**Astérisque**

Volume: 341;
2012;
113 pp;
Softcover

MSC: Primary 35; 37;
**Print ISBN: 978-2-85629-335-5
Product Code: AST/341**

List Price: $45.00

Individual Member Price: $36.00

# 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.