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Dispersive decay for the 1D Klein-Gordon equation with variable coefficient nonlinearities


Author: Jacob Sterbenz
Journal: Trans. Amer. Math. Soc. 368 (2016), 2081-2113
MSC (2010): Primary 35L70
DOI: https://doi.org/10.1090/tran/6478
Published electronically: May 6, 2015
MathSciNet review: 3449234
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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.


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Additional Information

Jacob Sterbenz
Affiliation: Department of Mathematics, University of California San Diego, La Jolla, California 92093-0112
Email: jsterbenz@math.ucsd.edu

DOI: https://doi.org/10.1090/tran/6478
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.
Article copyright: © Copyright 2015 American Mathematical Society