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Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities. (English) Zbl 1475.35305

Summary: 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35P25 Scattering theory for PDEs
35C08 Soliton solutions
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