Boundary-driven patterns in elongated convex domains

Abstract

We consider the heat equation in a smooth bounded convex domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ with nonlinear Neumann boundary condition ${\partial }_{\hat{I}}½u=\hat{I}»(u-u^3)$. Stable non-constant stationary solutions do not exist when $\hat{I}\text{c◯}$ is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of $\hat{I}\text{c◯}$ is fixed and its diameter is sufficiently large, then there exists $\hat{I}»>0$ for which the problem admits such a solution. The result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.

This work contains a significant error in the calculations, which compromises the conclusions presented