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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The singular limit of a vector-valued reaction-diffusion process

Authors: Lia Bronsard and Barbara Stoth
Journal: Trans. Amer. Math. Soc. 350 (1998), 4931-4953
MSC (1991): Primary 35B25, 35K57
MathSciNet review: 1443865
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Abstract: We study the asymptotic behaviour of the solution to the vector–valued reaction–diffusion equation \begin{equation*}\varepsilon {\partial _{t}}\varphi -\varepsilon \triangle \varphi + {\frac {1}{\varepsilon }} \tilde W_{,\varphi } (\varphi ) = 0 \quad \text { in } \Omega _{T}, \end{equation*} where $\varphi _{\varepsilon }=\varphi :\Omega _{T}:=(0,T)\times \Omega \longrightarrow \mathbf {R}^{2}$. We assume that the the potential $\tilde W$ depends only on the modulus of $\varphi$ and vanishes along two concentric circles. We present a priori estimates for the solution $\varphi$, and, in the spatially radially symmetric case, we show rigorously that in the singular limit as $\varepsilon \to 0$, two phases are created. The interface separating the bulk phases evolves by its mean curvature, while $\varphi$ evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.

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Additional Information

Lia Bronsard
Affiliation: Department of Mathematics, McMaster University, Hamilton, Ont. L8S 4K1, Canada

Barbara Stoth
Affiliation: IAM, Universität Bonn, 53115 Bonn, Deutschland

Received by editor(s): November 17, 1995
Received by editor(s) in revised form: October 15, 1996
Article copyright: © Copyright 1998 American Mathematical Society