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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Periodic traveling waves
and locating oscillating patterns
in multidimensional domains


Authors: Nicholas D. Alikakos, Peter W. Bates and Xinfu Chen
Journal: Trans. Amer. Math. Soc. 351 (1999), 2777-2805
MSC (1991): Primary 35B10, 35B25, 35B40, 35K57
Published electronically: March 1, 1999
MathSciNet review: 1467460
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Abstract | References | Similar Articles | Additional Information

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
Email: peter@math.byu.edu

Xinfu Chen
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: xinfu+@pitt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02134-0
PII: S 0002-9947(99)02134-0
Keywords: Periodic traveling waves, stability, singular perturbation, asymptotic behavior
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.
Article copyright: © Copyright 1999 American Mathematical Society