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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The singular limit of a vector-valued reaction-diffusion process
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by Lia Bronsard and Barbara Stoth PDF
Trans. Amer. Math. Soc. 350 (1998), 4931-4953 Request permission

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
  • Email: bronsard@math.mcmaster.ca
  • Barbara Stoth
  • Affiliation: IAM, Universität Bonn, 53115 Bonn, Deutschland
  • Email: bstoth@iam.uni-bonn.de
  • Received by editor(s): November 17, 1995
  • Received by editor(s) in revised form: October 15, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4931-4953
  • MSC (1991): Primary 35B25, 35K57
  • DOI: https://doi.org/10.1090/S0002-9947-98-02020-0
  • MathSciNet review: 1443865