Global existence of solutions for the drift-diffusion system with large initial data in (B) over dot∞,∞-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 (B) over dot(infinity,infinity)(-2)(R-d).