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 | α − 1 u |u|^{\alpha -1}u , where α ∈ [ 1 , d d − 2 ] \alpha \in \left [1,\frac {d}{d-2}\right ] , in dimension d ≥ 3 d\geq 3 with initial data in M p , p ′ 1 ( R d ) × M p , p ′ ( R d ) M_{p, p’}^{1}(\mathbb {R}^d)\times M_{p,p’}(\mathbb {R}^d) for p p sufficiently close to 2 2 . The proof is an application of the high-low method described by Bourgain in [Global solutions of nonlinear Schrödinger equations, American Mathematical Society, Providence, RI, 1999] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces.