Cahn-Hilliard equations with singular potential, reaction term and pure phase initial datum

Maurizio Grasselli; Luca Scarpa; Andrea Signori

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arXiv · 2404.12113

Cahn-Hilliard equations with singular potential, reaction term and pure phase initial datum

Maurizio Grasselli·Luca Scarpa·Andrea Signori

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Abstract

We consider local and nonlocal Cahn-Hilliard equations with constant mobility and singular potentials including, e.g., the Flory-Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. In other words, the separation process takes place even in presence of a pure phase, provided that it is triggered by a convenient reaction term. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we generalize the previously-mentioned concept by examining the essential assumptions required for the reaction term to apply the new strategy. Also, we explore the scenario involving the nonlocal Cahn-Hilliard equation and provide illustrative examples that contextualize within our abstract framework.

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Maurizio Grasselli, Luca Scarpa, Andrea Signori. 2024-04-18. Cahn-Hilliard equations with singular potential, reaction term and pure phase initial datum. https://arxiv.org/abs/2404.12113

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