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doi: 10.3934/dcdss.2020008

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

1. 

Institute of Mathematics, Polish Academy of Sciences, Warsaw, Śniadeckich 8, 00-656, Poland

2. 

OxPDE, Mathematical Institute, University of Oxford, UK

3. 

Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria. Avda. Los Castros s/n, Santander, Spain

Received  May 2017 Published  January 2019

Fund Project: JB was supported by the National Science Centre, Poland (NCN) grant 'SONATA' 2016/21/D/ST1/03085. RGB was partially supported by the Grant MTM2014-59488-P from the Ministerio de Economía y Competitividad (MINECO, Spain)

In this paper we consider a $ d $-dimensional ($ d = 1, 2 $) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order $ \alpha \in (0, 2) $. We prove uniform in time boundedness of its solution in the supercritical range $ \alpha>d\left(1-c\right) $, where $ c $ is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for $ \|u(t)-u_\infty\|_{L^\infty}\rightarrow0 $, where $ u_\infty\equiv 1 $ is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.

Citation: Jan Burczak, Rafael Granero-Belinchón. Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020008
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show all references

References:
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P. Aceves-Sanchez and L. Cesbron, Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation, arXiv preprint, arXiv: 1606.07939, 2016.

[2]

P. Aceves-Sanchez and A. Mellet, Asymptotic analysis of a Vlasov-Boltzmann equation with anomalous scaling, arXiv preprint, arXiv: 1606.01023, 2016.

[3]

Y. AscasibarR. Granero-Belinchón and J. M. Moreno, An approximate treatment of gravitational collapse, Physica D: Nonlinear Phenomena, 262 (2013), 71-82. doi: 10.1016/j.physd.2013.07.010.

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R. AtkinsonC. RhodesD. Macdonald and R. Anderson, Scale-free dynamics in the movement patterns of jackals, Oikos, 98 (2002), 134-140. doi: 10.1034/j.1600-0706.2002.980114.x.

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F. BartumeusF. PetersS. PueyoC. Marrasé and J. Catalan, Helical Lévy walks: adjusting searching statistics to resource availability in microzooplankton, Proceedings of the National Academy of Sciences, 100 (2003), 12771-12775. doi: 10.1073/pnas.2137243100.

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N. BellomoA. BellouquidY. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

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A. BellouquidJ. Nieto and L. Urrutia, About the kinetic description of fractional diffusion equations modeling chemotaxis, Mathematical Models and Methods in Applied Sciences, 26 (2016), 249-268. doi: 10.1142/S0218202516400029.

[8]

P. BilerT. CieślakG. Karch and J. Zienkiewicz, Local criteria for blowup in two-dimensional chemotaxis models, Discrete Contin. Dyn. Syst., 37 (2017), 1841-1856. doi: 10.3934/dcds.2017077.

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P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, Journal of Evolution equations, 10 (2010), 247–262. doi: 10.1007/s00028-009-0048-0.

[10]

P. Biler, G. Karch and P. Laurençot, Blowup of solutions to a diffusive aggregation model, Nonlinearity, 22 (2009), 1559. doi: 10.1088/0951-7715/22/7/003.

[11]

P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Methods Appl. Sci., 32 (2009), 112-126. doi: 10.1002/mma.1036.

[12]

A. Blanchet, On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher, Séminaire Laurent Schwartz -EDP et applications, 2013, 26pp.

[13]

A. Blanchet, J. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbb{R}^2$, Communications on Pure and Applied Mathematics, 61 (2008), 1449–1481. doi: 10.1002/cpa.20225.

[14]

N. Bournaveas and V. Calvez, The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923–935. doi: 10.1088/0951-7715/23/4/009.

[15]

J. Burczak and R. Granero-Belinchón, Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux, Topological Methods in Nonlinear Analysis, 47 (2016), 369–387.

[16]

J. Burczak and R. Granero-Belinchón, Critical Keller-Segel meets Burgers on $\mathbb{S}^1$: large-time smooth solutions, Nonlinearity, 29 (2016), 3810-3836. doi: 10.1088/0951-7715/29/12/3810.

[17]

J. Burczak and R. Granero-Belinchón, Global solutions for a supercritical drift-diffusion equation, Advances in Mathematics, 295 (2016), 334–367. doi: 10.1016/j.aim.2016.03.011.

[18]

J. Burczak and R. Granero-Belinchón, On a generalized doubly parabolic Keller-Segel system in one spatial dimension, Mathematical Models and Methods in the Applied Sciences, 26 (2016), 111-160. doi: 10.1142/S0218202516500044.

[19]

J. Burczak and R. Granero-Belinchón, Suppression of blow up by a logistic source in $2$ d Keller-Segel system with fractional dissipation, Journal of Differential Equations, 263 (2017), 6115-6142. doi: 10.1016/j.jde.2017.07.007.

[20]

J. Burczak, R. Granero-Belinchón and G. K. Luli, On the generalized Buckley-Leverett equation, Journal of Mathematical Physics, 57 (2016), 041501, 20 pp. doi: 10.1063/1.4945786.

[21]

F. A. Chalub, P. A. Markowich, B. Perthame, and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, In Nonlinear Differential Equation Models, Springer, 142 (2004), 123–141. doi: 10.1007/s00605-004-0234-7.

[22]

M. A. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Applied Mathematics Letters, 57 (2016), 1-6. doi: 10.1016/j.aml.2015.12.001.

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B. J. Cole, Fractal time in animal behaviour: The movement activity of drosophila, Animal Behaviour, 50 (1995), 1317-1324. doi: 10.1016/0003-3472(95)80047-6.

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A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Communications in Mathematical Physics, 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.

[25]

L. CorriasB. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, Comptes Rendus Mathematique, 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0.

[26]

J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $\Bbb R^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616. doi: 10.1016/j.crma.2004.08.011.

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