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; these terms replace, respectively, the constant mobility, the linear Laplace operator and a C2 potential, 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 ≥ p > 1. In the critical case 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.
ON A GENERALIZED CAHN-HILLIARD MODEL WITH p-LAPLACIAN
Strani M.
2022-01-01
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; these terms replace, respectively, the constant mobility, the linear Laplace operator and a C2 potential, 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 ≥ p > 1. In the critical case 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.
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