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

@article{Lindblad2020AsymptoticsF1,
  title={Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities},
  author={Hans Lindblad and Jonas L{\"u}hrmann and Avy Soffer},
  journal={arXiv: Analysis of PDEs},
  year={2020}
}
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… Expand
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