Asymptotic behavior of pulsating fronts and entire solutions of reaction–advection–diffusion equations in periodic media

Abstract

This paper is concerned with the reaction–advection–diffusion equations with bistable nonlinearity in periodic media. Assume that the equation has three equilibria: an unstable equilibrium θ and two stable equilibria 0 and 1. It is known that there exist different pulsating fronts connecting any two of those three equilibria. In this paper we first study the exponential behavior of the fronts when they approach their stable limiting states. Then, we establish three different types of pulsating entire solutions for the equation. To establish the existence of entire solutions, we consider combinations of any two of those different pulsating fronts and construct appropriate sub- and supersolutions.