Stability of standing waves for monostable reaction–convection equations in a large bounded domain with boundary conditions

It is well known that the standing wave u0 for the KPP type convection–diffusion equation is stable if the perturbations of the initial data are in the weighted function spaces proposed by Sattinger. We study boundary conditions so that in a large finite domain, there is a stable standing wave near u0. The standing wave may not be monotone, and the stability is proved by pseudo exponential dichotomies that are weighted both in the spatial variable ξ and in the dual variable s to the time t.