Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences
December 27, 2023
January 23, 2024
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This research paper revolves around investigating the phenomenon of quenching in reaction-diffusion systems and highlighting its significance. The primary focus is on analyzing a specific type of parabolic singular reaction-diffusion model that incorporates positive Dirichlet boundary conditions. The objective is to establish the sufficiency of certain conditions for quenching to occur within a finite time frame and to demonstrate the global existence of solutions. The novelty of this paper lies in the simplicity of the conditions imposed on the nonlinearity. This simplicity allows us to choose it from a wide range of possibilities, thus facilitating the application of the model to numerous singular reaction-diffusion phenomena. To bolster our findings, we will present various real-world applications in the fields of bioengineering and life sciences, showcasing the practical relevance of quenching phenomena. Finally, the paper ends with a conclusion and some potential future perspectives for further research in this area.
Mesbahi, S., Djemai, S., & Saffidine, K. I. (2024). Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences. Engineering And Technology Journal, 9(01), 3366–3371. https://doi.org/10.47191/etj/v9i01.17
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F. Rouabah, D. Dadache, N. Haddaoui. Thermophysical and Mechanical Properties of Polystyrene: Influence of Free Quenching. International Scholarly Research Notices 2012 (Article ID 161364) (2012) 8 pages. DOI: 10.540 2/2012/ 161364.
T. Salin. Quenching and blow-up problems for reaction diffusion equations. Helsinki University of Technology Institute of Mathematics Research Reports A466, 2004. ISBN: 951-22-6943-0.
D. Suckart et al., Experimental and simulative investigation of flame--wall interactions and quenching in spark-ignition engines. Automot. Engine Technol. 2, 25-38 (2017). DOI: 10.1007/ s41104-016-0015-z.
Q. Wang. Quenching Phenomenon for a Parabolic MEMS Equation. Chin. Ann. Math. Ser. B 39 (2018) 129-144. DOI: 10.1007/s11401-018-1056-6.
F. A. Williams, Progress in Scale Modeling, Springer Science+ Business Media B.V, 2008. ISBN: 978-1-4020-8681-6.
L. Zhou, et al., A reaction-diffusion model of ROS-induced ROS release in a mitochondrial network. PLOS computational biology, 6(1), e1000657, 2010. DOI: 10.1371/journal.pcbi.10 00 657.
X. Zhu, Y. Ge, Adaptive ADI numerical analysis of 2D quenching-type reaction: diffusion equation with convection term, Mathematical Problems in Engineering, 2020. DOI: 10.1155/2020/8161804.
B. Barka et al. Thermophysical Behavior of Polycarbonate: Effect of Free Quenching above and below the Glass Transition Temperature. Advanced Materials Research 1174 (2022) 123-136. DOI: 10.4028/p-a258c5.
H. Berestycki et al., Quenching and Propagation in KPP Reaction-Diffusion Equations with a Heat Loss. Arch. Rational Mech. Anal. 178, 57-80 (2005). DOI: 10.1007/s00205-005-0367-4.
C. Y. Chan. Recent advances in quenching phenomena, Proceedings of Dynamic Systems and Applications, Vol. 2 (Atlanta, GA, 1995) Dynamic, Atlanta, GA, 1996, 107-113.
S. Chenna et al., Mechanisms and mathematical modeling of ROS production by the mitochondrial electron transport chain, American Journal of Physiology: Cell Physiology, Am J Physiol Cell Physiol 323:C69--C83,2022. DOI: 10.1152/ajp cell.00455.2021.
M. Cross, H. Greenside, Pattern formation and dynamics in nonequilibrium systems, Cambridge University Press, 2009. DOI: 10.1017/CBO97805 11627200.
I. de Bonis. Singular elliptic and parabolic problems: existence and regularity of solutions, Diss. Ph D Thesis, Sapienza University of Rome, 2015.
L. L. Hench, J. R. Jones, Bioactive Glasses: Frontiers and Challenges. Front Bioeng Biotechnol. 2015, 3:194. DOI: 10.3389/fbioe. 2015.00194.
S. A. Hussain et al., Fluorescence Resonance Energy Transfer (FRET) senso, Sci. Letts. J. 2015, 4: 119, arXiv:1408.6559v1 [physics.bio-ph]. DOI: 10.48550/arXiv.1408.6559.
R. H. Ji, C. Y. Qu, L. D. Wang, Simultaneous and nonsimult- aneous quenching for coupled parabolic system. Appl. Anal. 94 (2) (2015) 233-250. DOI: 10.1080/00036811.2014.887694.
J. R. Jones, Review of bioactive glass: From Hench to hybrids, Acta Biomaterialia, 9 (2013) 4457-4486. DOI: 10.1016/j.actb io.2012.08.023.
H. Kawarada. On solutions of initial-boundary problem for u_t=u_xx+1/(1-u) , Publ. Res. Inst. Math. Sci. 10 (1974/75), 729-736. DOI: 10.2977/prims/1195191889.
S. C. Kundu, Silk biomaterials for tissue engineering and regenerative medicine, Elsevier, 2014. ISBN: 9780857097064.
B. Liščić et al., Quenching Theory and Technology, CRC Press, New York. 2010. DOI: 10.1201/9781420009163.
S. Mesbahi. On the existence of weak solutions for a class of singular reaction-diffusion systems. Hacet. J. Math. Stat. 51 (3) (2022) 757-774. DOI: 10.15672/hujms.936018.
S. Mesbahi. Analyse mathématique de systèmes de réaction-diffusion quasi-linéaires. Editions Universitaires Européennes, 2019. ISBN-13: 978-6138492917.
J. D. Murray, Mathematical Biology I: An Introduction, 3rd ed. Springer-Verlag, New York, 2002.
J. D. Murray, Mathematical Biology II: Spatial Models and Biochemical Applications, 3rd ed. Springer-Verlag, New York, 2002.
H. Pei, Z. Li. Quenching for a parabolic system with general singular terms. J. Nonlinear Sci. Appl. 9 (8) (2016) 5281-5290. DOI: 10.22436/ jnsa.009.08.14.
D. L. Purich, Enzyme kinetics: catalysis and control: A reference of theory and best-practice methods, Elsevier, 2010. ISBN: 9780123809254.
F. Rouabah, D. Dadache, N. Haddaoui. Thermophysical and Mechanical Properties of Polystyrene: Influence of Free Quenching. International Scholarly Research Notices 2012 (Article ID 161364) (2012) 8 pages. DOI: 10.540 2/2012/ 161364.
T. Salin. Quenching and blow-up problems for reaction diffusion equations. Helsinki University of Technology Institute of Mathematics Research Reports A466, 2004. ISBN: 951-22-6943-0.
D. Suckart et al., Experimental and simulative investigation of flame--wall interactions and quenching in spark-ignition engines. Automot. Engine Technol. 2, 25-38 (2017). DOI: 10.1007/ s41104-016-0015-z.
Q. Wang. Quenching Phenomenon for a Parabolic MEMS Equation. Chin. Ann. Math. Ser. B 39 (2018) 129-144. DOI: 10.1007/s11401-018-1056-6.
F. A. Williams, Progress in Scale Modeling, Springer Science+ Business Media B.V, 2008. ISBN: 978-1-4020-8681-6.
L. Zhou, et al., A reaction-diffusion model of ROS-induced ROS release in a mitochondrial network. PLOS computational biology, 6(1), e1000657, 2010. DOI: 10.1371/journal.pcbi.10 00 657.
X. Zhu, Y. Ge, Adaptive ADI numerical analysis of 2D quenching-type reaction: diffusion equation with convection term, Mathematical Problems in Engineering, 2020. DOI: 10.1155/2020/8161804.
