双极量子流体动力学模型的代数时间衰减
Algebraic time-decay for the bipolar quantum hydrodynamic model
摘要：
本文研究了半导体双极量子流体力学模型（QHD）在$\mathbb{R}^3$中的初值问题。证明了该方程的强解在时间上全局存在，并以$t\to+\infty$的代数速率趋于渐近解。并且，我们证明了线性化双极QHD系统的整体解从上到下都以代数衰减率随时间衰减。这意味着双极QHD系统一般不能得到指数时间衰减率，这与单极QHD模型不同（在单极QHD模型中，整体解以指数时间衰减率趋向于平衡态），这主要是由两个载流子之间的非线性耦合和抵消引起的。此外，它也表明，非线性色散不影响长时间的渐近行为，这由产品引起的代数时间衰减率的双极流体动力学模型的解决方案在半经典极限。
英文摘要：
The initial value problem is considered in the present paper for bipolar quantum hydrodynamic model for semiconductors (QHD) in $\mathbb{R}^3$. We prove that the unique strong solution exists globally in time and tends to the asymptotical state with an algebraic rate as $t\to+\infty$. And, we show that the global solution of linearized bipolar QHD system decays in time at an algebraic decay rate from both above and below. This means in general, we can not get exponential time-decay rate for bipolar QHD system, which is different from the case of unipolar QHD model (where global solutions tend to the equilibrium state at an exponential time-decay rate) and is mainly caused by the nonlinear coupling and cancelation between two carriers. Moreover, it is also shown that the nonlinear dispersion does not affect the long time asymptotic behavior, which by product gives rise to the algebraic time-decay rate of the solution of the bipolar hydrodynamical model in the semiclassical limit.
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