Recent results for the logarithmic Keller-Segel-Fisher/KPP System
Abstract
We consider a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. It is a 2 by 2 system describing the interaction of cells and a chemical signal. We study Cauchy problem with finite initial data, i.e., without the commonly used smallness assumption on initial perturbations around a constant ground state. We survey a sequence of recent results by the authors on the existence of global-in-time solution, long-time behavior, vanishing coefficient limit and optimal time decay rates of the solution.
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References
Keller, E.F. and Segel, L.A., Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol. 30, 235-248, (1971).
https://doi.org/10.1016/0022-5193(71)90051-8
Levine, H.A. and Sleeman, B.D., A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57, 683-730, (1997).
https://doi.org/10.1137/S0036139995291106
Levine, H.A., Sleeman, B.D. and Nilsen-Hamilton, M., A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis, Math. Biosci. 168, 71-115, (2000).
https://doi.org/10.1016/S0025-5564(00)00034-1
Li, D., Pan, R. and Zhao, K. Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity 28, 2181-2210, (2015).
https://doi.org/10.1088/0951-7715/28/7/2181
Martinez, V., Wang, Z. and Zhao, K., Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J. 67, 1383-1424, (2018).
https://doi.org/10.1512/iumj.2018.67.7394
Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13, 115-162 (1959).
Othmer, H. and Stevens, A., Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57, 1044-1081, (1997).
https://doi.org/10.1137/S0036139995288976
Rugamba, J. and Zeng, Y., Pointwise Asymptotic Behavior of a Chemotaxis Model, Math.Comput. Appl., 23 Paper No. 56, 12 pp. (2018).
https://doi.org/10.3390/mca23040056
Shizuta, Y. and Kawashima, S., Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14, 249-275 (1985).
https://doi.org/10.14492/hokmj/1381757663
Zeng, Y., Global existence theory for general hyperbolic-parabolic balance laws with application, J. Hyperbolic Differ. Equ. 14, 359-391, (2017).
https://doi.org/10.1142/S0219891617500126
Zeng, Y., Lp decay for general hyperbolic-parabolic systems of balance laws, Discrete Contin. Dyn. Syst. Ser. A 38, 363-396, (2018).
https://doi.org/10.3934/dcds.2018018
Zeng, Y., Asymptotic behavior of solutions to general hyperbolic-parabolic systems of balance laws in multi space dimensions, Pure Appl. Math. Q., special issue in honor of Prof. Chi-Wang Shu's 60th birthday, 14, 161-192 (2018).
https://doi.org/10.4310/PAMQ.2018.v14.n1.a6
Zeng, Y., Lp time asymptotic decay for general hyperbolic-parabolic balance laws with applications, Submitted.
Zeng, Y., Hyperbolic-parabolic balance laws: asymptotic behavior and a chemotaxis model, Comm. Appl. Anal., (2017 SEARCDE Proceedings), to appear.
Zeng, Y. and Zhao, K., On the logarithmic Keller-Segel-Fisher/KPP system, Discrete Contin. Dyn. Syst. Ser. A, 39, 5365-5402 (2019).
https://doi.org/10.3934/dcds.2019220
Zeng, Y. and Zhao, K., Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. J. Differential Equations, to appear.
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