Rubén Caballero, José Valero (Centro de Investigación Operativa, Universidad Miguel Hernandez de Elche), Alexandre N. Carvalho (Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo) and Pedro Marín (Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla)
Abstract: In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.