Global existence and decay to equilibrium for some crystal surface models
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866916575993397248 |
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| author | Granero-Belinchón, Rafael Magliocca, Martina |
| author_facet | Granero-Belinchón, Rafael Magliocca, Martina |
| contents | In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations $$ \partial_t u=Δe^{-Δu}, $$ $$ \partial_t u=-u^2Δ^2(u^3). $$ These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97, 281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_09645 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Global existence and decay to equilibrium for some crystal surface models Granero-Belinchón, Rafael Magliocca, Martina Analysis of PDEs In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations $$ \partial_t u=Δe^{-Δu}, $$ $$ \partial_t u=-u^2Δ^2(u^3). $$ These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97, 281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces. |
| title | Global existence and decay to equilibrium for some crystal surface models |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1804.09645 |