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doi: 10.3934/dcdsb.2022075
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Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production

Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

*Corresponding author: Yuya Tanaka

Received  December 2021 Revised  February 2022 Early access April 2022

Fund Project: The second author is supported by JSPS KAKENHI Grant Number 21K03278

This paper deals with finite-time blow-up of solutions to the quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production,
$ \begin{align*} \begin{cases} u_t = \Delta u^m - \chi \nabla \cdot (u^\alpha \nabla v) + \lambda u - \mu u^\kappa, \quad &x \in \Omega, \ t>0, \\ 0 = \Delta v - \overline{M_\ell}(t) + u^\ell, \quad &x \in \Omega, \ t>0, \end{cases} \end{align*} $
where
$ \Omega: = B_R(0) \subset \mathbb{R}^n \ (n \in \mathbb{N}) $
be a ball with some
$ R>0 $
and
$ m\ge1 $
,
$ \chi>0 $
,
$ \alpha\ge1 $
,
$ \lambda>0 $
,
$ \mu>0 $
,
$ \kappa>1 $
,
$ \ell>0 $
as well as
$ \overline{M_\ell}(t) $
is the average of
$ u^\ell $
over
$ \Omega $
. As to the corresponding system with nondegenerate diffusion, finite-time blow-up has been obtained under the condition that
$ \alpha-\ell>\max\left\{\overline{m} +\frac{2}{n}\kappa, \kappa\right\} $
, where
$ \overline{m}: = \max\{m,0\} $
in a previous paper [26], which is based a work by Fuest [7]. The purpose of this paper is to establish finite-time blow-up for the above degenerate chemotaxis system within a concept of weak solutions with a moment inequality leading to blow-up.
Citation: Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022075
References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

T. Black, M. Fuest and J. Lankeit, Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems, Z. Angew. Math. Phys., 72 (2021), Paper No. 96, 23pp. doi: 10.1007/s00033-021-01524-8.

[3]

X. Cao, Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.

[4]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[5]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[6]

S. Frassu and G. Viglialoro, Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation, Acta Appl. Math., 171 (2021), Paper No. 19, 20 pp. doi: 10.1007/s10440-021-00386-6.

[7]

M. Fuest, Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 16, 17pp. doi: 10.1007/s00030-021-00677-9.

[8]

T. HashiraS. Ishida and T. Yokota, Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type, J. Differential Equations, 264 (2018), 6459-6485.  doi: 10.1016/j.jde.2018.01.038.

[9]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[11]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

[12]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type, J. Differential Equations, 252 (2012), 1421-1440.  doi: 10.1016/j.jde.2011.02.012.

[13]

S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569.

[14]

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.

[15]

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.

[16]

J. Lankeit, Infinite time blow-up of many solutions to a general quasilinear parabolic–elliptic Keller–Segel system, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 233-255.  doi: 10.3934/dcdss.2020013.

[17]

J. Lankeit and M. Winkler, Facing low regularity in chemotaxis systems, Jahresber. Dtsch. Math.-Ver., 122 (2020), 35-64.  doi: 10.1365/s13291-019-00210-z.

[18]

D. Liu and Y. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ. Ser. B, 31 (2016), 379-388.  doi: 10.1007/s11766-016-3386-z.

[19]

N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller–Segel system, preprint.

[20]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller–Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[22]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[23]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.

[24]

Z. SzymańskaC. Morales-RodrigoM. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[25]

Y. Tanaka, Blow-up in a quasilinear parabolic–elliptic Keller–Segel system with logistic source, Nonlinear Anal. Real World Appl., 63 (2022), Paper No. 103396, 29 pp. doi: 10.1016/j.nonrwa.2021.103396.

[26]

Y. Tanaka, Boundedness and finite-time blow-up in a quasilinear parabolic–elliptic chemotaxis system with logistic source and nonlinear production, J. Math. Anal. Appl., 506 (2022), Paper No. 125654, 29 pp. doi: 10.1016/j.jmaa.2021.125654.

[27]

Y. Tanaka, G. Viglialoro and T. Yokota, Remarks on two connected papers about Keller–Segel systems with nonlinear production, J. Math. Anal. Appl., 501 (2021), Paper No. 125188, 5 pp. doi: 10.1016/j.jmaa.2021.125188.

[28]

Y. Tanaka and T. Yokota, Blow-up in a parabolic–elliptic Keller–Segel system with density-dependent sublinear sensitivity and logistic source, Math. Methods Appl. Sci., 43 (2020), 7372-7396.  doi: 10.1002/mma.6475.

[29]

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.

[30]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[31]

M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[32]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.

[33]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[34]

M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e.

[35]

M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[36]

M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities, J. Differential Equations, 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019.

[37]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[38]

D. E. WoodwardR. TysonM. R. MyerscoughJ. D. MurrayE. O. Budrene and H. C. Berg, Spatio-temporal patterns generated by salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189. 

[39]

H. YiC. MuG. Xu and P. Dai, A blow-up result for the chemotaxis system with nonlinear signal production and logistic source, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2537-2559.  doi: 10.3934/dcdsb.2020194.

[40]

J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

[41]

J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi-linear chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 97 (2017), 414-421.  doi: 10.1002/zamm.201600166.

show all references

References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

T. Black, M. Fuest and J. Lankeit, Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic–elliptic Keller–Segel systems, Z. Angew. Math. Phys., 72 (2021), Paper No. 96, 23pp. doi: 10.1007/s00033-021-01524-8.

[3]

X. Cao, Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.

[4]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[5]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[6]

S. Frassu and G. Viglialoro, Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation, Acta Appl. Math., 171 (2021), Paper No. 19, 20 pp. doi: 10.1007/s10440-021-00386-6.

[7]

M. Fuest, Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 16, 17pp. doi: 10.1007/s00030-021-00677-9.

[8]

T. HashiraS. Ishida and T. Yokota, Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type, J. Differential Equations, 264 (2018), 6459-6485.  doi: 10.1016/j.jde.2018.01.038.

[9]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[10]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[11]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

[12]

S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type, J. Differential Equations, 252 (2012), 1421-1440.  doi: 10.1016/j.jde.2011.02.012.

[13]

S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.  doi: 10.3934/dcdsb.2013.18.2569.

[14]

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.

[15]

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.

[16]

J. Lankeit, Infinite time blow-up of many solutions to a general quasilinear parabolic–elliptic Keller–Segel system, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 233-255.  doi: 10.3934/dcdss.2020013.

[17]

J. Lankeit and M. Winkler, Facing low regularity in chemotaxis systems, Jahresber. Dtsch. Math.-Ver., 122 (2020), 35-64.  doi: 10.1365/s13291-019-00210-z.

[18]

D. Liu and Y. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ. Ser. B, 31 (2016), 379-388.  doi: 10.1007/s11766-016-3386-z.

[19]

N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller–Segel system, preprint.

[20]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[21]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller–Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[22]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[23]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.

[24]

Z. SzymańskaC. Morales-RodrigoM. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[25]

Y. Tanaka, Blow-up in a quasilinear parabolic–elliptic Keller–Segel system with logistic source, Nonlinear Anal. Real World Appl., 63 (2022), Paper No. 103396, 29 pp. doi: 10.1016/j.nonrwa.2021.103396.

[26]

Y. Tanaka, Boundedness and finite-time blow-up in a quasilinear parabolic–elliptic chemotaxis system with logistic source and nonlinear production, J. Math. Anal. Appl., 506 (2022), Paper No. 125654, 29 pp. doi: 10.1016/j.jmaa.2021.125654.

[27]

Y. Tanaka, G. Viglialoro and T. Yokota, Remarks on two connected papers about Keller–Segel systems with nonlinear production, J. Math. Anal. Appl., 501 (2021), Paper No. 125188, 5 pp. doi: 10.1016/j.jmaa.2021.125188.

[28]

Y. Tanaka and T. Yokota, Blow-up in a parabolic–elliptic Keller–Segel system with density-dependent sublinear sensitivity and logistic source, Math. Methods Appl. Sci., 43 (2020), 7372-7396.  doi: 10.1002/mma.6475.

[29]

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.

[30]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[31]

M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[32]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.

[33]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[34]

M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e.

[35]

M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[36]

M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller–Segel systems with bounded sensitivities, J. Differential Equations, 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019.

[37]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045.

[38]

D. E. WoodwardR. TysonM. R. MyerscoughJ. D. MurrayE. O. Budrene and H. C. Berg, Spatio-temporal patterns generated by salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189. 

[39]

H. YiC. MuG. Xu and P. Dai, A blow-up result for the chemotaxis system with nonlinear signal production and logistic source, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2537-2559.  doi: 10.3934/dcdsb.2020194.

[40]

J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

[41]

J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi-linear chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 97 (2017), 414-421.  doi: 10.1002/zamm.201600166.

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