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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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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