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September  2021, 26(9): 5095-5100. doi: 10.3934/dcdsb.2020334

Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity

Xiangdong Zhao

School of Mathematics, Liaoning Normal University, Dalian 116029, China

Received  March 2020 Revised  September 2020 Published  September 2021 Early access  November 2020

Fund Project: The author is supported by the Doctoral Scientific Research Foundation of Liaoning Normal University grant No. 203070091907

We consider a chemotaxis system with singular sensitivity and logistic-type source: $ u_t = \Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k $, $ v_t = \epsilon\Delta v-v+u $ in a smooth bounded domain $ \Omega\subset\mathbb{R}^n $ with $ \chi,r,\mu,\epsilon>0 $, $ k>1 $ and $ n\ge 2 $. It is proved that the system possesses a globally bounded classical solution when $ \epsilon+\chi<1 $. This shows that the diffusive coefficient $ \epsilon $ of the chemical substance $ v $ properly small benefits the global boundedness of solutions, without the restriction on the dampening exponent $ k>1 $ in logistic source.

Keywords: Chemotaxis, Singular sensitivity, Boundedness.
Mathematics Subject Classification: Primary: 92C17; Secondary: 35Q92, 35K55.
Citation: 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
References:
[1]

J. Ahn, Global well-posedness and asymptotic stabilization for chemotaxis system with singal-dependent sensitivity, J. Differential Equations, 266 (2019), 6866-6904.  doi: 10.1016/j.jde.2018.11.015.

[2]

M. Aida, K. Osaka, T. Tsujikawa and A. Yagi, Chemotaxis and growth system with sigular sensitivity function, Nonlinear Anal. Real World Appl., 6 (2005), 323-336.  doi: 10.1016/j.nonrwa.2004.08.011.

[3]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[4] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. 
[5]

M. Ding, W. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.

[6]

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.

[7]

K. Fujie and T. Senba, Global existence and boundedness of radial solution to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.  doi: 10.1088/0951-7715/29/8/2417.

[8]

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.

[9]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.  doi: 10.1016/S0022-247X(02)00147-6.

[10]

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.

[11]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 33pp.

[12]

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.

[13]

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

[14]

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. 

[15]

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.

[16]

X. D. Zhao and S. N. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., (2017), 68.

[17]

X. D. Zhao and S. Zheng, Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source, J. Differential Equations, 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.

[18]

X. D. Zhao and S. Zheng, Asymptotic behavior to a chemotaxis consumption system with singular sensitivity, Math. Methods. Appl. Sci., 41 (2018), 2615-2624.  doi: 10.1002/mma.4762.

show all references

References:
[1]

J. Ahn, Global well-posedness and asymptotic stabilization for chemotaxis system with singal-dependent sensitivity, J. Differential Equations, 266 (2019), 6866-6904.  doi: 10.1016/j.jde.2018.11.015.

[2]

M. Aida, K. Osaka, T. Tsujikawa and A. Yagi, Chemotaxis and growth system with sigular sensitivity function, Nonlinear Anal. Real World Appl., 6 (2005), 323-336.  doi: 10.1016/j.nonrwa.2004.08.011.

[3]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.

[4] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. 
[5]

M. Ding, W. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.

[6]

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.

[7]

K. Fujie and T. Senba, Global existence and boundedness of radial solution to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.  doi: 10.1088/0951-7715/29/8/2417.

[8]

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.

[9]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.  doi: 10.1016/S0022-247X(02)00147-6.

[10]

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.

[11]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 33pp.

[12]

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.

[13]

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

[14]

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. 

[15]

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.

[16]

X. D. Zhao and S. N. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., (2017), 68.

[17]

X. D. Zhao and S. Zheng, Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source, J. Differential Equations, 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.

[18]

X. D. Zhao and S. Zheng, Asymptotic behavior to a chemotaxis consumption system with singular sensitivity, Math. Methods. Appl. Sci., 41 (2018), 2615-2624.  doi: 10.1002/mma.4762.

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