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

Yuriy Golovaty; Anna Marciniak-Czochra; Mariya Ptashnyk

doi:10.3934/cpaa.2012.11.229

Article Contents

2012, Volume 11, Issue 1: 229-241. Doi: 10.3934/cpaa.2012.11.229

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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
2.
University of Heidelberg, Interdisciplinary Center for Scientific Computing (IWR), Institute of Applied Mathematics and BIOQUANT, Im Neuenheimer Feld 267, 69120 Heidelberg
3.
Department of Mathematics I, RWTH Aachen University, Wüllnerstr. 5b, 52056 Aachen

Received: February 28, 2010

Revised: December 31, 2010

Published: December 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57, 35J57; Secondary: 92B99.

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References

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