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Large time behavior of solution to quasilinear chemotaxis system with logistic source
College of Mathematics and Information, China West Normal University, Nanchong 637009, China |
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u \nabla v)+\mu u- \mu u^{r}, \, \, \, &x\in\Omega, \, \, \, t>0, \\ \tau v_{t} = \Delta v-v+u, &x\in\Omega, \, \, \, t>0, \end{array} \right. \end{eqnarray*} $ |
$ \Omega\subset\mathbb{R}^{n} $ |
$ \tau\in\{0, 1\} $ |
$ \chi>0 $ |
$ \mu>0 $ |
$ r\geq2 $ |
$ D(u) $ |
$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $ |
$ \mu>\frac{\chi^{2}}{16} $ |
$ r\geq2 $ |
$ (1, 1) $ |
References:
[1] |
T. Cieálak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
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T. Cieálak and M. Winkler,
Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
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E. Galakhov, O. Salieva and J. I. Tello,
On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[4] |
X. He and S. N. 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. |
[5] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[6] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[7] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal. TMA, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[8] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[10] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[11] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[12] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[13] |
K. Lin and C. L. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[14] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[15] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[16] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433.
|
[17] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[18] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[19] |
Y. S. Tao and M. Winkler,
Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.
doi: 10.1016/j.jde.2015.07.019. |
[20] |
Y. S. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
Y. S. 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. |
[22] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[23] |
G. Viglialoro and T. E. Woolley,
Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3023-3045.
doi: 10.3934/dcdsb.2017199. |
[24] |
L. C. Wang, C. L. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[25] |
Z. A. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, arXiv: 1510.07204v1. |
[26] |
L. C. Wang, C. L. Mu, X. G. Hu and P. Zheng,
Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[27] |
L. C. Wang, J. Zhang, C. L. Mu and X. G. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.
|
[28] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[29] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[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,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[32] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[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,
How far can chemtactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[35] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[36] |
M. Winkler,
Global asymptotic stability of constant equilibria ina fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[37] |
J. Zhao, C. L. Mu, L. C. Wang and K. Lin, A quasilinear parabolic-elliptic chemotaxis-growth system with nonlinear secretion, Appl. Anal., (2018).
doi: 10.1080/00036811.2018.1489955. |
[38] |
J. S. Zheng and Y. F. Wang,
Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source, Comput. Math. Appl., 72 (2016), 2604-2619.
doi: 10.1016/j.camwa.2016.09.020. |
[39] |
P. Zheng, C. L. Mu and X. G. Hu,
Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 2299-2323.
doi: 10.3934/dcds.2015.35.2299. |
show all references
References:
[1] |
T. Cieálak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[2] |
T. Cieálak and M. Winkler,
Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[3] |
E. Galakhov, O. Salieva and J. I. Tello,
On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[4] |
X. He and S. N. 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. |
[5] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[6] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[7] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal. TMA, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[8] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[10] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[11] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[12] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[13] |
K. Lin and C. L. Mu,
Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.
doi: 10.3934/dcds.2016018. |
[14] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[15] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[16] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433.
|
[17] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[18] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[19] |
Y. S. Tao and M. Winkler,
Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.
doi: 10.1016/j.jde.2015.07.019. |
[20] |
Y. S. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[21] |
Y. S. 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. |
[22] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[23] |
G. Viglialoro and T. E. Woolley,
Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3023-3045.
doi: 10.3934/dcdsb.2017199. |
[24] |
L. C. Wang, C. L. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[25] |
Z. A. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, arXiv: 1510.07204v1. |
[26] |
L. C. Wang, C. L. Mu, X. G. Hu and P. Zheng,
Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401.
doi: 10.1016/j.jde.2017.11.019. |
[27] |
L. C. Wang, J. Zhang, C. L. Mu and X. G. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.
|
[28] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[29] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[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,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.
doi: 10.3934/dcdsb.2017135. |
[32] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[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,
How far can chemtactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[35] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[36] |
M. Winkler,
Global asymptotic stability of constant equilibria ina fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[37] |
J. Zhao, C. L. Mu, L. C. Wang and K. Lin, A quasilinear parabolic-elliptic chemotaxis-growth system with nonlinear secretion, Appl. Anal., (2018).
doi: 10.1080/00036811.2018.1489955. |
[38] |
J. S. Zheng and Y. F. Wang,
Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source, Comput. Math. Appl., 72 (2016), 2604-2619.
doi: 10.1016/j.camwa.2016.09.020. |
[39] |
P. Zheng, C. L. Mu and X. G. Hu,
Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 2299-2323.
doi: 10.3934/dcds.2015.35.2299. |
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