# Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres

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*J.-M. Delort*

The Hamiltonian \(\int_X(\lvert{\partial_t u}\rvert^2 +
\lvert{\nabla u}\rvert^2 + \mathbf{m}^2\lvert{u}\rvert^2)\,dx\),
defined on functions on \(\mathbb{R}\times X\), where
\(X\) is a compact manifold, has critical points which are
solutions of the linear Klein-Gordon equation.

The author considers perturbations of this Hamiltonian, given by
polynomial expressions depending on first order derivatives of
\(u\). The associated PDE is then a quasi-linear
Klein-Gordon equation. The author shows that, when \(X\) is the
sphere, and when the mass parameter \(\mathbf{m}\) is outside
an exceptional subset of zero measure, smooth Cauchy data of small
size \(\epsilon\) give rise to almost global solutions,
i.e. solutions defined on a time interval of length
\(c_N\epsilon^{-N}\) for any \(N\). Previous results
were limited either to the semi-linear case (when the perturbation of
the Hamiltonian depends only on \(u\)) 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.

#### Table of Contents

# Table of Contents

## Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres

- Cover Cover11 free
- Title page i2 free
- Chapter 0. Introduction 18 free
- Chapter 1. Statement of the main theorem 714 free
- Chapter 2. Symbolic calculus 2330
- Chapter 3. Quasi-linear Birkhoff normal forms method 4148
- Chapter 4. Proof of the main theorem 5360
- A. Appendix 7784
- Bibliography 7986
- Back Cover Back Cover192