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### Open Access Journals

$ 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.

$\delta u_{t t}-\Delta u +u_t = F(u, \nabla u),$

$t = 0 : u = \epsilon u_0(x), u_t = \epsilon u_1(x),$

where $\delta$ and $\epsilon$ are small positive parameters. Moreover, we consider the related singular perturbated problem as $\delta\to 0$ and show that the perturbated term $\delta u_{t t}$ has an appreciable effect only for a short times.

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