# American Institute of Mathematical Sciences

January  2016, 36(1): 151-169. doi: 10.3934/dcds.2016.36.151

## Stabilization in a chemotaxis model for tumor invasion

 1 Department of Mathematics, Tokyo University of Science, Tokyo 162-8601, Japan 2 Center for the Advancement of Higher Education, Faculty of Engineering, Kinki University, Takayaumenobe 1, Higashihiroshimashi, Hiroshima 739-2116 3 Institut für Mathematik, Universität Paderborn, 33098 Paderborn 4 Department of Mathematics, Science University of Tokyo, 26 Wakamiya-cho, Shinjuku-ku, Tokyo 162-8601

Received  August 2014 Revised  April 2015 Published  June 2015

This paper deals with the chemotaxis system $\begin{cases} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in \Omega, \ t>0, \\ v_t=\Delta v + wz, \qquad x\in \Omega, \ t>0, \\ w_t=-wz, \qquad x\in \Omega, \ t>0, \\ z_t=\Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{cases}$ in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$, that has recently been proposed as a model for tumor invasion in which the role of an active extracellular matrix is accounted for.
It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding initial-boundary value problem of Neumann type possesses a global solution which is bounded. Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain spatially homogeneous equilibrium in the sense that as $t\to\infty$,
$u(x,t)\to \overline{u_0}$ ,    $v(x,t) \to \overline{v_0} + \overline{w_0}$,    $w(x,t) \to 0$    and     $z(x,t) \to \overline{u_0}$,     uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$, $\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$    and    $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.
Citation: Kentarou Fujie, Akio Ito, Michael Winkler, Tomomi Yokota. Stabilization in a chemotaxis model for tumor invasion. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 151-169. doi: 10.3934/dcds.2016.36.151
##### References:
 [1] A. R. A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. BIOL. IMA J., 22 (2005), 163-186. doi: 10.1093/imammb/dqi005. [2] 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. [3] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439. doi: 10.3934/nhm.2006.1.399. [4] A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6. [5] K. Fujie, A. Ito and T. Yokota, Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type, Adv. Math. Sci. Appl., 24 (2014), 67-84. [6] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. [7] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. [8] T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480. [9] K. Kang, A. Stevens and J. J. L. Velázquez, Qualitative behavior of a Keller-Segel model with non-diffusive memory, Commun. Partial Differ. Equations, 35 (2010), 245-274. doi: 10.1080/03605300903473400. [10] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [11] O. A. Ladyzenskaja, 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. [12] G. Liţcanu and C. Morales-Rodrigo, Asymptotic behaviour of global solutions to a model of cell invasion, Math. Mod. Meth. Appl. Sci., 20 (2010), 1721-1758. doi: 10.1142/S0218202510004775. [13] 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. [14] C. Morales-Rodrigo, Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Modelling, 47 (2008), 604-613. doi: 10.1016/j.mcm.2007.02.031. [15] N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional Keller-Segel system, preprint. [16] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433. [17] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. [18] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007. [19] 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. [20] Z. Szymańska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425. [21] Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69. doi: 10.1016/j.jmaa.2008.12.039. [22] 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. [23] 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. [24] 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. [25] 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. [26] 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. [27] 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. [28] M. Winkler, 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. [29] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.

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##### References:
 [1] A. R. A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. BIOL. IMA J., 22 (2005), 163-186. doi: 10.1093/imammb/dqi005. [2] 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. [3] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439. doi: 10.3934/nhm.2006.1.399. [4] A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6. [5] K. Fujie, A. Ito and T. Yokota, Existence and uniqueness of local classical solutions to modified tumor invasion models of Chaplain-Anderson type, Adv. Math. Sci. Appl., 24 (2014), 67-84. [6] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. [7] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. [8] T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480. [9] K. Kang, A. Stevens and J. J. L. Velázquez, Qualitative behavior of a Keller-Segel model with non-diffusive memory, Commun. Partial Differ. Equations, 35 (2010), 245-274. doi: 10.1080/03605300903473400. [10] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [11] O. A. Ladyzenskaja, 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. [12] G. Liţcanu and C. Morales-Rodrigo, Asymptotic behaviour of global solutions to a model of cell invasion, Math. Mod. Meth. Appl. Sci., 20 (2010), 1721-1758. doi: 10.1142/S0218202510004775. [13] 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. [14] C. Morales-Rodrigo, Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Modelling, 47 (2008), 604-613. doi: 10.1016/j.mcm.2007.02.031. [15] N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional Keller-Segel system, preprint. [16] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433. [17] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. [18] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007. [19] 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. [20] Z. Szymańska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425. [21] Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69. doi: 10.1016/j.jmaa.2008.12.039. [22] 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. [23] 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. [24] 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. [25] 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. [26] 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. [27] 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. [28] M. Winkler, 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. [29] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
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