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Spike Dynamics for a Reaction-Diffusion SystemMathematical Physics & Probability
|Speaker: ||Michael Ward, University of British Columbia.|
|Location: ||693 Kerr|
|Start time: ||Fri, Jan 29 1999, 4:10PM|
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)
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
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).