Asymptotic stability and the hair-trigger effect in Cauchy problem of the flux-limited Keller-Segel system with logistic source

De-Ji-Xiang-Mao; Jing Li; Yifu Wang

doi:10.3934/dcdsb.2024138

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2025, Volume 30, Issue 5: 1517-1532. Doi: 10.3934/dcdsb.2024138

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Asymptotic stability and the hair-trigger effect in Cauchy problem of the flux-limited Keller-Segel system with logistic source

1.
College of Science, Minzu University of China, 100081 Beijing, China
2.
School of Mathematics and Statistics, Beijing Institute of Technology, 100081 Beijing, China

^*Corresponding author: Jing Li
^*Corresponding author: Jing Li

Received: April 2024

Revised: August 2024

Early access: October 12, 2024

Published online: October 12, 2024

The research of Jing Li is supported by NSFC grant No. 12171498, 12071030. The research of Yifu Wang is supported by NSFC grant No. 12071030.

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Abstract

This paper was concerned with Cauchy problem of the parabolic-parabolic flux-limited Keller-Segel system with logistic source. We discussed the global existence and global boundedness of the classical solution. By constructing auxiliary functions with quasi-linear structures, we can directly obtained the persistence and the asymptotic stability of the positive constant equilibria for strictly positive initial datum. Moreover, for any initial datum satisfying $ \int_{B(x, \delta)}\ln u_0(s)ds\in L^\infty(\mathbb R^N) $ for some $ \delta>0 $, the hair-trigger effect was detected by constructing the localized Lyapunov type functional.

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Mathematics Subject Classification: Primary: 35K45, 35K57; Secondary: 92C17.

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References

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