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Article Contents

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

• 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.

Mathematics Subject Classification: Primary 35Q99, 35B65.

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