Research ArticleOpen AccessGoogle Scholar indexed
Conservation Laws and Particular Solutions for a Keller-Segel Model
Department of Mathematics, Shanghai University, Shanghai, China
Department of Mathematics, Shanghai University, Shanghai, China
- 1 Department of Mathematics, Shanghai University, Shanghai, China
- 2 Department of Mathematics, Shanghai University, Shanghai, China
Advances in Pure Mathematics·Volume 15 (2025)·Pages 107–118·Published 28 February 2025·DOI10.4236/apm.2025.152005
Copy link · social · email
Abstract
This work focuses on a Keller-Segel chemotaxis model, with an emphasis on its conservation laws. Through a new approach combined with the multiplier method, called the mixed method , we obtain conservation vectors that are related and unrelated to symmetric information. In addition, some exact solutions with particular forms are obtained according to the method of conservation laws . These particular solutions are different from the group-invariant solutions.
KeywordsKeller-Segel ModelConservation LawsMixed MethodExact Solutions
- Keller, E.F. and Segel, L.A. (1970) Segel, Initiation of Slime Mold Aggregation Viewed as an Instability. Journal of Theoretical Biology , 26, 399-415. https://doi.org/10.1016/0022-5193(70)90092-5
- Keerthana, N., Saranya, R. and Annapoorani, N. (2024) Dynamics and Diffusion Limit of Traveling Waves in a Two-Species Chemotactic Model with Logarithmic Sensitivity. Mathematics and Computers in Simulation , 222, 311-329. https://doi.org/10.1016/j.matcom.2023.08.035
- Rosen, G. (1978) Steady-State Distribution of Bacteria Chemotactic Toward Oxygen. Bulletin of Mathematical Biology , 40, 671-674.
- Rosen, G. (1983) Theoretical Significance of the Condition in Bacterial Chemotaxis. Bulletin of Mathematical Biology , 45, 151-153.
- Corrias, L., Perthame, B. and Zaag, H. (2003) A Chemotaxis Model Motivated by Angiogenesis. Comptes Rendus de l ’ Académie des Sciences - Series I , 336, 141-146. https://doi.org/10.1016/S1631-073X(02)00008-0
- Fontelos, M.A., Friedman, A. and Hu, B. (2002) Mathematical Analysis of a Model for the Initiation of Angiogenesis. SIAM Journal on Mathematical Analysis , 33, 1330-1355. https://doi.org/10.1137/S0036141001385046
- Wang, Z.A. (2013) Mathematics of Traveling Waves in Chemotaxis-Review Paper Discrete Contin. Discrete and Continuous Dynamical Systems - B , 18, 601-641. https://doi.org/10.3934/dcdsb.2013.18.601
- Jin, H.Y., Li, J.Y. and Wang, Z.A. (2013) Asymptotic Stability of Traveling Waves of a Chemotaxis Model with Singular Sensitivity. Journal of Differential Equations , 255, 193-219. https://doi.org/10.1016/j.jde.2013.04.002
- Sleeman, B.D. and Levine, H.A. (1997) A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks. Siam Journal on Applied Mathematic , 57, 683-730. https://doi.org/10.1137/S0036139995291106
- Olver, P. (1986) Applications of Lie Groups to Differential Equations. Springer. https://doi.org/10.1007/978-1-4684-0274-2
- Ruggieri, M. and Speciale, M.P. (2017) On the Construction of Conservation Laws: A Mixed Approach. Journal of Mathematical Physics , 58, Article 023510. https://doi.org/10.1063/1.4976189
- Ruggieri, M. and Speciale, M.P. (2017) Speciale, Conservation Laws by Means of a New Mixed Method. International Journal of Non - Linear Mechanics , 95, 327-332. https://doi.org/10.1016/j.ijnonlinmec.2017.07.010
- Anco, S.C. and Bluman, G. (1996) Derivation of Conservation Laws from Nonlocal Symmetries of Differential Equations. Journal of Mathematical Physics , 37, 2361–2375. https://doi.org/10.1063/1.531515