The asymptotic behaviour of solutions for the following reaction-diffusion equation
is investigated. There and are positive constants, is a given function from , and the nonlinear function satisfies the following conditions: , , , with for and for .
First the existence of a unique solution of equation (1) with the initial data (2) is shown, which establishes the existence of a dynamical system such that in . Later the asymptotic compactness of is shown. To overcome the difficulty of the lack of compactness of the Sobolev embeddings in , the author approaches by a bounded domain and uses the compactness of the embeddings in bounded domains. The main result states that problem (1) (2) has a global attractor in . At the end the finite dimensionality of this global attractor is studied.