This paper concerns with the Cauchy problem of the 1-D bipolar hydrodynamic model for semiconductors, a system of Euler-Poisson equations with time-dependent damping effects $-J(1+t{)}^{-\lambda }$ and $-K(1+t{)}^{-\lambda }$ for $-1<\lambda <1$. Here, we consider a more physical case that allows the two pressure functions can be different and the doping profile can be non-zero. Different from the previous study [Li HT, Li JY, Mei. M, et al. Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping. J Math Anal Appl. 2019;437:1081-1121] which considered two identical pressure functions and zero doping profile, the asymptotic profiles of the solutions to this model are constant states rather than the nonlinear diffusion waves. When the initial perturbation around the constant states are sufficiently small in the sense of $L^2$, by means of the time-weighted energy method, we prove the global existence and uniqueness of the smooth solutions to the Cauchy problem, and obtain the optimal convergence rates of the solutions toward the constant states.

本文涉及半导体的一维双极流体动力学模型的柯西问题，这是一个具有时间相关阻尼效应的欧拉-泊松方程组$-杰(\mathrm{1个}+吨{)}^{-\lambda }$和$-钾(\mathrm{1个}+吨{)}^{-\lambda }$为了$-\mathrm{1个}<\lambda <\mathrm{1个}$. 在这里，我们考虑一个更物理的情况，允许两个压力函数可以不同并且掺杂分布可以是非零的。不同于以往的研究 [Li HT, Li JY, Mei. 金属。具有时间相关阻尼的双极欧拉-泊松方程解的渐近行为。J 数学肛门应用。2019;437:1081-1121] 考虑了两个相同的压力函数和零掺杂分布，该模型解的渐近分布是恒定状态而不是非线性扩散波。当围绕恒定状态的初始扰动在以下意义上足够小时${\mathrm{大号}}^{\mathrm{2个}}$，利用时间加权能量法，我们证明了柯西问题的光滑解的全局存在性和唯一性，并获得了解对恒定状态的最优收敛速度。