bona-Mahony equation. We rst present the existence of the global weak attractor in H 2 per for this equation. And then by an energy equation we show that the global weak attractor is actually the global strong attractor in H 2 per. In this note, we consider the following Benjamin-Bona-Mahony equation: (1) with the initial condition u(x; 0) = u 0 (x); x 2 ; (2) and the periodic boundary condition = (0; L) and u is-periodic; (3) where is a positive constant, f : R ! R is a C 1 function and g(x) 2 L 2 per (() with zero mean on. This equation, which incorporates nonlinear dispersive and dissipative eeects, has been proposed as a model for the propagation of long waves. In the case of f(u) = u + (1=2)u 2 , equation (1) has been investigated by many authors such as Benjamin, Bona and Mahony 1]; Bona and Dougalis 2]; Albert 3], and the references therein. Our aim of this paper is to derive the existence of a nite dimensional global attractor for the system (1){(3) in H 2 per ((). Since the dynamical system S(t) deened by equation (1) is not compact in H 2 per (() we cannot construct the global attractor by the usual method introduced by Temam 4]. Here we rst apply the techniques developed by Ghidaglia 5] to show the existence of global weak attractor for equation (1), and then by energy estimates we prove that the weak attractor is actually the strong attractor in H 2 per ((). By the Galerkin method, we can easily deduce the following existence results (see 6]). Theorem 1. Assume that u 0 2 H k per ((); k = 1; 2. Then problem (1){(3) possesses a unique solution u(t) deened on R + such that u(t) 2 L 1 ?