Existence of solutions to a new model of biological pattern formation |
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Authors: | M Alber HGE Hentschel SA Newman |
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Institution: | a Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA b Department of Physics, Emory University, Atlanta, GA 30322, USA c Institute of Fundamental Technological Research, Swietokrzyska 21, 00-049 Warsaw, Poland d Department of Cell Biology and Anatomy, New York Medical College, Valhalla, NY 10595, USA |
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Abstract: | In this paper we study the existence of classical solutions to a new model of skeletal development in the vertebrate limb. The model incorporates a general term describing adhesion interaction between cells and fibronectin, an extracellular matrix molecule secreted by the cells, as well as two secreted, diffusible regulators of fibronectin production, the positively-acting differentiation factor (“activator”) TGF-β, and a negatively-acting factor (“inhibitor”). Together, these terms constitute a pattern forming system of equations. We analyze the conditions guaranteeing that smooth solutions exist globally in time. We prove that these conditions can be significantly relaxed if we add a diffusion term to the equation describing the evolution of fibronectin. |
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