On the Existence of Patterns for a Diffusion equation on a Convex Domain with Nonlinear Boundary Reaction

We consider a diffusion equation on a domain Ω with a cubic reaction at the boundary. It is known that there are no patterns when the domain Ω is a ball, but the existence of such patterns is still unkown in the more general case in which Ω is convex. The goal of this paper is to present numerical evidence of the existence of nonconstant stable equilibria when Ω is the unit square. These patterns are found by continuation of families of unstable equilibria that bifurcate from constant solutions.