On explicit lower bounds and blow-up times in a model of chemotaxis

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2015, 2015(special): 409-417. doi: 10.3934/proc.2015.0409

On explicit lower bounds and blow-up times in a model of chemotaxis

Maria Antonietta Farina ^1, , Monica Marras ^2, and Giuseppe Viglialoro ^2,

1.	Department of Mathematics and Computer Science, University of Cagliari, V. Ospedale 72, 09124. Cagliari, Italy
2.	Department of Mathematics and Computer Science, University of Cagliari, V. le Merello 92, 09123. Cagliari, Italy, Italy

Received September 2014 Revised October 2014 Published November 2015

Abstract
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Under a Creative Commons license

This paper is concerned with a parabolic Keller-Segel system in $\mathbb{R}^n$, with $n=2$ or $3$, under Neumann boundary conditions. First, important theoretical and general results dealing with lower bounds for blow-up time estimates are summarized and analyzed. Next, a resolution method is proposed and used to both compute the real blow-up times of such unbounded solutions and analyze and discuss some of their properties.

Keywords: Blow-up time, finite-element method, Keller-Segel system., finite-difference method, lower bounds, nonlinear parabolic systems.

Mathematics Subject Classification: 35B44, 35K55, 65M06, 65M60, 82C22, 92C1.

Citation: 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

References:

[1]	G. Acosta, R. G: Duran and J.D. Rossi, An adaptive time step procedure for a parabolic problem with blow-up, Computing., 68 (2002), 343-373
[2]	C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, J. Comput. Appl. Math., {97} (1998), 3-22
[3]	M.C. Carrisi, A further condition in the extended macroscopic approach to relativistic gases, Int. J. Pure Appl. Math., 67 (2011), 259-289
[4]	M.C. Carrisi and S. Mignemi, Snyder-de Sitter model from two-time physics, Phys. Rev. D., 82, no. 105031 (2010), 5 pages
[5]	J. M. Díaz Moreno, C. García Vázquez, M. T. González Montesinos, F. Ortegón Gallego and G. Viglialoro, Mathematical modeling of heat treatment for a steering rack including mechanical effects, J. Numer. Math., 20(3-4) (2012), 215-232.
[6]	F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuda, FreeFem++ (Third Edition, Version 3.19), Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, http://www.freefem.org/ff++/
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Deutsche Math. Verein., 105 (2003), 103-165
[8]	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
[9]	E. F. Keller and A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415
[10]	S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, Springer-Verlag, 2003
[11]	M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim., 32 (2011), 453- 468
[12]	M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic systems, Discret. Contin. Dyn. Syst., Suppl., (2011), 1025-1031
[13]	M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discret. Contin. Dyn. Syst., 32 (2012), 4001-4014
[14]	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), 297-306
[15]	M. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, 2015, (Madrid, 2014), 809-816.
[16]	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
[17]	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
[18]	G. Viglialoro and J. Murcia, A singular elliptic problem related to the membrane equilibrium equations, Int. J. Comput. Math., 90(10) (2013), 2185-2196.
[19]	G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67 (2014), 1223-1232

show all references

References:

[1]	G. Acosta, R. G: Duran and J.D. Rossi, An adaptive time step procedure for a parabolic problem with blow-up, Computing., 68 (2002), 343-373
[2]	C. Bandle and H. Brunner, Blowup in diffusion equations: A survey, J. Comput. Appl. Math., {97} (1998), 3-22
[3]	M.C. Carrisi, A further condition in the extended macroscopic approach to relativistic gases, Int. J. Pure Appl. Math., 67 (2011), 259-289
[4]	M.C. Carrisi and S. Mignemi, Snyder-de Sitter model from two-time physics, Phys. Rev. D., 82, no. 105031 (2010), 5 pages
[5]	J. M. Díaz Moreno, C. García Vázquez, M. T. González Montesinos, F. Ortegón Gallego and G. Viglialoro, Mathematical modeling of heat treatment for a steering rack including mechanical effects, J. Numer. Math., 20(3-4) (2012), 215-232.
[6]	F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuda, FreeFem++ (Third Edition, Version 3.19), Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, http://www.freefem.org/ff++/
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Deutsche Math. Verein., 105 (2003), 103-165
[8]	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
[9]	E. F. Keller and A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415
[10]	S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, Springer-Verlag, 2003
[11]	M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim., 32 (2011), 453- 468
[12]	M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic systems, Discret. Contin. Dyn. Syst., Suppl., (2011), 1025-1031
[13]	M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discret. Contin. Dyn. Syst., 32 (2012), 4001-4014
[14]	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), 297-306
[15]	M. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, 2015, (Madrid, 2014), 809-816.
[16]	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
[17]	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
[18]	G. Viglialoro and J. Murcia, A singular elliptic problem related to the membrane equilibrium equations, Int. J. Comput. Math., 90(10) (2013), 2185-2196.
[19]	G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67 (2014), 1223-1232

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