December  2017, 22(10): 3663-3669. doi: 10.3934/dcdsb.2017147

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

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author

Received  December 2016 Revised  December 2016 Published  April 2017

Fund Project: Supported by the National Natural Science Foundation of China (11101060,11671066) and by the Fundamental Research Funds for the Central Universities (DUT16LK24)

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.

Citation: Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147
References:
[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.  Google Scholar

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P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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