Stability of stationary solutions for semilinear heat equations with concave nonlinearity

Abstract

This paper is concerned with the stability analysis of stationary solutions of the Cauchy–Dirichlet problem for some semilinear heat equation with concave nonlinearity. The instability of sign-changing solutions is verified under some variational assumption. Moreover, the exponential stability of the positive stationary solution at an optimal rate is proved by exploiting a super–subsolution method as well as the parabolic regularity theory. The base of our analysis relies on the linearization of the equation at each stationary solution and spectral analysis of the corresponding linearized operator. The main difficulties reside in the singularity of the linearized operator due to the concave nonlinearity.