J. Melenk, C. Xenophontos, L. Oberbroeckling:

"Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: a road map";

in: "ASC Report 30/2011", issued by: Institute for Analysis and scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< \varepsilon \le \mu \le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $\varepsilon$ and $% \mu$. We show how one can construct full asymptotic expansions together with error bounds that cover the complete range $0 < \varepsilon \leq \mu \leq 1$.

For the present case of analytic input data, we present derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.

http://www.asc.tuwien.ac.at/preprint/2011/asc30x2011.pdf

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