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

Annalaura Stingo (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

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 $ε → 0$. 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 $\langle x \rangle^ {−1}$ 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 $t → +∞$.

Mots clés: Global solution of quasi-linear Klein-Gordon equations; Klainerman vector fields; Semiclassical Analysis

Domaines: Equations aux dérivées partielles [math.AP]

Annalaura Stingo. Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data. 2015. ⟨hal-01172523⟩