  The initial-value problem for the drift-diffusion equation arising from the
model of semiconductor device simulations is studied. The dissipation on this
equation is given by the fractional Laplacian. When the exponent of the
fractional Laplacian is large, large-time behavior of solutions is known.
However, when the exponent is small, the perturbation methods used in the
preceding works would not work. Large-time behavior of solutions to the
drift-diffusion equation with small exponent is discussed. Particularly, the
asymptotic expansion of solutions with high-order is derived.
