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

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 $$ \frac{u(x^0+r\cdot)}{r} \to 0 \textrm{as} r\to 0.$$ 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.