Existence and characterization of attractors for a nonlocal reaction-diffusion equation with an energy functional

Abstract

In this paper we study a nonlocal reaction-di¤usion equation in which the di¤usion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by de ning a multivalued semi ow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satis ed). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories.