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Description

Many classes of singularly perturbed reaction-diffusion systems possess localized solutions where the gradient of the solution is very large only in the vicinity of certain points in the domain. An example of such a problem where spikes occur is the Geirer-Meinhardt (GM) activator-inhibitor model of morphogenesis. Of interest is to characterize the equilibria, the stability, and the dynamics of spike-layer patterns. Results of this nature are given for the GM model for the case of an infinite inhibitor diffusivity and for a finite inhibitor diffusivity. For the case of an infinite inhibitor diffusivity, the GM model reduces to a non-local problem for the activator concentration. By studying the spectrum of the linearization it is shown that the non-local term leads to the existence of metastable behavior for a one-spike solution in a multi-dimensional domain. An explicit characterization of the metastable dynamics is given and is confirmed in the case of one-spatial dimension by numerical computations. The dynamics of a spike attached to the boundary of a multi-dimensional domain are also described. The case of a finite inhibitor diffusivity is studied in a one-dimensional domain. It is shown that there are a sequence of critical values $D_n$ of the inhibitor diffusivity $D$ for which an $n$-spike equilibrium solution is stable if $DD_n$. An explicit formula for $D_n$ is given. The dynamics of an $n$-spike solution are then described. Finally, we show that many of the results given here for the GM model also hold for other reaction-diffusion equations including the non-local Allen-Cahn equation from materials science, a problem in microwave heating etc.

Some of this work is joint with David Iron (graduate student at UBC), and Prof. Juncheng Wei (Chinese University of Hong Kong).