Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping
Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping
复制标题
具有瞬态阻尼的双极欧拉-泊松方程解的渐近行为
DOI：
10.1016/j.jmaa.2019.01.010
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发表时间：
2019
影响因子：
1.3
通讯作者：
Zhang Kaijun
中科院分区：
文献类型：
--
作者：
Li Haitong;Li Jingyu;Mei Ming;Zhang Kaijun
本文研究了半导体器件双极流体力学模型的一维欧拉-泊松方程，当−1< λ< 1时，阻尼效应为−J (1+ t) λ，其中λ>时，阻尼效应为时间逐渐退化，λ< 0时，阻尼效应为时间逐渐增强。这种阻尼效应使水动力系统具有时间渐近的非线性扩散现象，或弱或强。在技术观测的基础上，采用时间加权能量法（权值选择得很巧妙），证明了当扩散波周围的初始扰动足够小时，系统存在唯一的全局光滑解，该解时间渐近收敛于相应的扩散波。根据λ在（- 1,17）和（1,7,1）中的不同取值，分别用代数形式O （t−34 (1+ λ)）和O (t−(1−λ)表示收敛速度，其中λ= 17为临界点，临界点处的收敛速度为O （t−6,7 ln (t)）。在初始扰动为l2可积的情况下，所有这些收敛速率都是最优的。其中，当λ= 17时，收敛速度最快，即原系统在临界点处的渐近轮廓最佳。
In this paper, we study the one-dimensional Euler–Poisson equations of bipolar hydrodynamic model for semiconductor device with time-dependent damping effect− J (1+ t) λ for− 1< λ< 1, where the damping effect is time-gradually-degenerate for λ> 0, and time-gradually-enhancing for λ< 0. Such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time-asymptotic-weakly or strongly. Based on technical observation, and by using the time-weighted energy method, where the weights are artfully chosen, we prove that the system admits a unique global smooth solution, which time-asymptotically converges to the corresponding diffusion wave, when the initial perturbation around the diffusion wave is small enough. The convergence rates are specified in the algebraic forms O (t− 3 4 (1+ λ)) and O (t−(1− λ)) according to different values of λ in (− 1, 1 7) and (1 7, 1), respectively, where λ= 1 7 is the critical point, and the convergence rate at the critical point is O (t− 6 7 ln t). All these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are L 2-integrable. Particularly, when λ= 1 7, the convergence rate is the fastest, namely, the asymptotic profile of the original system at the critical point is the best.