Application of asymptotic methods to the question of stability in stationary solution with discontinuity on a curve

This article is considering the stability property of the solution with inner layer for singularly perturbed stationary equation with Neumann boundary conditions. The right-hand side is assumed to have discontinuity on some arbitrary curve $h(t)$. Stability analysis is performed by obtaining the first non-zero coefficient of the series for eigenvalue and eigenfunction from the Sturm–Liouville problem. Theory of the asymptotic approximations is used in order to construct them.