Stable Transition Layers to Singularly Perturbed Spatially Inhomogeneous Allen-Cahn Equation

We use the variational concept of Γ-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation u = ε Δu + u[a (x) − u ], for (t, x) ∈ ℝ × Ω, where Ω ⊂ ℝ and a is positive, supplied with no-flux boundary condition. Let γ ⊂ Ω be a smooth simple closed curve and N(γ) a narrow tubular neighborhood of γ. Roughly speaking, the sufficient condition found for existence of such solutions relates the geometric profile of the function a in N(γ) to the signed curvature of γ .