On a generalized Cahn-Hilliard model with p-Laplacian
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916249769869312 |
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| author | Folino, Raffaele Rios, Luis Fernando Lopez Strani, Marta |
| author_facet | Folino, Raffaele Rios, Luis Fernando Lopez Strani, Marta |
| contents | A generalized Cahn-Hilliard model in a bounded interval of the real line with no-flux boundary conditions is considered. The label "generalized" refers to the fact that we consider a concentration dependent mobility, the $p$-Laplace operator with $p>1$ and a double well potential of the form $F(u)=\frac{1}{2θ}|1-u^2|^θ$, with $θ>1$; these terms replace, respectively, the constant mobility, the linear Laplace operator and the $C^2$ potential satisfying $F"(\pm1)>0$, which are typical of the standard Cahn-Hilliard model. After investigating the associated stationary problem and highlighting the differences with the standard results, we focus the attention on the long time dynamics of solutions when $θ\geq p>1$. In the $critical$ $θ=p>1$, we prove $exponentially$ $slow$ $motion$ of profiles with a transition layer structure, thus extending the well know results of the standard model, where $θ=p=2$; conversely, in the $supercritical$ case $θ>p>1$, we prove $algebraic$ $slow$ $motion$ of layered profiles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_11173 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On a generalized Cahn-Hilliard model with p-Laplacian Folino, Raffaele Rios, Luis Fernando Lopez Strani, Marta Analysis of PDEs A generalized Cahn-Hilliard model in a bounded interval of the real line with no-flux boundary conditions is considered. The label "generalized" refers to the fact that we consider a concentration dependent mobility, the $p$-Laplace operator with $p>1$ and a double well potential of the form $F(u)=\frac{1}{2θ}|1-u^2|^θ$, with $θ>1$; these terms replace, respectively, the constant mobility, the linear Laplace operator and the $C^2$ potential satisfying $F"(\pm1)>0$, which are typical of the standard Cahn-Hilliard model. After investigating the associated stationary problem and highlighting the differences with the standard results, we focus the attention on the long time dynamics of solutions when $θ\geq p>1$. In the $critical$ $θ=p>1$, we prove $exponentially$ $slow$ $motion$ of profiles with a transition layer structure, thus extending the well know results of the standard model, where $θ=p=2$; conversely, in the $supercritical$ case $θ>p>1$, we prove $algebraic$ $slow$ $motion$ of layered profiles. |
| title | On a generalized Cahn-Hilliard model with p-Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2202.11173 |