A Note on Lower Bound Lifespan Estimates for Semi-linear Wave/Klein–Gordon Equations Associated with the Harmonic Oscillator

Abstract

In this paper, we show that for almost every \(m>0\), the solution to the semi-linear Klein–Gordon equation associated with the harmonic oscillator, with small initial data, exists over a longer time interval than the one given by local existence theory, using the normal form method. A similar result for the quadratic wave equation is also obtained.

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Correspondence to Qidi Zhang.

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The work was supported by the Fundamental Research Funds for the Central Universities and NSFC 11601154.

Communicated by Amin Esfahani.

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Zhang, Q., Zheng, L. A Note on Lower Bound Lifespan Estimates for Semi-linear Wave/Klein–Gordon Equations Associated with the Harmonic Oscillator. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-020-00486-3

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Keywords

  • Harmonic oscillator
  • Wave equations
  • Klein–Gordon equations
  • Long time existence

Mathematics Subject Classification

  • 35L70
  • 35J15