Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities

Lindblad, Hans; Soffer, Avy

doi:10.1090/S0002-9947-2014-06455-6

Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities
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Trans. Amer. Math. Soc. 367 (2015), 8861-8909 Request permission

Abstract:

We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case where the worst of this resonant behavior is absent, we prove $L^\infty$ scattering as well as a certain kind of strong smoothness for the solution at time-like infinity with the help of several new normal-form transformations. Some explicit examples are also given which suggest qualitatively different behavior in the case where the strongest cubic resonances are present.

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