This paper mainly focuses on the dynamics behavior of a three-component chemotaxis system on alopecia areata
$ \begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta{u}-\chi_1\nabla\cdot(u\nabla{w})+w-\mu_1u^2, &x\in\Omega, t>0, \\ v_t = \Delta{v}-\chi_2\nabla\cdot(v\nabla{w})+w+ruv-\mu_2v^2, &x\in \Omega, t>0, \\ w_t = \Delta{w}+u+v-w, &x\in \Omega, t>0, \\ \frac{\partial{u}}{\partial{\nu}} = \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = 0, &x\in \partial \Omega, t>0, \\ u(x, 0) = u_0(x), \ v(x, 0) = v_0(x), \ w(x, 0) = w_0(x), &x\in \Omega, \ \end{array} \right. \end{equation*} $
where $ \Omega\subset\mathbb{R}^n $ $ (n \geq 4) $ is a bounded convex domain with smooth boundary $ \partial\Omega $, the parameters $ \chi_i $, $ \mu_i $ $ (i = 1, 2) $, and $ r $ are positive. We show that this system exists a globally bounded classical solution if $ \mu_i\; (i = 1, 2) $ is large enough. This result extends the corresponding results which were obtained by Lou and Tao (JDE, 2021) to the higher-dimensional case.
Citation: Wenjie Zhang, Lu Xu, Qiao Xin. Global boundedness of a higher-dimensional chemotaxis system on alopecia areata[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 7922-7942. doi: 10.3934/mbe.2023343
This paper mainly focuses on the dynamics behavior of a three-component chemotaxis system on alopecia areata
$ \begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta{u}-\chi_1\nabla\cdot(u\nabla{w})+w-\mu_1u^2, &x\in\Omega, t>0, \\ v_t = \Delta{v}-\chi_2\nabla\cdot(v\nabla{w})+w+ruv-\mu_2v^2, &x\in \Omega, t>0, \\ w_t = \Delta{w}+u+v-w, &x\in \Omega, t>0, \\ \frac{\partial{u}}{\partial{\nu}} = \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = 0, &x\in \partial \Omega, t>0, \\ u(x, 0) = u_0(x), \ v(x, 0) = v_0(x), \ w(x, 0) = w_0(x), &x\in \Omega, \ \end{array} \right. \end{equation*} $
where $ \Omega\subset\mathbb{R}^n $ $ (n \geq 4) $ is a bounded convex domain with smooth boundary $ \partial\Omega $, the parameters $ \chi_i $, $ \mu_i $ $ (i = 1, 2) $, and $ r $ are positive. We show that this system exists a globally bounded classical solution if $ \mu_i\; (i = 1, 2) $ is large enough. This result extends the corresponding results which were obtained by Lou and Tao (JDE, 2021) to the higher-dimensional case.
| [1] |
A. Dobreva, R. Paus, N. Cogan, Toward predicting the spatio-temporal dynamics of alopecia areata lesions using partial differential equation analysis, Bull. Math. Biol., 82 (2020), 1–32. https://doi.org/10.1007/s11538-020-00707-0 doi: 10.1007/s11538-020-00707-0
|
| [2] | A. Gilhar, A. Etzioni, R. Paus, Alopecia areata, N. Engl. J. Med., 366 (2012), 1515–1525. https://doi.org/10.1056/NEJMra1103442 |
| [3] |
A. Luster, J. Ravetch, Biochemical characterization of a gamma interferon-inducible cytokine (IP-10), J. Exp. Med., 166 (1987), 1084–1097. https://doi.org/10.1084/jem.166.4.1084 doi: 10.1084/jem.166.4.1084
|
| [4] |
Y. Lou, Y. Tao, The role of local kinetics in a three-component chemotaxis model for Alopecia Areata, J. Differ. Equation, 305 (2021), 401–427. https://doi.org/10.1016/j.jde.2021.10.020 doi: 10.1016/j.jde.2021.10.020
|
| [5] |
Y. Tao, D. Xu, Combined effects of nonlinear proliferation and logistic damping in a three-component chemotaxis system for alopecia areata, Nonlinear Anal. Real World Appl., 66 (2022), 103517. https://doi.org/10.1016/j.nonrwa.2022.103517 doi: 10.1016/j.nonrwa.2022.103517
|
| [6] |
E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399–415. https://doi.org/10.1016/0022-5193(70)90092-5 doi: 10.1016/0022-5193(70)90092-5
|
| [7] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equation, 248 (2010), 2889–2905. https://doi.org/10.1016/j.jde.2010.02.008 doi: 10.1016/j.jde.2010.02.008
|
| [8] |
G. Arumugam, J. Tyagi, Keller-Segel chemotaxis models: A review, Acta Appl. Math., 171 (2021), 1–82. https://doi.org/10.1007/s10440-020-00374-2 doi: 10.1007/s10440-020-00374-2
|
| [9] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748–767. https://doi.org/10.1016/j.matpur.2013.01.020 doi: 10.1016/j.matpur.2013.01.020
|
| [10] |
T. Xiang, On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model, Discrete Contin. Dyn. Syst., 34 (2014), 4911–4946. https://doi.org/10.3934/dcds.2014.34.4911 doi: 10.3934/dcds.2014.34.4911
|
| [11] |
T. Hillen, A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783–1801. https://doi.org/10.1002/mma.569 doi: 10.1002/mma.569
|
| [12] |
H. Jin, T. Xiang, Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller-Segel model, C. R. Math. Acad. Sci. Paris, 356 (2018), 875–885. https://doi.org/10.13140/RG.2.2.33597.36325 doi: 10.13140/RG.2.2.33597.36325
|
| [13] |
K. Osaki, T. Tsujikawa, A. Yagi, M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119–144. https://doi.org/10.1016/S0362-546X(01)00815-X doi: 10.1016/S0362-546X(01)00815-X
|
| [14] |
E. Nakaguchi, K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2627–2646. https://doi.org/10.3934/dcdsb.2013.18.2627 doi: 10.3934/dcdsb.2013.18.2627
|
| [15] |
K. Lin, C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025–5046. https://doi.org/10.3934/dcds.2016018 doi: 10.3934/dcds.2016018
|
| [16] |
T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172–1200. https://doi.org/10.1016/j.jmaa.2017.11.022 doi: 10.1016/j.jmaa.2017.11.022
|
| [17] |
M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial. Differ. Equation, 35 (2010), 1516–1537. https://doi.org/10.1080/03605300903473426 doi: 10.1080/03605300903473426
|
| [18] |
M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Am. Math. Soc., 369 (2017), 3067–3125. http://dx.doi.org/10.1090/tran/6733 doi: 10.1090/tran/6733
|
| [19] |
M. Winkler, Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components, J. Evol. Equation, 18 (2018), 1267–1289. https://doi.org/10.1007/s00028-018-0440-8 doi: 10.1007/s00028-018-0440-8
|
| [20] |
J. Zheng, An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion, J. Differ. Equation, 267 (2019), 2385–2415. https://doi.org/10.1016/j.jde.2019.03.013 doi: 10.1016/j.jde.2019.03.013
|
| [21] |
J. Zheng, A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, J. Differ. Equation, 272 (2021), 164-202. https://doi.org/10.1016/j.jde.2020.09.029 doi: 10.1016/j.jde.2020.09.029
|
| [22] |
J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differ. Equation, 259 (2015), 120-140. https://doi.org/10.1016/j.jde.2015.02.003 doi: 10.1016/j.jde.2015.02.003
|
| [23] |
J. Zheng, Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux, Calc. Var. Partial Differ. Equations, 61 (2022), 52. https://doi.org/10.1007/s00526-021-02164-6 doi: 10.1007/s00526-021-02164-6
|
| [24] |
Y. Ke, J. Zheng, An optimal result for global existence in a three-dimensional Keller-Segel-Navier-Stokes system involving tensor-valued sensitivity with saturation, Calc. Var. Partial Differ. Equations, 58 (2019), 1–27. https://doi.org/10.1007/s00526-019-1568-2 doi: 10.1007/s00526-019-1568-2
|
| [25] |
N. Bellomo, A. Bellouquid, Y. Tao, 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. https://doi.org/10.1142/S021820251550044X doi: 10.1142/S021820251550044X
|
| [26] |
Y. Tian, D. Li, C. Mu, Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production, C. R. Math. Acad. Sci. Paris, 357 (2019), 513–519. https://doi.org/10.1016/j.crma.2019.05.010 doi: 10.1016/j.crma.2019.05.010
|
| [27] |
C. Mu, W. Tao, Stabilization and pattern formation in chemotaxis models with acceleration and logistic source, Math. Biosci. Eng., 20 (2023), 2011–2038. https://doi.org/10.3934/mbe.2023093 doi: 10.3934/mbe.2023093
|
| [28] |
Y. Tao, M. Winkler, Taxis-driven Formation of Singular Hotspots in a May–Nowak Type Model for Virus Infection, SIAM J. Math. Anal., 53 (2021), 1411–1433. https://doi.org/10.1137/20M1362851 doi: 10.1137/20M1362851
|
| [29] |
T. Alzahrani, E. Raluca, T. Dumitru, Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci., 310 (2019), 76–95. https://doi.org/10.1016/j.mbs.2018.12.018 doi: 10.1016/j.mbs.2018.12.018
|
| [30] |
N. Bellomo, K. Painter, Y. Tao, M. Winkler, Occurrence vs. Absence of taxis-driven instabilities in a may-nowak model for virus infection, SIAM J. Appl. Math., 79 (2019), 1990–2010. https://doi.org/10.1137/19M1250261 doi: 10.1137/19M1250261
|
| [31] |
Y. Tao, M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555–2573. https://doi.org/10.1007/s00033-015-0541-y doi: 10.1007/s00033-015-0541-y
|
| [32] |
B. Hu, Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111–2128. https://doi.org/10.1142/S0218202516400091 doi: 10.1142/S0218202516400091
|
| [33] |
Y. Tao, M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229–4250. https://doi.org/10.1137/15M1014115 doi: 10.1137/15M1014115
|
| [34] |
D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equation, 215 (2005), 52–107. https://doi.org/10.1016/j.jde.2004.10.022 doi: 10.1016/j.jde.2004.10.022
|
| [35] |
Y. Tao, Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1–36. https://doi.org/10.1142/S0218202512500443 doi: 10.1142/S0218202512500443
|
| [36] |
K. Fujie, A. Ito, M. Winkler, T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169. https://doi.org/10.3934/dcds.2016.36.151 doi: 10.3934/dcds.2016.36.151
|
| [37] |
P. Lions, Résolution de problemes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335–353. https://doi.org/10.1007/BF00249679 doi: 10.1007/BF00249679
|
| [38] |
R. Kowalczyk, Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379–398. https://doi.org/10.1016/j.jmaa.2008.01.005 doi: 10.1016/j.jmaa.2008.01.005
|
| [39] | X. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553–583. http://www.jstor.org/stable/26318163 |
| [40] |
M. Porzio, V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differ. Equation, 103 (1993), 146–178. https://doi.org/10.1006/jdeq.1993.1045 doi: 10.1006/jdeq.1993.1045
|
| [41] |
Y. Tao, M. Winkler, Boundedness and stabilization in a population model with cross-diffusion for one species, Proc. Lond. Math. Soc., 119 (2019), 1598–1632. https://doi.org/10.1112/plms.12276 doi: 10.1112/plms.12276
|
| [42] | G. Lieberman, Second order parabolic differential equations, World Sci., 1996 (1996). https://doi.org/10.1142/3302 |
Wenjie Zhang, Lu Xu, Qiao Xin. Global boundedness of a higher-dimensional chemotaxis system on alopecia areata[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 7922-7942. doi: 10.3934/mbe.2023343
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