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, □u + u = P(u, ∂tu, ∂xu; ∂t∂xu, ∂ _x ² u), where P is a homogeneous polynomial of degree three, and with smooth Cauchy data of size ε → 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 hxi ^{− 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 → +∞.