Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

Long-Time Dispersive Estimates for Perturbations of a Kink Solution of One-Dimensional Cubic Wave Equations

“…Let us first discuss some results that are known concerning equations of the form (1.96) either in the case of potentials without bound states, or for equations of that form with V D 0 but where the nonlinearities have coefficients that are non-constant functions of x, as on the right-hand side of (1.97). Such results have been proved by Kopylova [53] for linear Klein-Gordon equations in a moving frame and, in the nonlinear case, by Lindblad and Soffer [66], Lindblad, Lührmann and Soffer [60,61], Lindblad, Lührmann, Schlag and Soffer [59], Sterbenz [81]. Very recently, Germain and Pusateri [33] obtained the most general result in that framework.…”

“…Finally, let us mention that for nonlinearities with coefficients that are rapidly enough decaying in x, Lindblad, Lührmann and Soffer [60] (in the case V Á 0) and Lindblad, Lührmann, Schlag and Soffer [59] (for generic potentials) could show that a dispersive bound like (1.99) does not hold in general, and has to be replaced by the product of the right-hand side with a logarithmic loss.…”

On The large Time Asymptotics of Klein-Gordon type equations with General Data-I

“…[32,15,25,45] and large literature on NLS, KdV and more. and on small data and long range type interactions [13,14,28,29,27,21,23,22] In contrast, the new approach of Liu-Soffer [30,31] is based on proving a-priory estimates on the full dynamics, which hold in a suitably defined domains of the extended phase-space. That is, one proves propagation estimates in domains exterior to the support of the interaction.…”

Kink dynamics under odd perturbations for (1+1)-scalar field models with one internal mode

“…When a resonance exists, rather than an internal model, the situation is much more delicate. Ways to deal with such situation with some generality were recently found in [34,35,36]. Last, we refer to [4] for cases where the Fermi golden rule does not hold.…”