Asymptotic Behavior Of The Solution Of The Hydrodynamic Model With Nonlinear Damping Terms

In this paper,we discuss the Cauchy problem of the one-dimensional Euler-Poisson equations with nonlinear damping terms,the model is a one-dimensional isentropic semiconductor fluid dynamics model,or HD model,which is formed by coupling the Euler equations containing mass and momentum equations and the Poisson equation that characterizes the electric field effect.First of all,we introduces some important research progress on the unipolar HD model(Euler-Poisson equations with relaxation terms)in the past ten years,including one-dimensional isentropic,non-isentropic models and multi–dimensional isentropic,non-isentropic models of various initial boundary value problems,the well-posedness of the solution,large time behavior and various asymptotic limits.Secondly,we discussed the one-dimensional steady state Euler-Poisson equations with nonlinear damping terms,Through the Schauder fixed point theorem,the existence of its steady-state solution is proved.The extreme value principle is used to prove the uniqueness of the steady-state solution.Finally,it is proved that the time-dependent solution exponentially converges to the steady state solution.The method is based on energy estimation.