# A Quasi-Linear Birkhoff Normal Forms Method. Application to the Quasi-Linear Klein-Gordon Equation on \(\mathbb{S}^{1}\)

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*J.-M. Delort*

A publication of the Société Mathématique de France

Consider a nonlinear Klein-Gordon equation on the unit circle, with smooth data of size \(\epsilon \to 0\). A solution \(u\) which, for any \(\kappa \in \mathbb{N}\), may be extended as a smooth solution on a time-interval \(]-c_\kappa \epsilon ^{-\kappa },c_\kappa \epsilon ^{-\kappa }[\) for some \(c_\kappa >0\) and for \(0<\epsilon <\epsilon _\kappa \), is called an almost global solution. It is known that when the nonlinearity is a polynomial depending only on \(u\), and vanishing at order at least \(2\) at the origin, any smooth small Cauchy data generate, as soon as the mass parameter in the equation stays outside a subset of zero measure of \(\mathbb{R}_+^*\), an almost global solution, whose Sobolev norms of higher order stay uniformly bounded. The goal of this book is to extend this result to general Hamiltonian quasi-linear nonlinearities. These are the only Hamiltonian nonlinearities that depend not only on \(u\) but also on its space derivative. To prove the main theorem, the author develops a Birkhoff normal form method for quasi-linear equations.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in Birkhoff normal forms, quasi-linear Hamiltonian equations, almost global existence, and Klein-Gordon equations.