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The self-similar asymptotics for solutions to the drift-diffusion equationwith fractional dissipation, coupled to the Poisson equation, is analyzed inthe whole space. It is shown that in the subcritical and supercritical cases,the solutions converge to the fractional heat kernel with algebraic rate. Theproof is based on the entropy method and leads to a decay rate in the$L^1(\mathbb{R}^d)$ norm. The technique is applied to other semilinearequations with fractional dissipation.

Large-time asymptotics of a fractional drift-diffusion-Poisson system via the entropy method