Optimal decay rate of the bipolar Euler-Poisson system with damping in $\mathbb{R}^3$

By rewriting a bipolar Euler-Poisson equations with damping into an Euler equation with damping coupled with an Euler-Poisson equation with damping, and using a new spectral analysis, we obtain the optimal decay results of the solutions in $L^2$-norm, which improve theose in \cite{Li3, Wu3}. More precisely, the velocities $u_1,u_2$ decay at the $L^2-$rate $(1+t)^{-{5}{4}}$, which is faster than the normal $L^2-$rate $(1+t)^{-{3}{4}}$ for the Heat equation and the Navier-Stokes equations. In addition, the disparity of two densities $\rho_1-\rho_2$ and the disparity of two velocities $u_1-u_2$ decay at the $L^2$-rate $(1+t)^{-2}$.