American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations
  • CPAA Home
  • This Issue
  • Next Article
    An effective design method to produce stationary chemical reaction-diffusion patterns
January  2012, 11(1): 209-228. doi: 10.3934/cpaa.2012.11.209

The singular limit of a haptotaxis model with bistable growth

Elisabeth Logak 1, and  Chao Wang 1,

1. 

University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, Cergy-Pontoise, F-95000, France, France

Received  March 2010 Revised  October 2010 Published  September 2011

We consider a model for haptotaxis with bistable growth and study its singular limit. This yields an interface motion where the normal velocity of the interface depends on the mean curvature and on some nonlocal haptotaxis term. We prove the result for general initial data after establishing a result about generation of interface in a small time.
Keywords: haptotaxis, comparison principle., free boundary problem, Reaction-diffusion system.
Mathematics Subject Classification: 35K57, 35B50, 35R35, 35Q92, 92C1.
Citation: Elisabeth Logak, Chao Wang. The singular limit of a haptotaxis model with bistable growth. Communications on Pure & Applied Analysis, 2012, 11 (1) : 209-228. doi: 10.3934/cpaa.2012.11.209
References:
[1]

M. Alfaro, The singular limit of a chemotaxis-growth system with general initial data,, Adv. Differential Equations, 11 (2006), 1227.   Google Scholar

[2]

M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system,, J. Differential Equations, 245 (2008), 505.   Google Scholar

[3]

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.   Google Scholar

[4]

A. Bonami, D. Hilhorst, E. Logak and M. Mimura, Singular limit of a chemotaxis-growth model,, Adv. Differential Equations, 6 (2001), 1173.   Google Scholar

[5]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity,, Networks and Heterogeneous Media, 1 (2006), 399.   Google Scholar

[6]

X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling,, J. Math. Anal. Appl., 162 (1992), 350.   Google Scholar

[7]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169.   Google Scholar

[8]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimension,, Milan. J. Math., 72 (2004), 1.   Google Scholar

[9]

E. Dibenedetto, "Degenerate parabolic equations,", Springer-Verlag, (1993).   Google Scholar

[10]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models,, Journal of Computational and Applied Mathematics, 224 (2009), 168.   Google Scholar

[11]

A. Marciniak-Czochra and M. Ptashnyk, Boundnedness of solutions of a haptotaxis model,, Mathematical Models and Methods in Applied Sciences \textbf{20} (2010), 20 (2010), 449.   Google Scholar

[12]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source,, Journal Appl. Math. Anal., 354 (2009), 60.   Google Scholar

[13]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion,, Nonlinearity, 21 (2008), 2221.   Google Scholar

[14]

C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model,, SIAM J. Math. Anal., 38 (2007), 1694.   Google Scholar

show all references

References:
[1]

M. Alfaro, The singular limit of a chemotaxis-growth system with general initial data,, Adv. Differential Equations, 11 (2006), 1227.   Google Scholar

[2]

M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system,, J. Differential Equations, 245 (2008), 505.   Google Scholar

[3]

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.   Google Scholar

[4]

A. Bonami, D. Hilhorst, E. Logak and M. Mimura, Singular limit of a chemotaxis-growth model,, Adv. Differential Equations, 6 (2001), 1173.   Google Scholar

[5]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity,, Networks and Heterogeneous Media, 1 (2006), 399.   Google Scholar

[6]

X. Chen and F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling,, J. Math. Anal. Appl., 162 (1992), 350.   Google Scholar

[7]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169.   Google Scholar

[8]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimension,, Milan. J. Math., 72 (2004), 1.   Google Scholar

[9]

E. Dibenedetto, "Degenerate parabolic equations,", Springer-Verlag, (1993).   Google Scholar

[10]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models,, Journal of Computational and Applied Mathematics, 224 (2009), 168.   Google Scholar

[11]

A. Marciniak-Czochra and M. Ptashnyk, Boundnedness of solutions of a haptotaxis model,, Mathematical Models and Methods in Applied Sciences \textbf{20} (2010), 20 (2010), 449.   Google Scholar

[12]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source,, Journal Appl. Math. Anal., 354 (2009), 60.   Google Scholar

[13]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion,, Nonlinearity, 21 (2008), 2221.   Google Scholar

[14]

C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model,, SIAM J. Math. Anal., 38 (2007), 1694.   Google Scholar

[1]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039
[2]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128
[3]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020033
[4]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[5]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[6]

Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369
[7]

Mihaela Negreanu, J. Ignacio Tello. On a comparison method to reaction-diffusion systems and its applications to chemotaxis. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2669-2688. doi: 10.3934/dcdsb.2013.18.2669
[8]

Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23
[9]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[10]

Wanli Yang, Jie Sun, Su Zhang. Analysis of optimal boundary control for a three-dimensional reaction-diffusion system. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 325-344. doi: 10.3934/naco.2017021
[11]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[12]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[13]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[14]

Michele V. Bartuccelli, S.A. Gourley, Y. Kyrychko. Comparison and convergence to equilibrium in a nonlocal delayed reaction-diffusion model on an infinite domain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1015-1026. doi: 10.3934/dcdsb.2005.5.1015
[15]

Chiun-Chuan Chen, Li-Chang Hung. An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1503-1521. doi: 10.3934/dcdsb.2018054
[16]

Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191
[17]

Xin Li, Xingfu Zou. On a reaction-diffusion model for sterile insect release method with release on the boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2509-2522. doi: 10.3934/dcdsb.2012.17.2509
[18]

Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020054
[19]

Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
[20]

Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences