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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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Asymptotic stability for odd perturbations of the stationary kink in the variable-speed $\phi ^4$ model
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by Stanley Snelson PDF
Trans. Amer. Math. Soc. 370 (2018), 7437-7460 Request permission

Abstract:

We consider the $\phi ^4$ model in one space dimension with propagation speeds that are small deviations from a constant function. In the constant-speed case, a stationary solution called the kink is known explicitly, and the recent work of Kowalczyk, Martel, and Muñoz established the asymptotic stability of the kink with respect to odd perturbations in the natural energy space. We show that a stationary kink solution exists also for our class of nonconstant propagation speeds, and extend the asymptotic stability result by taking a perturbative approach to the method of Kowalczyk, Martel, and Muñoz. This requires an understanding of the spectrum of the linearization around the variable-speed kink.
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Additional Information
  • Stanley Snelson
  • Affiliation: Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
  • Address at time of publication: Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, Florida 32901
  • MR Author ID: 966432
  • Email: ssnelson@fit.edu
  • Received by editor(s): April 24, 2017
  • Published electronically: June 20, 2018
  • Additional Notes: The author was partially supported by NSF grant DMS-1246999.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 7437-7460
  • MSC (2010): Primary 35L71
  • DOI: https://doi.org/10.1090/tran/7300
  • MathSciNet review: 3841854