趋化系统解的全局存在性与有界性——献给陈化教授 70 寿辰

摘要/Abstract

摘要：

该文主要考虑了在光滑有界区域 $\Omega\subset\mathbb{R}^n\,(n\geqslant1)$ 上具有齐次 Neumann 边界条件的一类运动依赖于信号的趋化模型$$\begin{equation*} \begin{cases} u_t=\Delta(\gamma (v)u)+\rho u-\mu u^\alpha,&x\in\Omega,\,t>0,\\ v_t=\Delta v-v+u^\beta,&x\in\Omega,\,t>0 \end{cases} \end{equation*}$$的解的全局存在性与有界性. 当 $\rho$, $\mu>0$, $\alpha> 1$, $\beta>0$ 满足适当的条件, 且运动函数 $\gamma\in C^3((0,+\infty))$ 满足一定条件时, 该文证明了上述方程对所有足够光滑的初始值都存在一个全局古典解. 这改进了 [Lv W B, Wang Q Y. Proc Roy Soc Edinburgh, 2021, 151(2): 821-841], [Tao X Y, Fang Z B. Z Angew Math Phys, 2022, 73(3): Art 123] 中得到的结果.

关键词: 全局存在性, 趋化性, 有界性, 广义 Logistic 项

Abstract:

This paper is concerned with the global existence for a class of Keller-Segel model $$\begin{equation*} \begin{cases} u_t=\Delta(\gamma (v)u)+\rho u-\mu u^\alpha,&x\in\Omega,\,t>0,\\ v_t=\Delta v-v+u^\beta,&x\in\Omega,\,t>0, \end{cases} \end{equation*}$$ under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^n\,(n\geqslant1)$. It is proved that for $\rho\in\mathbb{R},\,\mu>0$, $\alpha> 1$, $\beta>0$ satisfying certain additional relations, and under suitable assumptions on the motility function $\gamma$, the system admits a global classical solution for all sufficiently smooth initial data. This result improves recent ones established in [Lv W B, Wang Q Y. Proc Roy Soc Edinburgh, 2021, 151(2): 821-841], [Tao X Y, Fang Z B. Z Angew Math Phys, 2022, 73(3): Art 123].

Key words: global existence, chemotaxis, boundedness, general logistic source

中图分类号:

O175.23

郭雅平, 李佳琳, 吕文斌. 趋化系统解的全局存在性与有界性——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 535-551.

Yaping Guo, Jialin Li, Wenbin Lyu. Global Existence for a Class of Chemotaxis Systems with Signal-Dependent Motility and Generalized Logistic Source[J]. Acta mathematica scientia,Series A, 2026, 46(2): 535-551.

参考文献 49

[1]	Amann H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems.//Schmeisser H J, Triebel H. Function Spaces, Differential Operators and Nonlinear Analysis. Wiesbaden: Vieweg+Teubner Verlag, 1993, 133: 9--126
[2]	Desvillettes L, Kim Y J, Trescases A, Yoon C W. A logarithmic chemotaxis model featuring global existence and aggregation. Nonlinear Anal Real World Appl, 2019, 50: 562-582
[3]	Ding M Y, Wang W, Zhou S L, Zheng S N. Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production. J Differential Equations, 2020, 268(11): 6729-6777
[4]	Fujie K, Jiang J. Boundedness of classical solutions to a degenerate Keller--Segel type model with signal-dependent motilities. Acta Appl Math, 2021, 176(1): Art 3
[5]	Fujie K, Jiang J. Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities. Calc Var Partial Differential Equations, 2021, 60(3): Art 92
[6]	Horstmann D, Winkler M. Boundedness vs. blow-up in a chemotaxis system. J Differential Equations, 2005, 215(1): 52-107
[7]	Ishida S, Seki K, Yokota T. Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains. J Differential Equations, 2014, 256(8): 2993-3010
[8]	Ishida S, Yokota T. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete Contin Dyn Syst Ser S, 2020, 13(2): 212-232
[9]	Jin H Y, Kim Y J, Wang Z A. Boundedness, stabilization, and pattern formation driven by density-suppressed motility. SIAM J Appl Math, 2018, 78(3): 1632-1657
[10]	Jin H Y, Wang Z A. Critical mass on the Keller-Segel system with signal-dependent motility. Proc Amer Math Soc, 2020, 148(11): 4855-4873
[11]	Keller E F, Segel L A. Initiation of slime mold aggregation viewed as an instability. J Theoret Biol, 1970, 26(3): 399-415
[12]	Lankeit J. Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. J Differential Equations, 2015, 258(4): 1158-1191
[13]	Liu D M, Tao Y S. Boundedness in a chemotaxis system with nonlinear signal production. Appl Math J Chinese Univ, 2016, 31(4): 379-388
[14]	Liu Z R, Xu J. Large time behavior of solutions for density-suppressed motility system in higher dimensions. J Math Anal Appl, 2019, 475(2): 1596-1613
[15]	Liu C, Fu X, Liu L, et al. Sequential establishment of stripe patterns in an expanding cell population. Science, 2011, 334: 238-241
[16]	Lv W B. Global existence for a class of chemotaxis-consumption systems with signal-dependent motility and generalized logistic source. Nonlinear Anal Real World Appl, 2020, 56: Art 103160
[17]	Lv W B, Wang Q. A chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source: Global existence and asymptotic stabilization. J Math Anal Appl, 2020, 488(2): Art 124108
[18]	Lv W B, Wang Q Y. Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source. Z Angew Math Phys, 2020, 71(2): Art 53
[19]	Lv W B, Wang Q Y. Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evol Equ Control Theory, 2021, 10(1): 25-36
[20]	Lv W B, Wang Q Y. An $n$-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization. Proc Roy Soc Edinburgh, 2021, 151(2): 821-841
[21]	Lyu W B Weak solutions to a class of signal-dependent motility Keller-Segel systems with superlinear damping. J Differential Equations, 2022, 312: 209-236
[22]	Lyu W B, Wang Z A. Global classical solutions for a class of reaction-diffusion system with density-suppressed motility. Electron Res Arch, 2022, 30(3): 995-1015
[23]	Nakaguchi E, Osaki K. Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation. Nonlinear Anal, 2011, 74(1): 286-297
[24]	Tao X Y, Fang Z B. Global boundedness and stability in a density-suppressed motility model with generalized logistic source and nonlinear signal production. Z Angew Math Phys, 2022, 73(3): Art 123
[25]	Tao Y S, Winkler M. A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source. SIAM J Math Anal, 2011, 43(2): 685-704
[26]	Tao Y S, Winkler M. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J Differential Equations, 2012, 252(1): 692-715
[27]	Tao Y S, Winkler M. Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system. Math Models Methods Appl Sci, 2017, 27(9): 1645-1683
[28]	Tello J I, Winkler M. A chemotaxis system with logistic source. Comm Partial Differential Equations, 2007, 32(4-6): 849-877
[29]	Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer-Verlag, 1988
[30]	Viglialoro G. Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source. J Math Anal Appl, 2016, 439(1): 197-212
[31]	Wang J P, Wang M X. Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth. J Math Phys, 2019, 60(1): Art 011507
[32]	Wang W. Global boundedness of solutions resulting from both the self-diffusion and the logistic-type source. Z Angew Math Phys, 2019, 70(4): Art 99
[33]	Wang Z A, Zheng J S. Global boundedness of the fully parabolic Keller-Segel system with signal-dependent motilities. Acta Appl Math, 2021, 171(1): Art 25
[34]	Winkler M. Chemotaxis with logistic source: very weak global solutions and their boundedness properties. J Math Anal Appl, 2008, 348(2): 708-729
[35]	Winkler M. Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J Differential Equations, 2010, 248(12): 2889-2905
[36]	Winkler M. Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Comm Partial Differential Equations, 2010, 35(8): 1516-1537
[37]	Winkler M. Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J Math Anal Appl, 2011, 384(2): 261-272
[38]	Winkler M. Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Math Methods Appl Sci, 2011, 34(2): 176-190
[39]	Winkler M. Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system. J Math Pures Appl, 2013, 100(5): 748-767
[40]	Winkler M. Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation. Z Angew Math Phys, 2018, 69(2): Art 69
[41]	Winkler M. Can simultaneous density-determined enhancement of diffusion and cross-diffusion foster boundedness in Keller-Segel type systems involving signal-dependent motilities? Nonlinearity, 2020, 33(12): 6590-6623
[42]	Winkler M. The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^1$. Adv Nonlinear Anal, 2020, 9(1): 526-566
[43]	Winkler M. Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration-consumption interaction. Z Angew Math Phys, 2023, 74(1): Art 32
[44]	Winkler M.$L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation. Ann Sc Norm Super Pisa Cl Sci (5), 2023, 24(1): 141-172
[45]	Winkler M. A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion. Ann Inst H Poincaré C Anal Non Linéaire, 2024, 41(1): 95-127
[46]	Xiang T. Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system. J Math Phys, 2018, 59(8): Art 081502
[47]	Yan J L, Fuest M. When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions? Discrete Contin Dyn Syst, 2021, 26(8): 4093-4109
[48]	Yoon C, Kim Y J. Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion. Acta Appl Math, 2017, 149: 101-123
[49]	Zhuang M D, Wang W, Zheng S N. Boundedness in a fully parabolic chemotaxis system with logistic-type source and nonlinear production. Nonlinear Anal Real World Appl, 2019, 47: 473-483