Exponential asymptotic stability via Krein-Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems

Abstract

We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein-Rutman theorem.