Stable equilibria of a singularly perturbed reaction–diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface

Abstract

We use the variational concept of ${\Gamma}$-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation ${u_t=\epsilon^2 {\rm div}(k\nabla u)+f(u,x)}$, for ${(t,x)\in \mathbb{R}^+\times\Omega}$, where ${\Omega\subset\mathbb{R}^n}$, ${n\geq 1}$, supplied with no-flux boundary condition. The novelty here lies in the fact that the roots of the bistable function f are not isolated, meaning that the graphs of its roots are allowed to have contact or intersect each other along a Lipschitz-continuous (n − 1)-dimensional hypersurface ${\gamma \subset \Omega}$; across this hypersurface, the stable equilibria may have corners. The case of intersecting roots stems from the phenomenon known as exchange of stability which is characterized by ${f(\cdot,x)}$ having only two roots.