Boundary-driven patterns in elongated convex domains

We consider the heat equation in a smooth bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with nonlinear Neumann boundary condition ${\mathrm{\partial }}_{\nu }u=\lambda (u-u^3)$. Stable non-constant stationary solutions do not exist when $\mathrm{\Omega }$ is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of $\mathrm{\Omega }$ is fixed and its diameter is sufficiently large, then there exists $\lambda >0$ 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.