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

 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.

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 [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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##### 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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