Tomás Caraballo

Univ. Sevilla, Spain

Title: Recent results on non-local deterministic and stochastic models

Abstract: The aims of this talk is to report on some recent results on non-local partial differential equations (both deterministic and stochastic) that have been obtained, by our research group during the last decade, and which are inspired in the nice investigation that Michel chipot and his collaborators had previously carried out from the 90’s on. First of all, we were able to analyze non-local non-autonomous reaction-diffusion equations by exploiting the techniques of pullback attractors, which has proven very appropriate for the asymptotic behavior of non-autonomous dynamical systems. Next, we considered non-local problems containing some hereditary characteristics (delay, memory), and performed a complete analysis concerning well-posedness, local stability, and global stability (existence of attractors). A model concerned with non-local diffusion and memory deserves a special treatment since we could establish the appropriate framework for studying its asymptotic behavior, instead of the one provided by Dafermo’s transformation which usually had been used for this kind of problems. But also, we have analyzed the more realistic models containing some kind of randomness or stochasticity. We have developed a theory for the treatment of non-local models containing different kinds of noise, and have made use of the powerful theories of random dynamical systems, mean weak random attractors, approximation by colored noise, etc to obtain interesting properties about the differences with the deterministic counterparts. Our deepest thanks to Michel for having been the inspiration of a fruitful research line of our research group.