Some new well-posedness results for the Klein-Gordon-Schrödinger system

Abstract

We consider the Cauchy problem for the 2D and 3D Klein-Gordon-Schrödinger system. In 2D we show local well-posedness for Schrödinger data in H^s and wave data in H^σ x H^{σ-1} for s=-1/4 + and σ= -1/2, whereas ill-posedness holds for s<- 1/4 or σ<-1/2, and global well-posedness for s\ge 0 and s- 1/2 \le σ< s+ 3/2. In 3D we show global well-posedness for s \ge 0, s - 1/2 < σ\le s+1. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schrödinger system by Colliander, Holmer and Tzirakis.

19 pages. Some typos corrected. Final version to be published in Differential and Integral Equations