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

Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity

Abstract / Introduction Related Papers Cited by
  • This paper considers the parabolic-parabolic Keller-Segel system with nonlinear sensitivity $u_t=\Delta u-\nabla (u^{\alpha}\nabla v)$, $v_t=\Delta v-v+u$, subject to homogeneous Neumann boundary conditions with smooth and bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq1$. It is proved that if $\alpha\geq\max\{1,\frac{2}{n}\}$, then the solutions are globally bounded, and both the components $u$ and $v$ decay to the same constant steady state $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u_0(x) dx$ exponentially in the $L^\infty$-norm provided both $\|u_0\|_{L^{q^{\ast}}(\Omega)}$ and $\|\nabla v_0\|_{L^{p^{\ast}}(\Omega)}$ small enough with $q^{\ast}=\frac{n\alpha K}{n+K}$, $p^{\ast}=\frac{n\alpha K}{n+K-n\alpha}$, $K\in[n,2n\alpha-n]\cap ((\alpha-1)n,\infty)$.
    Mathematics Subject Classification: Primary: 35K55, 35B35, 35B40; Secondary: 92C17.

    Citation:

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

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

    [2]

    X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.doi: 10.3934/dcds.2015.35.1891.

    [3]

    T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.doi: 10.1016/j.jde.2012.01.045.

    [4]

    T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.doi: 10.1007/s10440-013-9832-5.

    [5]

    T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.doi: 10.1016/j.jde.2014.12.004.

    [6]

    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.

    [7]

    M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.

    [8]

    D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.doi: 10.1017/S0956792501004363.

    [9]

    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.

    [10]

    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.

    [11]

    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.

    [12]

    T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.

    [13]

    T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.doi: 10.1155/S1025583401000042.

    [14]

    T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.

    [15]

    K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.

    [16]

    Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.doi: 10.1016/j.jde.2011.08.019.

    [17]

    M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.doi: 10.1016/j.jde.2010.02.008.

    [18]

    M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.doi: 10.1002/mma.1146.

    [19]

    M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.doi: 10.1016/j.matpur.2013.01.020.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(185) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return