\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A degenerate $p$-Laplacian Keller-Segel model

Abstract Related Papers Cited by
  • This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < \frac{3d}{d+1}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(3-p)}{p}}$ norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.
    Mathematics Subject Classification: Primary: 35K65, 35K92, 92C17.

    Citation:

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

    T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598.

    [2]

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

    [3]

    S. Bian, J.-G. Liu and C. Zou, Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$, Kinet. Relat. Models, 7 (2014), 9-28.doi: 10.3934/krm.2014.7.9.

    [4]

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

    [5]

    F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.

    [6]

    F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523.doi: 10.2307/1970660.

    [7]

    L. Chen, J.-G. Liu and J. 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.

    [8]

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

    [9]

    L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire}, 15 (1998), 493-516.doi: 10.1016/S0294-1449(98)80032-2.

    [10]

    E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.doi: 10.1007/978-1-4612-0895-2.

    [11]

    E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1 < p < 2$, Arch. Rational Mech. Anal., 111 (1990), 225-290.doi: 10.1007/BF00400111.

    [12]

    P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310.doi: 10.1007/BF02392210.

    [13]

    D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.

    [14]

    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.

    [15]

    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.

    [16]

    O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva and S. Smith, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I., 1968.

    [17]

    J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.

    [18]

    E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.doi: 10.1090/gsm/014.

    [19]

    J.-G. Liu and J. Wang, A note on $L^{\infty}$-bound and uniqueness to a degenerate Keller-Segel model, Acta Appl. Math., 142 (2016), 173-188.doi: 10.1007/s10440-015-0022-5.

    [20]

    G. J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038-1041.doi: 10.1073/pnas.50.6.1038.

    [21]

    G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346.doi: 10.1215/S0012-7094-62-02933-2.

    [22]

    B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.doi: 10.1007/s10492-004-6431-9.

    [23]

    B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007.

    [24]

    Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.

    [25]

    Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type, J. Differential Equations, 250 (2011), 3047-3087.doi: 10.1016/j.jde.2011.01.016.

    [26]

    G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.doi: 10.1007/BF02418013.

    [27]

    J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, The Clarendon Press, Oxford University Press, Oxford, 2007.

    [28]

    Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.doi: 10.1142/9789812799791.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(204) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return