Infinite-Time Blow-up Arising in a Mean Curvature Flow
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916867455582208 |
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| author | Chen, Xinfu Lou, Bendong Wang, Xiaoliu Yuan, Lixia |
| author_facet | Chen, Xinfu Lou, Bendong Wang, Xiaoliu Yuan, Lixia |
| contents | We consider a mean curvature flow in a cylinder with Robin boundary conditions, which can be used to model the interface motion in singular limit problems of the Allen-Cahn equation with nonlinear boundary conditions. It was shown in \cite{LWY} that the planar curvature flow converges to a translating Grim Reaper with {\it finite speed} and {\it fixed profile}. In this paper we study the high dimensional problem, and show surprisingly different features caused by the dimension: a radial flow $u(|x|,t)$ propagates at {\it exponential asymptotic speed}, both the gradient $|Du|$ (everywhere except for the center) and the instantaneous speed $u_t$ (everywhere) also increase to infinity exponentially as $t\to \infty$. Due to the lack of uniform-in-time $C^0, C^1$ and $C^2$ estimates, the equation is asymptotically degenerate, we will use a new approach (that is, the zero number argument) to prove the conclusions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_07831 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Infinite-Time Blow-up Arising in a Mean Curvature Flow Chen, Xinfu Lou, Bendong Wang, Xiaoliu Yuan, Lixia Differential Geometry 35B40, 53A05, 53E10 We consider a mean curvature flow in a cylinder with Robin boundary conditions, which can be used to model the interface motion in singular limit problems of the Allen-Cahn equation with nonlinear boundary conditions. It was shown in \cite{LWY} that the planar curvature flow converges to a translating Grim Reaper with {\it finite speed} and {\it fixed profile}. In this paper we study the high dimensional problem, and show surprisingly different features caused by the dimension: a radial flow $u(|x|,t)$ propagates at {\it exponential asymptotic speed}, both the gradient $|Du|$ (everywhere except for the center) and the instantaneous speed $u_t$ (everywhere) also increase to infinity exponentially as $t\to \infty$. Due to the lack of uniform-in-time $C^0, C^1$ and $C^2$ estimates, the equation is asymptotically degenerate, we will use a new approach (that is, the zero number argument) to prove the conclusions. |
| title | Infinite-Time Blow-up Arising in a Mean Curvature Flow |
| topic | Differential Geometry 35B40, 53A05, 53E10 |
| url | https://arxiv.org/abs/2302.07831 |