4.5 Article

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

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ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
卷 241, 期 3, 页码 1459-1527

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SPRINGER
DOI: 10.1007/s00205-021-01675-y

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资金

  1. NSF [DMS-1500925, DMS-1954707, DMS-1600749]
  2. Simons Foundation [638955]
  3. [NSFC11671163]

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This study investigates the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. It discovers a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions, leading to a novel type of modified scattering behavior. Sharp decay estimates and asymptotics are established in both resonant and non-resonant cases.
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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