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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. Google Scholar

[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 Google Scholar

[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 Google Scholar

[4]

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

[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 Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[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 Google Scholar

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. Google Scholar

[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 Google Scholar

[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 Google Scholar

[4]

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

[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 Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[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 Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[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 Google Scholar

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