Uniqueness of monostable pulsating wave fronts for time periodic reaction-diffusion equations

Keywords: Reaction–diffusion equations Pulsating wave fronts KPP and monostable nonlinearities Uniqueness a b s t r a c t We establish the uniqueness of pulsating wave fronts for reaction–diffusion equations in time periodic media with monostable nonlinearities. For the Kolmogorov–Petrovsky– Piskunov (KPP) type nonlinearity, this result provides a complete classification of all types of KPP pulsating fronts. Environmental heterogeneities are always present in natural phenomena, such as fluid convection effects in combustion, inhomogeneous porous structures in transport of solutes, noise effects in biology, and deposition processes [11]. For instance , Berestycki and Nirenberg [2] gave a detailed discussion on traveling wave fronts in cylinders and later Berestycki and Hamel [3] investigated front propagation in periodically varying media, which generalized the traveling wave theory in a class of periodic domains. The interested reader can also find general theory for spreading speeds and traveling waves in Weinberger [10]. Particularly, time periodic media have been widely used to model spatial propagation or spreading of biological invasions and disease spread when the effect of seasonal variation is concerned. Nadin and Rossi [7] investigated the propagation phenomena for reaction–diffusion equation with the heterogeneous KPP monostable nonlinearity depending on time. There are many papers devoted to this field and the readers can refer to Alikakos et al. [1], Zhao [12], Shen [9] and Jin and Zhao [6]. Recently, Hamel [4] investigated a kind of convection–reaction–diffusion equation in periodic media and established the decay rate of pulsating fronts near unstable equilibrium under some suitable conditions. Moreover, the author gave some qualitative properties of pulsating fronts of the following time periodic equation