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September/October 2022 On a generalized Cahn--Hilliard model with $p$-Laplacian
Raffaele Folino, Luis F. López Ríos, Marta Strani
Adv. Differential Equations 27(9/10): 647-682 (September/October 2022).

Abstract

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 $F(u)=\frac{1}{2\theta}|1-u^2|^\theta$, with $\theta > 1$; 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 $\theta\geq p > 1$. In the critical case $\theta= p > 1$, we prove exponentially slow motion of profiles with a transition layer structure, thus extending the well know results of the standard model, where $\theta=p=2$; conversely, in the supercritical case $\theta > p > 1$, we prove algebraic slow motion of layered profiles.

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Raffaele Folino. Luis F. López Ríos. Marta Strani. "On a generalized Cahn--Hilliard model with $p$-Laplacian." Adv. Differential Equations 27 (9/10) 647 - 682, September/October 2022.

Information

Published: September/October 2022
First available in Project Euclid: 2 June 2022

Subjects:
Primary: 35B36 , 35B40 , 82C26

Rights: Copyright © 2022 Khayyam Publishing, Inc.

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Vol.27 • No. 9/10 • September/October 2022
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