# American Institute of Mathematical Sciences

February  2021, 41(2): 813-847. doi: 10.3934/dcds.2020301

## Asymptotic dynamics of a system of conservation laws from chemotaxis

 1 Department of Mathematics, Nanchang University, Nanchang 330031, China 2 School of Mathematics, South China University of Technology, Guangzhou 510640, China 3 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China 4 Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

* Corresponding author: Kun Zhao

Received  August 2019 Revised  January 2020 Published  August 2020

This paper is devoted to the analytical study of the long-time asymptotic behavior of solutions to the Cauchy problem of a system of conservation laws in one space dimension, which is derived from a repulsive chemotaxis model with singular sensitivity and nonlinear chemical production rate. Assuming the $H^2$-norm of the initial perturbation around a constant ground state is finite and using energy methods, we show that there exists a unique global-in-time solution to the Cauchy problem, and the constant ground state is globally asymptotically stable. In addition, the explicit decay rates of the solutions to the chemically diffusive and non-diffusive models are identified under different exponent ranges of the nonlinear chemical production function.

Citation: Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
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##### References:
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