Lindblad, Hans; Lührmann, Jonas; Soffer, Avy Decay and asymptotics for the one-dimensional Klein-Gordon equation with variable coefficient cubic nonlinearities. (English) Zbl 1455.35021 SIAM J. Math. Anal. 52, No. 6, 6379-6411 (2020). Summary: 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. Cited in 20 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35L71 Second-order semilinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations 35P25 Scattering theory for PDEs 35Q51 Soliton equations 35Q56 Ginzburg-Landau equations Keywords:long-range scattering; asymptotic stability of kink solutions PDFBibTeX XMLCite \textit{H. Lindblad} et al., SIAM J. Math. Anal. 52, No. 6, 6379--6411 (2020; Zbl 1455.35021) Full Text: DOI arXiv References: [1] T. Candy and H. Lindblad, Long range scattering for the cubic Dirac equation on \(\mathbb{R}^{1+1} \), Differential Integral Equations, 31 (2018), pp. 507-518. · Zbl 1463.35422 [2] G. Chen and F. Pusateri, The 1D Nonlinear Schrödinger Equation with a Weighted L1 Potential, preprint, arXiv:1912.10949, 2019. [3] J.-M. Delort, Modified Scattering for Odd Solutions of Cubic Nonlinear Schrödinger Equations with Potential in Dimension One, preprint, hal-01396705, 2016. [4] J.-M. 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