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

TL;DR

The paper addresses existence of weak solutions for local and nonlocal Cahn–Hilliard equations with singular potentials and reaction terms when the initial state is a pure phase, a setting where mass is not conserved. It develops a λ-approximation framework and a sharp mean-value (MZ sharp) analysis to control the singular nonlinearity, enabling uniform estimates and a rigorous limit passage to obtain solutions. The results cover both local and nonlocal formulations, provide uniqueness in the local case with phase-independent reaction terms, and connect the theory to tumor-growth and inpainting models. This work broadens the applicability of CH models with singular potentials by allowing initial pure phases and reaction-driven mass evolution, with potential extensions to more complex multi-physics systems.

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.

Theorems & Definitions (16)

• Proposition 2.4
• Theorem 2.5: local case
• Theorem 2.6: nonlocal case
• Remark 2.7
• Theorem 2.8: local case
• Theorem 2.9: nonlocal case
• Lemma 3.1
• proof : Proof of Lemma \ref{['lem:mean']}
• Remark 3.2
• proof : Proof of Theorem \ref{['THM:UNI:LOCAL']}