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Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities


Authors: Hans Lindblad and Avy Soffer
Journal: Trans. Amer. Math. Soc. 367 (2015), 8861-8909
MSC (2010): Primary 35Qxx
DOI: https://doi.org/10.1090/S0002-9947-2014-06455-6
Published electronically: December 3, 2014
MathSciNet review: 3403074
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Abstract: We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case where the worst of this resonant behavior is absent, we prove $ L^\infty $ scattering as well as a certain kind of strong smoothness for the solution at time-like infinity with the help of several new normal-form transformations. Some explicit examples are also given which suggest qualitatively different behavior in the case where the strongest cubic resonances are present.


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Additional Information

Hans Lindblad
Affiliation: Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, Maryland 21218
Email: lindblad@math.jhu.edu

Avy Soffer
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854
Email: soffer@math.rutgers.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06455-6
Keywords: Nonlinear KG equation, long-range scattering, normal form analysis
Received by editor(s): August 13, 2013
Received by editor(s) in revised form: April 6, 2014
Published electronically: December 3, 2014
Additional Notes: The first author was partially supported by NSF grant DMS–1237212
The second author was partially supported by NSF grant DMS–1201394
Article copyright: © Copyright 2014 American Mathematical Society