## 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.## References

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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