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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), 1227-1260.  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), 505-565.  Google Scholar

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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. Google Scholar

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A. Bonami, D. Hilhorst, E. Logak and M. Mimura, Singular limit of a chemotaxis-growth model, Adv. Differential Equations, 6 (2001), 1173-1218.  Google Scholar

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M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439.  Google Scholar

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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), 350-362.  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), 169-205.  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), 1-28.  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), 168-181.  Google Scholar

[11]

A. Marciniak-Czochra and M. Ptashnyk, Boundnedness of solutions of a haptotaxis model, Mathematical Models and Methods in Applied Sciences 20 (2010), 449-476.  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), 60-69.  Google Scholar

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Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.  Google Scholar

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C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.  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), 1227-1260.  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), 505-565.  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), 163-186. Google Scholar

[4]

A. Bonami, D. Hilhorst, E. Logak and M. Mimura, Singular limit of a chemotaxis-growth model, Adv. Differential Equations, 6 (2001), 1173-1218.  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), 399-439.  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), 350-362.  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), 169-205.  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), 1-28.  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), 168-181.  Google Scholar

[11]

A. Marciniak-Czochra and M. Ptashnyk, Boundnedness of solutions of a haptotaxis model, Mathematical Models and Methods in Applied Sciences 20 (2010), 449-476.  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), 60-69.  Google Scholar

[13]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.  Google Scholar

[14]

C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.  Google Scholar

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