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

Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces

Abstract Related Papers Cited by
  • In this paper, we study two types of generalized Keller-Segel system of chemotaxis. We establish the global existence and uniqueness of solutions to the semilinear Keller-Segel system of doubly parabolic type and the nonlinear nonlocal type Keller-Segel system with data in Besov spaces. Moreover, we prove the stability of solution to the first type. Our main tools are the $L^p-L^q$ estimates for $e^{-t(-\triangle)^{\theta/2}}$ in Besov spaces and the perturbation of linearization.
    Mathematics Subject Classification: Primary: 35K55, 35K15; Secondary: 92C17, 92B05.

    Citation:

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

    J. Bergh and J. L$\ddoto$fstr$\ddoto$m, "Interpolation Spaces: An Introduction," Springer, Heidelberg, 1976.

    [2]

    P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.

    [3]

    P. Biler and G. Karch, Blow up of solutions to generalized Keller-Segel model, J. Evol. Equ., 2 (2010), 247-262.doi: doi:10.1007/s00028-009-0048-0.

    [4]

    P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quatratic evolution problems, SIAM J. APPL. MATH., 59 (1998), 845-869.doi: doi:10.1137/S0036139996313447.

    [5]

    P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Meth. Appl. Sci., 32 (2009), 112-126.doi: doi:10.1002/mma.1036.

    [6]

    V. Calvez, B. Perthame and M. Sharifi Tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, Contemporary Math., 429 (2007), 45-62 Stochastic analysis and pde; Chen, Gui-Qiang (ed.) et al., AMS.

    [7]

    L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in $\mathbbR^n$, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 745-750.

    [8]

    C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.doi: doi:10.1088/0951-7715/19/12/010.

    [9]

    D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutch. Math.-Verein., 105 (2003), 103-165.

    [10]

    E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.doi: doi:10.1016/0022-5193(71)90050-6.

    [11]

    H. Kozono and Y. Sugiyama, The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ and its application to self-similar solutions, Inidana Univ. Math. J., 57 (2008), 1467-1500.doi: doi:10.1512/iumj.2008.57.3316.

    [12]

    H. Kozono and Y. Sugiyama, Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differ. Equations, 247 (2009), 1-32.doi: doi:10.1016/j.jde.2009.03.027.

    [13]

    H. Kozono, T. Ogawa and Y. Taniuchi, Navier-Stokes equations in the Besov space near $L^\infty$ and BMO, Kyushu J. Math., 57 (2003), 303-324.doi: doi:10.2206/kyushujm.57.303.

    [14]

    P. G. Lemarié-Rieusset, "Recent Development in the Navier-Stokes Problem," Chapman & Hall/CRC Press, Boca Raton, 2002.doi: doi:10.1201/9781420035674.

    [15]

    D. Li, J. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, to appear in Rev.Mat. Iberoam., 1 (2010), 295-332.

    [16]

    Y. Meyer, Wavelets, paraproducts and Navier-Stokes equations, Current developments in mathematics, 1996 (Cambridge, MA), 105-212, Internat. Press, MA, 1997.

    [17]

    C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal. TMA, 68 (2008), 461-484.doi: doi:10.1016/j.na.2006.11.011.

    [18]

    J. D. Murray, "Spatial Models and Biomedical Applications, in: Mathematical Biology," II, Third ed., in: Interdiscip. Appl. Math., vol. 18, Springer- Verlag, New York, 2003.

    [19]

    T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," de Gruyter Series in Nonlinear Analysis and Applications, vol. 3 (Berlin: Walter de Gruyter, 1996).

    [20]

    E. M. Stein, "Harmonic Analysis: Real-Varible Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, New Jersey, 1993.

    [21]

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

    [22]

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

    [23]

    H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, 1978.

    [24]

    G. Wu and J. Yuan, Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces, J. Math. Anal. Appl., 340 (2008), 1326-1335.doi: doi:10.1016/j.jmaa.2007.09.060.

    [25]

    Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl., 356 (2009), 642-658.

    [26]

    Z. Zhai, Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces, Nonlinear Analysis TMA, 72 (2010), 3173-3189.doi: doi:10.1016/j.na.2009.12.011.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(126) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return