American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 809-816. doi: 10.3934/proc.2015.0809

Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system

Monica Marras 1, , Stella Vernier Piro 2, and  Giuseppe Viglialoro 1,

1. 

Department of Mathematics and Computer Science, University of Cagliari, V. le Merello 92, 09123. Cagliari
2. 

Department of Mathematics and Computer Science, University of Cagliari, V. Ospedale 72, 09124. Cagliari, Italy

Received  September 2014 Revised  February 2015 Published  November 2015

This paper deals with a parabolic-parabolic Keller-Segel system, modeling chemotaxis, with time dependent coefficients. We consider non-negative solutions of the system which blow up in finite time $t^*$ and an explicit lower bound for $t^*$ is derived under sufficient conditions on the coefficients and the spatial domain.
Keywords: blow-up time, Nonlinear parabolic systems, chemotaxis..
Mathematics Subject Classification: Primary: 35K55, 35B44. Secondary: 92C1.
Citation: Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro. Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system. Conference Publications, 2015, 2015 (special) : 809-816. doi: 10.3934/proc.2015.0809
References:
[1]

M. A. Farina, M. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, 2015, (Madrid, 2014), 409-417.

[2]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein. 105, (3) (2003), 103-165

[3]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II, Jahresber. Deutsch. Math.-Verein, 106 (2004), 51-69

[4]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215, (1) (2005), 52-107

[5]

W. Jager and S. Luckhaus, On explosion of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329, (2) (1992), 819-824

[6]

E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415

[7]

M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Discrete Contin. Dyn. Syst., Suppl. (2007), 704-712

[8]

M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim., 32 (2011), 453- 468

[9]

M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system, Discrete Contin. Dyn. Syst., Suppl. (2011), 1025-1031

[10]

M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete Contin. Dyn. Syst., 32, (11) (2012), 4001-4014

[11]

M. Marras, S. Vernier-Piro and G. Viglialoro, Estimate from below of blow-up time in a parabolic system with gradient term, Int. J. Pure Appl. Math., 93, (2) (2014), 297-306

[12]

L. E. Payne, G.A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II, Nonlinear Analysis-Theor., 73 (2010), 971-978

[13]

L. E. Payne and G.A. Philippin, Blow-up Phenomena for a Class of Parabolic Systems with Time Dependent Coefficients, Appl. Math., 3 (2012), 325-330

[14]

L. E. Payne and P.W. Schaefer, Lower bound for blow-up time in parabolic problems under Neumann conditions, Appl. Anal., 85 (2006), 1301-1311

[15]

L. E. Payne and J.C. Song, Lower bound for blow-up in a model of chemotaxis, J. Math. Anal. Appl., 385 (2012), 672-676

[16]

N. Mizoguchi and Ph. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel System, Ann. I. H. Poincaré-AN., 31 (2014), 851-875

[17]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529

[18]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equations, 252, (3) (2012), 2520-2543.

[19]

G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67, (2014), 1223-1232

[20]

M. Winkler, Finite time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100, (5) (2013), 748-767

show all references

References:
[1]

M. A. Farina, M. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, 2015, (Madrid, 2014), 409-417.

[2]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein. 105, (3) (2003), 103-165

[3]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II, Jahresber. Deutsch. Math.-Verein, 106 (2004), 51-69

[4]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215, (1) (2005), 52-107

[5]

W. Jager and S. Luckhaus, On explosion of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329, (2) (1992), 819-824

[6]

E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415

[7]

M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Discrete Contin. Dyn. Syst., Suppl. (2007), 704-712

[8]

M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim., 32 (2011), 453- 468

[9]

M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system, Discrete Contin. Dyn. Syst., Suppl. (2011), 1025-1031

[10]

M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete Contin. Dyn. Syst., 32, (11) (2012), 4001-4014

[11]

M. Marras, S. Vernier-Piro and G. Viglialoro, Estimate from below of blow-up time in a parabolic system with gradient term, Int. J. Pure Appl. Math., 93, (2) (2014), 297-306

[12]

L. E. Payne, G.A. Philippin and S. Vernier-Piro, Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II, Nonlinear Analysis-Theor., 73 (2010), 971-978

[13]

L. E. Payne and G.A. Philippin, Blow-up Phenomena for a Class of Parabolic Systems with Time Dependent Coefficients, Appl. Math., 3 (2012), 325-330

[14]

L. E. Payne and P.W. Schaefer, Lower bound for blow-up time in parabolic problems under Neumann conditions, Appl. Anal., 85 (2006), 1301-1311

[15]

L. E. Payne and J.C. Song, Lower bound for blow-up in a model of chemotaxis, J. Math. Anal. Appl., 385 (2012), 672-676

[16]

N. Mizoguchi and Ph. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel System, Ann. I. H. Poincaré-AN., 31 (2014), 851-875

[17]

Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529

[18]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equations, 252, (3) (2012), 2520-2543.

[19]

G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67, (2014), 1223-1232

[20]

M. Winkler, Finite time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100, (5) (2013), 748-767

[1]

Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025
[2]

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, 2022  doi: 10.3934/dcdsb.2022075
[3]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[4]

Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[5]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[6]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[7]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[8]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[9]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[10]

Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[11]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194
[12]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[13]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013
[14]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
[15]

Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
[16]

Maria Antonietta Farina, Monica Marras, Giuseppe Viglialoro. On explicit lower bounds and blow-up times in a model of chemotaxis. Conference Publications, 2015, 2015 (special) : 409-417. doi: 10.3934/proc.2015.0409
[17]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[19]

María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic and Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024
[20]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

 Impact Factor: 

Tools

Metrics

  • PDF downloads (201)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy