Acta Applicandae Mathematicae

, Volume 164, Issue 1, pp 1–19 | Cite as

Uniform in Time \(L^{\infty }\)-Estimates for Nonlinear Aggregation-Diffusion Equations

  • Jose A. Carrillo
  • Jinhuan WangEmail author


We derive uniform in time \(L^{\infty }\)-bound for solutions to an aggregation-diffusion model with attractive-repulsive potentials or fully attractive potentials. We analyze two cases: either the repulsive nonlocal term dominates over the attractive part, or the diffusion term dominates over the fully attractive nonlocal part. When the repulsive part of the potential has a weaker singularity (\(2-n\leq B< A\leq 2\)), we use the classical approach by the Sobolev and Young inequalities together with differential iterative inequalities to prove that solutions have the uniform in time \(L^{\infty }\)-bound. When the repulsive part of the potential has a stronger singularity (\(-n< B<2-n \leq A\leq 2\)), we show the uniform bounds by utilizing properties of fractional operators. We also show uniform bounds in the purely attractive case \(2-n\leq A\leq 2\) within the diffusion dominated regime.


Aggregation-diffusion equations Global in time uniform estimates Stroock-Varopoulos inequality 



JAC was partially supported by the EPSRC grant number EP/P031587/1. JW is partially supported by Program for Liaoning Excellent Talents in University (Grant No. LJQ2015041) and Key Project of Education Department of Liaoning Province (Grant No. LZD201701). The authors warmly thank the referees for the very valuable comments and suggestions that allowed us to improve this paper. We also thank Edoardo Mainini for pointing us out an important reference.


  1. 1.
    Bian, S., Liu, J.-G.: Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent \(m>0\). Commun. Math. Phys. 323(3), 1017–1070 (2013) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blanchet, A., Carrillo, J.A., Laurençot, P.: Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions. Calc. Var. Partial Differ. Equ. 35(2), 133–168 (2009) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blanchet, A., Dolbeault, J., Perthame, B.: Two dimensional Keller-Segel model in \(\mathbb{R}^{2}\): optimal critical mass and qualitative properties of the solution. Electron. J. Differ. Equ. 2006(44), 1–33 (2006) (electronic) Google Scholar
  4. 4.
    Burger, M., Capasso, V., Morale, D.: On an aggregation model with long and short range interactions. Nonlinear Anal., Real World Appl. 8(3), 939–958 (2007) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caffarelli, L., Soria, F., Vázquez, J.L.: Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. 15(5), 1701–1746 (2013) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Caffarelli, L., Vazquez, J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202(2), 537–565 (2011) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Calvez, V., Carrillo, J.A.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86, 155–175 (2006) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Calvez, V., Carrillo, J.A., Hoffmann, F.: Equilibria of homogeneous functionals in the fair-competition regime. Nonlinear Anal. 159, 85–128 (2017) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Calvez, V., Carrillo, J.A., Hoffmann, F.: The Geometry of Diffusing and Self-Attracting Particles in a One-Dimensional Fair-Competition Regime, pp. 1–71. Springer, Cham (2017) zbMATHGoogle Scholar
  10. 10.
    Carrillo, J.A., Hittmeir, S., Volzone, B., Yao, Y.: Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics (2016). arXiv:1603.07767
  11. 11.
    Carrillo, J.A., Hoffmann, F., Mainini, E., Volzone, B.: Ground states in the diffusion-dominated regime (2017). arXiv:1705.03519
  12. 12.
    Cotsiolis, A., Tavoularis, N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295(1), 225–236 (2004) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kowalczyk, R.: Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305, 566–588 (2005) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lieb, E.H., Loss, M.: Analysis Graduate Studies in Math., vol. 14. second edition. Amer. Math. Soc., Providence, RI (2001) zbMATHGoogle Scholar
  17. 17.
    Liu, J.-G., Wang, J.: A note on \(L^{\infty }\)-bound and uniqueness to a degenerate Keller-Segel model. Acta Appl. Math. 142, 173–188 (2016) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sugiyama, Y.: Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems. Differ. Integral Equ. 19(8), 841–876 (2006) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sugiyama, Y.: Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models. Adv. Differ. Equ. 12(2), 121–144 (2007) Google Scholar
  20. 20.
    Sugiyama, Y.: The global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis. Differ. Integral Equ. 20(2), 133–180 (2007) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Topaz, C.M., Bertozzi, A.L., Lewis, M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68, 1601–1623 (2006) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of MathematicsLiaoning UniversityShenyangP.R. China
  2. 2.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations