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Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. (English) Zbl 1264.35034

Summary: 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 [A. Marciniak-Czochra and M. Kimmel, “Reaction-diffusion model of early carcinogenesis: the effects of influx of mutated cells”, Math. Model. Nat. Phenom. 3, No. 7, 90–114 (2008)] 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.

MSC:

35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
35J57 Boundary value problems for second-order elliptic systems
92C15 Developmental biology, pattern formation
35B36 Pattern formations in context of PDEs
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