Existence of a Degenerate Singularity in the High Activation Energy Limit of a Reaction-Diffusion Equation

We consider the singular perturbation problem where , β is a Lipschitz continuous function such that β > 0 in (0, 1), β ≡ 0 outside (0, 1) and . We construct an example exhibiting a degenerate singularity as ε k ↘ 0. More precisely, there is a sequence of solutions u ε k → u as k → ∞, and there exists x 0 ∈ ∂{u > 0} such that Known results suggest that this singularity must be 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.