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Transactions of the American Mathematical Society

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Asymptotic stability for odd perturbations of the stationary kink in the variable-speed $\phi ^4$ model

Author: Stanley Snelson
Journal: Trans. Amer. Math. Soc. 370 (2018), 7437-7460
MSC (2010): Primary 35L71
Published electronically: June 20, 2018
MathSciNet review: 3841854
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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

Received by editor(s): April 24, 2017
Published electronically: June 20, 2018
Additional Notes: The author was partially supported by NSF grant DMS-1246999.
Article copyright: © Copyright 2018 American Mathematical Society