Seminario del Departamento de
Ecuaciones Diferenciales y Análisis Numérico

Fecha : 14 de julio de 2014
Hora  : 10:30
Lugar : Seminario del Departamento (Fac. de Matemáticas, 3a. planta, módulo 34)

Silvia Sastre Gómez

(Universidad Complutense de Madrid)

Resumen

In this work we study the existence, uniqueness, comparison properties and asymptotic behaviour of the solutions of some nonlocal diffusion problems. All the problems in this work are set in metric measure spaces. These spaces include very different type of spaces, for example, open subsets in R^N, graphs, manifolds, multistructures or some fractal sets. First of all we study the solutions of the linear nonlocal diffusion problem. In particular we describe the asymptotic behaviour using spectral methods. After that we will study the nonlinear nonlocal diffusion problem with a local reaction. In particular, we prove weak and strong maximum principles and the existence of two extremal equilibria, which attract the asymptotic dynamics of the solutions. We also show how the lack of smoothing prevent us from proving the existence of a global attractor. After that, we consider a nonlinear term that is a nonlinear function of the average of the solution in a ball. In this case we prove that there are strong restrictions for the weak and strong maximum principles to hold. When these hold, we prove the existence of a global attractor.