A description of the global attractor for a class of reaction-diffusion systems with periodic solutions

We examine the autonomous reaction-diffusion system [GRAPHICS] with Dirichlet boundary conditions on (0, 1), where alpha, beta are real, alpha > 0, and g is C-1 and satisfies some conditions which we need in order to prove the existence of solutions. We construct a solution of (RD) for every initial value in L-2((0, 1)) X L-2((0, 1)), we show that this solution is uniquely determined and that the solution has C-infinity-smooth representatives for ail positive t. We determine the long time behaviour of each solution. In particular, we show that each solution of (RD) tends either to the zero solution or to a periodic orbit. We construct all periodic orbits and show that their number is always finite. It turns out that the global attractor is a finite union of subsets of L-2 X L-2, which are finite-dimensional manifolds, and the dynamics in these sets can be described completely