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

Abstract

We consider the singular perturbation problem $$\Delta u_{\epsilon }={\beta }_{\epsilon }(u_{\epsilon }),$$ where ${\beta }_{\epsilon }(s)=\frac{1}{\epsilon }\beta (\frac{s}{\epsilon })$, $\beta$ is a Lipschitz continuous function such that $\beta >0$ in $(0,1)$, $\beta \equiv 0$ outside $(0,1)$ and ${\int }_0^1\beta (s)ds=1/2$. We construct an example exhibiting a {\em degenerate singularity} as ${\epsilon }_k\searrow 0$. More precisely, there is a sequence of solutions $u_{{\epsilon }_k}\to u$ as $k\to \infty$, and there exists $x^0\in \partial \{u>0\}$ such that $$\frac{u(x^0+r\cdot )}{r}\to 0\text{as}r\to 0.$$ 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

References in corpus (1)

• Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem