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

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 establi...

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Bibliographic Details
Published inEuropean journal of applied mathematics pp. 1 - 26
Main Authors Grasselli, Maurizio, Scarpa, Luca, Signori, Andrea
Format Journal Article
LanguageEnglish
Published Cambridge University Press 27.05.2025
Subjects
35D30
35K35
35K86
35Q92
92C17
Cahn–Hilliard equation
existence of a weak solution
initial pure phase
nonlocal Cahn–Hilliard equation
reaction term
singular potential
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Summary: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. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we examine the essential assumptions required for the reaction term to ensure the existence of a weak solution. Also, we explore the scenario involving the nonlocal Cahn–Hilliard equation and provide some illustrative examples that contextualize within our abstract framework.
ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792525000166