On the fractional-in-time Keller-Segel model via Sonine kernels
DOI:
https://doi.org/10.12775/TMNA.2023.035Keywords
Sonine kernels, existence, asymptotic behavior, diffusion, chemotaxis aggregation, Keller-Segel modelAbstract
In this paper, we study the existence and asymptotic behavior to a diffusion system which is non-local in time. As consequence of our theorems we deduce new results for the fractional-in-time Keller-Segel model. Our approach is intimately related with the Sonine kernels.References
H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11 (1984), no. 4, 593–676.
A. Atangana and S.T.A. Badr, Analysis of the Keller–Segel model with a fractional derivative without singular kernel, Entropy 17 (2015), 4439–4453.
J. Azevedo, M. Bezerra, C. Cuevas and H. Soto, Well-posedness and asymptotic behavior for the fractional Keller–Segel system in critical Besov–Herz-type spaces, Math. Meth. Appl. Sci. 45 (2022), 6268–6287.
J. Azevedo, C. Cuevas, J. Dantas and C. Silva, On the fractional chemotaxis Navier–Stokes system in the critical spaces, Discrete Contin. Dyn. Syst. Ser. B 28 (2023), no. 1, 538–559.
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.
J. Azevedo, J.C. Pozo and A. Viana, Global solutions to the non-local Navier–Stokes equations, Discrete Contin. Dyn. Syst. Ser. B 27 (2022), no. 5, 2515–2335.
M. Bezerra, C. Cuevas, C. Silva, and H. Soto, On the fractional doubly parabolic Keller–Segel system modelling chemotaxis, Sci. China Math. 65 (2022), 1827–1874.
A. Blanchet, J.A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak–Keller–Segel mode in R2 , Comm. Pure Appl. Math. 61 (2008), no. 10, 1449–1481.
A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 44 (2006), 1–32.
N. Bournaveas and V. Calvez, The one-dimensional Keller–Segel model with fractional diffusion of cells, Nonlinearity 23 (2010), 923–925.
R. Carlone, A. Fiorenza and L. Tentarelli, The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions, J. Funct. Anal. 273 (2017), no. 3, 1258–1294.
P. Clément and J.A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal. 12 (1981), 514–534.
C. Cuevas, C. Silva and H. Soto, On the time-fractional Keller–Segel model for chemotaxis, Math. Meth. Appl. Sci. 43 (2020), 769–798.
A.M.A. El-Sayed, S.Z. Rida and A.A.M. Arafa, On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, Int. J. Nonlinear Sci. 7 (2009), no. 4, 485–492.
L. Guo, F. Gao and H. Zhan, Existence, uniqueness and L∞ -bound for weak solutions of a time fractional Keller–Segel system, Chaos Solitons Fractals 160 (2022), 112185.
M.A. Herrero and J.J.L. Velázquez, Chemptatic collapse for the Keller–Segel model, J. Math. Biol. 35 (1996), 177–194.
T. Hillen and K.J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183–217.
D. Horstmann, On the existence of radially symmetric blowup solutions for the Keller–Segel model, J. Math. Biol. 44 (2002), 463–478.
D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I, Jahresber. Dtsch. Math.-Ver. 105 (2003), no. 3, 103–165.
D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences II, Jahresber. Dtsch. Math.-Ver. 106 (2004), no. 2, 51–69.
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005), 52–107.
Z. Jiang and L. Wang, Mild solutions to the Cauchy problem for time-space fractional Keller–Segel–Navier–Stokes system (2022), arXiv: 2209.12171v2.
E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.
J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for timefractional and other non-local in time subdiffusion equations in Rd , Math. Ann. 366 (2016), 941–979.
A.N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl. 340 (2008), no. 1. 252–281.
S. Kumar, A. Kumar and I.K. Argyros, A new analysis for the Keller–Segel model of fractional order, Numer. Algorithms 75 (2017), 213–228.
T.A.M. Langlands and B.I. Henry, Fractional chemotaxis diffusion equations, Phys. Rev. E 81 (2010), no. 5, 1–12.
M. Naghibolhosseini, Estimation of outer-middle ear transmission using DPOAEs and fractional-order modeling of human middle ear, Ph.D. thesis, City University of New York, New York, 2015.
A.T. Nguyen, N.H. Tuan, and C. Yang, On Cauchy problem for fractional parabolicelliptic Keller–Segel model, Adv. Nonlinear Anal. 12 (2023), 97–116.
J.E. Pérez-López, D.A. Rueda-Gómez and E.J. Villamizar-Roa, Existence and uniqueness of global solutions for time fractional Keller–Segel system and fractional dissipation (2022), ArXiv: 2204.13813v1.
M.T. Widman, D. Emerson and C.C. Chiu, Modeling microbial chemotaxis in a diffusion gradient chamber, Biotechnol. Bioeng. 55 (1997), 191–205.
M. Zayernouri and A. Matzavinos, Fractional Adams–Bashforth/Moulton methods: An application to the fractional Keller–Segel chemotaxis system, J. Comput. Phys. 317 (2019), 1–14.
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Copyright (c) 2024 Masterson Costa, Claudio Cuevas, Clessius Silva, Herme Soto
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