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

Abstract

Let u be a solution to a quasi-linear Klein-Gordon equation in one-space dimension, $\Box u + u = P (u, ∂_t u, ∂_x u; ∂_t ∂_x u, ∂^2_x u)$ , where P is a homogeneous polynomial of degree three, and with smooth Cauchy data of size $e → 0$. 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 $t → +∞$.