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. FujieM. 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. HouC. LiuY. 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. LiT. 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. RebholzD. WangZ. WangC. 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. TaoL. 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.

show all references

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. FujieM. 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. HouC. LiuY. 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. LiT. 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. RebholzD. WangZ. WangC. 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. TaoL. 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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