Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

Large time behavior and diffusion limit for a system of balance laws from chemotaxis in multi-dimensions

Pages: 229 – 272



Tong Li (Department of Mathematics, University of Iowa, Iowa City, Ia., U.S.A.)

Dehua Wang (Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A.)

Fang Wang (School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan, China)

Zhi-An Wang (Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong)

Kun Zhao (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)


We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero.


system of balance laws, global well-posedness, long-time behavior, diffusion limit

2010 Mathematics Subject Classification

35K45, 35K55, 35K57, 35Q92, 92C15, 92C17

Received 15 October 2019

Accepted 17 August 2020

Published 24 March 2021