Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat

Weiwei Ding (1, 2) , Francois Hamel (1) , Xiao-Qiang Zhao (3)

1 I2M - Institut de Mathématiques de Marseille
2 School of Mathematical Sciences
3 Department of Mathematics and Statistics

This paper is concerned with the existence and qualitative properties of pulsating fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. We focus especially on the influence of the spatial period and, under various assumptions on the reaction terms and by using different types of arguments, we show several existence results when the spatial period is small or large. We also establish some properties of the set of periods for which there exist non-stationary fronts. Furthermore, we prove the existence of stationary fronts or non-stationary partial fronts at any period which is on the boundary of this set. Lastly, we characterize the sign of the front speeds and we show the global exponential stability of the non-stationary fronts for various classes of initial conditions.

Mots clés: bistable equations; periodic media; pulsating fronts; Reaction-diffusion equations

Domaines: Equations aux dérivées partielles [math.AP]

Weiwei Ding, Francois Hamel, Xiao-Qiang Zhao. Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat. Indiana University Mathematics Journal, 2017, 66, pp.1189-1265. ⟨10.1512/iumj.2017.66.6070⟩. ⟨hal-01052490⟩