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

Abstract: 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 $L<sup>2$-norm, which improve theose in \cite{Li3, Wu3}. More precisely, the velocities $u_1,u_2$ decay at the $L<sup>2-$rate $(1+t)<sup>{-{5}{4}}$, which is faster than the normal $L<sup>2-$rate $(1+t)<sup>{-{3}{4}}$ 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 $L<sup>2$-rate $(1+t)<sup>{-2}$.