Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations

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January 2012, 11(1): 229-241. doi: 10.3934/cpaa.2012.11.229

Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations

Yuriy Golovaty ^1, , Anna Marciniak-Czochra ^2, and Mariya Ptashnyk ^3,

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

Received March 2010 Revised January 2011 Published September 2011

Abstract
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In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous cells, whose proliferation is controlled by growth factors diffusing in the extracellular medium and binding to the cell surface. We extend the results on the existence of nonhomogenous stationary solutions obtained in [9] to a general Hill-type production function and full parameter set. Using spectral analysis and perturbation theory we derive conditions for the linearized stability of such spatial patterns.

Keywords: reaction-diffusion equations, Pattern formation, linearized stability, spectral analysis..

Mathematics Subject Classification: Primary: 35K57, 35J57; Secondary: 92B9.

Citation: Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

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.

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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