Large Time Behavior Of Bipolar Hydrodynamic Model For Semiconductors

In this paper, we study the large time behavior of the solutions for hydrodynamic model for semiconductor where (x, t) ∈ (0,1) x R+.This paper is divided into four parts. In chapter one, we study the physical back-ground of the bipolar hydrodynamic model for semiconductor. Then, we summarize the relating studying results, both home and abroad. In chapter two, we investigate the existence and uniqueness of steady solutions of bipolar hydrodynamic model for semi-conductor. At the same time, we get some estimates of the steady solutions. In this chapter, we use two different methods-Schauder fixed point method and the calculus of variations method to get the results. The theorem conditions of the Schauder fixed point method request stronger than the calculus of variations method. In chapter three, under the condition of smallness assumption of amplitude of the doping profile, we prove that the smooth small solutions of hydrodynamic model converge to the stationary solution and an exponential decay rate is also derived. Moreover, we get the similar result in drift-diffusion model. Last, using the triangle inequality, we build the exponential con-vergence relationship between the two smooth solutions. In chapter four, by means of the energy method and the entropy dissipation, we obtain the large time behavior framework for any uniformly bounded weak entropy solutions for bipolar hydrodynamic model. The solutions are shown to converge to the stationary solutions and an exponential decay rate is also derived. No smallness and regularity conditions are assumed and the doping profile are permitted to be of big variation.