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Mathematical analysis of a model of chemotaxis arising from morphogenesis
Author(s) -
Stinner Christian,
Tello J. Ignacio,
Winkler Michael
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1573
Subject(s) - mathematics , bounded function , domain (mathematical analysis) , convergence (economics) , ordinary differential equation , limit (mathematics) , boundary (topology) , partial differential equation , boundary value problem , mathematical analysis , diffusion , differential equation , physics , economic growth , economics , thermodynamics
We consider non‐negative solution couples ( u , v ) ofu t=u xx− χuvv xx− λu ,v t= 1 − v + u ,with positive parameters χ and λ , where the spatial domain is the interval (0,1). This system appears as a limit case of a model for morphogenesis proposed by Bollenbach et al . (Phys. Rev. E. 75 , 2007). Under suitable boundary conditions, modeling the presence of a morphogen source at x = 0, we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover, we prove the convergence of the solution to the unique steady state provided that χ is small and λ is large enough. Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements. Copyright © 2012 John Wiley & Sons, Ltd.