Collision and collapse of layers in a 1D scalar reaction–diffusion equation

This paper is devoted to the long time behavior of solutions to a scalar reaction–diffusion equation of 1-space dimension: ut=ϵ2uxx+u(u−a)(1−u), −∞<x<∞ (0<a<12,0<ϵ≪1). It is known that transition layers emerge for appropriate initial data and those move with speed O(ϵ) until they get close enough. One can also see that the layers annihilate in a finite time. In this paper with the aid of appropriate upper solutions and lower solutions we exhibit the collapse of the two facing layers by the collision and estimate the lifetime of the layers more precisely than previous works. The lower and upper solutions proposed here are explicitly given and play a key role in the study.