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Quasi-linear perturbations of Hamiltonian Klein-Gordon equations on spheres


About this Title

J.-M. Delort

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 234, Number 1103
ISBNs: 978-1-4704-0983-8 (print); 978-1-4704-2030-7 (online)
DOI: http://dx.doi.org/10.1090/memo/1103
Published electronically: July 28, 2014
Keywords:Hamiltonian quasi-linear Klein-Gordon equations, Almost global existence

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Statement of the main theorem
  • Chapter 3. Symbolic calculus
  • Chapter 4. Quasi-linear Birkhoff normal forms method
  • Chapter 5. Proof of the main theorem
  • A. Appendix

Abstract


The Hamiltonian , defined on functions on , where is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. We consider perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of . The associated PDE is then a quasi-linear Klein-Gordon equation. We show that, when is the sphere, and when the mass parameter is outside an exceptional subset of zero measure, smooth Cauchy data of small size give rise to almost global solutions, i.e. solutions defined on a time interval of length for any . Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on ) or to the one dimensional problem. The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus.

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