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Boundedness in a three-component quasilinear chemotaxis system on Alopecia Areata

  • *Corresponding author: Qiao Xin

    *Corresponding author: Qiao Xin

This work is supported by the National Natural Science Foundation of China (No.12261092), Yili Normal University's "High-level Talent" Program of Distinguished Professor of Academic Integrity (No.YSXSJS22005) and the Scientific Research and Innovation Team in Yili Normal University (No.CXZK2021018)

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  • In this paper, we consider a three component quasilinear chemotaxis system for alopecia areata

    $ \begin{equation*} \begin{cases} u_t = \nabla\cdot(D_1(u)\nabla u)-\chi_1\nabla\cdot(S_1(u)\nabla w)+w-\mu_1u^{\gamma_1}, &x\in\Omega,t>0,\\ v_t = \nabla\cdot(D_2(v)\nabla v)-\chi_2\nabla\cdot(S_2(v)\nabla w)+w+ruv-\mu_2v^{\gamma_2},&x\in\Omega,t>0,\\ w_t = \Delta w+u+v-w, &x\in\Omega,t>0, \end{cases} \end{equation*} $

    in a smoothly bounded domain $ \Omega\subset\mathbb{R}^n(n\geq1) $ with Neumman boundary conditions, where parameters $ \chi_i,\mu_i\; (i = 1,2) $ and $ r $ are positive. The functions $ D_i(\cdot) $ and $ S_i(\cdot) $ belong to $ C^2 $ satisfying $ D_i(s)\ge(s+1)^{\alpha_i} $ and $ S_i(s)\le s(s+1)^{\beta_i-1} $ with $ \alpha_i,\beta_i\in \mathbb{ R} $ for all $ s\ge0 $ and $ i = 1,2 $. We study the global boundedness of classical solutions existing without any further restrictions on the size of system parameters in two cases: (i) both the diffusion and the logistic damping balance the cross-diffusion; (ii) the logistic damping inhibits the cross-diffusion. Those results not only extend the existing results by Xu (JMAA, 2023), but also draw some new conclusions.

    Mathematics Subject Classification: Primary: 35A01, 35K59, 35Q92, 92C17.

    Citation:

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  • [1] G. Arumugam and J. Tyagi, Keller-Segel chemotaxis models: A review, Acta Appl. Math., 171 (2021), 1-82.  doi: 10.1007/s10440-020-00374-2.
    [2] N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.
    [3] X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.  doi: 10.1002/mma.2992.
    [4] T. Cieślak and C. Stinner, Finite-time blow-up and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.
    [5] T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.
    [6] M. Ding and W. Wang, Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684.  doi: 10.3934/dcdsb.2018328.
    [7] A. DobrevaR. Paus and N. Cogan, Toward predicting the spatio-temporal dynamics of alopecia areata lesions using partial differential equation analysis, Bull. Math. Biol., 82 (2020), 1-32.  doi: 10.1007/s11538-020-00707-0.
    [8] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.
    [9] S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equ., 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.
    [10] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.
    [11] X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503.
    [12] Y. Lou and Y. Tao, The role of local kinetics in a three-component chemotaxis model for alopecia areta, J. Differ. Equ., 305 (2021), 401-427.  doi: 10.1016/j.jde.2021.10.020.
    [13] K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.
    [14] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 
    [15] W. Shan and P. Zheng, Boundedness and asymptotic behavior in a quasilinear chemotaxis system for alopecia areata, Nonlinear Anal. RWA., 72 (2023), 103858.  doi: 10.1016/j.nonrwa.2023.103858.
    [16] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.
    [17] Y. Tao and D. Xu, Combined effects of nonlinear proliferation and logistic damping in a three-component chemotaxis system for alopecia areata, Nonlinear Anal. RWA., 66 (2022), 103517.  doi: 10.1016/j.nonrwa.2022.103517.
    [18] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Bull. Amer. Math. Soc., 21 (1989), 196-198. 
    [19] L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.
    [20] W. Wang, A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source, J. Math. Anal. Appl., 477 (2019), 488-522.  doi: 10.1016/j.jmaa.2019.04.043.
    [21] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differ. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.
    [22] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.
    [23] M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, J. Differ. Equ., 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019.
    [24] M. Winkler, A family of mass-critical Keller-Segel systems, Proc. Lond. Math. Soc., 124 (2022), 133-181.  doi: 10.1112/plms.12425.
    [25] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differ. Equ., 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032.
    [26] J. Xie, A new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source, J. Math. Anal. Appl., 496 (2021), 124784.  doi: 10.1016/j.jmaa.2020.124784.
    [27] L. XuQ. Xin and H. Yang, Boundedness in a three-component chemotaxis system with nonlinear diffusion for alopecia areata, J. Math. Anal. Appl., 520 (2023), 126893.  doi: 10.1016/j.jmaa.2022.126893.
    [28] W. ZhangL. Xu and Q. Xin, Global boundedness of a higher-dimensional chemotaxis system on alopecia areata, Math. Biosci. Eng., 20 (2023), 7922-7942.  doi: 10.3934/mbe.2023343.
    [29] Y. Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.
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