Stable equilibria to a singularly perturbed reaction–diffusion equation in a degenerated heterogeneous environment

Abstract

We address the problem ut=ϵ2Δu+f(u,x) in Ω⊂Rn(n≥1) under boundary condition ∂νu=0 where f(u,x)=−(u−a(x))(u−θ(x))(u−b(x)), θ(x)=[a(x)+b(x)]/2 and a≤b in Ω. The novelty here lies in the fact that the roots of f are allowed to degenerate in the sense that a=θ=b in Ω∖D where D⊂Ω is such that D=D1∪D2, D1‾∩D2‾=∅, D1 and D2 are non-empty open connected sets with Lipschitz-continuous boundaries and a<b in D. For ϵ small, we prove existence of four families of stable stationary solutions uϵ approaching the roots of f in the topology of L1. Our approach is variational and based on Γ-convergence theory.