THE ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO THE CHEMOTAXIS MODEL IN THE CRITICAL FRAMEWORK

Weixuan Shi

doi:10.11948/20210128

2022 Volume 12 Issue 4

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Weixuan Shi. THE ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO THE CHEMOTAXIS MODEL IN THE CRITICAL FRAMEWORK[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1371-1385. doi: 10.11948/20210128

Citation:

Weixuan Shi. THE ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO THE CHEMOTAXIS MODEL IN THE CRITICAL FRAMEWORK[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1371-1385. doi: 10.11948/20210128

THE ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO THE CHEMOTAXIS MODEL IN THE CRITICAL FRAMEWORK

Weixuan Shi^1, ,

1.
School of Science, Jiangnan University, Wuxi, 214122, China

Corresponding author: Email: wxshi_0610@jiangnan.edu.cn(W.X. Shi)
Fund Project: The author was supported by the National Natural Science Foundation of China (12101263) and the Fundamental Research Funds for the Central Universities (JUSRP121047)

Abstract

Abstract

The Keller-Segel model is an effective mathematical model (derived by Keller and Segel), which is used to describe the phenomenon of chemotaxis in biological sciences. The purpose of this paper is to investigate the asymptotic behavior of solutions in the $ L^p $ framework by developing the pure energy approach (independent of spectral analysis). Precisely, a new low-frequency regularity of initial data is posted, which enables us to establish the Lyapunov-type inequality in time for energy norms. As a result, the large-time behavior of strong solutions near constant equilibrium can be obtained. The proof crucially depends on non standard product estimates and interpolations. It's worth noting that the smallness requirement of low frequencies is no longer needed.
- Chemotaxis model /
- critical Besov spaces /
- decay estimates /
- pure energy approach
MSC: 35Q92, 35M31, 92C17, 35B40

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References

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