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On the global wellposedness of the Klein-Gordon equation for initial data in modulation spaces (1910.07413v1)
Published 16 Oct 2019 in math.AP
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 $d\geq3$ with initial data in $M_{p, p'}{1}(\mathbb{R}d)\times M_{p,p'}(\mathbb{R}d)$ 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.