American Institute of Mathematical Sciences

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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  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, 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

[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

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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

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[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

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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

[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

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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

[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

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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