A blow-up result for the chemotaxis system with nonlinear signal production and logistic source

Hong Yi; Chunlai Mu; Guangyu Xu; Pan Dai

doi:10.3934/dcdsb.2020194

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doi: 10.3934/dcdsb.2020194

A blow-up result for the chemotaxis system with nonlinear signal production and logistic source

Hong Yi ^, , Chunlai Mu , Guangyu Xu and Pan Dai

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

^* Corresponding author

Received October 2019 Revised March 2020 Published June 2020

In this paper we consider the following chemotaxis-growth system with nonlinear signal production and logistic source

$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi\nabla\cdot( u\nabla v)+\lambda u-\mu u^{\alpha}, \quad &x\in \Omega, t>0, \\ 0 = \Delta v-\mu (t)+f(u), \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}f(u(\cdot, t)), \quad &x\in \Omega, t>0, \ \end{array} \right. \end{eqnarray*} $

with homogeneous Neumann boundary conditions in the ball

$ \Omega = B_R(0)\subset \mathbb{R}^n $

for

$ n\geq 1 $

and

$ R>0 $

, where

$ \chi, \lambda, \mu>0 $

$ \alpha>1 $

, and

$ f $

is an appropriate regular function satisfying

$ f(u)\geq ku^{\kappa} $

for all

$ u\geq1, \kappa>0 $

with some constants

$ k>0 $

. If the number

$ \kappa $

and

$ \alpha $

satisfy

$ \begin{equation*} \kappa+1>\alpha\left(\frac{2}{n}+1\right), \end{equation*} $

then there exists appropriate initial data such that the corresponding solution

$ (u, v) $

of the system blow up in finite time. This result extends the blow-up result of the chemotaxis model without logistic cell kinetics in [45]. Apparently, for the case

$ \kappa = 1 $

, this provides a rigorous detection for blow-up of solution in spaces-dimensions three and four.

Keywords: Chemotaxis, finite-time blow up, initial value problems for second-order parabolic equations, logistic source, nonlinear signal production.

Mathematics Subject Classification: Primary: 92C17, 35B44; Secondary: 35K15, 35K55, 35J60.

Citation: Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020194

References:

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[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. Google Scholar
[5]	T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009. Google Scholar
[6]	M. Fuest, Finite-time blow-up in a two-dimensional Keller-Segel system with an environmental dependent logistic source, Nonlinear Anal. Real World Appl., 52 (2020), 14pp. doi: 10.1016/j.nonrwa.2019.103022. Google Scholar
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[13]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[14]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar
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[21]	X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503. Google Scholar
[22]	Y. Li, Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production, J. Math. Anal. Appl., 480 (2019), 18pp. doi: 10.1016/j.jmaa.2019.123376. Google Scholar
[23]	G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4), 148 (1987), 77–99. doi: 10.1007/BF01774284. Google Scholar
[24]	K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046. doi: 10.3934/dcds.2016018. Google Scholar
[25]	K. Lin, C. Mu and H. Zhong, A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions, J. Math. Anal. Appl., 464 (2018), 435-455. doi: 10.1016/j.jmaa.2018.04.015. Google Scholar
[26]	P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719. doi: 10.1007/BF02461550. Google Scholar
[27]	M. R. Myerscough, P. K. Maini and K. J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. doi: 10.1006/bulm.1997.0010. Google Scholar
[28]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. Google Scholar
[29]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042. Google Scholar
[30]	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. Google Scholar
[31]	K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar
[32]	T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367. doi: 10.4310/MAA.2001.v8.n2.a9. Google Scholar
[33]	Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. Google Scholar
[34]	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. Google Scholar
[35]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. Google Scholar
[36]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar
[37]	L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789. Google Scholar
[38]	L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. Google Scholar
[39]	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. Google Scholar
[40]	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. Google Scholar
[41]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. Google Scholar
[42]	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. Google Scholar
[43]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748–767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar
[44]	M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), 40pp. doi: 10.1007/s00033-018-0935-8. Google Scholar
[45]	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. Google Scholar
[46]	M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x. Google Scholar
[47]	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. Google Scholar
[48]	T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200. doi: 10.1016/j.jmaa.2017.11.022. Google Scholar
[49]	Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484. doi: 10.1007/s00033-015-0532-z. Google Scholar
[50]	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. Google Scholar
[51]	P. Zheng, C. Mu and X. Hu, Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 2299-2323. doi: 10.3934/dcds.2015.35.2299. Google Scholar

show all references

References:

[1]	N. Bellomo, A. Belloquid, 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), 1663-1764. doi: 10.1142/S021820251550044X. Google Scholar
[2]	X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188. doi: 10.1016/j.jmaa.2013.10.061. Google Scholar
[3]	X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330. doi: 10.1002/mma.2992. Google Scholar
[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. Google Scholar
[5]	T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009. Google Scholar
[6]	M. Fuest, Finite-time blow-up in a two-dimensional Keller-Segel system with an environmental dependent logistic source, Nonlinear Anal. Real World Appl., 52 (2020), 14pp. doi: 10.1016/j.nonrwa.2019.103022. Google Scholar
[7]	H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106. Google Scholar
[8]	E. Galakhov, O. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647. doi: 10.1016/j.jde.2016.07.008. Google Scholar
[9]	X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982. doi: 10.1016/j.jmaa.2015.12.058. Google Scholar
[10]	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. Google Scholar
[11]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar
[12]	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. Google Scholar
[13]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[14]	W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar
[15]	K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017. Google Scholar
[16]	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. Google Scholar
[17]	R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588. doi: 10.1016/j.jmaa.2004.12.009. Google Scholar
[18]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968. Google Scholar
[19]	J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527. doi: 10.3934/dcdsb.2015.20.1499. Google Scholar
[20]	P. Laurençot and N. Mizoguchi, Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal Non Linéaire, 34 (2017), 197-220. doi: 10.1016/j.anihpc.2015.11.002. Google Scholar
[21]	X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503. Google Scholar
[22]	Y. Li, Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production, J. Math. Anal. Appl., 480 (2019), 18pp. doi: 10.1016/j.jmaa.2019.123376. Google Scholar
[23]	G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4), 148 (1987), 77–99. doi: 10.1007/BF01774284. Google Scholar
[24]	K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046. doi: 10.3934/dcds.2016018. Google Scholar
[25]	K. Lin, C. Mu and H. Zhong, A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions, J. Math. Anal. Appl., 464 (2018), 435-455. doi: 10.1016/j.jmaa.2018.04.015. Google Scholar
[26]	P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719. doi: 10.1007/BF02461550. Google Scholar
[27]	M. R. Myerscough, P. K. Maini and K. J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. doi: 10.1006/bulm.1997.0010. Google Scholar
[28]	T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. Google Scholar
[29]	T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042. Google Scholar
[30]	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. Google Scholar
[31]	K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar
[32]	T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367. doi: 10.4310/MAA.2001.v8.n2.a9. Google Scholar
[33]	Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. Google Scholar
[34]	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. Google Scholar
[35]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. Google Scholar
[36]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003. Google Scholar
[37]	L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789. Google Scholar
[38]	L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007. Google Scholar
[39]	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. Google Scholar
[40]	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. Google Scholar
[41]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. Google Scholar
[42]	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. Google Scholar
[43]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9), 100 (2013), 748–767. doi: 10.1016/j.matpur.2013.01.020. Google Scholar
[44]	M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), 40pp. doi: 10.1007/s00033-018-0935-8. Google Scholar
[45]	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. Google Scholar
[46]	M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x. Google Scholar
[47]	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. Google Scholar
[48]	T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200. doi: 10.1016/j.jmaa.2017.11.022. Google Scholar
[49]	Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484. doi: 10.1007/s00033-015-0532-z. Google Scholar
[50]	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. Google Scholar
[51]	P. Zheng, C. Mu and X. Hu, Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 2299-2323. doi: 10.3934/dcds.2015.35.2299. Google Scholar

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American Institute of Mathematical Sciences

A blow-up result for the chemotaxis system with nonlinear signal production and logistic source

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American Institute of Mathematical Sciences

A blow-up result for the chemotaxis system with nonlinear signal production and logistic source

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