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February  2005, 12(2): 233-242. doi: 10.3934/dcds.2005.12.233

On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data

Yang Han 1,

1. 

Department of Mathematics, South-west Jiaotong University, Chengdu 610031, China

Received  May 2003 Revised  June 2004 Published  December 2004

An interaction equations of the complex scalar nucleon field and real scalar meson field is considered. we show that the Cauchy problem of the Klein-Gordon-Schrödinger system

$ iu_{t}+u_{x x}=-uv, $

$ v_{t t}-v_{x x}+v=|u|^2,$

$u(0, x)= u_0(x), v(0, x)= v_0(x), v_t(0, x)= v_1(x)$

is locally well-posed for weak initial data $(u_0, v_0, v_1)\in H^s\times H^{s-1/2}\times H^{s-3/2}$ with $s\geq 0$. We use the analogous method for estimate the nonlinear couple terms developed by Bourgain and refined by Kenig, Ponce and Vega.

Keywords: Klein-Gordon-Schrödinger system, rough data., well-posedness.
Mathematics Subject Classification: Primary 35Q99, 35B6.
Citation: Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233
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