Existence of a degenerate singularity in the high activation energy limit of a reaction-diffusion equation

Abstract

We consider the singular perturbation problem $$ Δu_ε=β_ε(u_ε), $$ where $β_ε(s)=\frac{1}εβ(\frac{s}ε)$, $β$ is a Lipschitz continuous function such that $β>0$ in $(0, 1)$, $β\equiv 0$ outside $(0, 1)$ and $\int_0^1β(s) ds={1/2}$. We construct an example exhibiting a {\em degenerate singularity} as $ε_k\searrow 0$. More precisely, there is a sequence of solutions $u_{ε_k}\to u$ as $k\to \infty$, and there exists $x^0\in\partial\{u>0\}$ such that $$ \frac{u(x^0+r\cdot)}{r} \to 0 \textrm{as} r\to 0.$$ Known results suggest that this singularity must be {\em unstable}, which makes it hard to capture analytically and numerically. Our result answers a question raised by Jean-Michel Roquejoffre at the FBP'08 in Stockholm.

17 pages, 5 figures