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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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Periodic traveling waves and locating oscillating patterns in multidimensional domains
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by Nicholas D. Alikakos, Peter W. Bates and Xinfu Chen PDF
Trans. Amer. Math. Soc. 351 (1999), 2777-2805 Request permission

Abstract:

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in $\mathbb {R}^n$, when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.
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Additional Information
  • Nicholas D. Alikakos
  • Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300; Department of Mathematics, University of Athens, Panestimiopolis, Greece 15784
  • Email: alikakos@utk.edu, nalikako@atlas.uoa.gr
  • Peter W. Bates
  • Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
  • MR Author ID: 32495
  • Email: peter@math.byu.edu
  • Xinfu Chen
  • Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
  • MR Author ID: 261335
  • Email: xinfu+@pitt.edu
  • Received by editor(s): March 23, 1995
  • Received by editor(s) in revised form: February 18, 1997
  • Published electronically: March 1, 1999
  • Additional Notes: The first author was partially supported by the National Science Foundation Grant DMS–9306229, the Science Alliance, and the NATO Scientific Affairs Division CRG930791.
    The second author was partially supported by the National Science Foundation Grant DMS–9305044, and the NATO Scientific Affairs Division CRG 930791.
    The third author partially supported by the National Science Foundation Grant DMS–9404773, and the Alfred P. Sloan Research Fellowship.
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 2777-2805
  • MSC (1991): Primary 35B10, 35B25, 35B40, 35K57
  • DOI: https://doi.org/10.1090/S0002-9947-99-02134-0
  • MathSciNet review: 1467460