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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Traveling wave dynamics for Allen-Cahn equations with strong irreversibility
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by Goro Akagi, Christian Kuehn and Ken-Ichi Nakamura PDF
Trans. Amer. Math. Soc. 375 (2022), 3173-3238 Request permission


Constrained gradient flows are studied in fracture mechanics to describe strongly irreversible (or unidirectional) evolution of cracks. The present paper is devoted to a study on the long-time behavior of non-compact orbits of such constrained gradient flows. More precisely, traveling wave dynamics for a one-dimensional fully nonlinear Allen-Cahn type equation involving the positive-part function is considered. Main results of the paper consist of a construction of a one-parameter family of degenerate traveling wave solutions (even identified when coinciding up to translation) and exponential stability of such traveling wave solutions with some basin of attraction, although they are unstable in a usual sense.
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Additional Information
  • Goro Akagi
  • Affiliation: Mathematical Institute and Graduate School of Science, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
  • MR Author ID: 725232
  • ORCID: 0000-0001-7686-652X
  • Email:
  • Christian Kuehn
  • Affiliation: Department of Mathematics, Technical University of Munich, Boltzmannstraße 3, D-85748 Garching bei München, Germany
  • MR Author ID: 875693
  • ORCID: 0000-0002-7063-6173
  • Email:
  • Ken-Ichi Nakamura
  • Affiliation: Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa 920-1192, Japan
  • MR Author ID: 613791
  • ORCID: 0000-0001-9930-4001
  • Email:
  • Received by editor(s): July 30, 2020
  • Received by editor(s) in revised form: September 24, 2021
  • Published electronically: February 4, 2022
  • Additional Notes: The first author was supported by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation and by JSPS KAKENHI Grant Number JP21KK0044, JP21K18581, JP20H01812, JP18K18715, JP16H03946 and JP20H00117, JP17H01095. The second author acknowledges support via a Lichtenberg Professorship of the VolkswagenStiftung
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 3173-3238
  • MSC (2020): Primary 35C07; Secondary 35R35, 47J35
  • DOI:
  • MathSciNet review: 4402659