We study the drift-diffusion equation with fractional dissipation \((-\varDelta )^{\theta /2}\) arising from a model of semiconductors. First, we prove the existence and uniqueness of the small solution to the corresponding stationary problem in the whole space. Moreover, it is proved that the unique solution of non-stationary problem exists globally in time and decays exponentially, if initial data are suitably close to the stationary solution and the stationary solution is sufficiently small.

中文翻译：

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分数耗散的漂移扩散模型平稳解的渐近稳定性

我们研究了由半导体模型引起的具有分数耗散\（（-\ varDelta）^ {\ theta / 2} \）的漂移扩散方程。首先，我们证明了整个空间中对应平稳问题的小解的存在性和唯一性。此外，证明了：如果初始数据适当地接近固定解并且固定解足够小，则非平稳问题的唯一解在时间上全局存在并且呈指数衰减。