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| Morphogen is a protein substance which by the mechanism of positional signalling determines the process of cell differentiation. To model morphogen transport, a PDE-ODE system of reaction-diffusion type was proposed in [Huf]. The system consists of 1 parabolic semilinear PDE, posed on a rectangle $\Omega_h=I\times(0,h)$, which is coupled via nonlinear reaction terms, with 1 semilinear PDE and 3 ODE's posed on $I\times\{0\}$. We prove that the system has a unique steady state and that the evolution problem is well-posed in the appropriately chosen function setting. Moreover we analize limit of solutions as $h\to0$. The presence of singular source term (a Dirac delta) in the boundary condition for the 1st equation, causes problems with regularity in the ODE part of the sytem.
\begin{biblio}
\biblioitem [Huf] L. Hufnagel, J. Kreuger, S. M. Cohen, B. I. Shraiman, On the role of glypicans in the process of morphogen gradient formation, Dev. Biol., Vol. 300, (2006).
\biblioitem [Mal] M. Ma{\l}ogrosz, A model of morphogen transport in the presence of glypicans I, Non. Anal. TMA, Vol. 83, (2013)
\end{biblio} |
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