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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. Bellomo, A. Bellouquid, Y. 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. Hashira, S. 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. Ishida, K. 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. Nagai, T. 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. Shigesada, K. 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. Szymańska, C. Morales-Rodrigo, M. 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. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. O. Budrene and H. C. Berg, Spatio-temporal patterns generated by salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189. [39] H. Yi, C. Mu, G. 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. Bellomo, A. Bellouquid, Y. 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. Hashira, S. 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. Ishida, K. 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. Nagai, T. 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. Shigesada, K. 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. Szymańska, C. Morales-Rodrigo, M. 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. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. O. Budrene and H. C. Berg, Spatio-temporal patterns generated by salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189. [39] H. Yi, C. Mu, G. 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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