Palestra – Multivalued semiflows generated by reaction-diffusion equations and the structure of their global attractors.

“Multivalued semiflows generated by reaction-diffusion equations and the structure of their global attractors.”

Prof. Dr. José Valero Cuadra

Universidad Miguel Hernández de Elche – Espanha.

Abstract: The theory of attractors for reaction-diffusion equations has been developed intensively over the last 40 years, including the study of their precise structure in several particular situations, specially in the one dimensional case.

 Describing the dynamics inside the global attractor becomes incresingly difficult when we consider equations in which uniqueness of solutions of the Cauchy problem fails. We study a scalar reaction-diffusion equation for which the non-linear term satisfies some growth and dissipativity assumptions ensuring global existence of solutions for the Cauchy problem, but not uniqueness. In this framework it is possible to define several multivalued semiflows generated by weak, regular and strong solutions of the equation and study the structure of the global attractor. More, precisely, we are able to prove that the attractor can be characterized as the union of all unstable manifolds of the set of stationary points.

 In the one dimensional case, for equations of the Chafee-Infante type, we are studying also whether a dynamically gradient structure of the attractor is true. More precisely, our hypothesis is that it consists of the stationary points and all the heteroclinic connections between them.