Decay of the solution to the bipolar Euler-Poisson system with damping in $\mathbb{R}^3$

We construct the global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in $\mathbb{R}^3$ when $H^3$ norm of the initial data is small. If further, the $\dot{H}^{-s}$ norm ($0\leq s<3/2)$ or $\dot{B}_{2,\infty}^{-s}$ norm ($0<s\leq3/2$) of the initial data is bounded, we give the optimal decay rates of the solution. As a byproduct, the decay results of the $L^p-L^2$ ($1\leq p\leq2$) type hold without the smallness of the $L^p$ norm of the initial data. In particular, we deduce that $\|\nabla^k(\rho_1-\rho_2)\|_{L^2} \sim(1+t)^{-5/4-\frac{k}{2}}$ and $\|\nabla^k(\rho_i-\bar{\rho},u_i,\nabla\phi)\|_{L^2} \sim(1+t)^{-3/4-\frac{k}{2}}$. We improve the decay results in Li and Yang \cite{Li3}(\emph{J.Differential Equations} 252(2012), 768-791), where they showed the decay rates as $\|\nabla^k(\rho_i-\bar{\rho})\|_{L^2} \sim(1+t)^{-3/4-\frac{k}{2}}$ and $\|\nabla^k(u_i,\nabla\phi)\|_{L^2} \sim(1+t)^{-1/4-\frac{k}{2}}$, when the $H^3\cap L^1$ norm of the initial data is small. Our analysis is motivated by the technique developed recently in Guo and Wang \cite{Guo}(\emph{Comm. Partial Differential Equations} 37(2012), 2165-2208) with some modifications.