Pulsating type entire solutions of monostable reaction–advection–diffusion equations in periodic excitable media

Abstract

We establish the existence of pulsating type entire solutions of reaction–advection–diffusion equations with monostable nonlinearities in a periodic framework. Here the nonlinearities include the classic KPP case. The pulsating type entire solutions are defined in the whole space and for all time t ∈ R . By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function, we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions, and approaching each other. The key techniques are to characterize the asymptotic behavior of the solutions as t → − ∞ in terms of appropriate subsolutions and supersolutions.