Global existence of solutions for the drift–diffusion system with large initial data in Ḃ−2∞,∞ (Rd)

In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in ${\dot{B}}_{\infty ,\infty }^{-2}\left(R^d\right)$.