Astérisque 2012; 113 pp; softcover Number: 341 ISBN10: 2856293352 ISBN13: 9782856293355 List Price: US$45 Member Price: US$36 Order Code: AST/341
 Consider a nonlinear KleinGordon 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 timeinterval \(]c_\kappa \epsilon ^{\kappa },c_\kappa \epsilon ^{\kappa }[\) 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 quasilinear 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 quasilinear 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, quasilinear Hamiltonian equations, almost global existence, and KleinGordon equations. Table of Contents  Introduction
 Almost global existence
 Symbolic calculus
 Composition and Poisson brackets
 Symplectic reductions
 Proof of almost global existence
 Bibliography
 Index
