The heterogeneous Allen-Cahn equation in a ball: Solutions with layers and spikes

Abstract

Let u∊ be a single layered radially symmetric unstable solution of the Allen-Cahn equation -∈²Δu=u(u-a(|x|))(1-u) over the unit ball with Neumann boundary conditions. Based on our estimate of the small eigenvalues of the linearized eigenvalue problem at u∊ when ∈ is small, we construct solutions of the form u∊ + v∊, with v∊ non-radially symmetric and close to zero in the unit ball except near one point x₀ such that |x₀| is close to a nondegenerate critical point of a(r). Such a solution has a sharp layer as well as a spike.