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An effective design method to produce stationary chemical reaction-diffusion patterns
The singular limit of a haptotaxis model with bistable growth
1. | University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, Cergy-Pontoise, F-95000, France, France |
References:
[1] |
M. Alfaro, The singular limit of a chemotaxis-growth system with general initial data,, Adv. Differential Equations, 11 (2006), 1227.
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[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.
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[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.
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[5] |
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity,, Networks and Heterogeneous Media, 1 (2006), 399.
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[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.
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[7] |
A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169.
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[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.
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[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.
|
[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.
|
[12] |
Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source,, Journal Appl. Math. Anal., 354 (2009), 60.
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[13] |
Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion,, Nonlinearity, 21 (2008), 2221.
|
[14] |
C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model,, SIAM J. Math. Anal., 38 (2007), 1694.
|
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.
|
[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.
|
[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.
|
[5] |
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity,, Networks and Heterogeneous Media, 1 (2006), 399.
|
[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.
|
[7] |
A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169.
|
[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.
|
[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.
|
[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.
|
[12] |
Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source,, Journal Appl. Math. Anal., 354 (2009), 60.
|
[13] |
Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion,, Nonlinearity, 21 (2008), 2221.
|
[14] |
C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model,, SIAM J. Math. Anal., 38 (2007), 1694.
|
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