Stability and \begin{document}$ L^{p}$\end{document} convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems

In the paper, we consider a three-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler-Poisson with electric field and relaxation term added to the momentum equations. We first construct the planar diffusion waves. Next we show the global existence of smooth solutions for the initial value problem of three-dimensional bipolar Euler-Poisson systems when the initial data are near the planar diffusive waves. Finally, we also establish the \begin{document}$ L^p(p∈[2,+∞])$\end{document} convergence rates of the solutions toward the planar diffusion waves. A frequency decomposition, approximate Green function and delicate energy method are used to prove our results.