Alfaro, Matthieu and Garcke, Harald and Hilhorst, Danielle and Matano, Hiroshi and Schaetzle, Reiner (2010) Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 140. pp. 673-706. ISSN 0308-2105,
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We consider the spatially inhomogeneous and anisotropic reaction-diffusion equation u(t) = m(s)(-1) div[m(x)a(p)(x, del u)] + epsilon(-2) f(u), involving a small parameter epsilon > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order epsilon(2) vertical bar ln epsilon vertical bar the unique weak solution u(epsilon) develops a steep transition layer that separates the regions {u(epsilon) approximate to 0} and {u(epsilon) approximate to 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as epsilon -> 0, the solution u(epsilon) converges almost everywhere (a.e.) to 0 in Omega(-)(t) and 1 in Omega(+)(t), where Omega(-)(t) and Omega(+)(t) are sub-domains of Omega separated by an interface P(t), whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order epsilon.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | NONCONVEX VARIATIONAL-PROBLEMS; GEOMETRIC EVOLUTION-EQUATIONS; REACTION-DIFFUSION-EQUATIONS; SINGULAR LIMIT; PERTURBATIONS; PROPAGATION; |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Harald Garcke |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 24 Aug 2020 08:37 |
| Last Modified: | 24 Aug 2020 08:37 |
| URI: | https://pred.uni-regensburg.de/id/eprint/25609 |
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