Rubén Caballero, José Valero (Centro de Investigación Operativa, Universidad Miguel Hernandez de Elche) and Pedro Marín (Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla)
Abstract: In this paper we study a nonlocal reaction–diffusion equation in which the diffusion 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 defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). 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.