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Akagi, G.* ; Efendiyev, M.A.

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

Proc. R. Soc. Edinburgh, Sect. A, DOI: 10.1017/prm.2024.97 (2024)
DOI
Free by publisher: Publ. Version/Full Text online available 01/2026
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.
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Publication type Article: Journal article
Document type Scientific Article
Corresponding Author
Keywords Lyapunov Stability Of Equilibria ; Non-isolated Equilibria ; Obstacle Problem ; Strongly Irreversible Allen-cahn Equation ; Variational Inequality; Evolution; Approximation; Model; Existence; Damage
ISSN (print) / ISBN 0308-2105
e-ISSN 1473-7124
Journal Proceedings of the Royal Society of Edinburgh, Section A: Mathematics
Publisher Cambridge Univ. Press
Publishing Place Edinburgh Bldg, Shaftesbury Rd, Cb2 8ru Cambridge, England
Non-patent literature Publications
Reviewing status Peer reviewed
Institute(s) Institute of Computational Biology (ICB)
Grants Carl Friedrich von Siemens Foundation
Alexander von Humboldt Foundation
JSPS KAKENHI