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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

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

Author(s): Scipio Cuccagna
Journal: Trans. Amer. Math. Soc. 360 (2008), 2581-2614.
MSC (2000): Primary 35L70, 37K40, 35B40
Posted: November 28, 2007
MathSciNet review: 2373326
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Abstract | References | Similar articles | Additional information

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.

References:

[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 $ W^{k,p}$ 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: 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
Posted: November 28, 2007
Additional Notes: This research was fully supported by a special grant from the Italian Ministry of Education, University and Research.
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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