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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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On asymptotic stability in 3D of kinks for the $\phi ^4$ model
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by Scipio Cuccagna PDF
Trans. Amer. Math. Soc. 360 (2008), 2581-2614 Request permission

Abstract:

We add to a kink, which is a 1 dimensional structure, two transversal directions. We then check its asymptotic stability with respect to compactly supported perturbations in 3D and a time evolution under a Nonlinear Wave Equation (NLW). The problem is inspired by work by Jack Xin on asymptotic stability in dimension larger than 1 of fronts for reaction diffusion equations. The proof involves a separation of variables. The transversal variables are treated as in work on Nonlinear Klein Gordon Equation (NLKG) originating from Klainerman and from Shatah in a particular elaboration due to Delort et al. The longitudinal variable is treated by means of a result by Weder on dispersion for Schrödinger operators in 1D.
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Additional Information
  • Scipio Cuccagna
  • Affiliation: Dipartimento di Scienze e Metodi per l’Ingegneria, Università di Modena e Reggio Emilia, Padiglione Morselli, via Amendola 2, Reggio Emilia 42100, Italy
  • Email: cuccagna.scipio@unimore.it
  • Received by editor(s): June 28, 2004
  • Received by editor(s) in revised form: November 30, 2005, and March 5, 2006
  • Published electronically: November 28, 2007
  • Additional Notes: This research was fully supported by a special grant from the Italian Ministry of Education, University and Research.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 2581-2614
  • MSC (2000): Primary 35L70, 37K40, 35B40
  • DOI: https://doi.org/10.1090/S0002-9947-07-04356-5
  • MathSciNet review: 2373326