Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data

Let u be a solution to a quasi-linear Klein-Gordon equation in one-space 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 global-in-time 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 $t \rightarrow +\infty$.