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Global dynamics above the ground state energy for the one-dimensional NLKG equation. (English) Zbl 1263.35002
In this investigation, the authors obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the line with focusing nonlinearity \({|u|^{p-1}u, p > 5}\), provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case [K. Nakanishi and W. Schlag, J. Differ. Equations 250, No. 5, 2299–2333 (2011; Zbl 1213.35307)], the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, the writers establish local dispersive estimates for the perturbed linear Klein-Gordon equation, similar to those of T. Mizumachi [J. Math. Kyoto Univ. 48, No. 3, 471–497 (2008; Zbl 1175.35138)]. The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35L70 Second-order nonlinear hyperbolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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