- Previous Article
- DCDS-B Home
- This Issue
-
Next Article
Numerical simulation of chemotaxis models on stationary surfaces
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity
1. | Department of Applied Mathematics, Dong Hua University, Shanghai 200051 |
It is proved that under the assumptions that $0\not\equiv u_0\in C^0(\bar{\Omega})$ and $v_0\in C^1(\bar{\Omega})$ are nonnegative and that $\alpha<\frac{4}{n+2}$, the classical solutions to the above system are uniformly-in-time bounded. Moreover, it is shown that for any given $(u_0, v_0)$, the corresponding solution $(u, v)$ converges to $(\bar{u}_0, \bar{u}_0)$ as time goes to infinity, where $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.
References:
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
D. G. Aronson, The porous medium equation. Nonlinear diffusion problems, Lect. 2nd 1985 Sess. C. I. M. E., Montecatini Terme/Italy 1985, Lect. Notes Math., 1224 (1986), 1-46.
doi: 10.1007/BFb0072687. |
[3] |
M. Chuai, W. Zeng, X. Yang, V. Boychenko, J. A. Glazier and C. J. Weijer, Cell movement during chick primitive streak formation, Dev. Biol., 296 (2006), 137-149.
doi: 10.1016/j.ydbio.2006.04.451. |
[4] |
T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 437-446.
doi: 10.1016/j.anihpc.2009.11.016. |
[5] |
T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publ., 81 (2008), Polish Acad. Sci., Warsaw, 105-117.
doi: 10.4064/bc81-0-7. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, New York, 1969. |
[7] |
M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Saptially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit, Euro. J. Neuroscicen, 19 (2004), 831-844. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. |
[10] |
M. A. Herrero and J. L. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683. |
[11] |
T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[12] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.- Verien, 105 (2003), 103-165. |
[13] |
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[14] |
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. |
[15] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.2307/2153966. |
[16] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[17] |
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. |
[18] |
P. L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal., 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[19] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and alzheimer's disease senile plague: is there a connection? Bull. Math. Biol., 65 (2003), 673-730. |
[20] |
T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. of Inequal. & Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[21] |
K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543. |
[22] |
B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion-existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270.
doi: 10.1088/0951-7715/24/4/012. |
[23] |
Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[24] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[25] |
Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2534.
doi: 10.1016/j.jde.2011.07.010. |
[26] |
Y. Tao and M. Winkler, Locally bounded global soutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[27] |
M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[28] |
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. |
[29] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 99 (2013).
doi: 10.1016/j.matpur.2013.01.020. |
show all references
References:
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
D. G. Aronson, The porous medium equation. Nonlinear diffusion problems, Lect. 2nd 1985 Sess. C. I. M. E., Montecatini Terme/Italy 1985, Lect. Notes Math., 1224 (1986), 1-46.
doi: 10.1007/BFb0072687. |
[3] |
M. Chuai, W. Zeng, X. Yang, V. Boychenko, J. A. Glazier and C. J. Weijer, Cell movement during chick primitive streak formation, Dev. Biol., 296 (2006), 137-149.
doi: 10.1016/j.ydbio.2006.04.451. |
[4] |
T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 437-446.
doi: 10.1016/j.anihpc.2009.11.016. |
[5] |
T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publ., 81 (2008), Polish Acad. Sci., Warsaw, 105-117.
doi: 10.4064/bc81-0-7. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart & Winston, New York, 1969. |
[7] |
M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Saptially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit, Euro. J. Neuroscicen, 19 (2004), 831-844. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. |
[10] |
M. A. Herrero and J. L. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683. |
[11] |
T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[12] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.- Verien, 105 (2003), 103-165. |
[13] |
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[14] |
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. |
[15] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.2307/2153966. |
[16] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[17] |
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. |
[18] |
P. L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal., 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[19] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and alzheimer's disease senile plague: is there a connection? Bull. Math. Biol., 65 (2003), 673-730. |
[20] |
T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. of Inequal. & Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[21] |
K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543. |
[22] |
B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion-existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270.
doi: 10.1088/0951-7715/24/4/012. |
[23] |
Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[24] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[25] |
Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2534.
doi: 10.1016/j.jde.2011.07.010. |
[26] |
Y. Tao and M. Winkler, Locally bounded global soutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[27] |
M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[28] |
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. |
[29] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 99 (2013).
doi: 10.1016/j.matpur.2013.01.020. |
[1] |
Aichao Liu, Binxiang Dai, Yuming Chen. Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021306 |
[2] |
Hai-Yang Jin, Tian Xiang. Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3071-3085. doi: 10.3934/dcdsb.2017197 |
[3] |
Runlin Hu, Pan Zheng, Zhangqin Gao. Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022018 |
[4] |
Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147 |
[5] |
Xiangdong Zhao. Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5095-5100. doi: 10.3934/dcdsb.2020334 |
[6] |
Chun Huang. Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop. Electronic Research Archive, 2021, 29 (5) : 3261-3279. doi: 10.3934/era.2021037 |
[7] |
Yilong Wang, Zhaoyin Xiang. Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1953-1973. doi: 10.3934/dcdsb.2016031 |
[8] |
Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic and Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034 |
[9] |
Miaoqing Tian, Shujuan Wang, Xia Xiao. Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022071 |
[10] |
Runlin Hu, Pan Zheng. On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022041 |
[11] |
Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 |
[12] |
Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 191-221. doi: 10.3934/dcdsb.2019178 |
[13] |
Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503 |
[14] |
Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078 |
[15] |
Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 |
[16] |
Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216 |
[17] |
Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 |
[18] |
Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170 |
[19] |
Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231 |
[20] |
Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]