Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866909206261530624 |
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| author | Folino, Raffaele |
| author_facet | Folino, Raffaele |
| contents | The aim of this paper is to prove that, for specific initial data $(u_0,u_1)$ and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen-Cahn equation on the interval $[a,b]$ shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the "energy approach" proposed by Bronsard and Kohn [8], if $\varepsilon\ll 1$ is the diffusion coefficient, we show that in a time scale of order $\varepsilon^{-k}$ nothing happens and the solution maintains the same number of transitions of its initial datum $u_0$. The novelty consists mainly in the role of the initial velocity $u_1$, which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen-Cahn equation with relaxation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1510_07168 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension Folino, Raffaele Analysis of PDEs The aim of this paper is to prove that, for specific initial data $(u_0,u_1)$ and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen-Cahn equation on the interval $[a,b]$ shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the "energy approach" proposed by Bronsard and Kohn [8], if $\varepsilon\ll 1$ is the diffusion coefficient, we show that in a time scale of order $\varepsilon^{-k}$ nothing happens and the solution maintains the same number of transitions of its initial datum $u_0$. The novelty consists mainly in the role of the initial velocity $u_1$, which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen-Cahn equation with relaxation. |
| title | Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1510.07168 |