We study the existence and large-time behavior of globally defined entropy solutions to one dimensional bipolar hydrodynamic model for semiconductors on bounded interval. By taking advantage of vanishing viscosity method and compensated compactness framework, we construct approximate solutions uniformly bounded in both space $x$ and time $t$. Compared with the paper, Huang and Li (2009), a restrictive assumption on the uniform bound with respect to time $t$ of entropy solutions is removed on bounded interval by introducing modified Riemann invariants and invariant region theory, which is a main novelty of this paper. Also, we extend the result for $\gamma \in (1,3]$ in Huang et al. (2018) to the whole range of physical adiabatic exponent $\gamma \in (1,\infty )$ by using an elementary inequality. Furthermore, we extend the large-time behavior of weak solutions in Huang and Li (2009) from zero doping profile to non-flat one. It is shown that the entropy solution converges to the stationary solution exponentially in time under appropriate condition on doping profile. Some sharp energy estimates are derived to overcome the difficulties due to the coupling and cancellation effect between the difference of densities.

我们研究有界区间上半导体的一维双极流体动力学模型的全局定义熵解的存在和长时间行为。利用消失粘度法和补偿的紧致度框架，我们构造了在两个空间中均匀有界的近似解$X$ 和时间 $Ť$。与本文相比，Huang和Li（2009）对时间的统一界线作了限制性假设$Ť$通过引入修正的黎曼不变量和不变区域理论，去除了有界区间上的熵解，这是本文的主要创新之处。此外，我们将结果扩展为$\gamma \in （\mathrm{1个}，3]$在黄等人。（2018）到整个物理绝热指数范围$\gamma \in （\mathrm{1个}，\infty ）$通过使用基本不等式 此外，我们将Huang和Li（2009）中弱解的长时间行为从零掺杂分布扩展到非平坦分布。结果表明，在掺杂分布的适当条件下，熵解随时间呈指数收敛于平稳解。为了克服由于密度差之间的耦合和抵消效应而引起的困难，导出了一些精确的能量估计。