-
Previous Article
Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation
- DCDS-B Home
- This Issue
-
Next Article
Intermittent dispersal population model with almost period parameters and dispersal delays
Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion
1. | Department of Applied Mathematics Chongqing University of Posts, and Telecommunications, Chongqing 400065 |
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] |
S. Aznavoorian, M. L. Stracke, H. Krutzsch, E. Schiffmann and L. A. Liotta, Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells, J. Cell Biol., 110 (1990), 1427-1438.
doi: 10.1083/jcb.110.4.1427. |
[3] |
D. Besser, P. Verde, Y. Nagamine and F. Blasi, Signal transduction and u-PA/u-PAR system, Fibrinolysis, 10 (1996), 215-237.
doi: 10.1016/S0268-9499(96)80018-X. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), p67, arXiv:1501.05383.
doi: 10.1007/s00033-015-0601-3. |
[5] |
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. |
[6] |
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[7] |
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[8] |
T. Cieślak 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. |
[9] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[10] |
K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71.
doi: 10.1016/j.na.2014.06.017. |
[11] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[12] |
T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[13] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103-165. |
[14] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69. |
[15] |
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. |
[16] |
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: 10.1090/S0002-9947-1992-1046835-6. |
[17] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[18] |
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. |
[19] |
Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), arXiv:1508.05846v1.
doi: 10.1088/0951-7715/29/5/1564. |
[20] |
L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737. |
[21] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[22] |
B. Perthame, Transport Equations in Biology, Birkhäser-BaselVerlag, Switzerland, 2007. |
[23] |
Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.
doi: 10.1016/j.jmaa.2008.12.039. |
[24] |
Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv:1407.7382v1, 2014. |
[25] |
Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.
doi: 10.1088/0951-7715/21/10/002. |
[26] |
Y. Tao and M. Wang, A combined chemotaxis-haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558.
doi: 10.1137/090751542. |
[27] |
Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 1 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[28] |
Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[29] |
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. |
[30] |
Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239.
doi: 10.1088/0951-7715/27/6/1225. |
[31] |
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[32] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol.2, North-Holland, Amsterdam, 1977. |
[33] |
L. Wang, C. 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. |
[34] |
Y. F. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[35] |
Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.
doi: 10.1137/110853972. |
[36] |
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. |
[37] |
M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[38] |
M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[39] |
J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
[40] |
P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.
doi: 10.1016/j.jmaa.2014.11.031. |
[41] |
P. Zheng, C. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1737-1757.
doi: 10.3934/dcds.2016.36.1737. |
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] |
S. Aznavoorian, M. L. Stracke, H. Krutzsch, E. Schiffmann and L. A. Liotta, Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells, J. Cell Biol., 110 (1990), 1427-1438.
doi: 10.1083/jcb.110.4.1427. |
[3] |
D. Besser, P. Verde, Y. Nagamine and F. Blasi, Signal transduction and u-PA/u-PAR system, Fibrinolysis, 10 (1996), 215-237.
doi: 10.1016/S0268-9499(96)80018-X. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), p67, arXiv:1501.05383.
doi: 10.1007/s00033-015-0601-3. |
[5] |
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. |
[6] |
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[7] |
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[8] |
T. Cieślak 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. |
[9] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[10] |
K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71.
doi: 10.1016/j.na.2014.06.017. |
[11] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[12] |
T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[13] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103-165. |
[14] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69. |
[15] |
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. |
[16] |
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: 10.1090/S0002-9947-1992-1046835-6. |
[17] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[18] |
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. |
[19] |
Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), arXiv:1508.05846v1.
doi: 10.1088/0951-7715/29/5/1564. |
[20] |
L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737. |
[21] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[22] |
B. Perthame, Transport Equations in Biology, Birkhäser-BaselVerlag, Switzerland, 2007. |
[23] |
Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.
doi: 10.1016/j.jmaa.2008.12.039. |
[24] |
Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv:1407.7382v1, 2014. |
[25] |
Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.
doi: 10.1088/0951-7715/21/10/002. |
[26] |
Y. Tao and M. Wang, A combined chemotaxis-haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558.
doi: 10.1137/090751542. |
[27] |
Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 1 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[28] |
Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[29] |
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. |
[30] |
Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239.
doi: 10.1088/0951-7715/27/6/1225. |
[31] |
J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[32] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol.2, North-Holland, Amsterdam, 1977. |
[33] |
L. Wang, C. 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. |
[34] |
Y. F. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[35] |
Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.
doi: 10.1137/110853972. |
[36] |
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. |
[37] |
M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[38] |
M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[39] |
J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
[40] |
P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.
doi: 10.1016/j.jmaa.2014.11.031. |
[41] |
P. Zheng, C. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1737-1757.
doi: 10.3934/dcds.2016.36.1737. |
[1] |
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 |
[2] |
Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069 |
[3] |
Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026 |
[4] |
Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324 |
[5] |
Youshan Tao, Michael Winkler. A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2047-2067. doi: 10.3934/cpaa.2019092 |
[6] |
Changchun Liu, Pingping Li. Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1399-1419. doi: 10.3934/cpaa.2020070 |
[7] |
Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3825-3849. doi: 10.3934/cpaa.2021133 |
[8] |
Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258 |
[9] |
Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328 |
[10] |
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 |
[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] |
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 |
[13] |
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 |
[14] |
Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064 |
[15] |
Fuchen Zhang, Xiaofeng Liao, Chunlai Mu, Guangyun Zhang, Yi-An Chen. On global boundedness of the Chen system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1673-1681. doi: 10.3934/dcdsb.2017080 |
[16] |
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 |
[17] |
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 |
[18] |
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 |
[19] |
Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064 |
[20] |
Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 321-328. doi: 10.3934/dcdss.2020018 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]