We consider the Nernst-Planck-type drift-diffusion equation with fractional dissipation. For the initial-value problem of this equation, the well-posedness, the time-global existence, and the decay of solutions were already shown. When the dissipation operator is given by the Laplacian, the asymptotic expansion of the solution as $t\to\infty$ was obtained in a previous paper. We also derive the asymptotic expansion of the solution to the drift-diffusion equation with the fractional Laplacian.
Masakazu Yamamoto. "Large-time behavior of solutions to the drift-diffusion equation with fractional dissipation." Differential Integral Equations 25 (7/8) 731 - 758, July/August 2012. https://doi.org/10.57262/die/1356012661