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doi: 10.3934/dcdsb.2022118
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Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

2. 

College of Mathematics and Statistics, Yili Normal University, Yining 835000, China

* Corresponding author: Lu Xu

Received  September 2021 Revised  April 2022 Early access June 2022

Fund Project: The first author is supported by Natural Science Foundation of Xinjiang Autonomous Region under grant 2022D01C335 and Scientific Research Program of the Higher Education Institution of XinJiang [No.XJEDU2021Y043]. The second author is supported by NSFC under grants 11771062 and 11971082, the Fundamental Research Funds for the Central Universities under grant 2019CDJCYJ001 and 2020CDJQY-Z001, Chongqing Key Laboratory of Analytic Mathematics and Applications, Science and Technology Research Program of Chongqing Municipal Educational Commission(No.KJZD-M201900501)

The paper deals with the quasilinear parabolic-parabolic-ODE and parabolic-elliptic-ODE chemotaxis system with nonlinear signal secretion for multiple sclerosis and Boló's concentric sclerosis, respectively. Under appropriate assumptions on parameter, we study the global boundedness and asymptotic behavior of classical solutions to the problem. Our results improve or extend some results in [13] and [12].

Citation: Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022118
References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.

[2]

E. BilottaF. GarganoV. GiuntaM. C. LombardoP. Pantano and M. Sammartino, Axisymmetric solutions for a chemotaxis model of multiple sclerosis, Ric. Mat., 68 (2019), 281-294.  doi: 10.1007/s11587-018-0406-8.

[3]

C. F. Brosnan and C. S. Raine, Mechanisms of immune injury in multiple sclerosis, Brain Pathol., 6 (1996), 243-257. 

[4]

V. Calvez and R. H. Khonsari, Mathematical description of concentric demyelination in the human brain: Self-organization models, from Liesegang rings to chemotaxis, Math. Comput. Model., 47 (2008), 726-742.  doi: 10.1016/j.mcm.2007.06.011.

[5]

X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.  doi: 10.1016/j.jmaa.2013.10.061.

[6]

X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141.

[7]

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. Diff. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[8]

L. Desvillettes and V. Giunta, Existence and regularity for a chemotaxis model involved in the modeling of multiple sclerosis, Ric. Mat., 70 (2017), 99-113.  doi: 10.1007/s11587-020-00495-8.

[9]

L. Desvillettes, V. Giunta, J. Morgan and B. Q. Tang, Global well-posedness and nonlinear stability of a chemotaxis system modeling multiple sclerosis, preprint, 2020, arXiv: 2009.13131.

[10]

M. DingW. WangS. Zhou and S. Zheng, Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Diff. Equ., 268 (2020), 6729-6777.  doi: 10.1016/j.jde.2019.11.052.

[11]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Amer. Math. Soc., 2010. doi: 10.1090/gsm/019.

[12]

X. Hu and S. Fu, Global boundedness and stability for a chemotaxis model of Boló's concentric sclerosis, Math. Biosci. Eng., 17 (2020), 5134-5146.  doi: 10.3934/mbe.2020277.

[13]

X. HuS. Fu and S. Ai, Global asymptotic behavior of solutions for a parabolic-parabolic-ODE chemotaxis system modeling multiple sclerosis, J. Diff. Equ., 269 (2020), 6875-6898.  doi: 10.1016/j.jde.2020.05.020.

[14]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[15]

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.

[16]

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.

[17]

R. H. Khonsari and V. Calvez, The origins of concentirc demyelination: Self-organization in the human brain, PLoS ONE, 2 (2007), e150.

[18]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[19]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Diff. Equ., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[20]

M. C. LombardoR. BarresiE. BilottaF. GarganoP. Pantano and M. Sammartino, Demyelination patterns in a mathematical model of multiple sclerosis, J. Math. Biol., 75 (2017), 373-417.  doi: 10.1007/s00285-016-1087-0.

[21]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Diff. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.

[22]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction-repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689.  doi: 10.1142/S0218202520500517.

[23]

G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differ. Equ., 268 (2020), 4320-4373. doi: 10.1016/j.jde.2019.10.027.

[24]

Q. TangQ. Xin and C. Mu, Boundedness of the higher-dimensional quasilinear chemotaxis system with generalized logistic source, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 713-722.  doi: 10.1007/s10473-020-0309-0.

[25]

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.

[26]

Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.

[27]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[28]

G. Viglialoro and T. E. Woolley, Boundedness in a parabolic-elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source, Math. Methods Appl. Sci., 41 (2018), 1809-1824.  doi: 10.1002/mma.4707.

[29]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[30]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic Source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[31]

M. Winkler, Attractiveness of constant states in logistic-type Keller-Segel systems involving subquadratic growth restrictions, Adv. Nonlinear Stud., 20 (2020), 795-817.  doi: 10.1515/ans-2020-2107.

[32]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Diff. Equ., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[33]

M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e.

[34]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.

[35]

M. Winkler and K. C. Djie, Boundedness and finite-ime collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[36]

Y. Wang and X. Zhang, On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 321-328.  doi: 10.3934/dcdss.2020018.

[37]

J. Zhao, Large time behavior of solution to quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 40 (2020), 1737-1755.  doi: 10.3934/dcds.2020091.

[38]

J. ZhaoC. MuL. Wang and K. Lin, A quasilinear parabolic-elliptic chemotaxis-growth system with nonlinear secretion, Appl. Anal., 99 (2020), 86-102.  doi: 10.1080/00036811.2018.1489955.

show all references

References:
[1]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.

[2]

E. BilottaF. GarganoV. GiuntaM. C. LombardoP. Pantano and M. Sammartino, Axisymmetric solutions for a chemotaxis model of multiple sclerosis, Ric. Mat., 68 (2019), 281-294.  doi: 10.1007/s11587-018-0406-8.

[3]

C. F. Brosnan and C. S. Raine, Mechanisms of immune injury in multiple sclerosis, Brain Pathol., 6 (1996), 243-257. 

[4]

V. Calvez and R. H. Khonsari, Mathematical description of concentric demyelination in the human brain: Self-organization models, from Liesegang rings to chemotaxis, Math. Comput. Model., 47 (2008), 726-742.  doi: 10.1016/j.mcm.2007.06.011.

[5]

X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.  doi: 10.1016/j.jmaa.2013.10.061.

[6]

X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141.

[7]

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. Diff. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[8]

L. Desvillettes and V. Giunta, Existence and regularity for a chemotaxis model involved in the modeling of multiple sclerosis, Ric. Mat., 70 (2017), 99-113.  doi: 10.1007/s11587-020-00495-8.

[9]

L. Desvillettes, V. Giunta, J. Morgan and B. Q. Tang, Global well-posedness and nonlinear stability of a chemotaxis system modeling multiple sclerosis, preprint, 2020, arXiv: 2009.13131.

[10]

M. DingW. WangS. Zhou and S. Zheng, Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Diff. Equ., 268 (2020), 6729-6777.  doi: 10.1016/j.jde.2019.11.052.

[11]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Amer. Math. Soc., 2010. doi: 10.1090/gsm/019.

[12]

X. Hu and S. Fu, Global boundedness and stability for a chemotaxis model of Boló's concentric sclerosis, Math. Biosci. Eng., 17 (2020), 5134-5146.  doi: 10.3934/mbe.2020277.

[13]

X. HuS. Fu and S. Ai, Global asymptotic behavior of solutions for a parabolic-parabolic-ODE chemotaxis system modeling multiple sclerosis, J. Diff. Equ., 269 (2020), 6875-6898.  doi: 10.1016/j.jde.2020.05.020.

[14]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[15]

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.

[16]

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.

[17]

R. H. Khonsari and V. Calvez, The origins of concentirc demyelination: Self-organization in the human brain, PLoS ONE, 2 (2007), e150.

[18]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[19]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Diff. Equ., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[20]

M. C. LombardoR. BarresiE. BilottaF. GarganoP. Pantano and M. Sammartino, Demyelination patterns in a mathematical model of multiple sclerosis, J. Math. Biol., 75 (2017), 373-417.  doi: 10.1007/s00285-016-1087-0.

[21]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Diff. Equ., 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.

[22]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction-repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689.  doi: 10.1142/S0218202520500517.

[23]

G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differ. Equ., 268 (2020), 4320-4373. doi: 10.1016/j.jde.2019.10.027.

[24]

Q. TangQ. Xin and C. Mu, Boundedness of the higher-dimensional quasilinear chemotaxis system with generalized logistic source, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 713-722.  doi: 10.1007/s10473-020-0309-0.

[25]

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.

[26]

Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.

[27]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[28]

G. Viglialoro and T. E. Woolley, Boundedness in a parabolic-elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source, Math. Methods Appl. Sci., 41 (2018), 1809-1824.  doi: 10.1002/mma.4707.

[29]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[30]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic Source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[31]

M. Winkler, Attractiveness of constant states in logistic-type Keller-Segel systems involving subquadratic growth restrictions, Adv. Nonlinear Stud., 20 (2020), 795-817.  doi: 10.1515/ans-2020-2107.

[32]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Diff. Equ., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[33]

M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e.

[34]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.

[35]

M. Winkler and K. C. Djie, Boundedness and finite-ime collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[36]

Y. Wang and X. Zhang, On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 321-328.  doi: 10.3934/dcdss.2020018.

[37]

J. Zhao, Large time behavior of solution to quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 40 (2020), 1737-1755.  doi: 10.3934/dcds.2020091.

[38]

J. ZhaoC. MuL. Wang and K. Lin, A quasilinear parabolic-elliptic chemotaxis-growth system with nonlinear secretion, Appl. Anal., 99 (2020), 86-102.  doi: 10.1080/00036811.2018.1489955.

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