# American Institute of Mathematical Sciences

April  2022, 27(4): 2065-2075. doi: 10.3934/dcdsb.2021122

## Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source

 1 College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404100, China 2 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

* Corresponding author: Liangwei Wang

Received  October 2020 Revised  February 2021 Published  April 2022 Early access  April 2021

In this paper, we consider the following chemotaxis-consumption model with porous medium diffusion and singular sensitivity
 \begin{align*} \left\{ \begin{aligned} &u_{t} = \Delta u^{m}-\chi \mathrm{div}(\frac{u}{v}\nabla v)+\mu u(1-u), \\ &v_{t} = \Delta v-u^{r}v, \end{aligned}\right. \end{align*}
in a bounded domain
 $\Omega\subset\mathbb R^N$
(
 $N\ge 2$
) with zero-flux boundary conditions. It is shown that if
 $r<\frac{4}{N+2}$
, for arbitrary case of fast diffusion (
 $m\le 1$
) and slow diffusion
 $(m>1)$
, this problem admits a locally bounded global weak solution. It is worth mentioning that there are no smallness restrictions on the initial datum and chemotactic coefficient.
Citation: Langhao Zhou, Liangwei Wang, Chunhua Jin. Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2065-2075. doi: 10.3934/dcdsb.2021122
##### References:
 [1] 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. [2] K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Analysis, 109 (2014), 56-71.  doi: 10.1016/j.na.2014.06.017. [3] Q. Hou, C. Liu, Y. Wang and Z. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: One-dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.  doi: 10.1137/17M112748X. [4] Q. Hou and Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl. (9), 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008. [5] C. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.  doi: 10.1016/j.jde.2017.06.034. [6] C. Jin, Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1675-1688.  doi: 10.3934/dcdsb.2018069. [7] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8. [8] J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations, 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007. [9] E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0. [10] E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012. [11] J. Li, T. Li and Z. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.  doi: 10.1142/S0218202514500389. [12] J. Lankeit and G. Viglialoro, Global existence and boundedness of solutions to a chemotaxis-consumption model with singular sensitivity, Acta Appl. Math., 167 (2020), 75-97.  doi: 10.1007/s10440-019-00269-x. [13] H. Li and K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-308.  doi: 10.1016/j.jde.2014.09.014. [14] T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.  doi: 10.1007/BF00160334. [15] L. Rebholz, D. Wang, Z. Wang, C. Zerfas and K. Zhao, Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 3789-3838.  doi: 10.3934/dcds.2019154. [16] Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180. [17] Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin, Dyn. Syst. Ser. B, 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821. [18] G. Viglialoro, Global existence in a two-dimensional chemotaxis-consumption model with weakly singular sensitivity, Applied Mathematics Letters, 91 (2019), 121-127.  doi: 10.1016/j.aml.2018.12.012. [19] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [20] M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differential Equations, 264 (2018), 2310-2350.  doi: 10.1016/j.jde.2017.10.029. [21] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190.  doi: 10.1002/mma.1346. [22] Z. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601. [23] J. Yan and Y. Li, Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity, Nonlinear Analysis, 176 (2018), 288-302.  doi: 10.1016/j.na.2018.06.016. [24] X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 2, 13pp. doi: 10.1007/s00033-016-0749-5.

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##### References:
 [1] 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. [2] K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Analysis, 109 (2014), 56-71.  doi: 10.1016/j.na.2014.06.017. [3] Q. Hou, C. Liu, Y. Wang and Z. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: One-dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.  doi: 10.1137/17M112748X. [4] Q. Hou and Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl. (9), 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008. [5] C. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.  doi: 10.1016/j.jde.2017.06.034. [6] C. Jin, Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1675-1688.  doi: 10.3934/dcdsb.2018069. [7] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8. [8] J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations, 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007. [9] E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0. [10] E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012. [11] J. Li, T. Li and Z. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.  doi: 10.1142/S0218202514500389. [12] J. Lankeit and G. Viglialoro, Global existence and boundedness of solutions to a chemotaxis-consumption model with singular sensitivity, Acta Appl. Math., 167 (2020), 75-97.  doi: 10.1007/s10440-019-00269-x. [13] H. Li and K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-308.  doi: 10.1016/j.jde.2014.09.014. [14] T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.  doi: 10.1007/BF00160334. [15] L. Rebholz, D. Wang, Z. Wang, C. Zerfas and K. Zhao, Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 3789-3838.  doi: 10.3934/dcds.2019154. [16] Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180. [17] Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin, Dyn. Syst. Ser. B, 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821. [18] G. Viglialoro, Global existence in a two-dimensional chemotaxis-consumption model with weakly singular sensitivity, Applied Mathematics Letters, 91 (2019), 121-127.  doi: 10.1016/j.aml.2018.12.012. [19] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [20] M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differential Equations, 264 (2018), 2310-2350.  doi: 10.1016/j.jde.2017.10.029. [21] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190.  doi: 10.1002/mma.1346. [22] Z. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601. [23] J. Yan and Y. Li, Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity, Nonlinear Analysis, 176 (2018), 288-302.  doi: 10.1016/j.na.2018.06.016. [24] X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 2, 13pp. doi: 10.1007/s00033-016-0749-5.
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