# American Institute of Mathematical Sciences

## 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
 $$$\begin{cases} c_{1t} = \Delta c_1-\chi_1\nabla\cdot(c_1\nabla v)-\mu_{\rm EMT}c_1+\mu_1c_1(r_1-c_1^\kappa-c_2-v),\\ c_{2t} = \Delta c_2-\chi_2\nabla\cdot(c_2\nabla v)+\mu_{\rm EMT}c_1+\mu_2c_2(r_2-c_1-c_2^\kappa-v),\\ \tau m_t = \Delta m+c_1+c_2-m,\\ v_t = -mv+\eta v(1-c_1-c_2-v) \end{cases}\nonumber$$$
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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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. 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.  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. 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.  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. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. 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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. 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.  Google Scholar [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.  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. 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.  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. 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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. 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.  Google Scholar [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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar [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.  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. 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.  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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