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Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

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Abstract

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Hans Lindblad

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Jonas Lührmann

  3. Mathematics Department, Rutgers University, New Brunswick, NJ, 08903, USA

    Avy Soffer

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  1. Hans Lindblad
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  2. Jonas Lührmann
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  3. Avy Soffer
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Correspondence to Avy Soffer.

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Communicated by N. Masmoudi.

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H. Lindblad was supported in part by NSF grant DMS-1500925 and by Simons Foundation Collaboration grant 638955. J. Lührmann was supported in part by NSF grant DMS-1954707 during the completion of this work. A. Soffer was supported in part by NSF grant DMS-1600749 and by NSFC11671163. Part of this work was conducted while the last two authors were visiting Central China Normal University, Wuhan, China.

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Lindblad, H., Lührmann, J. & Soffer, A. Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities. Arch Rational Mech Anal 241, 1459–1527 (2021). https://doi.org/10.1007/s00205-021-01675-y

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