Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities

Abstract

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.

45 pages. Minor Revisions. To appear in Arch. Ration. Mech. Anal

References in corpus (5)

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• Asymptotic stability of Landau solutions to Navier-Stokes system
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Cited by in corpus (4)

• Quadratic Klein-Gordon equations with a potential in one dimension
• On $1$d quadratic Klein-Gordon equations with a potential and symmetries
• Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel
• Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation