Motion of interfaces for a damped hyperbolic Allen-Cahn equation
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arXiv
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| Format: | Preprint |
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2018
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| author | Folino, Raffaele Lattanzio, Corrado Mascia, Corrado |
| author_facet | Folino, Raffaele Lattanzio, Corrado Mascia, Corrado |
| contents | Consider the Allen-Cahn equation $u_t=\varepsilon^2Δu-F'(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient $\varepsilon\to0^+$, and it is well known that, if the initial datum $u(\cdot,0)$ takes the values $+1$ and $-1$ in the regions $Ω_+$ and $Ω_-$, then the "interface" connecting $Ω_+$ and $Ω_-$ moves with normal velocity equal to the sum of its principal curvatures, i.e. the interface moves by mean curvature flow.
This paper concerns with the motion of the inteface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of $\mathbb{R}^n$, for $n=2$ or $n=3$. In particular, we focus the attention on radially simmetric solutions, studying in detail the differences with the classic parabolic case, and we prove that, under appropriate assumptions on the initial data $u(\cdot,0)$ and $u_t(\cdot,0)$, the interface moves by mean curvature as $\varepsilon\to0^+$ also in the hyperbolic framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1802_05038 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Motion of interfaces for a damped hyperbolic Allen-Cahn equation Folino, Raffaele Lattanzio, Corrado Mascia, Corrado Analysis of PDEs Consider the Allen-Cahn equation $u_t=\varepsilon^2Δu-F'(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient $\varepsilon\to0^+$, and it is well known that, if the initial datum $u(\cdot,0)$ takes the values $+1$ and $-1$ in the regions $Ω_+$ and $Ω_-$, then the "interface" connecting $Ω_+$ and $Ω_-$ moves with normal velocity equal to the sum of its principal curvatures, i.e. the interface moves by mean curvature flow. This paper concerns with the motion of the inteface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of $\mathbb{R}^n$, for $n=2$ or $n=3$. In particular, we focus the attention on radially simmetric solutions, studying in detail the differences with the classic parabolic case, and we prove that, under appropriate assumptions on the initial data $u(\cdot,0)$ and $u_t(\cdot,0)$, the interface moves by mean curvature as $\varepsilon\to0^+$ also in the hyperbolic framework. |
| title | Motion of interfaces for a damped hyperbolic Allen-Cahn equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1802.05038 |