The Neumann problem in an irregular domain

Abstract

We consider ‘patterns’ stability for the reaction-diffusion equation with Neumann boundary conditions in an irregular domain in $R^N$, $N\ge 2$, the model example being two convex regions connected by a small ‘hole’ in their boundaries. By patterns we mean solutions having an interface, i.e. a transition layer between two constants. It is well known that in 1D domains and in many 2D domains, patterns are unstable for this equation. We show that, unlike the 1D case, but as in 2D dumbbell domains, stable patterns exist. In a more general way, we prove invariance of stability properties for steady states when a sequence of domains ${\Omega }_n$ converges to our limit domain $\Omega$ in the sense of Mosco. We illustrate the theoretical results by numerical simulations of evolving and persisting interfaces.