The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment

In this paper we study the dynamics of a single transition layer of a solution to a spatially inhomogeneous bistable reaction diffusion equation in one space dimension. The spatial inhomogeneity is given by a function $a(x)$. In particular, we consider the case where $a(x)$ is identically zero on an interval $I$ and study the dynamics of the transition layer on $I$. In this case the dynamics of the transition layer on $I$ becomes so-called very slow dynamics. In order to analyze such a dynamics, we construct an attractive local invariant manifold giving the dynamics of the transition layer and we derive an equation describing the flow on the manifold. We also give applications of our results to two well known nonlinearities of bistable type.