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

Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

Abstract Related Papers Cited by
  • For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension $2$, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d\geq 3$.
    Mathematics Subject Classification: Primary: 35K65, 34C10; Secondary: 92C17.

    Citation:

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

    A. Blanchet, E. Carlen and J. A. CarrilloFunctional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, preprint, arXiv:1009.0134.

    [2]

    A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model, Comm. Pure Appl. Math., 61 (2008), 1449-1481.

    [3]

    A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.

    [4]

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

    [5]

    P.-H. Chavanis and C. Sire, Anomalous diffusion and collapse of self-gravitating Langevin particles in $D$ dimensions, Phys. Rev. E, 69 (2004), 016116.

    [6]

    J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $R^2$, C. R. Math. Acad. Sci. Paris, 339 (2004), 611-616.

    [7]

    P. L. Felmer, A. Quaas, M. Tang and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 105-119.

    [8]

    R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. J. Math., 2 (1931), 259-288.

    [9]

    M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.

    [10]

    M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.

    [11]

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

    [12]

    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.

    [13]

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

    [14]

    M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $R^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.

    [15]

    E. H. Lieb and M. Loss, "Analysis,'' Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, second ed., 2001.

    [16]

    P. M. Lushnikov, Critical chemotactic collapse, Phys. Lett. A, 374 (2010), 1678-1685.

    [17]

    T. Nagai, Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean Math. Soc., 37 (2000), 721-733.

    [18]

    Y. Naito and T. Suzuki, Self-similarity in chemotaxis systems, Colloq. Math., 111 (2008), 11-34.

    [19]

    C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.

    [20]

    J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.

    [21]

    C. Sire and P.-H. Chavanis, Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions, Phys. Rev. E, 66 (2002), 046133.

    [22]

    C. Sire and P.-H. Chavanis, Critical dynamics of self-gravitating Langevin particles and bacterial populations, Phys. Rev. E, 78 (2008), 061111.

    [23]

    D. Slepčev and M. C. Pugh, Selfsimilar blowup of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738.

    [24]

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

    [25]

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

    [26]

    T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, I, generation of the weak solution, Adv. Differential Equations, 14 (2009), 433-476.

    [27]

    M. Tang, Uniqueness of positive radial solutions for $\Delta u - u + u^p=0$ on an annulus, J. Differential Equations, 189 (2003), 148-160.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(101) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return