
-
Previous Article
Ebola: Impact of hospital's admission policy in an overwhelmed scenario
- MBE Home
- This Issue
-
Next Article
Modeling the control of infectious diseases: Effects of TV and social media advertisements
Early and late stage profiles for a chemotaxis model with density-dependent jump probability
1. | School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China |
2. | School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China |
3. | Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada |
4. | Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada |
In this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, this model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain, we first estimate the expanding speed of tumour region as $O(t^{β})$ for $ 0 < β < \frac{1}{2}$. Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c>0$ is also obtained.
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and 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.
doi: 10.1142/S021820251550044X. |
[2] |
A. A. Blumberg, Logistic growth rate functions, J. Theor. Biol., 21 (1968), 42-44. Google Scholar |
[3] |
H. M. Byrne, Dissecting cancer through mathematics: from the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221-230. Google Scholar |
[4] |
M. A. Chaplain and A. R. Anderson, Mathematical modelling of tissue invasion, in Cancer Model. Simul., Chapman and Hall/CRC, London, (2003), 269-297. |
[5] |
Y. H. Du, F. Quirós and M. L. Zhou, Logarithmic corrections in Fisher-KPP type Porous Medium Equations, preprint, arXiv: 1806.02022. Google Scholar |
[6] |
P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanisms, Nature Reviews Cancer, 3 (2003), 362-374. Google Scholar |
[7] |
K. Fujie, A. Ito, M. Winkler and T. Yokota,
Stabilization in a chemotaxis model for tumor invasion, Disc. Cont. Dyn. Syst., 36 (2016), 151-169.
doi: 10.3934/dcds.2016.36.151. |
[8] |
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. Google Scholar |
[9] |
W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous population, J. Theors. Biol., 52 (1975), 441-457. Google Scholar |
[10] |
M. E. Gurtin and R. C. MacCamy,
On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.
doi: 10.1016/0025-5564(77)90062-1. |
[11] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[12] |
T. Hillen and K. J. Painter,
A user's guide to pde models in a chemotaxis, J. Math. Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[13] |
C. H. Jin,
Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.
doi: 10.1016/j.jde.2017.06.034. |
[14] |
C. H. Jin,
Boundedness and global solvability to a chemotaxis-haptotaixs model with slow and fast diffusion, Disc. Cont. Dyn. Syst., 23 (2018), 1675-1688.
doi: 10.3934/dcdsb.2018069. |
[15] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biology, 30 (1971), 225-234. Google Scholar |
[16] |
R. D. Leek, The role of tumour associated macrophages in breast cancer angiogenesis, Ph.D thesis, Oxford Brookes University, Oxford, 1999. Google Scholar |
[17] |
D. Li, C. Mu and P. Zheng,
Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion, Math. Models Methods Appl. Sci., 28 (2018), 1413-1451.
doi: 10.1142/S0218202518500380. |
[18] |
J. Liu and Y. F. Wang,
A quasilinear chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, Math. Methods Appl. Sci., 40 (2017), 2107-2121.
doi: 10.1002/mma.4126. |
[19] |
Y. Lou and M. Winkler,
Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Part. Diff. Equ., 40 (2015), 1905-1941.
doi: 10.1080/03605302.2015.1052882. |
[20] |
P. Lu, V. Weaver and Z. Werb, The extracellular matrix: a dynamic niche in cancer progression, J. Cell Biol., 196 (2012), 395-406. Google Scholar |
[21] |
H. McAneney and S. F. C. O'Rourke, Investigation of various growth mechanisms of solid tumour growth within the linear quadratic model for radiotherapy, Phys. Med. Biol., 52 (2007), 1039-1054. Google Scholar |
[22] |
M. Mei, H. Y. Peng and Z. A. Wang,
Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 5168-5191.
doi: 10.1016/j.jde.2015.06.022. |
[23] |
Y. Mimura,
The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion, J. Differential Equations, 263 (2017), 1477-1521.
doi: 10.1016/j.jde.2017.03.020. |
[24] |
J. D. Murry,
Mathematical Biology I: An Introduction, Springer, New York, USA, 2002. |
[25] |
A. Okubo and S. A. Levin,
Diffusion and Ecological Problems: Modern Perspectives, Springer Science Business Media, 2013.
doi: 10.1007/978-1-4757-4978-6. |
[26] |
M. E. Orme and M. A. J. Chaplain, A mathematical model of vascular tumour growth and invasion, Math. Comput. Modelling, 23 (1996), 43-60. Google Scholar |
[27] |
M. R. Owen, H. M. Byrne and C. E. Lewis,
Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites, J. Theor. Biol., 226 (2004), 377-391.
doi: 10.1016/j.jtbi.2003.09.004. |
[28] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quart., 10 (2002), 501-543.
|
[29] |
K. J. Painter and J. A. Sherratt,
Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[30] |
B. G. Sengers, C. P. Please and R. O. C. Oreffo, Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration, J. R. Soc. Interface, 4 (2007), 1107-1117. Google Scholar |
[31] |
J. A. Sherratt,
On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion, Math. Model. Nat. Phenom, 5 (2010), 64-79.
doi: 10.1051/mmnp/20105505. |
[32] |
J. A. Sherratt and M. A. J. Chaplain,
A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.
doi: 10.1007/s002850100088. |
[33] |
J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. London B, 241 (1990), 29-35. Google Scholar |
[34] |
M. J. Simpson, R. E. Baker and S. W. McCue, Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models, Phys. Rev. E, 83 (2011), 021901. Google Scholar |
[35] |
M. J. Simpson, D. C. Zhang, M. Mariani, K. A. Landman and D. F. Newgreen, Cell proliferation drives neural crest cell invasion of the intestine, Dev. Biol., 302 (2007), 553-568. Google Scholar |
[36] |
A. Stevens and H. G. Othmer,
Aggregation, blowup, and collapse: The abc's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (2001), 1044-1081.
doi: 10.1137/S0036139995288976. |
[37] |
Z. Szymańska, C. M. Rodrigo, M. Lachowicz and M. A. Chaplain,
Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.
doi: 10.1142/S0218202509003425. |
[38] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-705.
doi: 10.1137/100802943. |
[39] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[40] |
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. |
[41] |
M. E. Turner, B. A. Blumenstein and J. L. Sebaugh, A generalization of the logistic law of growth, Biometrics, 25 (1969), 577-580. Google Scholar |
[42] |
H. A. S. Van den Brenk, Studies in restorative growth processes in mammalian wound healing, Br. J. Surg., 43 (1956), 525-550. Google Scholar |
[43] |
J. L. Vàzquez,
The Porous Medium Equation: Mathematical Theory, Oxford Univ. Press, Oxford, 2007. |
[44] |
L. Von Bertalanffy, Quantitative laws in metabolism and growth, Q. Rev. Biol., 32 (1957), 217-231. Google Scholar |
[45] |
Y. F. Wang,
Boundedness in a multi-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Appl. Math. Lett., 59 (2016), 122-126.
doi: 10.1016/j.aml.2016.03.019. |
[46] |
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. |
[47] |
Y. F. Wang and Y. Y. Ke,
Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher demensions, J. Differential Equations, 260 (2016), 6960-6988.
doi: 10.1016/j.jde.2016.01.017. |
[48] |
Z. A. Wang, M. Winkler and D. Wrzosek,
Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297.
doi: 10.1088/0951-7715/24/12/001. |
[49] |
Z. A. Wang, Z. Y. Xiang and P. Yu,
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.
doi: 10.1016/j.jde.2015.09.063. |
[50] |
Z. A. Wang, M. Winkler and D. Wrzosek,
Global regularity vs. infinite-time singularity formation in chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.
doi: 10.1137/110853972. |
[51] |
M. Winkler and K. C. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Analysis, 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[52] |
M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151, arXiv: 1704.05648.
doi: 10.1016/j.jde.2018.01.027. |
[53] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[54] |
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. |
[55] |
T. P. Witelski,
Segregation and mixing in degenerate diffusion in population dynamics, J. Math. Biol., 35 (1997), 695-712.
doi: 10.1007/s002850050072. |
[56] |
Z. Wu, J. Zhao, J. Yin and H. Li,
Nonlinear Diffusion Equations, World Scientific Publishing Co. Pvt. Ltd., 2001.
doi: 10.1142/9789812799791. |
[57] |
T. Y. Xu, S. M. Ji, M. Mei and J. X. Yin, Global existence of solutions to a chemotaxis-haptotaxis model with density-dependent jump probability and quorum-sensing mechanisms, Math. Meth. Appl. Sci., 41 (2018), 4208-4226. Google Scholar |
[58] |
T. Y. Xu, S. M. Ji, M. Mei and J. X. Yin,
Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion, J. Differential Equations, 265 (2018), 4442-4485.
doi: 10.1016/j.jde.2018.06.008. |
[59] |
A. Zhigun, C. Surulescu and A. Uatay,
Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), 1-29.
doi: 10.1007/s00033-016-0741-0. |
show all references
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and 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.
doi: 10.1142/S021820251550044X. |
[2] |
A. A. Blumberg, Logistic growth rate functions, J. Theor. Biol., 21 (1968), 42-44. Google Scholar |
[3] |
H. M. Byrne, Dissecting cancer through mathematics: from the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221-230. Google Scholar |
[4] |
M. A. Chaplain and A. R. Anderson, Mathematical modelling of tissue invasion, in Cancer Model. Simul., Chapman and Hall/CRC, London, (2003), 269-297. |
[5] |
Y. H. Du, F. Quirós and M. L. Zhou, Logarithmic corrections in Fisher-KPP type Porous Medium Equations, preprint, arXiv: 1806.02022. Google Scholar |
[6] |
P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanisms, Nature Reviews Cancer, 3 (2003), 362-374. Google Scholar |
[7] |
K. Fujie, A. Ito, M. Winkler and T. Yokota,
Stabilization in a chemotaxis model for tumor invasion, Disc. Cont. Dyn. Syst., 36 (2016), 151-169.
doi: 10.3934/dcds.2016.36.151. |
[8] |
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. Google Scholar |
[9] |
W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous population, J. Theors. Biol., 52 (1975), 441-457. Google Scholar |
[10] |
M. E. Gurtin and R. C. MacCamy,
On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.
doi: 10.1016/0025-5564(77)90062-1. |
[11] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[12] |
T. Hillen and K. J. Painter,
A user's guide to pde models in a chemotaxis, J. Math. Biology, 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[13] |
C. H. Jin,
Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.
doi: 10.1016/j.jde.2017.06.034. |
[14] |
C. H. Jin,
Boundedness and global solvability to a chemotaxis-haptotaixs model with slow and fast diffusion, Disc. Cont. Dyn. Syst., 23 (2018), 1675-1688.
doi: 10.3934/dcdsb.2018069. |
[15] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biology, 30 (1971), 225-234. Google Scholar |
[16] |
R. D. Leek, The role of tumour associated macrophages in breast cancer angiogenesis, Ph.D thesis, Oxford Brookes University, Oxford, 1999. Google Scholar |
[17] |
D. Li, C. Mu and P. Zheng,
Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion, Math. Models Methods Appl. Sci., 28 (2018), 1413-1451.
doi: 10.1142/S0218202518500380. |
[18] |
J. Liu and Y. F. Wang,
A quasilinear chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, Math. Methods Appl. Sci., 40 (2017), 2107-2121.
doi: 10.1002/mma.4126. |
[19] |
Y. Lou and M. Winkler,
Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Part. Diff. Equ., 40 (2015), 1905-1941.
doi: 10.1080/03605302.2015.1052882. |
[20] |
P. Lu, V. Weaver and Z. Werb, The extracellular matrix: a dynamic niche in cancer progression, J. Cell Biol., 196 (2012), 395-406. Google Scholar |
[21] |
H. McAneney and S. F. C. O'Rourke, Investigation of various growth mechanisms of solid tumour growth within the linear quadratic model for radiotherapy, Phys. Med. Biol., 52 (2007), 1039-1054. Google Scholar |
[22] |
M. Mei, H. Y. Peng and Z. A. Wang,
Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 5168-5191.
doi: 10.1016/j.jde.2015.06.022. |
[23] |
Y. Mimura,
The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion, J. Differential Equations, 263 (2017), 1477-1521.
doi: 10.1016/j.jde.2017.03.020. |
[24] |
J. D. Murry,
Mathematical Biology I: An Introduction, Springer, New York, USA, 2002. |
[25] |
A. Okubo and S. A. Levin,
Diffusion and Ecological Problems: Modern Perspectives, Springer Science Business Media, 2013.
doi: 10.1007/978-1-4757-4978-6. |
[26] |
M. E. Orme and M. A. J. Chaplain, A mathematical model of vascular tumour growth and invasion, Math. Comput. Modelling, 23 (1996), 43-60. Google Scholar |
[27] |
M. R. Owen, H. M. Byrne and C. E. Lewis,
Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites, J. Theor. Biol., 226 (2004), 377-391.
doi: 10.1016/j.jtbi.2003.09.004. |
[28] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quart., 10 (2002), 501-543.
|
[29] |
K. J. Painter and J. A. Sherratt,
Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[30] |
B. G. Sengers, C. P. Please and R. O. C. Oreffo, Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration, J. R. Soc. Interface, 4 (2007), 1107-1117. Google Scholar |
[31] |
J. A. Sherratt,
On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion, Math. Model. Nat. Phenom, 5 (2010), 64-79.
doi: 10.1051/mmnp/20105505. |
[32] |
J. A. Sherratt and M. A. J. Chaplain,
A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.
doi: 10.1007/s002850100088. |
[33] |
J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. London B, 241 (1990), 29-35. Google Scholar |
[34] |
M. J. Simpson, R. E. Baker and S. W. McCue, Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models, Phys. Rev. E, 83 (2011), 021901. Google Scholar |
[35] |
M. J. Simpson, D. C. Zhang, M. Mariani, K. A. Landman and D. F. Newgreen, Cell proliferation drives neural crest cell invasion of the intestine, Dev. Biol., 302 (2007), 553-568. Google Scholar |
[36] |
A. Stevens and H. G. Othmer,
Aggregation, blowup, and collapse: The abc's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (2001), 1044-1081.
doi: 10.1137/S0036139995288976. |
[37] |
Z. Szymańska, C. M. Rodrigo, M. Lachowicz and M. A. Chaplain,
Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.
doi: 10.1142/S0218202509003425. |
[38] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-705.
doi: 10.1137/100802943. |
[39] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[40] |
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. |
[41] |
M. E. Turner, B. A. Blumenstein and J. L. Sebaugh, A generalization of the logistic law of growth, Biometrics, 25 (1969), 577-580. Google Scholar |
[42] |
H. A. S. Van den Brenk, Studies in restorative growth processes in mammalian wound healing, Br. J. Surg., 43 (1956), 525-550. Google Scholar |
[43] |
J. L. Vàzquez,
The Porous Medium Equation: Mathematical Theory, Oxford Univ. Press, Oxford, 2007. |
[44] |
L. Von Bertalanffy, Quantitative laws in metabolism and growth, Q. Rev. Biol., 32 (1957), 217-231. Google Scholar |
[45] |
Y. F. Wang,
Boundedness in a multi-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Appl. Math. Lett., 59 (2016), 122-126.
doi: 10.1016/j.aml.2016.03.019. |
[46] |
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. |
[47] |
Y. F. Wang and Y. Y. Ke,
Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher demensions, J. Differential Equations, 260 (2016), 6960-6988.
doi: 10.1016/j.jde.2016.01.017. |
[48] |
Z. A. Wang, M. Winkler and D. Wrzosek,
Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297.
doi: 10.1088/0951-7715/24/12/001. |
[49] |
Z. A. Wang, Z. Y. Xiang and P. Yu,
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.
doi: 10.1016/j.jde.2015.09.063. |
[50] |
Z. A. Wang, M. Winkler and D. Wrzosek,
Global regularity vs. infinite-time singularity formation in chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.
doi: 10.1137/110853972. |
[51] |
M. Winkler and K. C. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Analysis, 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[52] |
M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151, arXiv: 1704.05648.
doi: 10.1016/j.jde.2018.01.027. |
[53] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[54] |
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. |
[55] |
T. P. Witelski,
Segregation and mixing in degenerate diffusion in population dynamics, J. Math. Biol., 35 (1997), 695-712.
doi: 10.1007/s002850050072. |
[56] |
Z. Wu, J. Zhao, J. Yin and H. Li,
Nonlinear Diffusion Equations, World Scientific Publishing Co. Pvt. Ltd., 2001.
doi: 10.1142/9789812799791. |
[57] |
T. Y. Xu, S. M. Ji, M. Mei and J. X. Yin, Global existence of solutions to a chemotaxis-haptotaxis model with density-dependent jump probability and quorum-sensing mechanisms, Math. Meth. Appl. Sci., 41 (2018), 4208-4226. Google Scholar |
[58] |
T. Y. Xu, S. M. Ji, M. Mei and J. X. Yin,
Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion, J. Differential Equations, 265 (2018), 4442-4485.
doi: 10.1016/j.jde.2018.06.008. |
[59] |
A. Zhigun, C. Surulescu and A. Uatay,
Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), 1-29.
doi: 10.1007/s00033-016-0741-0. |

[1] |
Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021011 |
[2] |
Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017 |
[3] |
Jakub Kantner, Michal Beneš. Mathematical model of signal propagation in excitable media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 935-951. doi: 10.3934/dcdss.2020382 |
[4] |
Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020377 |
[5] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[6] |
Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103 |
[7] |
Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279 |
[8] |
Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 |
[9] |
Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561 |
[10] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020400 |
[11] |
Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265 |
[12] |
Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035 |
[13] |
Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020053 |
[14] |
Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020316 |
[15] |
Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160 |
[16] |
Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021009 |
[17] |
P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 |
[18] |
H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020433 |
[19] |
Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 |
[20] |
Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3657-3682. doi: 10.3934/dcds.2020051 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]