On the global well-posedness and scattering of the 3D Klein–Gordon–Zakharov system

• Published in: Calculus of variations and partial differential equations Vol. 63; no. 1
• Main Authors: Cheng, Xinyu, Xu, Jiao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Fields (mathematics); Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; Well posed problems; 35L52; 35L05
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-023-02620-5

In this paper we are interested in the global well-posedness of the 3D Klein–Gordon–Zakharov equations with small non-compactly supported initial data. We show the uniform boundedness of the energy for the global solution without any compactness assumptions on the initial data. The main novelty of our proof is to apply a modified Alinhac’s ghost weight method together with a newly developed normal-form type estimate to remedy the lack of the space-time scaling vector field; moreover, we give a clear description of the smallness conditions on the initial data.