American Institute of Mathematical Sciences

Advanced
February  2020, 13(2): 203-209. doi: 10.3934/dcdss.2020011

Global asymptotic stability in a chemotaxis-growth model for tumor invasion

Kentarou Fujie ,

Department of Mathematics, Tokyo University of Science, Tokyo 162-8601, Japan

* Corresponding author: Kentarou Fujie

Received  May 2017 Revised  December 2017 Published  January 2019

This paper presents global existence and asymptotic behavior of solutions to the chemotaxis-growth system
$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + ru -\mu u^\alpha, \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{array} \right. $
in a smoothly bounded domain
$ \Omega \subset \mathbb{R}^n $
,
$ n \le 3 $
, where
$ r>0 $
,
$ \mu>0 $
 and
$ \alpha>1 $
. Without the logistic source
$ ru-\mu u^\alpha $
, the stabilization of this system has been shown by Fujie, Ito, Winkler and Yokota (2016), whereas especially about asymptotic behavior, the logistic source disturbs applying this method directly. In the present paper, a way out of this difficulty is introduced and the asymptotic behavior of solutions to the system with logistic source is precisely determined.
Keywords: Chemotaxis, asymptotic behavior, tumor invasion.
Mathematics Subject Classification: Primary: 35B40, 35Q92; Secondary: 92C17.
Citation: Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011
References:
[1]

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

[2]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[3]

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

[4]

E. Feireisl, P. Laurençot and H. Petzeltová, On convergence to equilibria for the Keller-Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569.  doi: 10.1016/j.jde.2007.02.002.  Google Scholar

[5]

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.  Google Scholar

[6]

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.   Google Scholar

[7]

K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.  Google Scholar

[8]

K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.  Google Scholar

[9]

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.   Google Scholar

[10]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[12]

B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., Theory Methods Appl., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.   Google Scholar

[22]

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.  Google Scholar

[23]

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

[24]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc.(JEMS), 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.  Google Scholar

[25]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[26]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.  Google Scholar

[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.  Google Scholar

[28]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[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.  Google Scholar

[30]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

show all references

References:
[1]

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

[2]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[3]

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

[4]

E. Feireisl, P. Laurençot and H. Petzeltová, On convergence to equilibria for the Keller-Segel chemotaxis model, J. Differential Equations, 236 (2007), 551-569.  doi: 10.1016/j.jde.2007.02.002.  Google Scholar

[5]

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.  Google Scholar

[6]

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.   Google Scholar

[7]

K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.  Google Scholar

[8]

K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.  Google Scholar

[9]

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.   Google Scholar

[10]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[12]

B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[17]

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.  Google Scholar

[18]

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

[19]

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.  Google Scholar

[20]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., Theory Methods Appl., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.   Google Scholar

[22]

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.  Google Scholar

[23]

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

[24]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc.(JEMS), 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.  Google Scholar

[25]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[26]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.  Google Scholar

[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.  Google Scholar

[28]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[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.  Google Scholar

[30]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[1]

Kentarou Fujie, Akio Ito, Michael Winkler, Tomomi Yokota. Stabilization in a chemotaxis model for tumor invasion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 151-169. doi: 10.3934/dcds.2016.36.151
[2]

Junde Wu, Shihe Xu. Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020018
[3]

Yuanyuan Liu, Youshan Tao. Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 465-475. doi: 10.3934/dcdsb.2017021
[4]

Janet Dyson, Eva Sánchez, Rosanna Villella-Bressan, Glenn F. Webb. An age and spatially structured model of tumor invasion with haptotaxis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 45-60. doi: 10.3934/dcdsb.2007.8.45
[5]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[6]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437
[7]

Risei Kano. The existence of solutions for tumor invasion models with time and space dependent diffusion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 63-74. doi: 10.3934/dcdss.2014.7.63
[8]

Gülnihal Meral, Christian Stinner, Christina Surulescu. On a multiscale model involving cell contractivity and its effects on tumor invasion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 189-213. doi: 10.3934/dcdsb.2015.20.189
[9]

Risei Kano, Akio Ito. The existence of time global solutions for tumor invasion models with constraints. Conference Publications, 2011, 2011 (Special) : 774-783. doi: 10.3934/proc.2011.2011.774
[10]

Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721
[11]

Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193
[12]

Sandesh Athni Hiremath, Christina Surulescu, Anna Zhigun, Stefanie Sonner. On a coupled SDE-PDE system modeling acid-mediated tumor invasion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2339-2369. doi: 10.3934/dcdsb.2018071
[13]

Rafał Celiński, Andrzej Raczyński. Asymptotic profile of solutions to a certain chemotaxis system. Communications on Pure & Applied Analysis, 2020, 19 (2) : 911-922. doi: 10.3934/cpaa.2020041
[14]

Mykhailo Potomkin. Asymptotic behavior of thermoviscoelastic Berger plate. Communications on Pure & Applied Analysis, 2010, 9 (1) : 161-192. doi: 10.3934/cpaa.2010.9.161
[15]

Hunseok Kang. Asymptotic behavior of a discrete turing model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 265-284. doi: 10.3934/dcds.2010.27.265
[16]

Akisato Kubo, Hiroki Hoshino, Katsutaka Kimura. Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion model. Conference Publications, 2015, 2015 (special) : 733-744. doi: 10.3934/proc.2015.0733
[17]

Doan Duy Hai, Atsushi Yagi. Longtime behavior of solutions to chemotaxis-proliferation model with three variables. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3957-3974. doi: 10.3934/dcds.2012.32.3957
[18]

Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091
[19]

Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183
[20]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (155)
  • HTML views (663)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences