Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems (Q406683)

In this paper, the author study the spatially heterogeneous reaction-diffusion equation \[ \partial _{t}u=\partial _{xx}u+f(x,u),\quad x\in \mathbb{R},t>0, \] where \(f\) satisfies proper continuity conditions is of the heterogeneous KPP type. When \(f(x,u)\) is periodic with respect to \(x\in \mathbb{R}\), the long-time behavior of the corresponding initial value problem is investigated by the pulsating traveling front with minimal wave speed. When \(f(x,u)\) is asymptotically periodic with respect to \(x\in \mathbb{R}\), the author introduces an auxiliary equation that is periodic in \(x\in \mathbb{R}.\) With the help of the pulsating traveling front with minimal wave speed of the auxiliary equation, the Cauchy problem is considered.