On the nonlocal parabolic-elliptic Keller-Segel model in bounded domains

Dinh-Ke Tran; Thi-Thu Tran; Van-Tuan Tran

doi:10.3934/cpaa.2024064

Article Contents

2024, Volume 23, Issue 11: 1730-1747. Doi: 10.3934/cpaa.2024064

This issue Previous Article Time-periodic solutions to the non-isothermal model for compressible nematic liquid crystals in $ \mathbb{R}^3 $ Next Article On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart

On the nonlocal parabolic-elliptic Keller-Segel model in bounded domains

1.
Department of Mathematics, Hanoi National University of Education, Xuan Thuy, Cau Giay, Hanoi, Vietnam
2.
Department of Mathematics, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam

^*Corresponding author: Dinh-Ke Tran
^*Corresponding author: Dinh-Ke Tran

Received: March 2024

Revised: June 2024

Early access: August 5, 2024

Published online: August 5, 2024

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We deal with the nonlocal parabolic-elliptic model of Keller-Segel system in bounded domains, where the nonlinearity expressing the cell growth and death is taken into account. Our aim is to establish a global well-posedness result with small data and prove the Hölder regularity for the obtained mild solution. The approach used in our work is to employ a relation between the Hilbert scales generated by the Neumann Laplacian and fractional Sobolev spaces, the fixed point argument and some regularity estimates of resolvents.

Keywords:

Mathematics Subject Classification: Primary: 35B65, 35C15, 35R11, 45K05; Secondary: 92C17.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	G. Arumugam and J. Tyagi, Keller-Segel chemotaxis models: A review, Acta Appl. Math., 171 (2021), Paper No. 6, 82 pp. doi: 10.1007/s10440-020-00374-2.
[2]	J. Azevedo, C. Cuevas and E. Henriquez, Existence and asymptotic behaviour for the time-fractional Keller-Segel model for chemotaxis, Math. Nachr., 292 (2019), 462-480. doi: 10.1002/mana.201700237.
[3]	P. K. Bhattacharyya, Distributions. Generalized Functions with Applications in Sobolev Spaces, De Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012. doi: 10.1515/9783110269291.
[4]	M. Bonforte, Y. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767. doi: 10.3934/dcds.2015.35.5725.
[5]	P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535. doi: 10.1137/0512045.
[6]	N. V. Dac, T. D. Ke and L. T. P. Thuy, On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities, Discrete Contin. Dyn. Syst. S., 16 (2023), 2883-2901. doi: 10.3934/dcdss.2023071.
[7]	G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.
[8]	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[9]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.
[10]	T. D. Ke, N. N. Thang and L. T. P. Thuy, Regularity and stability analysis for a class of semilinear nonlocal differential equations in Hilbert spaces, J. Math. Anal. Appl., 483 (2020), 123655. doi: 10.1016/j.jmaa.2019.123655.
[11]	T. D. Ke and L. T. P. Thuy, Nonlocal final value problem governed by semilinear anomalous diffusion equations, Evol. Equ. Control Theory, 9 (2020), 891-914. doi: 10.3934/eect.2020038.
[12]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[13]	E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.
[14]	E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[15]	A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281. doi: 10.1016/j.jmaa.2007.08.024.
[16]	L. Li and J. G. Liu, Some compactness criteria for weak solutions of time fractional PDEs, SIAM J. Math. Anal., 50 (2018), 3963-3995. doi: 10.1137/17M1145549.
[17]	L. Li, J. G. Liu and L. Wang, Cauchy problems for Keller-Segel type time-space fractional diffusion equation, J. Differential Equations, 265 (2018), 1044-1096. doi: 10.1016/j.jde.2018.03.025.
[18]	J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I., Translated from the French by P. Kenneth, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.
[19]	Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548. doi: 10.1016/j.jmaa.2010.08.048.
[20]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300.
[21]	J. D. Murray, Mathematical Biology II. Spatial Models and Biomedical Applications, Springer, Berlin, 2003. doi: 10.1007/b98869.
[22]	J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics 87, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.
[23]	S. G. Samko and R. P. Cardoso, Integral equations of the first kind of Sonine type, Int. J. Math. Math. Sci., 57 (2003), 3609-3632. doi: 10.1155/S0161171203211455.
[24]	N. A. Tuan, N. H. Tuan and C. Yang, On Cauchy problem for fractional parabolic-elliptic Keller-Segel model, Adv. Nonlinear Anal., 12 (2023), 97-116. doi: 10.1515/anona-2022-0256.
[25]	V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239. doi: 10.1137/130941900.