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Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations
1. | Department of Mechanics and Mathematics, Franko Lviv National University, Lviv 79000, Ukraine |
2. | University of Heidelberg, Interdisciplinary Center for Scientific Computing (IWR), Institute of Applied Mathematics and BIOQUANT, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany |
3. | Department of Mathematics I, RWTH Aachen University, Wüllnerstr. 5b, 52056 Aachen, Germany |
References:
[1] |
V. I. Arnold, "Ordinary Differential Equations," MIT Press, Cambridge, 1978. |
[2] |
K. I. Chueh, C. Conley and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Ind. Univ. Math. J., 26 (1977), 373-392.
doi: 10.1512/iumj.1977.26.26029. |
[3] |
A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), 1-36.
doi: 10.1016/S0167-2789(98)00180-8. |
[4] |
A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507.
doi: 10.1512/iumj.2001.50.1873. |
[5] |
D. Henry, "Geomertic Theory of Semilinear Parabolic Equations," Springer-Verlag, 1981. |
[6] |
T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York Inc, 1966. |
[7] |
A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Mod. Meth. Appl. Sci., 17 (2007), 1693-1719.
doi: 10.1142/S0218202507002443. |
[8] |
A. Marciniak-Czochra and M. Kimmel, Dynamics of growth and signalling along linear and surface structures in very early tumours, Comp. Math. Meth. Med., 7 (2006), 189-213.
doi: 10.1080/10273660600969091. |
[9] |
A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: The effects of influx of mutated cells, Math. Model. Nat.Phenom., 3 (2008), 90-114.
doi: 10.1051/mmnp:2008043. |
[10] |
J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003. |
[11] |
P. K. Maini, In "On Growth and Form. Spatio-Temporal Pattern Formation in Biology," John Wiley & Sons, 1999. |
[12] |
F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Springer-Verlag, Berlin, 1994. |
[13] |
J. Smoller, "Shock-Waves and Reaction-Diffusion Equations," Springer-Verlag, New York Heidelberg Berlin, 1994. |
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A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[15] |
J. Wei, On the interior spike layer solutions for some singular perturbation problems, Proc. Royal Soc. Edinb., 128A (1998), 849-874. |
show all references
References:
[1] |
V. I. Arnold, "Ordinary Differential Equations," MIT Press, Cambridge, 1978. |
[2] |
K. I. Chueh, C. Conley and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Ind. Univ. Math. J., 26 (1977), 373-392.
doi: 10.1512/iumj.1977.26.26029. |
[3] |
A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), 1-36.
doi: 10.1016/S0167-2789(98)00180-8. |
[4] |
A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507.
doi: 10.1512/iumj.2001.50.1873. |
[5] |
D. Henry, "Geomertic Theory of Semilinear Parabolic Equations," Springer-Verlag, 1981. |
[6] |
T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York Inc, 1966. |
[7] |
A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Mod. Meth. Appl. Sci., 17 (2007), 1693-1719.
doi: 10.1142/S0218202507002443. |
[8] |
A. Marciniak-Czochra and M. Kimmel, Dynamics of growth and signalling along linear and surface structures in very early tumours, Comp. Math. Meth. Med., 7 (2006), 189-213.
doi: 10.1080/10273660600969091. |
[9] |
A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: The effects of influx of mutated cells, Math. Model. Nat.Phenom., 3 (2008), 90-114.
doi: 10.1051/mmnp:2008043. |
[10] |
J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003. |
[11] |
P. K. Maini, In "On Growth and Form. Spatio-Temporal Pattern Formation in Biology," John Wiley & Sons, 1999. |
[12] |
F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Springer-Verlag, Berlin, 1994. |
[13] |
J. Smoller, "Shock-Waves and Reaction-Diffusion Equations," Springer-Verlag, New York Heidelberg Berlin, 1994. |
[14] |
A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[15] |
J. Wei, On the interior spike layer solutions for some singular perturbation problems, Proc. Royal Soc. Edinb., 128A (1998), 849-874. |
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