Singüler hassasiyet ve lojistik terimi ile birlikte tek tür ve iki kimyasal içeren Kemotaksi sisteminin Global varlığı

Yıl 2025, Cilt: 15 Sayı: 3, 1080 - 1088, 01.09.2025

Mustafa Ekici

https://doi.org/10.21597/jist.1641070

Öz

Bu araştırma makalesi, düzgün sınırlı bir alanda tek tür ve iki kimyasal singüler hassasiyet içeren bir parabolik-eliptik-eliptik kemotaksis sistemine ait çözümlerin küresel varlığıyla ilgilidir. Bu çalışmada, yukarıda belirtilen sistemin pozitif çözümlerinin global varlığını inceledik ve belirtilen IVP probleminin bazı parametre koşulları altında global klasik bir çözümün varlığını kanıtladık.

Anahtar Kelimeler

Kemotaksi , çok türlü sistem , singüler hassasiyet , global varlık

Global Existence of Solutions to a One-Species and Two-Chemicals Chemotaxis System with Singular Sensitivity and Logistic Source

Öz

The research paper investigates the global existence of solutions to a parabolic-elliptic-elliptic chemotaxis system that describes the interaction of a single species with two chemical substances in a smoothly bounded domain. The system under consideration incorporates singular sensitivity functions, which pose significant analytical challenges.In this study, the global existence of positive classical solutions under specific parameter constraints is established. The paper employs rigorous mathematical analysis to derive sufficient conditions that ensure the well-posedness of the associated initial value problem (IVP).The results contribute to the theoretical understanding of chemotaxis models with singular sensitivities, which are relevant in biological and ecological contexts.

Anahtar Kelimeler

Chemotaxis , multi-species system , singular sensitivity , global existence

Kaynakça

• Annamalai, B., Venugopal, P. (2024). Finite element analysis of the nonlocal diffusion effect in a two-species chemotaxis system. Discrete and Continuous Dynamical Systems-S, doi: 10.3934/dcdss.2024164.
• Au, V. V. (2024). Multispecies Lotka–Volterra competition–diffusion system with forcing terms depending on the variables. Mathematical Methods in the Applied Sciences, 47(18), 14767-14795.
• Bellomo, N., Bellouquid, A., Tao Y., & Winkler, M. (2015). Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25, 1663-1763.
• Biler, P. (1999). Global solutions to some parabolic-elliptic systems of chemotaxis, Advanced Mathematics and Applications, 9, 347-359.
• Black, T. (2020). Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity, Discrete and Continuous Dynamical Systems S, 13, 119-37.
• Bucur, V., & Vasiev, D. B. (2024). Formation of stationary periodic patterns in a model of two competing populations with chemotaxis. arXiv preprint arXiv:2411.00724.
• Fujie, K., Winkler, M., & Yokota, T. (2014). Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Analysis, 109, 56-71.
• Fujie, K., Winkler, M., & Yokota, T. (2015). Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal dependent sensitivity, Mathematical Methods in the Applied Sciences, 38(6), 1212-1224.
• Fujie, K., & Senba, T. (2016). Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete and Continuous Dynamical Systems B, 21(1), 81-102.
• Hillen, T., & Painter, K. (2009). A user’s guide to PDE models for chemotaxis, Journal of Theoretical Biology, 58, 183-217.
• Horstmann, D. (2004). From 1970 until present: the Keller-Segel model in chemotaxis, Jahresber DMV, 106, 51-69.
• Keller E.F., & Segel, L.A. (1970). Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26, 399-415.
• Keller, E.F., & Segel, L.A. (1971). Traveling bans of chemotactic bacteria: a theoretical analysis. J. Theor. Biol., 30, 377-380.
• Kurt, H.I. (2025). Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting, Journal of Differential Equations, 416(2), 1429-1461.
• Kurt, H.I., & Shen, W. (2021). Finite-time blow-up prevention by logistic source in chemotaxis models with singular sensitivity in any dimensional setting, SIAM Journal on Mathematical Analysis, 53(1), 973-1003.
• Kurt, H.I., & Shen, W. (2023). Chemotaxis models with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions, Nonlinear Analysis: Real World Applications, 69, 103762-27.
• Kurt, H.I., & Shen, W. (2023). Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence, Journal of Differential Equations, 355, 248-295.
• Kurt, H.I., & Shen, W. (2024). Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics, Discrete and Continuous Dynamical Systems, 44, No. 4, pp. 882-904.
• Kurt, H.I., Shen, W., & Xue, S. (2024). Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source, Mathematical Models and Methods in Applied Sciences, 34(9), , 1649-1700.
• Le, M. (2025). Boundedness in a chemotaxis system with weakly singular sensitivity in dimension two with arbitrary sub-quadratic degradation sources, Journal of Mathematical Analysis and Applications, 542, p. 128803.
• Li, Y., Mu, C., & Pan, X. (2024). Boundedness and asymptotic stability in a predator-prey system with density-dependent motilities. Discrete and Continuous Dynamical Systems-B, 29(5), 2192-2212.
• Nagai, T. & Senba, T. (1998). Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Advances in Mathematical Sciences and Applications, 8, 145-156.
• Zhang, J., Mu, C., & Tu, X. (2023). Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source, Zeitschrift für angewandte Mathematik und Physik, 74, 229.
• Zhao, X., & Zheng, S. (2017). Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Zeitschrift fur Angewandte Mathematik und Physik, 68, 2.
• Zhao, X. (2020). Boundedness to a logistic chemotaxis system with singular sensitivity, 1, arXiv: 2003.03016.
• Zhao, X. (2023). Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two, Nonlinearity, 36, 3909-3938.
• Zhao, X. (2023). Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, Czechoslovak Mathematical Journal, 1-25.