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Global dynamics above the ground state energy for the one-dimensional NLKG equation

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Abstract

In this paper we 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 (Nakanishi and Schlag in Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, preprint, 2010), 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, we establish local dispersive estimates for the perturbed linear Klein–Gordon equation, similar to those of Mizumachi (J Math Kyoto Univ 48(3):471–497, 2008). The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator.

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Authors and Affiliations

  1. Bâtiment des Mathématiques, Station 8, 1015, Lausanne, Switzerland

    J. Krieger

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    K. Nakanishi

  3. Department of Mathematics, The University of Chicago, Chicago, IL, 60615, USA

    W. Schlag

Authors
  1. J. Krieger
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  2. K. Nakanishi
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  3. W. Schlag
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Corresponding author

Correspondence to W. Schlag.

Additional information

W. Schlag was supported by the NSF, DMS-0617854, and a Guggenheim fellowship.

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Krieger, J., Nakanishi, K. & Schlag, W. Global dynamics above the ground state energy for the one-dimensional NLKG equation. Math. Z. 272, 297–316 (2012). https://doi.org/10.1007/s00209-011-0934-3

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  • DOI: https://doi.org/10.1007/s00209-011-0934-3

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