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

We address the problem ut=ϵ2Δu+f(u,x)ut=ϵ2Δu+f(u,x) in Ω⊂RnΩ⊂Rn(n≥1)(n≥1) under boundary condition ∂νu=0∂νu=0 where f(u,x)=−(u−a(x))(u−θ(x))(u−b(x))f(u,x)=−(u−a(x))(u−θ(x))(u−b(x)), θ(x)=[a(x)+b(x)]/2θ(x)=[a(x)+b(x)]/2 and a≤ba≤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=ba=θ=b in Ω∖DΩ∖D where D⊂ΩD⊂Ω is such that D=D1∪D2D=D1∪D2, D1‾∩D2‾=∅, D1D1 and D2D2 are non-empty open connected sets with Lipschitz-continuous boundaries and a