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Allen–Cahn equation with strong irreversibility

Published online by Cambridge University Press:  16 July 2018

GORO AKAGI  and
MESSOUD EFENDIEV
GORO AKAGI
Affiliation:
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85748 Garching, Germany email: goro.akagi@tohoku.ac.jp
MESSOUD EFENDIEV
Affiliation:
Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany email: messoud.efendiyev@helmholtz-muenchen.de
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Abstract

This paper is concerned with a fully non-linear variant of the Allen–Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. The main purposes of this paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviours of solutions. More precisely, by deriving partial energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution u(x,t) converges to a solution of an elliptic obstacle problem as t → +∞.

Keywords

Strongly irreversible evolution equationAllen–Cahn equationobstacle parabolic problemglobal attractorω-limit setpartial energy-dissipation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 4 , August 2019 , pp. 707 - 755
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

† G. Akagi is supported in part by JSPS KAKENHI Grant Numbers JP16H03946, JP16K05199, JP17H01095, in part by the Alexander von Humboldt Foundation and in part by the Carl Friedrich von Siemens Foundation.

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