On asymptotic stability in 3D of kinks for the model

Author:
Scipio Cuccagna

Journal:
Trans. Amer. Math. Soc. **360** (2008), 2581-2614

MSC (2000):
Primary 35L70, 37K40, 35B40

DOI:
https://doi.org/10.1090/S0002-9947-07-04356-5

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.

**[AS]**Ablowitz, Segur,*Solitons and the Inverse Scattering Transform*, SIAM Studies in Appl. Math, SIAM, 1981. MR**642018 (84a:35251)****[D]**Delort,*Existence globale et comportamente asymptotique pour l'equation de Klein Gordon quasilineaire a donnees petites en dimension 1*, Ann. Scient. Ec. Norm. Sup.**34**(2001), 1-61.**[DFX]**Delort, Fang, Xue,*Global existence of small solutions for quadratic quasilinear Klein Gordon systems in two space dimensions*, preprint.**[K]**Klainerman,*Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimension*, Comm. Pure Appl. Math.**38**(1985), 631-641. MR**803252 (87e:35080)****[Ka]**Kapitula,*Multidimensional stability of planar travelling waves*, Trans. Amer. Math. Soc.**349**(1998), 257-269. MR**1360225 (97d:35104)****[Ko]**Kosecki,*The Unit Condition and Global Existence for a Class of Nonlinear Klein-Gordon Equation*, Jour. Diff. Eq**100**(1992), 257-268. MR**1194810 (93k:35178)****[GK]**Goldman, Krivchenkov,*Problems in quantum mechanics*, Dover.**[GP]**Georgiev, Popivanov,*Global solution to the two-dimensional Klein-Gordon equation*, Comm. Part. Diff. Eq.**16**(1991), 941-995. MR**1116850 (92g:35140)****[GSS]**Grillakis, Shatah, Strauss,*Stability theory of solitary waves in the presence of symmetry, I*, Jour. Funct. Anal.**74**(1987), 160-197. MR**901236 (88g:35169)****[H]**Henry,*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, 840, Springer, 1981. MR**610244 (83j:35084)****[HPW]**Henry, Perez, Wreszinski,*Stability theory for solitary-wave solutions of scalar field equations*, Comm. Math. Phys.**85**(1982), 351-361. MR**678151 (83m:35131)****[Ho]**Hörmander,*Lectures on Nonlinear Hyperbolic Differential Equations*, Springer, 1996.**[LX]**Levermore, Xin,*Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II*, Comm. Part. Diff. Eq.**17**(1992), 1901-1924. MR**1194744 (94c:35105)****[OTT]**Ozawa, Tsutaya, Tsutsumi,*Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimension*, Math. Z.**222**(1996), 341-362. MR**1400196 (97e:35112)****[Sh1]**Shatah,*Global existence of small solutions to nonlinear evolution equations*, J. Diff. Eq.**46**(1982), 409-425. MR**681231 (84g:35036)****[Sh2]**-,*Normal forms and quadratic nonlinear Klein Gordon equations*, Comm. Pure Appl. Math.**38**(1985), 685-696. MR**0803256 (87b:35160)****[So]**Sogge,*Lectures on nonlinear wave equations*, Monographs in Analysis, II, International Press, 1995. MR**1715192 (2000g:35153)****[T]**Taylor,*Pseudodifferential operators*, Princeton Math. Series, 34, Princeton Un. Press, 1981. MR**618463 (82i:35172)****[We]**Weder,*The continuity of the Schrödinger wave operators on the line*, Comm. Math. Phys.**208**(1999), 507-520. MR**1729096 (2001c:34178)****[X]**Xin,*Multidimensional stability of traveling waves in a bistable reaction-diffusion equation*, Comm. PDE**17**(1992), 1889-189.

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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:
https://doi.org/10.1090/S0002-9947-07-04356-5

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.