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Global boundedness for a $ \mathit{\boldsymbol{N}} $-dimensional two species cancer invasion haptotaxis model with tissue remodeling
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China |
2. | Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China |
$ \begin{equation} \begin{cases} c_{1t} = \Delta c_1-\chi_1\nabla\cdot(c_1\nabla v)-\mu_{\rm EMT}c_1+\mu_1c_1(r_1-c_1^\kappa-c_2-v),\\ c_{2t} = \Delta c_2-\chi_2\nabla\cdot(c_2\nabla v)+\mu_{\rm EMT}c_1+\mu_2c_2(r_2-c_1-c_2^\kappa-v),\\ \tau m_t = \Delta m+c_1+c_2-m,\\ v_t = -mv+\eta v(1-c_1-c_2-v) \end{cases}\nonumber \end{equation} $ |
$ \Omega\subset\mathbb{R}^N\;(N\geq1) $ |
$ c_1,c_2 $ |
$ m $ |
$ \chi_i,\mu_i,r_i>0\;(i = 1,2) $ |
$ \eta>0 $ |
$ \kappa\geq1 $ |
$ \tau\in\{0,1\} $ |
$ \mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty) $ |
$ \mu_{\rm EMT}\leq\mu_M $ |
$ \mu_M>0 $ |
$ \kappa = 1 $ |
$ N = 3 $ |
$ c_1 $ |
$ c_2 $ |
$ L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega) $ |
$ p>2 $ |
$ r_i $ |
$ \mu_M $ |
$ \kappa>1 $ |
$ N\geq1 $ |
$ c_1 $ |
$ c_2 $ |
$ L^1(\Omega) $ |
$ L^p(\Omega) $ |
$ p>1 $ |
$ L^\infty(\Omega) $ |
$ r_i $ |
$ \mu_M $ |
$ \kappa = 1 $ |
$ \kappa>1 $ |
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] |
A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson,
Mathematical modelling of tumour invasion and metastasis,, J. Theor. Med., 2 (2000), 129-154.
doi: 10.1080/10273660008833042. |
[3] |
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. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), Art. 11, 13 pp.
doi: 10.1007/s00033-015-0601-3. |
[5] |
M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of tissue invasion, in Cancer Modelling and Simulation, Chapman & Hall/CRC Math. Biol. Med. Ser., Chapman & Hall/CRC, Boca Raton, FL, (2003), 269–297. |
[6] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[7] |
Z. Chen and Y. Tao,
Large-data solutions in a three-dimensional chemotaxis-haptotaxis System with remodeling of non-diffusible attractant: The role of sub-linear production of diffusible signal, Acta Appl. Math., 163 (2019), 129-143.
doi: 10.1007/s10440-018-0216-8. |
[8] |
F. Dai and B. Liu,
Optimal control and pattern formation for a haptotaxis model of solid tumor invasion, J. Franklin Inst., 356 (2019), 9364-9406.
doi: 10.1016/j.jfranklin.2019.08.039. |
[9] |
F. Dai and B. Liu, Global boundedness of classical solutions to a two species cancer invasion haptotaxis model with tissue remodeling, J. Math. Anal. Appl., 483 (2020), 123583, 33pp.
doi: 10.1016/j.jmaa.2019.123583. |
[10] |
F. Dai and B. Liu,
Asymptotic stability in a quasilinear chemotaxis-haptotaxis model with general logistic source and nonlinear signal production, J. Differential Equations, 269 (2020), 10839-10918.
doi: 10.1016/j.jde.2020.07.027. |
[11] |
F. Dai and B. Liu, Global solvability and optimal control to a haptotaxis cancer invasion model with two cancer cell species, Appl. Math. Optim., 2020.
doi: 10.1007/s00245-020-09712-0. |
[12] |
J. Giesselmann, N. Kolbe, M. Lukáčová-Medvid'ová and N. Sfakianakis,
Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model, Discrete contin. Dyn. Syst. Ser. B, 23 (2018), 4397-4431.
doi: 10.3934/dcdsb.2018169. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[14] |
D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. |
[15] |
N. Hellmann, N. Kolbe and N. Sfakianakis,
A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 397-412.
doi: 10.1007/s00574-016-0147-9. |
[16] |
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. |
[17] |
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. |
[18] |
C. Jin,
Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source, Discrete Contin. Dyn. Syst., 38 (2018), 3547-3566.
doi: 10.3934/dcds.2018150. |
[19] |
C. Jin,
Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bull. Lond. Math. Soc., 50 (2018), 598-618.
doi: 10.1112/blms.12160. |
[20] |
Y. Ke and J. Zheng,
A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, Nonlinearity, 31 (2018), 4602-4620.
doi: 10.1088/1361-6544/aad307. |
[21] |
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. |
[22] |
O. A. Ladyžzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, , Amer. Math. Soc. Transl., Vol. 23, Providence, RI, 1968. |
[23] |
G. Liţcanu and C. Morales-Rodrigo,
Asymptotic behaviour of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.
doi: 10.1142/S0218202510004775. |
[24] |
S. A. Mani, W. Guo and M. J. Liao,
The epithelial-mesenchymal transition generates cells with properties of stem cells, Cell, 133 (2008), 704-715.
doi: 10.1016/j.cell.2008.03.027. |
[25] |
A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
doi: 10.1142/S0218202510004301. |
[26] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[27] |
P. Y. H. Pang and Y. Wang,
Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 263 (2017), 1269-1292.
doi: 10.1016/j.jde.2017.03.016. |
[28] |
P. Y. H. Pang and Y. Wang,
Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci., 28 (2018), 2211-2235.
doi: 10.1142/S0218202518400134. |
[29] |
N. Sfakianakis, N. Kolbe, N. Hellmann and M. Lukáčová-Medvid'ová,
A multiscale approach to the migration of cancer stem cells: mathematical modelling and simulations, Bull. Math. Biol., 79 (2017), 209-235.
doi: 10.1007/s11538-016-0233-6. |
[30] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[31] |
Y. Tao,
Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl., 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[32] |
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. |
[33] |
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. |
[34] |
Y. Tao and M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1067-1084.
doi: 10.1017/S0308210512000571. |
[35] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[36] |
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. |
[37] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[38] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047-2067.
doi: 10.3934/cpaa.2019092. |
[39] |
Y. Tao and G. Zhu,
Global solution to a model of tumor invasion, Appl. Math. Sci., 1 (2007), 2385-2398.
|
[40] |
C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[41] |
Y. 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. |
[42] |
M. Winker,
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. |
[43] |
S. Wu, J. Wang and J. Shi,
Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.
doi: 10.1142/S0218202518400158. |
[44] |
J. Zheng and Y. Wang,
Boundedness of solutions to a quasilinear chemotaxis-haptotaxis model, Comput. Math. Appl., 71 (2016), 1898-1909.
doi: 10.1016/j.camwa.2016.03.014. |
[45] |
J. Zheng,
Boundedness of solution of a higher-dimensional parabolic-ODE-parabolic chemotaxis-haptotaxis model with generalized logistic source, Nonlinearity, 30 (2017), 1987-2009.
doi: 10.1088/1361-6544/aa675e. |
[46] |
J. Zheng and Y. Ke,
Large time behavior of solutions to a fully parabolic chemotaxis-haptotaxis model in $N$ dimensions, J. Differential Equations, 266 (2019), 1969-2018.
doi: 10.1016/j.jde.2018.08.018. |
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] |
A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson,
Mathematical modelling of tumour invasion and metastasis,, J. Theor. Med., 2 (2000), 129-154.
doi: 10.1080/10273660008833042. |
[3] |
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. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), Art. 11, 13 pp.
doi: 10.1007/s00033-015-0601-3. |
[5] |
M. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of tissue invasion, in Cancer Modelling and Simulation, Chapman & Hall/CRC Math. Biol. Med. Ser., Chapman & Hall/CRC, Boca Raton, FL, (2003), 269–297. |
[6] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[7] |
Z. Chen and Y. Tao,
Large-data solutions in a three-dimensional chemotaxis-haptotaxis System with remodeling of non-diffusible attractant: The role of sub-linear production of diffusible signal, Acta Appl. Math., 163 (2019), 129-143.
doi: 10.1007/s10440-018-0216-8. |
[8] |
F. Dai and B. Liu,
Optimal control and pattern formation for a haptotaxis model of solid tumor invasion, J. Franklin Inst., 356 (2019), 9364-9406.
doi: 10.1016/j.jfranklin.2019.08.039. |
[9] |
F. Dai and B. Liu, Global boundedness of classical solutions to a two species cancer invasion haptotaxis model with tissue remodeling, J. Math. Anal. Appl., 483 (2020), 123583, 33pp.
doi: 10.1016/j.jmaa.2019.123583. |
[10] |
F. Dai and B. Liu,
Asymptotic stability in a quasilinear chemotaxis-haptotaxis model with general logistic source and nonlinear signal production, J. Differential Equations, 269 (2020), 10839-10918.
doi: 10.1016/j.jde.2020.07.027. |
[11] |
F. Dai and B. Liu, Global solvability and optimal control to a haptotaxis cancer invasion model with two cancer cell species, Appl. Math. Optim., 2020.
doi: 10.1007/s00245-020-09712-0. |
[12] |
J. Giesselmann, N. Kolbe, M. Lukáčová-Medvid'ová and N. Sfakianakis,
Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model, Discrete contin. Dyn. Syst. Ser. B, 23 (2018), 4397-4431.
doi: 10.3934/dcdsb.2018169. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[14] |
D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. |
[15] |
N. Hellmann, N. Kolbe and N. Sfakianakis,
A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix, Bull. Braz. Math. Soc. (N.S.), 47 (2016), 397-412.
doi: 10.1007/s00574-016-0147-9. |
[16] |
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. |
[17] |
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. |
[18] |
C. Jin,
Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source, Discrete Contin. Dyn. Syst., 38 (2018), 3547-3566.
doi: 10.3934/dcds.2018150. |
[19] |
C. Jin,
Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bull. Lond. Math. Soc., 50 (2018), 598-618.
doi: 10.1112/blms.12160. |
[20] |
Y. Ke and J. Zheng,
A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, Nonlinearity, 31 (2018), 4602-4620.
doi: 10.1088/1361-6544/aad307. |
[21] |
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. |
[22] |
O. A. Ladyžzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, , Amer. Math. Soc. Transl., Vol. 23, Providence, RI, 1968. |
[23] |
G. Liţcanu and C. Morales-Rodrigo,
Asymptotic behaviour of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.
doi: 10.1142/S0218202510004775. |
[24] |
S. A. Mani, W. Guo and M. J. Liao,
The epithelial-mesenchymal transition generates cells with properties of stem cells, Cell, 133 (2008), 704-715.
doi: 10.1016/j.cell.2008.03.027. |
[25] |
A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
doi: 10.1142/S0218202510004301. |
[26] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[27] |
P. Y. H. Pang and Y. Wang,
Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 263 (2017), 1269-1292.
doi: 10.1016/j.jde.2017.03.016. |
[28] |
P. Y. H. Pang and Y. Wang,
Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci., 28 (2018), 2211-2235.
doi: 10.1142/S0218202518400134. |
[29] |
N. Sfakianakis, N. Kolbe, N. Hellmann and M. Lukáčová-Medvid'ová,
A multiscale approach to the migration of cancer stem cells: mathematical modelling and simulations, Bull. Math. Biol., 79 (2017), 209-235.
doi: 10.1007/s11538-016-0233-6. |
[30] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[31] |
Y. Tao,
Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl., 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[32] |
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. |
[33] |
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. |
[34] |
Y. Tao and M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1067-1084.
doi: 10.1017/S0308210512000571. |
[35] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[36] |
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. |
[37] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[38] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047-2067.
doi: 10.3934/cpaa.2019092. |
[39] |
Y. Tao and G. Zhu,
Global solution to a model of tumor invasion, Appl. Math. Sci., 1 (2007), 2385-2398.
|
[40] |
C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[41] |
Y. 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. |
[42] |
M. Winker,
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. |
[43] |
S. Wu, J. Wang and J. Shi,
Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci., 28 (2018), 2275-2312.
doi: 10.1142/S0218202518400158. |
[44] |
J. Zheng and Y. Wang,
Boundedness of solutions to a quasilinear chemotaxis-haptotaxis model, Comput. Math. Appl., 71 (2016), 1898-1909.
doi: 10.1016/j.camwa.2016.03.014. |
[45] |
J. Zheng,
Boundedness of solution of a higher-dimensional parabolic-ODE-parabolic chemotaxis-haptotaxis model with generalized logistic source, Nonlinearity, 30 (2017), 1987-2009.
doi: 10.1088/1361-6544/aa675e. |
[46] |
J. Zheng and Y. Ke,
Large time behavior of solutions to a fully parabolic chemotaxis-haptotaxis model in $N$ dimensions, J. Differential Equations, 266 (2019), 1969-2018.
doi: 10.1016/j.jde.2018.08.018. |
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