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March  2022, 21(3): 927-942. doi: 10.3934/cpaa.2022004

Global generalized solutions of a haptotaxis model describing cancer cells invasion and metastatic spread

School of Mathematics, Southeast University, Nanjing 211189, China

* Corresponding author

Received  May 2021 Revised  October 2021 Published  March 2022 Early access  December 2021

Fund Project: The authors are supported in part by the National Natural Science Foundation of China (No. 11671079, No. 11701290, No. 11601127 and No. 11171063), the Natural Science Foundation of Jiangsu Province (No. BK20170896), the Postgraduate Research and Practice Innovation Program of Jiangsu Province(No.KYCX21_0077)

In this paper, we consider the following haptotaxis model describing cancer cells invasion and metastatic spread
$\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \chi \nabla \cdot (u\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{v_t} = {d_v}\Delta v - \xi \nabla \cdot (v\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{m_t} = {d_m}\Delta m + u - m,}&{x \in \Omega ,\;t > 0,}\\{{w_t} = - \left( {{\gamma _1}u + m} \right)w,}&{x \in \Omega ,\;t > 0,}\end{array}} \right.}&{(0.1)}\end{array}$
where
$ \Omega\subset \mathbb{R}^3 $
is a bounded domain with smooth boundary and the parameters
$ \chi, \xi, d_{v}, d_{m},\gamma_{1}>0 $
. Under homogeneous boundary conditions of Neumann type for
$ u $
,
$ v $
,
$ m $
and
$ w $
, it is proved that, for suitable smooth initial data
$ (u_0, v_0, m_0, w_0) $
, the corresponding Neumann initial-boundary value problem possesses a global generalized solution.
Citation: Meng Liu, Yuxiang Li. Global generalized solutions of a haptotaxis model describing cancer cells invasion and metastatic spread. Communications on Pure and Applied Analysis, 2022, 21 (3) : 927-942. doi: 10.3934/cpaa.2022004
References:
[1]

X. Cao, Boundedness in a three-dimensional chemotaxi–Chaptotaxis model, Z. Angew. Math. Phys., 67 (2016), 13 pp. doi: 10.1007/s00033-015-0601-3.

[2]

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.

[3]

L. C. FranssenT. LorenziA. E. F. Burgess and M. A. J. Chaplain, A mathematical framework for modelling the metastatic spread of cancer, Bull. Math. Biol., 81 (2019), 1965-2010.  doi: 10.1007/s11538-019-00597-x.

[4]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[5]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[6]

C. Jin, Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread, J. Differ. Equ., 269 (2020), 3987-4021.  doi: 10.1016/j.jde.2020.03.018.

[7]

H. Y. Jin and T. Xiang, Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model, Math. Models Methods Appl. Sci., 31 (2021), 1373-1417.  doi: 10.1142/S0218202521500287.

[8]

J. Lankeit and M. Winkler, Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary, preprint, arXiv: 2103.07232, 2021.

[9]

G. Litcanu and C. Morales-Rodrig, Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.  doi: 10.1142/S0218202510004775.

[10]

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.

[11]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851–875. doi: 10.1016/j.anihpc.2013.07.007.

[12]

C. Morales-Rodrigo, Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Model., 47 (2008), 604-613.  doi: 10.1016/j.mcm.2007.02.031.

[13]

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.

[14]

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.

[15]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041.

[16]

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.

[17]

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.

[18]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[19]

C. Walker and G.F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Ana., 38 (2006/07), 1694-1713.  doi: 10.1137/060655122.

[20]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[21]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

show all references

References:
[1]

X. Cao, Boundedness in a three-dimensional chemotaxi–Chaptotaxis model, Z. Angew. Math. Phys., 67 (2016), 13 pp. doi: 10.1007/s00033-015-0601-3.

[2]

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.

[3]

L. C. FranssenT. LorenziA. E. F. Burgess and M. A. J. Chaplain, A mathematical framework for modelling the metastatic spread of cancer, Bull. Math. Biol., 81 (2019), 1965-2010.  doi: 10.1007/s11538-019-00597-x.

[4]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[5]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[6]

C. Jin, Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread, J. Differ. Equ., 269 (2020), 3987-4021.  doi: 10.1016/j.jde.2020.03.018.

[7]

H. Y. Jin and T. Xiang, Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model, Math. Models Methods Appl. Sci., 31 (2021), 1373-1417.  doi: 10.1142/S0218202521500287.

[8]

J. Lankeit and M. Winkler, Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary, preprint, arXiv: 2103.07232, 2021.

[9]

G. Litcanu and C. Morales-Rodrig, Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.  doi: 10.1142/S0218202510004775.

[10]

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.

[11]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851–875. doi: 10.1016/j.anihpc.2013.07.007.

[12]

C. Morales-Rodrigo, Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Model., 47 (2008), 604-613.  doi: 10.1016/j.mcm.2007.02.031.

[13]

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.

[14]

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.

[15]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.  doi: 10.1016/j.jmaa.2011.02.041.

[16]

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.

[17]

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.

[18]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[19]

C. Walker and G.F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Ana., 38 (2006/07), 1694-1713.  doi: 10.1137/060655122.

[20]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[21]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

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