On a study of a solution with a transition layer for a bistable reaction diffusion equation with a heterogeneous environment

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In this research 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 of the space variable. In particular, we consider the case where this function is identically zero on an interval and study the dynamics of the transition layer on the interval. In this case the dynamics of the transition layer on the interval 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.