Existence of solution of a free boundary problem for reaction-diffusion systems
- Authors: Younes G.A.1,2, El Khatib N.3, Volpert V.A.4
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Affiliations:
- Institut Camille Jordan
- University Lyon 1
- Lebanese American University
- Peoples’ Friendship University of Russia (RUDN University)
- Issue: Vol 68, No 4 (2022)
- Pages: 716-731
- Section: Articles
- URL: https://journals.rudn.ru/CMFD/article/view/33500
- DOI: https://doi.org/10.22363/2413-3639-2022-68-4-716-731
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Abstract
In this paper, we prove the existence of solution of a novel free boundary problem for reaction-diffusion systems describing growth of biological tissues due to cell influx and proliferation. For this aim, we transform it into a problem with fixed boundary, through a change of variables. The new problem thus obtained has space and time dependent coeffcients with nonlinear terms. We then prove the existence of solution for the corresponding linear problem, and deduce the existence of solution for the nonlinear problem using the xed point theorem. Finally, we return to the problem with free boundary to conclude the existence of its solution.
About the authors
G. A. Younes
Institut Camille Jordan; University Lyon 1
Author for correspondence.
Email: volpert@math.univ-lyon1.fr
Villeurbanne, France
N. El Khatib
Lebanese American University
Email: volpert@math.univ-lyon1.fr
Byblos, Lebanon
V. A. Volpert
Peoples’ Friendship University of Russia (RUDN University)
Email: volpert@math.univ-lyon1.fr
Moscow, Russia
References
- Bessonov N., Morozova N., Volpert V. Modeling of branching patterns in plants// Bull. Math. Biol. - 2008. - 70. - C. 868-893.
- Fok P.-W. Mathematical model of intimal thickening in atherosclerosis: vessel stenosis as a free boundary problem// J. Theor. Biol. - 2012. - 314. - C. 23-33.
- Islam H., Johnston P. R. A mathematical model for atherosclerotic plaque formation and arterial wall remodelling// ANZIAM J. - 2016. - 57. - C. C320-C345.
- Ladyzenskaja O. A., Solonnikov V. A., Ural’tseva N. N. Linear and Quasi-linear Equations of Parabolic Type. - Providence: Am. Math. Soc., 1968.
- Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. - Basel etc.: Springer, 1995.
- Silva T., Ja¨ger W., Neuss-Radu M., Sequeira A. Modeling of the early stage of atherosclerosis with emphasis on the regulation of the endothelial permeability// J. Theor. Biol. - 2020. - 496. - 110229.
- Tao Y., Guo Q. A free boundary problem modelling cancer radiovirotherapy// Math. Models Methods Appl. Sci. - 2007. - 17, № 8. - C. 1241-1259.
- Yousefnezhad M., Mohammadi S. A., Bozorgnia F. A free boundary problem for a predator-prey model with nonlinear prey-taxis// Appl. Math. - 2018. - 63. - C. 125-147.
