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

摘要

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 n_+ = n_- = over(n, ?), and obtain the optimal convergence rate toward the constant state (over(n, ?), 0, over(n, ?), 0). We accomplish the proofs by energy estimates and the decay rates of fundamental solutions of the heat-type equations.