American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2022004
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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 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 & Applied Analysis, 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.  Google Scholar [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.  Google Scholar [3] L. C. Franssen, T. Lorenzi, A. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] 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 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] L. C. Franssen, T. Lorenzi, A. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] 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 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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