A note on asymptotic behavior of solutions for the one-dimensional bipolar Euler–Poisson system

In this note, we consider a one-dimensional bipolar Euler–Poisson system (hydrodynamic model). This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. When n+≠n−, paper [I. Gasser, L. Hsiao, H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations 192 (2003) 326–359] discussed the asymptotic behavior of small smooth solutions to the Cauchy problem of the one-dimensional bipolar Euler–Poisson system. Subsequent to [I. Gasser, L. Hsiao, H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations 192 (2003) 326–359], we investigate the asymptotic behavior of solutions to the Cauchy problem with , and obtain the optimal convergence rate toward the constant state . We accomplish the proofs by energy estimates and the decay rates of fundamental solutions of the heat-type equations.