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

TL;DR

This paper investigates the long-time behavior of solutions to a fractional drift-diffusion-Poisson system, demonstrating convergence to the fractional heat kernel using the entropy method, with implications for other fractional dissipation equations.

Contribution

It introduces an entropy-based approach to analyze the asymptotic behavior of fractional drift-diffusion-Poisson systems, providing decay rates and extending to other semilinear equations.

Findings

Solutions converge to the fractional heat kernel in subcritical and supercritical cases.

Decay rates are established in the $L^1(R^d)$ norm.

Method is applicable to a broader class of fractional dissipation equations.

Abstract

The self-similar asymptotics for solutions to the drift-diffusion equation with fractional dissipation, coupled to the Poisson equation, is analyzed in the whole space. It is shown that in the subcritical and supercritical cases, the solutions converge to the fractional heat kernel with algebraic rate. The proof 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 semilinear equations with fractional dissipation.