The Pointwise Estimate Of Two Bipolar Semiconductor Equations

In this paper, we discuss two bipolarequations from semiconductor devices and plasmas. From the analysis of Green function and the elaborate energy estimates, we show the pointwise of smooth solutions for the initialvalue problem when the initialdata are small perturbation around the constant state. Then we obtain the optimal＄{L^p}({\mathbb{R}^3})＄convergence rate ofthe solutions.