doi: 10.3934/dcdsb.2021044

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

* Corresponding author: Bin Liu

Received  September 2020 Revised  December 2020 Published  February 2021

Fund Project: This work is supported by National Natural Science Foundation of China grant 11971185

This paper is concerned with the two species cancer invasion haptotaxis model with tissue remodeling
$ \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} $
in a bounded and smooth domain
$ \Omega\subset\mathbb{R}^N\;(N\geq1) $
with zero-flux boundary conditions for
$ c_1,c_2 $
and
$ m $
, where
$ \chi_i,\mu_i,r_i>0\;(i = 1,2) $
,
$ \eta>0 $
,
$ \kappa\geq1 $
,
$ \tau\in\{0,1\} $
, and
$ \mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty) $
is the epithelial-mesenchymal transition rate function such that
$ \mu_{\rm EMT}\leq\mu_M $
with some constant
$ \mu_M>0 $
. When
$ \kappa = 1 $
and
$ N = 3 $
, by rasing the coupled a priori estimates of
$ c_1 $
and
$ c_2 $
in the following way
$ L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega) $
with any
$ p>2 $
, it is shown that for some appropriately regular and small initial data, the associated initial-boundary value problem possesses a unique globally bounded classical solution for suitably small
$ r_i $
and
$ \mu_M $
. When
$ \kappa>1 $
and
$ N\geq1 $
, by rasing the coupled a priori estimates of
$ c_1 $
and
$ c_2 $
from
$ L^1(\Omega) $
to
$ L^p(\Omega) $
with any
$ p>1 $
, then to
$ L^\infty(\Omega) $
, it is proved that for any reasonably regular initial data, the corresponding initial-boundary value problem admits a unique globally bounded classical solution for arbitrary
$ r_i $
and
$ \mu_M $
. The result for
$ \kappa = 1 $
complements previously known one, and the result for
$ \kappa>1 $
is new.
Citation: Feng Dai, Bin Liu. Global boundedness for a $ \mathit{\boldsymbol{N}} $-dimensional two species cancer invasion haptotaxis model with tissue remodeling. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021044
References:
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N. BellomoA. BellouquidY. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. GiesselmannN. KolbeM. 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.  Google Scholar

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D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.  Google Scholar

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N. HellmannN. 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.  Google Scholar

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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.  Google Scholar

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S. IshidaK. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. A. ManiW. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[29]

N. SfakianakisN. KolbeN. 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.  Google Scholar

[30]

C. StinnerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

Y. Tao and G. Zhu, Global solution to a model of tumor invasion, Appl. Math. Sci., 1 (2007), 2385-2398.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

S. WuJ. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. R. A. AndersonM. A. J. ChaplainE. L. NewmanR. 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.  Google Scholar

[3]

N. BellomoA. BellouquidY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. GiesselmannN. KolbeM. 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.  Google Scholar

[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.  Google Scholar

[14]

D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.  Google Scholar

[15]

N. HellmannN. 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.  Google Scholar

[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.  Google Scholar

[17]

S. IshidaK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. A. ManiW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

N. SfakianakisN. KolbeN. 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.  Google Scholar

[30]

C. StinnerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

Y. Tao and G. Zhu, Global solution to a model of tumor invasion, Appl. Math. Sci., 1 (2007), 2385-2398.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

S. WuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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