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

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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

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

Received May 2003 Revised June 2004 Published December 2004

Abstract
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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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