On the global wellposedness of the Klein-Gordon equation for initial data in modulation spaces

Abstract

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u{|}^{\alpha -1}u$, where $\alpha \in \left[1,\frac{d}{d-2}\right]$, in dimension $d\ge 3$ with initial data in $M_{p,p^{\prime }}^1({\mathbb{R}}^d)\times M_{p,p^{\prime }}({\mathbb{R}}^d)$ for $p$ sufficiently close to $2$. The proof is an application of the high-low method described by Bourgain [1] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces.

12 pages

References in corpus (3)

• On existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M_{p,q}(\mathbb{R})$
• Strichartz estimate and nonlinear Klein-Gordon on non-trapping scattering space
• Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces $M_{p,q}^s({\mathbb{R}}^d)\cap M_{\infty ,1}({\mathbb{R}}^d)$