# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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## The singular limit of a vector-valued reaction-diffusion processHTML articles powered by AMS MathViewer

by Lia Bronsard and Barbara Stoth
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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