doi: 10.3934/dcdsb.2022009
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Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities

Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

*Corresponding author: Michael Winkler

Received  September 2021 Revised  December 2021 Early access January 2022

Fund Project: The author is supported by Deutsche Forschungsgemeinschaft (No. 411007140)

The chemotaxis system
$ \begin{array}{l}\left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(u) \nabla u \big) - \nabla \cdot \big( uS(x, u, v)\cdot \nabla v\big), \\ v_t = \Delta v -uv, \end{array} \right. \end{array} $
is considered in a bounded domain
$ \Omega\subset \mathbb{R}^n $
,
$ n\ge 2 $
, with smooth boundary.
It is shown that if
$ D: [0, \infty) \to [0, \infty) $
and
$ S: \overline{\Omega}\times [0, \infty)\times (0, \infty)\to \mathbb{R}^{n\times n} $
are suitably smooth functions satisfying
$ \begin{array}{l}D(u) \ge k_D u^{m-1} \qquad {\rm{for\; all}}\; u\ge 0 \end{array} $
and
$ \begin{array}{l}|S(x, u, v)| \le \frac{S_0(v)}{v^\alpha} \qquad {\rm{for\; all}}\; (x, u, v)\; \in \Omega\times (0, \infty)^2 \end{array} $
with some
$ \begin{array}{l}m>\frac{3n-2}{2n} \qquad {\rm{and}}\;\alpha\in [0, 1), \end{array} $
and with some
$ k_D>0 $
and nondecreasing
$ S_0: (0, \infty)\to (0, \infty) $
, then for all suitably regular initial data a corresponding no-flux type initial-boundary value problem admits a global bounded weak solution which actually is smooth and classical if
$ D(0)>0 $
.
Citation: Michael Winkler. Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022009
References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. 

[2]

J. AhnK. Kang and C. Yoon, Global classical solutions for chemotaxis-fluid systems in two dimensions, Math. Meth. Appl. Sci., 44 (2021), 2254-2264.  doi: 10.1002/mma.6838.

[3]

H.-W. Alt, Lineare Funktionalanalysis, Springer-Verlag, Berlin/Heidelberg, 2006.

[4]

T. Black, Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differential Equations, 265 (2018), 2296-2339.  doi: 10.1016/j.jde.2018.04.035.

[5]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Part. Differ. Eq., 55 (2016), 107.  doi: 10.1007/s00526-016-1027-2.

[6]

R. DuanA. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.

[7]

Z. Jia and Z. Yang, Large time behavior to a chemotaxis-consumption model with singular sensitivity and logistic source, Math. Meth. Appl. Sci., 44 (2021), 3630-3645.  doi: 10.1002/mma.6971.

[8]

Z. JiaZ. Yang and Q. Li, Global existence, boundedness and asymptotic behavior to a chemotaxis model with singular sensitivity and logistic source, Appl. Anal., 100 (2021), 1471-1486.  doi: 10.1080/00036811.2019.1646421.

[9]

H.-Y. JinJ. Li and Z.-A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.  doi: 10.1016/j.jde.2013.04.002.

[10]

Y. V. KalininL. Jiang and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448.  doi: 10.1016/j.bpj.2008.10.027.

[11]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 26 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[12]

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.

[13]

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.

[14]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.

[15]

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.

[16]

T. LiA. SuenM. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746.  doi: 10.1142/S0218202515500177.

[17]

T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.  doi: 10.1137/09075161X.

[18]

T. Li and Z.-A. Wang, Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.  doi: 10.1142/S0218202510004830.

[19]

D. Liu, Global classical solution to a chemotaxis consumption model with singular sensitivity, Nonlinear Anal. Real World Appl., 41 (2018), 497-508.  doi: 10.1016/j.nonrwa.2017.11.004.

[20]

D. Liu, Global solutions in a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production, J. Math. Phys., 61 (2020), 021503.  doi: 10.1063/1.5111650.

[21]

N. Mizoguchi and Ph. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.

[22]

H. G. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.

[23]

P. Y. H. PangY. Wang and J. Yin, Asymptotic profile of a two-dimensional chemotaxis-Navier-Stokes system with singular sensitivity and logistic source, Math. Models Methods Appl. Sci., 31 (2021), 577-618.  doi: 10.1142/S0218202521500135.

[24]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[25]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[26]

Y. TaoL. Wang and Z.-A. 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.

[27]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., Vol. 2, North-Holland, Amsterdam, 1977.

[28]

G. Viglialoro, Global existence in a two-dimensional chemotaxis-consumption model with weakly singular sensitivity, Appl. Math. Lett., 91 (2019), 121-127.  doi: 10.1016/j.aml.2018.12.012.

[29]

L. WangC. MuK. Lin and J. Zhao, Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Z. Angew. Math. Phys., 66 (2015), 1633-1648.  doi: 10.1007/s00033-014-0491-9.

[30]

L. WangC. Mu and S. Zhou, Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion, Z. Angew. Math. Phys., 65 (2014), 1137-1152.  doi: 10.1007/s00033-013-0375-4.

[31]

Y. Wang, Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity, Bound. Value Probl., (2016), 177. doi: 10.1186/s13661-016-0687-3.

[32]

Z.-A. WangZ. Xiang and P. Yu, Asymptotic dynamics in a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.

[33]

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.

[34]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[35]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Part. Differ. Eq., 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.

[36]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.

[37]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

[38]

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.

[39]

J. XingP. ZhengY. Xiang and H. Wang, On a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption, Z. Angew. Math. Phys., 72 (2021), 105.  doi: 10.1007/s00033-021-01534-6.

[40]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.

[41]

J. Yan and Y. Li, Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity, Nonlinear Anal., 176 (2018), 288-302.  doi: 10.1016/j.na.2018.06.016.

[42]

Q. Zhang and Y. Li, Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant, J. Math. Phys., 56 (2015), 081506.  doi: 10.1063/1.4929658.

[43]

X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), 2.  doi: 10.1007/s00033-016-0749-5.

[44]

X. Zhao and S. Zheng, Asymptotic behavior to a chemotaxis consumption system with singular sensitivity, Math. Meth. Appl. Sci., 41 (2018), 2615-2624.  doi: 10.1002/mma.4762.

[45]

X. Zhao and S. Zheng, Global existence and asymptotic behavior to a chemotaxis-consumption system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 42 (2018), 120-139.  doi: 10.1016/j.nonrwa.2017.12.007.

[46]

J. Zheng and Y. Wang, A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 669-686.  doi: 10.3934/dcdsb.2017032.

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. 

[2]

J. AhnK. Kang and C. Yoon, Global classical solutions for chemotaxis-fluid systems in two dimensions, Math. Meth. Appl. Sci., 44 (2021), 2254-2264.  doi: 10.1002/mma.6838.

[3]

H.-W. Alt, Lineare Funktionalanalysis, Springer-Verlag, Berlin/Heidelberg, 2006.

[4]

T. Black, Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differential Equations, 265 (2018), 2296-2339.  doi: 10.1016/j.jde.2018.04.035.

[5]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Part. Differ. Eq., 55 (2016), 107.  doi: 10.1007/s00526-016-1027-2.

[6]

R. DuanA. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.

[7]

Z. Jia and Z. Yang, Large time behavior to a chemotaxis-consumption model with singular sensitivity and logistic source, Math. Meth. Appl. Sci., 44 (2021), 3630-3645.  doi: 10.1002/mma.6971.

[8]

Z. JiaZ. Yang and Q. Li, Global existence, boundedness and asymptotic behavior to a chemotaxis model with singular sensitivity and logistic source, Appl. Anal., 100 (2021), 1471-1486.  doi: 10.1080/00036811.2019.1646421.

[9]

H.-Y. JinJ. Li and Z.-A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.  doi: 10.1016/j.jde.2013.04.002.

[10]

Y. V. KalininL. Jiang and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448.  doi: 10.1016/j.bpj.2008.10.027.

[11]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 26 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[12]

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.

[13]

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.

[14]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.

[15]

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.

[16]

T. LiA. SuenM. Winkler and C. Xue, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746.  doi: 10.1142/S0218202515500177.

[17]

T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.  doi: 10.1137/09075161X.

[18]

T. Li and Z.-A. Wang, Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.  doi: 10.1142/S0218202510004830.

[19]

D. Liu, Global classical solution to a chemotaxis consumption model with singular sensitivity, Nonlinear Anal. Real World Appl., 41 (2018), 497-508.  doi: 10.1016/j.nonrwa.2017.11.004.

[20]

D. Liu, Global solutions in a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production, J. Math. Phys., 61 (2020), 021503.  doi: 10.1063/1.5111650.

[21]

N. Mizoguchi and Ph. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.

[22]

H. G. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.

[23]

P. Y. H. PangY. Wang and J. Yin, Asymptotic profile of a two-dimensional chemotaxis-Navier-Stokes system with singular sensitivity and logistic source, Math. Models Methods Appl. Sci., 31 (2021), 577-618.  doi: 10.1142/S0218202521500135.

[24]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[25]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.  doi: 10.1016/j.jde.2011.07.010.

[26]

Y. TaoL. Wang and Z.-A. 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.

[27]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., Vol. 2, North-Holland, Amsterdam, 1977.

[28]

G. Viglialoro, Global existence in a two-dimensional chemotaxis-consumption model with weakly singular sensitivity, Appl. Math. Lett., 91 (2019), 121-127.  doi: 10.1016/j.aml.2018.12.012.

[29]

L. WangC. MuK. Lin and J. Zhao, Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Z. Angew. Math. Phys., 66 (2015), 1633-1648.  doi: 10.1007/s00033-014-0491-9.

[30]

L. WangC. Mu and S. Zhou, Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion, Z. Angew. Math. Phys., 65 (2014), 1137-1152.  doi: 10.1007/s00033-013-0375-4.

[31]

Y. Wang, Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity, Bound. Value Probl., (2016), 177. doi: 10.1186/s13661-016-0687-3.

[32]

Z.-A. WangZ. Xiang and P. Yu, Asymptotic dynamics in a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.

[33]

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.

[34]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[35]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Part. Differ. Eq., 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2.

[36]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.

[37]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

[38]

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.

[39]

J. XingP. ZhengY. Xiang and H. Wang, On a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption, Z. Angew. Math. Phys., 72 (2021), 105.  doi: 10.1007/s00033-021-01534-6.

[40]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.

[41]

J. Yan and Y. Li, Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity, Nonlinear Anal., 176 (2018), 288-302.  doi: 10.1016/j.na.2018.06.016.

[42]

Q. Zhang and Y. Li, Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant, J. Math. Phys., 56 (2015), 081506.  doi: 10.1063/1.4929658.

[43]

X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), 2.  doi: 10.1007/s00033-016-0749-5.

[44]

X. Zhao and S. Zheng, Asymptotic behavior to a chemotaxis consumption system with singular sensitivity, Math. Meth. Appl. Sci., 41 (2018), 2615-2624.  doi: 10.1002/mma.4762.

[45]

X. Zhao and S. Zheng, Global existence and asymptotic behavior to a chemotaxis-consumption system with singular sensitivity and logistic source, Nonlinear Anal. Real World Appl., 42 (2018), 120-139.  doi: 10.1016/j.nonrwa.2017.12.007.

[46]

J. Zheng and Y. Wang, A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 669-686.  doi: 10.3934/dcdsb.2017032.

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