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

Hans Lindblad, Jonas Lührmann, Avy Soffer

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)1459-1527
Number of pages69
JournalArchive For Rational Mechanics And Analysis
Volume241
Issue number3
DOIs
StatePublished - Sep 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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