Advanced Search
Article Contents
Article Contents

Parabolic elliptic type Keller-Segel system on the whole space case

Abstract Related Papers Cited by
  • This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is closely related to the Logarithmic Hardy-Littlewood-Sobolev inequality, which directly introduced the critical mass $8\pi$. While in the higher dimension case, it is related to the Hardy-Littlewood-Sobolev inequality. Therefore, a porous media type nonlinear diffusion has been introduced in order to balance the aggregation. We will review the critical exponents which were introduced in the literature, namely, the exponent $m=2-2/n$ which comes from the scaling invariance of the mass, and the exponent $m=2n/(n+2)$ which comes from the conformal invariance of the entropy. Finally a new result on the model with a general potential, inspired from the Hardy-Littlewood-Sobolev inequality, will be given.
    Mathematics Subject Classification: Primary: 35K65, 35B45, 35J20.


    \begin{equation} \\ \end{equation}
  • [1]

    J. Bedrossian, Intermediate Asymptotics for Critical and Supercritical Aggregation Equations and Patlak-Keller-Segel models, Comm. Math. Sci., 9 (2011), 1143-1161.doi: 10.4310/CMS.2011.v9.n4.a11.


    J. Bedrossian, N. Rodríguez and A. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714.doi: 10.1088/0951-7715/24/6/001.


    W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 138 (1993), 213-242.doi: 10.2307/2946638.


    S. Bian and J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m\geq 0$, Comm. Math. Phys., 323 (2013), 1017-1070.doi: 10.1007/s00220-013-1777-z.


    P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math., 66 (1993), 131-145.


    A. Blanchet, E. A. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal., 262 (2012), 2142-2230.doi: 10.1016/j.jfa.2011.12.012.


    A. Blanchet, On the Parabolic-Elliptic Patlak-Keller-Segel System in Dimension 2 and Higher, Séminaire Laurent Schwartz$-$EDP et applications, 1-26.


    A. Blanchet, J. A. Carrillo and P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), 133-168.doi: 10.1007/s00526-008-0200-7.


    A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $\mathbbR^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481.doi: 10.1002/cpa.20225.


    A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. Eqns., 44 (2006), 32 pp. (electronic).


    A. Blanchet and P. Laurençot, Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion, Comm. Pure Appl. Math., 11 (2012), 47-60.


    L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.doi: 10.1002/cpa.3160420304.


    E. Carlen, J. A. Carrillo and M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Nat. Acad. USA, 107 (2010), 19696-19701.doi: 10.1073/pnas.1008323107.


    E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $S^n$, Geom. Funct. Anal., 2 (1992), 90-104.doi: 10.1007/BF01895706.


    J. A. Carrillo, L. Chen, J.-G. Liu and J. Wang, A note on the subcritical two dimensional Keller-Segel system, Acta appl. math., 119 (2012), 43-55.doi: 10.1007/s10440-011-9660-4.


    L. Chen, J.-G. Liu and J. H. Wang, Multidimensional degenerate Keller-Segel system with critical diffusion exponent $2n/(n+2)$, SIAM J. Math. Anal., 44 (2012), 1077-1102.doi: 10.1137/110839102.


    L. Chen and J. H. Wang, Exact criterion for global existence and blow-up to a degenerate Keller-Segel system, Doc. Math., 19 (2014), 103-120.


    X. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.doi: 10.1007/s10440-013-9858-8.


    W. X. Chen and C. M. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. Journal, 63 (1991), 615-622.doi: 10.1215/S0012-7094-91-06325-8.


    S. Childress, Chemotactic collapse in two dimensions, Lect. Notes in Biomathematics, 55 (1984), 61-68.doi: 10.1007/978-3-642-45589-6_6.


    T. Cieślak and P. Laurencot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchwski-Poisson system, C. R. Acad. Sci. Paris Ser. I, 347 (2009), 237-242.doi: 10.1016/j.crma.2009.01.016.


    T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.doi: 10.1088/0951-7715/21/5/009.


    J. M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc., 4 (1991), 553-586.doi: 10.1090/S0894-0347-1991-1102579-6.


    R. DiPerna and A. Majda, Concentrations in regularizations for 2-D incompressible flow, Comm. Pure Appl. Math., 40 (1987), 301-345.doi: 10.1002/cpa.3160400304.


    J. Dolbeault, Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion, Math. Res. Lett., 18 (2011), 1037-1050.doi: 10.4310/MRL.2011.v18.n6.a1.


    J. Dolbeault, M. J. Esteban and G. Jankowiak, The Moser-Trudinger-Onofri inequality, preprint, arXiv:1403.5042.


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


    R. L. Frank and E. H. Lieb, A new rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Spectral theory, function spaces and inequalities, Oper. Theory Adv. Appl., Springer Basel, 219 (2012), 55-67.doi: 10.1007/978-3-0348-0263-5_4.


    H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124.


    H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.doi: 10.1002/mana.19981950106.


    B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Appl. Math., 34 (1981), 525-598.doi: 10.1002/cpa.3160340406.


    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.


    D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV, 105 (2003), 103-165.


    D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II, Jahresberichte der DMV, 106 (2004), 51-69.


    D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478.doi: 10.1007/s002850100134.


    D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.doi: 10.1016/j.jde.2004.10.022.


    S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440.doi: 10.1016/j.jde.2011.02.012.


    W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.doi: 10.1090/S0002-9947-1992-1046835-6.


    J. Jang, Nonlinear instability in gravitational Euler-Poisson systems for $\gamma=6/5$, Arch. Rational Mech. Anal., 188 (2008), 265-307.doi: 10.1007/s00205-007-0086-0.


    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.


    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.


    I. Kim and Y. Yao, The patlak-keller-segel model and its variations: Properties of solutions via maximum principle, SIAM J. Math. Anal., 44 (2012), 568-602.doi: 10.1137/110823584.


    R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.doi: 10.1016/j.jmaa.2008.01.005.


    E. H. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities, Ann. Math., 118 (1983), 349-374.doi: 10.2307/2007032.


    E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, $2^{nd}$ edition, American Mathematical Society Providence, Rhode Island, 2001.doi: 10.1090/gsm/014.


    J.-G. Liu and Z. P. Xin, Convergence of Vortex Methods for Weak Solution to the 2-D Euler Equations with Vortex Sheet Data, Comm. Pure Appl. Math., 48 (1995), 611-628.doi: 10.1002/cpa.3160480603.


    J.-G. Liu and Z. P. Xin, Convergence of point vortex method for 2-D vortex sheet, Math. Comp., 70 (2001), 595-606.doi: 10.1090/S0025-5718-00-01271-0.


    S. Luckhaus and Y. Sugiyama, Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems, M2AN Math. Model. Numer. Anal., 40 (2006), 597-621.doi: 10.1051/m2an:2006025.


    A. J. Majda, Remarks on Weak Solution for Vortex Sheets with a Distinguished Sign, Indiana Univ. Math. J., 42 (1993), 921-939.doi: 10.1512/iumj.1993.42.42043.


    T. Nagai and T. Senba, Global existence and blow-up of radial solutioins to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156.


    K. J. Painter and T. Hillen, Volume filling and quorum sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.


    C. S. Patlak, Random walk with persistenc and external bias, Bull. Math. Biol. Biophys., 15 (1953), 311-338.doi: 10.1007/BF02476407.


    B. Perthame, Transport Equations in Biology, Birkhaeuser Verlag, Basel-Boston-Berlin, 2007.


    G. Rein, Non-linear stability of gaseous stars, Arch. Rational Mech. Anal., 168 (2003), 115-130.doi: 10.1007/s00205-003-0260-y.


    T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstract and Applied Analysis, (2006), Article ID 23061, 21pp.doi: 10.1155/AAA/2006/23061.


    Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Diff. Int. Equa., 19 (2006), 841-876.


    Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Diff. Eqns., 12 (2007), 121-144.


    Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.doi: 10.1016/j.jde.2006.03.003.


    Y. Yao, Asymptotic behavior of radial solutions for critical Patlak-Keller-Segel model and an repulsive-attractive aggregation equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 81-101.doi: 10.1016/j.anihpc.2013.02.002.

  • 加载中

Article Metrics

HTML views() PDF downloads(102) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint