Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities

2019/07/23 by Lindblad, Hans, Luhrmann, Jonas, Soffer, Avy
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1907.09922

Abstract

We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein-Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic nonlinearities. Vector-field techniques to deal with the long-range nature of the cubic nonlinearity become problematic in the presence of variable coefficients. We introduce a novel approach based on pointwise-in-time local decay estimates for the Klein-Gordon propagator to overcome this impasse.

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