Boundary-driven patterns in elongated convex domains

We consider the heat equation in a smooth bounded convex domain $\Omega \subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_\nu u = \lambda (u - u^3)$. Stable non-constant stationary solutions do not exist when $\Omega$ is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of $\Omega$ 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.