Optimal convergence rates to nonlinear diffusion waves for the compressible Euler-Poisson system with damping

In this work, we study the 1-D isentropic bipolar hydrodynamic model. This model takes the form of compressible Euler-Poisson system with nonlinear damping added to the momentum equations. Under some smallness conditions, the solutions to the Cauchy problem of the system globally exist and convergence to the nonlinear diffusion waves, which are the corresponding solutions of nonlinear parabolic equations given by the Darcy's law with a specified initial data. The optimal convergence rates are obtained by Green function method when the initial perturbation is in L¹-space.