Global existence and asymptotics for quasilinear onedimensional KleinGordon equations with mildly decaying Cauchy data
Abstract
Let u be a solution to a quasilinear KleinGordon equation in onespace dimension, $\Box u + u = P (u, $\partial$\_t u, $\partial$\_x u; $\partial$\_t $\partial$\_x u, $\partial$^2\_x u)$ , where P is a homogeneous polynomial of degree three, and with smooth Cauchy data of size $\epsilon \rightarrow 0$. It is known that, under a suitable condition on the nonlinearity, the solution is globalintime for compactly supported Cauchy data. We prove in this paper that the result holds even when data are not compactly supported but just decaying as $\langle x \rangle^ {1}$ at infinity, combining the method of Klainerman vector fields with a semiclassical normal forms method introduced by Delort. Moreover, we get a one term asymptotic expansion for u when $t \rightarrow +\infty$.
 Publication:

arXiv eprints
 Pub Date:
 July 2015
 arXiv:
 arXiv:1507.02035
 Bibcode:
 2015arXiv150702035S
 Keywords:

 Mathematics  Analysis of PDEs