On reaction processes with hyperbolic or saturating diffusion

Abstract:

Reaction-diffusion equations are widely used to describe a variety of phenomena such as pattern formation and front propagation in biological, chemical and physical systems. In this talk, I consider different variations of the classic reaction-diffusion equations and, in particular, I focus the attention on two cases: first I present some hyperbolic variations obtained by substituting the Fick’s law with a relaxation law of Maxwell-Cattaneo type. Second, I consider the case when the linear diffusion is substituted by a strong saturating diffusion.

After a brief introduction and derivation of the models, the second part of the talk is devoted to the case of a balanced bistable reaction with the aim to extend the results on the metastable dynamics of the solutions to the one-dimensional Allen-Cahn equation. Precisely, I discuss the existence and persistence of metastable patterns both in the hyperbolic framework and in the case of strong saturation diffusion.