Propagation dynamics of a reaction-diffusion equation in a time-periodic shifting environment.

This paper concerns the nonautonomous reaction-diffusion equation u t = u x x + u g (t , x − c t , u) , t > 0 , x ∈ R , where c ∈ R is the shifting speed, and the time periodic nonlinearity u g (t , ξ , u) is asymptotically of KPP type as ξ → − ∞ and is negative as ξ → + ∞. Under a subhomogeneity condition, we show that there is c ⁎ > 0 such that a unique forced time periodic wave exists if and only if | c | < c ⁎ and it attracts other solutions in a certain sense according to the tail behavior of initial values. In the case where | c | ≥ c ⁎ , the propagation dynamics resembles that of the limiting system as ξ → ± ∞ , depending on the shifting direction. [ABSTRACT FROM AUTHOR]