Stabilization in a chemotaxis model for tumor invasion

Kentarou  Fujie; Akio Ito; Michael Winkler; Tomomi Yokota

doi:10.3934/dcds.2016.36.151

Article Contents

2016, Volume 36, Issue 1: 151-169. Doi: 10.3934/dcds.2016.36.151

This issue Previous Article Linearization of solution operators for state-dependent delay equations: A simple example Next Article Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations

Stabilization in a chemotaxis model for tumor invasion

1.
Department of Mathematics, Tokyo University of Science, Tokyo 162-8601
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: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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 $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$ .

Keywords:

Mathematics Subject Classification: Primary: 35B40, 35Q92; Secondary: 92C17.

Citation:

Related Papers

Cited by

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.