## Kink dynamics in the $\phi ^4$ model: Asymptotic stability for odd perturbations in the energy space

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- by Michał Kowalczyk, Yvan Martel and Claudio Muñoz PDF
- J. Amer. Math. Soc.
**30**(2017), 769-798 Request permission

## Abstract:

We consider a classical equation known as the $\phi ^4$ model in one space dimension. The kink, defined by $H(x)=\tanh (x/{\sqrt {2}})$, is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.## References

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## Additional Information

**Michał Kowalczyk**- Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- Email: kowalczy@dim.uchile.cl
**Yvan Martel**- Affiliation: CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
- MR Author ID: 367956
- Email: yvan.martel@polytechnique.edu
**Claudio Muñoz**- Affiliation: CNRS and Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
- MR Author ID: 806855
- Email: claudio.munoz@math.u-psud.fr, cmunoz@dim.uchile.cl
- Received by editor(s): July 9, 2015
- Received by editor(s) in revised form: July 18, 2016
- Published electronically: September 27, 2016
- Additional Notes: The first author was partially supported by Chilean research grants Fondecyt 1130126, Fondo Basal CMM-Chile and ERC 291214 BLOWDISOL. The author would like to thank Centre de mathématiques Laurent Schwartz at the Ecole Polytechnique and the Université Cergy-Pontoise where part of this work was done.

The second author was partially supported by ERC 291214 BLOWDISOL

The third author would like to thank the Laboratoire de Mathématiques d’Orsay where part of this work was completed. His work was partly funded by ERC 291214 BLOWDISOL, and Chilean research grants FONDECYT 1150202, Fondo Basal CMM-Chile, and Millennium Nucleus Center for Analysis of PDE NC130017 - © Copyright 2016 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**30**(2017), 769-798 - MSC (2010): Primary 35L71; Secondary 35Q51, 37K40
- DOI: https://doi.org/10.1090/jams/870
- MathSciNet review: 3630087