-
Previous Article
Area preserving geodesic curvature driven flow of closed curves on a surface
- DCDS-B Home
- This Issue
-
Next Article
A PDE model of intraguild predation with cross-diffusion
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 |
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 [
References:
[1] |
M. Aida, K. Osaki, T. Tsujikawa, A. 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. Fujie, M. 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. Fujie, M. 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. Zheng, C. Mu, X. 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. |
show all references
References:
[1] |
M. Aida, K. Osaki, T. Tsujikawa, A. 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. Fujie, M. 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. Fujie, M. 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. Zheng, C. Mu, X. 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. |
[1] |
Xiangdong Zhao. Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5095-5100. doi: 10.3934/dcdsb.2020334 |
[2] |
Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170 |
[3] |
Chun Huang. Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop. Electronic Research Archive, 2021, 29 (5) : 3261-3279. doi: 10.3934/era.2021037 |
[4] |
Guoqiang Ren, Heping Ma. Global existence in a chemotaxis system with singular sensitivity and signal production. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 343-360. doi: 10.3934/dcdsb.2021045 |
[5] |
Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463 |
[6] |
Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 |
[7] |
Jie Zhao. A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3487-3513. doi: 10.3934/dcdsb.2021193 |
[8] |
Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 |
[9] |
Youshan Tao. Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2705-2722. doi: 10.3934/dcdsb.2013.18.2705 |
[10] |
Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6155-6171. doi: 10.3934/dcdsb.2021011 |
[11] |
Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328 |
[12] |
Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 |
[13] |
Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168 |
[14] |
Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069 |
[15] |
Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81 |
[16] |
Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 |
[17] |
Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324 |
[18] |
Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3825-3849. doi: 10.3934/cpaa.2021133 |
[19] |
Miaoqing Tian, Shujuan Wang, Xia Xiao. Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022071 |
[20] |
Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022118 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]