Some Mathematical Results On The Semiconductor Partial Differential Equations

In this thesis we consider the problems of existence, asymptotic stability and asymptotic limit for the solutions to three types of semiconductor partial differential equations.The introduction to these problems is given in chapter 1.In chapter 2, we study the initial-boundary value problem of the one-dimensional(1-D) full bipolar hydrodynamic model for semiconductors. Based on the new idea of the regular perturbation, we prove the existence and uniqueness of a subsonic stationary solution by the theory of strongly elliptic system and Banach fixed point theorem. The exponentially asymptotic stability of the stationary solution is established through the energy method.In chapter 3, we consider the initial-boundary value problem of the 1-D bipolar isothermal quantum hydrodynamic model for semiconductors. In order to deal with the third order dispersive terms in the system, we adopt the nonlinear boundary conditions which mean the quantum effect vanishes on the boundary. Based on the new idea and approach developed in chapter 2, we show the existence and uniqueness of a subsonic stationary solution. The exponentially asymptotic stability and the semi-classical limit of the stationary solution, and the semi-classical limit of the global solution are established by energy method.In chapter 4, we study the Cauchy problem of a 1-D full hydrodynamic model for semiconductors, in which the recombination-generation effects between electrons and holes are taken into consideration. Although the recombination-generation phenomenon is of physical importance, the rigorous mathematical analysis on this phenomenon is rather few so far. Based on the idea of the regular perturbation again, we obtain the existence and uniqueness of a subsonic stationary solution on the whole real line Rthrough the theory of the ordinary differential equations and Banach fixed point theorem. The exponentially asymptotic stability of the stationary solution is proved by the delicate energy method. Our results successfully reveal the importance of the recombination-generation effects on the charge transport.