Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Suppression of blow up by mixing in generalized Keller–Segel system with fractional dissipation

Pages: 1413 – 1440



Binbin Shi (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Weike Wang (School of Mathematical Sciences and Institute of Natural Science, Shanghai Jiao Tong University, Shanghai, China)


In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller–Segel equation with a fractional dissipation and an additional mixing effect of advection by an incompressible flow. Under a suitable mixing condition on the advection, we study well-posedness of solution with large initial data. We establish the global $L^\infty$ estimate of the solution through nonlinear maximum principle, and obtain the global existence of classical solution.


generalized Keller–Segel system, mixing, fractional dissipation, suppression of blow up

2010 Mathematics Subject Classification

35A01, 35B45, 35Q92, 35R11

Received 21 June 2019

Accepted 8 March 2020

Published 23 September 2020