Limiting classification on linearized eigenvalue problems for 1-dimensional Allen–Cahn equation I—asymptotic formulas of eigenvalues

We are interested in classical linearized eigenvalue problems for one dimensional reaction–diffusion equations with small diffusion coefficient. Previously, authors have obtained the precise asymptotic formulas of all eigenvalues and eigenfunctions for f(u)=sin⁡uf(u)=sin⁡u, which suggest interesting structure of eigenpairs: existence of two special eigenpairs and the limiting classification of the others by special eigenpairs. In this paper we study the Allen–Cahn case f(u)=u−u3f(u)=u−u3 and obtain the asymptotic formulas of eigenvalues, which leads us to the limiting classification of eigenvalues.