DCDS
On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data
Yang Han
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
DCDS
A singular perturbed problem for semilinear wave equations with small parameter
Han Yang
In this paper we get a lower bound independent of $\delta$ on the life-span of classical solutions to the following Cauchy problem by using the global iteration method

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

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

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.

keywords: singular perturbation. Cauchy problem life-span of classical solutions

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