Controlled dynamics of a chemotaxis model with logarithmic sensitivity by physical boundary conditions

Ling Xue; Min Zhang; Kun Zhao; Xiaoming Zheng; Ling Xue; Min Zhang; Kun Zhao; Xiaoming Zheng

doi:10.3934/era.2022230

Electronic Research Archive

2022, Volume 30, Issue 12: 4530-4552. doi: 10.3934/era.2022230

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Research article

Controlled dynamics of a chemotaxis model with logarithmic sensitivity by physical boundary conditions

1.
College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China
2.
Department of Mathematics, Tulane University, LA 70118, USA
3.
Department of Mathematics, Central Michigan University, MI 48859, USA

Received: 15 February 2022 Revised: 04 June 2022 Accepted: 09 June 2022 Published: 17 October 2022

We study the global dynamics of large amplitude classical solutions to a system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, subject to time-dependent boundary conditions. The model is supplemented with $ H^2 $ initial data and unmatched boundary conditions at the endpoints of a one-dimensional interval. Under suitable assumptions on the boundary data, it is shown that classical solutions exist globally in time. Time asymptotically, the differences between the solutions and their corresponding boundary data converge to zero, as time goes to infinity. No smallness restrictions on the magnitude of the initial perturbations is imposed. Numerical simulations are carried out to explore some topics that are not covered by the analytical results.
- chemotaxis,
- dynamic boundary conditions,
- large data solution,
- long-time behavior
Citation: Ling Xue, Min Zhang, Kun Zhao, Xiaoming Zheng. Controlled dynamics of a chemotaxis model with logarithmic sensitivity by physical boundary conditions[J]. Electronic Research Archive, 2022, 30(12): 4530-4552. doi: 10.3934/era.2022230

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Abstract

We study the global dynamics of large amplitude classical solutions to a system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, subject to time-dependent boundary conditions. The model is supplemented with $ H^2 $ initial data and unmatched boundary conditions at the endpoints of a one-dimensional interval. Under suitable assumptions on the boundary data, it is shown that classical solutions exist globally in time. Time asymptotically, the differences between the solutions and their corresponding boundary data converge to zero, as time goes to infinity. No smallness restrictions on the magnitude of the initial perturbations is imposed. Numerical simulations are carried out to explore some topics that are not covered by the analytical results.

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