Abstract
Morphogen transport is a biological process, occurring in the tissue of living organisms, which is a determining step in cell differentiation. We present rigorous analysis of a simple model of this process, which is a system coupling parabolic PDE with ODE. We prove existence and uniqueness of solutions for both stationary and evolution problems. Moreover, we show that the solution converges exponentially to the equilibrium in C 1,α × C 0,α topology. We prove all results for arbitrary dimension of the domain. Our results improve significantly previously known results for the same model in the case of one-dimensional domain.
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Acknowledgments
The author would like to express his gratitude toward his PhD supervisors Philippe Laurençot and Dariusz Wrzosek for their constant encouragement and countless helpful remarks and toward Christoph Walker for discussions on interpolation techniques and multiplication in Sobolev spaces. The author was supported by the International Ph.D. Projects Program of Foundation for Polish Science operated within the Innovative Economy Operational Program 2007–2013 funded by European Regional Development Fund (Ph.D. Program: Mathematical Methods in Natural Sciences). Part of this research was carried out during author’s visit to the Institut de Mathématiques de Toulouse.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Małogrosz, M. Well-posedness and asymptotic behavior a multidimensional model of morphogen transport. J. Evol. Equ. 12, 353–366 (2012). https://doi.org/10.1007/s00028-012-0135-5
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DOI: https://doi.org/10.1007/s00028-012-0135-5