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arXiv · 2307.11456

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

Divyang G. Bhimani
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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 $$u_{tt}-\Delta u+u + ( |\cdot|^{-\gamma} \ast |u|^2)u=0$$ with initial data in modulation spaces $M^{p, p'}_1 \times M^{p,p}$ for $p\in \left(2, \frac{54 }{27-2\gamma} \right),$ $2<\gamma<3.$ 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).

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Divyang G. Bhimani. 2023-07-21. The Global well-posedness for Klein-Gordon-Hartree equation in modulation spaces. https://arxiv.org/abs/2307.11456

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