Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: 35B44, 35K55, 65M06, 65M60, 82C22, 92C17.

 Citation:

•  [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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