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

Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity

  • * Corresponding author

    * Corresponding author 
Supported by the National Natural Science Foundation of China (11101060,11671066) and by the Fundamental Research Funds for the Central Universities (DUT16LK24).
Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we study the global boundedness of solutions to the fully parabolic chemotaxis system with singular sensitivity:$u_t=\Delta u-\chi\nabla·(\frac{u}{v}\nabla v)$, $v_t=k\Delta v-v+u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^{n}$ ($n\ge 2$), where $\chi, \, k>0$. It is shown that the solution is globally bounded provided $0<\chi<\frac{-(k-1)+\sqrt{(k-1)^2+\frac{8k}{n}}}{2}$. This result removes the additional restriction of $n \le 8 $ in Zhao, Zheng [15] for the global boundedness of solutions.

    Mathematics Subject Classification: Primary:35B35, 35B40, 35K55;Secondary:92C17.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] M. AidaK. OsakiT. TsujikawaA. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal. Real World Appl., 6 (2005), 323-336.  doi: 10.1016/j.nonrwa.2004.08.011.
    [2] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. 
    [3] K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.
    [4] K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 81-102.  doi: 10.3934/dcdsb.2016.21.81.
    [5] K. Fujie and T. Senba, Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.  doi: 10.1088/0951-7715/29/8/2417.
    [6] K. FujieM. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71.  doi: 10.1016/j.na.2014.06.017.
    [7] K. FujieM. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. sci., 38 (2015), 1212-1224.  doi: 10.1002/mma.3149.
    [8] K. Fujie and T. Yokota, Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity, Appl. Math. Lett., 38 (2014), 140-143.  doi: 10.1016/j.aml.2014.07.021.
    [9] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.
    [10] J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489.
    [11] T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156. 
    [12] C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal. Real World Appl., 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006.
    [13] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346.
    [14] P. ZhengC. MuX. Hua and Q. Zhang, Global boundedness in a quasilinear chemotaxis system with signal-dependent sensitivity, J. Math. Anal. Appl., 428 (2015), 508-524.  doi: 10.1016/j.jmaa.2015.03.047.
    [15] X. Zhao and S. Zheng, Global boundedness of solutions in a parabolic-parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 443 (2016), 445-452.  doi: 10.1016/j.jmaa.2016.05.036.
  • 加载中
SHARE

Article Metrics

HTML views(130) PDF downloads(210) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return