Dispersive decay for the 1D Klein-Gordon equation with variable coefficient nonlinearities
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Abstract:
We study the 1D Klein-Gordon equation with variable coefficient nonlinearity. This problem exhibits an interesting resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case when only the cubic coefficients are variable we prove $L^\infty$ scattering and smoothness of the solution in weighted spaces with the help of both quadratic and cubic normal forms transformations. In the case of cubic interactions these normal forms appear to be novel.References
- Jean-Marc Delort, Erratum: “Global existence and asymptotic behavior for the quasilinear Klein-Gordon equation with small data in dimension 1” (French) [Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 1, 1–61; MR1833089], Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 2, 335–345 (French, with English and French summaries). MR 2245535, DOI 10.1016/j.ansens.2006.01.001
- Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
- Sergiu Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), no. 5, 631–641. MR 803252, DOI 10.1002/cpa.3160380512
- Hans Lindblad and Avy Soffer, A remark on long range scattering for the nonlinear Klein-Gordon equation, J. Hyperbolic Differ. Equ. 2 (2005), no. 1, 77–89. MR 2134954, DOI 10.1142/S0219891605000385
- Hans Lindblad and Avy Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys. 73 (2005), no. 3, 249–258. MR 2188297, DOI 10.1007/s11005-005-0021-y
- H. Lindblad and A. Soffer, Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities, Trans. Amer. Math. Soc., electronically published on December 3, 2014, DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06455-6.
- Nicholas Manton and Paul Sutcliffe, Topological solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. MR 2068924, DOI 10.1017/CBO9780511617034
- Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. MR 803256, DOI 10.1002/cpa.3160380516
Additional Information
- Jacob Sterbenz
- Affiliation: Department of Mathematics, University of California San Diego, La Jolla, California 92093-0112
- MR Author ID: 733516
- Email: jsterbenz@math.ucsd.edu
- Received by editor(s): July 31, 2013
- Received by editor(s) in revised form: September 1, 2013, April 28, 2014, and May 22, 2014
- Published electronically: May 6, 2015
- Additional Notes: The author was supported in part by NSF grant DMS-1001675.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 2081-2113
- MSC (2010): Primary 35L70
- DOI: https://doi.org/10.1090/tran/6478
- MathSciNet review: 3449234