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

 

On asymptotic stability in 3D of kinks for the $ \phi ^4$ model


Author: Scipio Cuccagna
Journal: Trans. Amer. Math. Soc. 360 (2008), 2581-2614
MSC (2000): Primary 35L70, 37K40, 35B40
Published electronically: November 28, 2007
MathSciNet review: 2373326
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04356-5
PII: S 0002-9947(07)04356-5
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.