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Lyapunov stability of non-isolated equilibria for strongly irreversible Allen–Cahn equations

Part of: Equations and inequalities involving nonlinear operators Topological dynamics Parabolic equations and systems

Published online by Cambridge University Press:  18 December 2024

Goro Akagi Open the ORCID record for Goro Akagi [Opens in a new window]  and
Messoud Efendiev
Goro Akagi*
Affiliation:
Mathematical Institute and Graduate School of Science, Tohoku University, Aoba, Sendai 980-8578, Japan ([email protected]) (corresponding author)
Messoud Efendiev
Affiliation:
Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstraße 1, 85764 Neunerberg, Germany ([email protected]; [email protected])
*
*Corresponding author.
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Abstract

The present article is concerned with the Lyapunov stability of stationary solutions to the Allen–Cahn equation with a strong irreversibility constraint, which was first intensively studied in [2] and can be reduced to an evolutionary variational inequality of obstacle type. As a feature of the obstacle problem, the set of stationary solutions always includes accumulation points, and hence, it is rather delicate to determine the stability of such non-isolated equilibria. Furthermore, the strongly irreversible Allen–Cahn equation can also be regarded as a (generalized) gradient flow; however, standard techniques for gradient flows such as linearization and Łojasiewicz–Simon gradient inequalities are not available for determining the stability of stationary solutions to the strongly irreversible Allen–Cahn equation due to the non-smooth nature of the obstacle problem.

Keywords

Lyapunov stability of equilibrianon-isolated equilibriaobstacle problemstrongly irreversible Allen–Cahn equationvariational inequality

MSC classification

Primary: 47J35: Nonlinear evolution equations
Secondary: 35K86: Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities 37B25: Lyapunov functions and stability; attractors, repellers
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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