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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

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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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D. Henry, "Geomertic Theory of Semilinear Parabolic Equations," Springer-Verlag, 1981.  Google Scholar

[6]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York Inc, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003.  Google Scholar

[11]

P. K. Maini, In "On Growth and Form. Spatio-Temporal Pattern Formation in Biology," John Wiley & Sons, 1999. Google Scholar

[12]

F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Springer-Verlag, Berlin, 1994.  Google Scholar

[13]

J. Smoller, "Shock-Waves and Reaction-Diffusion Equations," Springer-Verlag, New York Heidelberg Berlin, 1994.  Google Scholar

[14]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.  Google Scholar

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J. Wei, On the interior spike layer solutions for some singular perturbation problems, Proc. Royal Soc. Edinb., 128A (1998), 849-874.  Google Scholar

show all references

References:
[1]

V. I. Arnold, "Ordinary Differential Equations," MIT Press, Cambridge, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Henry, "Geomertic Theory of Semilinear Parabolic Equations," Springer-Verlag, 1981.  Google Scholar

[6]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York Inc, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. D. Murray, "Mathematical Biology," Springer-Verlag, 2003.  Google Scholar

[11]

P. K. Maini, In "On Growth and Form. Spatio-Temporal Pattern Formation in Biology," John Wiley & Sons, 1999. Google Scholar

[12]

F. Rothe, "Global Solutions of Reaction-Diffusion Systems," Springer-Verlag, Berlin, 1994.  Google Scholar

[13]

J. Smoller, "Shock-Waves and Reaction-Diffusion Equations," Springer-Verlag, New York Heidelberg Berlin, 1994.  Google Scholar

[14]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012.  Google Scholar

[15]

J. Wei, On the interior spike layer solutions for some singular perturbation problems, Proc. Royal Soc. Edinb., 128A (1998), 849-874.  Google Scholar

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