Diffusion and interfaces in pattern formation

Abstract

We discuss several qualitative properties of the solutions of reaction-diffusion systems and equations of the form $u_t = \epsilon^2 D \Delta u + f(u,x,\epsilon t)$, that are used in modeling pattern formation. We analyze the diffusion neutral and the diffusion dependent situations that, in the time autonomous case, are distinguished by considering the attractors of the shorted equation $u_t = f(u,x)$. We discuss the consequences of being in one or in the other of the two situations and present examples from developmental biology and from fluid mechanics.