# American Institute of Mathematical Sciences

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

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

 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.
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, (2015), 409. [2] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I,, Jahresber. Deutsch. Math.-Verein. 105, 105 (2003), 103. [3] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II,, Jahresber. Deutsch. Math.-Verein, 106 (2004), 51. [4] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. [5] W. Jager and S. Luckhaus, On explosion of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819. [6] E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. [7] M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems,, Discrete Contin. Dyn. Syst., (2007), 704. [8] M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions,, Num. Funct. Anal. Optim., 32 (2011). [9] M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system,, Discrete Contin. Dyn. Syst., (2011), 1025. [10] M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems,, Discrete Contin. Dyn. Syst., 32 (2012), 4001. [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 (2014), 297. [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. [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. [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. [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. [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. [17] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria,, J. Math. Anal. Appl., 381 (2011), 521. [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 (2012), 2520. [19] G. Viglialoro, On the blow-up time of a parabolic system with damping terms,, C. R. Acad. Bulg. Sci., 67 (2014), 1223. [20] M. Winkler, Finite time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.

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##### 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, (2015), 409. [2] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I,, Jahresber. Deutsch. Math.-Verein. 105, 105 (2003), 103. [3] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II,, Jahresber. Deutsch. Math.-Verein, 106 (2004), 51. [4] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. [5] W. Jager and S. Luckhaus, On explosion of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819. [6] E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. [7] M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems,, Discrete Contin. Dyn. Syst., (2007), 704. [8] M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions,, Num. Funct. Anal. Optim., 32 (2011). [9] M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system,, Discrete Contin. Dyn. Syst., (2011), 1025. [10] M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems,, Discrete Contin. Dyn. Syst., 32 (2012), 4001. [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 (2014), 297. [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. [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. [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. [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. [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. [17] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria,, J. Math. Anal. Appl., 381 (2011), 521. [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 (2012), 2520. [19] G. Viglialoro, On the blow-up time of a parabolic system with damping terms,, C. R. Acad. Bulg. Sci., 67 (2014), 1223. [20] M. Winkler, Finite time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.
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