The Asymptotic Behavior Of Solutions To The Drift-diffusion Equation

In recent years,the mathematical theory of the simulation of electro-kinetic behavior has been studied extensively.Many mathematicians and physicists are increasingly concerned and proposed many mathematical models of semiconductor materials and devices.Because it is needed about better research on the structure of a semiconductor device,furthermore,the dynamics of quantum and electric hole is also needed.Semiconductor model is becoming more and more important.The drift-diffusion equation is one of the most widely used for the macroscopic model.In this thesis,we study bipolar drift-diffusion model arising from the semiconductor device simulation.First,we mainly show the long-time behavior of solutions to the one-dimensional bipolar quantum drift-diffusion model in a bounded domain.That is,we prove the existence of the global attractor for the solution.Finally,we study the three-dimensional bipolar drift-diffusion model,which is the simplest macroscopic model describing the dynamics of the electron and the hole.Based on the results of the self-similar stability for the one-dimensional bipolar drift-diffusion equation,we show the stability of planar self-similar wave for the three-dimensional bipolar drift-diffusion model.Using the energy methods,we present the global existence of smooth solutions for the initial value problem of the three-dimensional bipolar drift-diffusion equation when the initial data are close to the planar self-similar wave.We also show that in large time,the solutions of the bipolar drift-diffusion equations tend to the planar self-similar wave,at an algebraic time-decay rate.