Long time asymptotics for the semiconductor Vlasov-Poisson-Boltzmann equations

Abstract

In this paper we analyze the long time behavior of solutions to the one-dimensional Vlasov–Poisson–Boltzmann (VPB) equations for semiconductors in unbounded domains when only one type of carriers (electrons) are considered. We prove that the distribution of electrons tends for large times to a steady state of the VPB equations with vanishing collision term and the same total charge as the initial data. In the proof of the main result, the conservation law of charge, the balance of energy and entropy inequalities are rigorously derived. An important argument in the proof is to use a Lyapunov-type functional related to these physical quantities.