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

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.

arXiv comment27 pages. Minor revisions. To appear in SIAM J. Math. Anal

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Cited by in corpus (5)

• Quadratic Klein-Gordon equations with a potential in one dimension
• The $1$d nonlinear Schrödinger equation with a weighted $L^1$ potential
• Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities
• Long-time asymptotics and stability for the sine-Gordon equation
• Bilinear estimates in the presence of a large potential and a critical NLS in 3d