Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

See all formats and pricing
More options …
Volume 19, Issue 7-8


Weakness and Mittag–Leffler Stability of Solutions for Time-Fractional Keller–Segel Models

Y. Zhou
  • Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, P.R. China
  • Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ J. Manimaran / L. Shangerganesh / A. Debbouche
Published Online: 2018-08-22 | DOI: https://doi.org/10.1515/ijnsns-2018-0035


We introduce a time-fractional Keller–Segel model with Dirichlet conditions on the boundary and Caputo fractional derivative for the time. The main result shows the existence theorem of the proposed model using the Faedo–Galerkin method with some compactness arguments. Moreover, we prove the Mittag–Leffler stability of solutions of the considered model.

Keywords: fractional PDE; weak solution; Keller–Segel model; Faedo–Galerkin method; Mittag–Leffler stability

MSC 2010: 35R11; 34A08; 35B35; 35D30


  • [1]

    A. Blanchet, J. A. Carrillo and Ph. Laurencot, Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial. Differ. Equ. 35 (2009), 133–168.

  • [2]

    R. Borsche, S. G\"ottlich, A. Klar and P. Scillen, The scalar Keller–Segel on networks, Math. Modes Methods Appl. Sci. 24 (2014), 221–247.

  • [3]

    R. Cherniha and M. Didovych, Exact solutions of the simplified Keller–Segel model, Commun. Nonlinear. Sci. Numer. Simulat. 18 (2013), 2960–2971.

  • [4]

    S. Fu, G. Huang and B. Adam, Instability in a generalized multi-species Keller–Segel chemotaxis model, Comput. Math. Appl. 72 (2016), 2280–2288.

  • [5]

    E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), 399–415.

  • [6]

    M. Negreanu and J. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal. 46 (2014), 3761–3781.

  • [7]

    M. Negreanu and J. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differ. Equ. 258 (2015), 1592–1617.

  • [8]

    L. Shangerganesh, N. Barani Balan and K. Balachandran, Weak-renormalized solutions for three species competition model in ecology, Int. J. Biomath. 7 (2014), 1450062 (24 pages).

  • [9]

    L. Shangerganesh, N. Barani Balan and K. Balachandran, Existence and uniqueness of solutions of degenerate chemotaxis system, Taiwanese J. Math. 18 (2014), 1605–1622.

  • [10]

    Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity, J. Differ. Equ. 252 (2012), 692–715.

  • [11]

    X. J. Yang and J. A. T. Machado, A new fractional operator of variable order: application in the description of anomalous diffusion, Phys. A: Stat. Mech. Appl. 481 (2017), 276–283.

  • [12]

    X. J. Yang, New general fractional-order rheological models with kernels of Mittag-Leffler functions, Rom. Rep. Phys. 69 (2017), 118.

  • [13]

    E. Ahmed, A. H. Hashis and F. A. Rihan, On fractional order cancer model, J. Fract. Calc. Appl. 3 (2012), 1–6.

  • [14]

    O. S. Iyiola and F. D. Zaman, A fractional diffusion equation model for cancer tumor, Am. Inst. Phys. (AIP) Adv. 4 (2014), 107121(17 pages).

  • [15]

    B. Ahmad, M. S. Alhothuali, H. H. Alsulami, M. Kirane and S. Timoshin, On a time fractional reaction-diffusion equation, Appl. Math. Comput. 257 (2015), 199–204.

  • [16]

    A. Alsaedi, M. Kirane and R. Lassoued, Global existence and asymptotic behavior for a time fractional reaction-diffusion system, Comput. Math. Appl. 73 (2017), 951–958.

  • [17]

    Y. L. Huang and C. H. Wu, Positive steady states of reaction-diffusion-advection competition models in periodic environment, J. Math. Anal. Appl. 453 (2017), 724–745.

  • [18]

    Z. Liu and S. Lü, Hermite Pseudospectral method for the time fractional diffusion equation with variable coefficients, Int. J. Nonlinear Sci. Numer. Simul. 18 (2017), 385–393.

  • [19]

    Y. Zhou, J. Manimaran, L. Shangerganesh and A. Debbouche, A class of time fractional reaction-diffusion equation with nonlocal boundary condition, Math. Methods Appl. Sci. 41 (2018), 2987–2999.

  • [20]

    J. Mu, B. Ahmad and S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Comput. Math. Appl. 73 (2017), 985–996.

  • [21]

    M. Yamamoto, Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations, J. Math. Anal. Appl. 460 (2018), 365–381.

  • [22]

    Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Comput. Math. Appl. 73 (2017), 874–891.

  • [23]

    Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl. 73 (2017), 1016–1027.

  • [24]

    I. Ameen and P. Novati, The solution of fractional order epidemic model by implicit Adams methods, Appl. Math. Modell. 43 (2017), 78–84.

  • [25]

    A. J. Arenas, G. González-Parra and B. M. Chen-Charpentier, Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order, Math. Comput. Simul. 121 (2016), 48–63.

  • [26]

    G. González-Parra, A. J. Arenas and B. M. Chen-Charpentier, A fractional order epidemic model for the simulation of outbreaks of influenza A (H1N1), Math. Methods Appl. Sci. 37 (2014), 2218–2226.

  • [27]

    B. I. Henry and S. L. Wearne, Fractional reaction-diffusion, Phys. A. 276 (2000), 448–455.

  • [28]

    A. Ibeas, M. Shafi, M. Ishfaq et al., Vaccination controllers for SEIR epidemic models based on fractional order dynamics, Biomed. Signal Proc. Control. 38 (2017), 136–142.

  • [29]

    S. Z. Rida, A. A. M. Arafa and Y. A. Gaber, Solution of the fractional epidemic model by L-ADM, Frac. Calc. Appl. 7 (2016), 189–195.

  • [30]

    A. A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations, Differ. Equ. 46 (2010), 660–666.

  • [31]

    A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, Elsevier, Amsterda, 2006.

  • [32]

    Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

  • [33]

    Y. Li, Y. Chen and I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamics systems, Automatica. 45 (2009), 1965–1969.

  • [34]

    I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, San Diego: Academic Press, 1999.

  • [35]

    O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A. 40 (2007), 6287–6303.

  • [36]

    R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979.

  • [37]

    F. Ren, F. Cao and J. Cao, Mittag–Leffler stability and generalized Mittag–Leffler stability of fractional-order gene regulatory networks, Neurocomputing. 160 (2015), 185–190.

  • [38]

    X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly compactness theorem, J. Math. Biol. 66 (2013), 1241–1266.

About the article

Received: 2018-02-04

Published Online: 2018-08-22

Published in Print: 2018-12-19

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 7-8, Pages 753–761, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2018-0035.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in