The global well-posedness for Klein-Gordon-Hartree equation in modulation spaces

Abstract

Modulation spaces have received considerable interest recently as it is the natural function spaces to consider low regularity Cauchy data for several nonlinear evolution equations. We establish global well-posedness for 3D Klein-Gordon-Hartree equation u(tt) - Delta u + u + (vertical bar center dot vertical bar(-gamma) * vertical bar u vertical bar(2))u = 0 with initial data in modulation spaces M-1(p,p ') x M-p,M-p ' for p is an element of (2, 54 /27-2 gamma), 2 < gamma < 3. We implement Bourgain's high-low frequency decomposition method to establish global well-posedness, which was earlier used for classical Klein-Gordon equation. This is the first result on low regularity for Klein-Gordon-Hartree equation with large initial data in modulation spaces (which do not coincide with Sobolev spaces). (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.