The Asymptotic Behavior Of The Solution For A One-dimensional Bipolar Quantum Hydrodynamic Model

In this thesis,the initial boundary value problem for the bipolar quantum hydrodynamic model of semiconductors in a one-dimensional bounded region is studied,which contains the dispersive term of the third derivative.In this thesis,we give the boundary conditions for the quantum term,corresponding to quantum effects that disappear on the boundary.In general,the doping profile in semiconductor device is not flat.In order to be more close to the physical reality,we consider the general non-constant doping profile in this thesis.While studying the unique existence of the stationary solution,we first consider some transformation of the steady-state equations:following the traditional way of dealing 3-rd order quantum hydrodynamic models,the steady-state equations is transformed into a second order elliptic system by integration.Next,by the Leray-Schauder fixed-point theorem we prove the existence of the stationary solution,and the positive upper and lower bounds of the stationary solution are derived by the essential truncation method.Furthermore,some important estimates of the stationary solution can be obtained.Then,the uniqueness of the stationary solution is shown by a delicate energy estimate.For the asymptotic behavior of the non-steady state solution,the local in time existence of the solution to the non-stationary problem and uniform a prior estimate are used to obtain the global existence of the non-steady state solution,and the solution exponentially converges to the stationary solution with suitable assumptions.