We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural restriction on the power of interactions. In the last part of this paper, we use special admissible pairs and Strichartz estimates to remove the restriction, thereby generalizing previous results and obtaining the well-posedness of the system.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap89-2-6,
author = {Changxing Miao and Youbin Zhu},
title = {The Cauchy problem for the coupled Klein-Gordon-Schr\"odinger system},
journal = {Annales Polonici Mathematici},
volume = {89},
year = {2006},
pages = {163-195},
zbl = {1105.35027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap89-2-6}
}
Changxing Miao; Youbin Zhu. The Cauchy problem for the coupled Klein-Gordon-Schrödinger system. Annales Polonici Mathematici, Tome 89 (2006) pp. 163-195. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap89-2-6/