ANZIAM  J.  47 (2006), 581-591
A note on the small-time development of the solution to a coupled, nonlinear, singular reaction-diffusion system

J. A. Leach
  Department of Mathematics
  University of Reading
  Berkshire RG6 6AX

In this paper, we consider a coupled, nonlinear, singular (in the sense that the reaction terms in the equations are not Lipschitz continuous) reaction-diffusion system, which arises from a model of fractional order chemical autocatalysis and decay, with positive initial data. In particular, we consider the cases when the initial data for the the dimensionless concentration of the autocatalyst, $\beta$, is of (a) $O(x^{-\lambda})$ or (b) $O(e^{-\sigma x})$ at large $x$ (dimensionless distance), where $\sigma>0$ and $\lambda$ are constants. While initially the dimensionless concentration of the reactant, $\alpha$, is identically unity, we establish, by developing the small- $t$ (dimensionless time) asymptotic structure of the solution, that the support of $\beta(x,t)$ becomes finite in infinitesimal time in both cases (a) and (b) above. The asymptotic form for the location of the edge of the support of $\beta$ as $t \to 0$ is given in both cases.
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