Asymptotic stability of stationary solutions to the drift-diffusion model with the fractional dissipation

Abstract

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.