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

Previous Article
Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric
PROC Home
This Issue
Next Article
Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations

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

Abstract
Related Papers

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, (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.*

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, (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.*

[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]	Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[3]	Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[4]	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
[5]	Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[6]	Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[7]	Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[8]	Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[9]	Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013
[10]	Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
[11]	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
[12]	Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11
[13]	María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic & Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024
[14]	José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[15]	Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449
[16]	Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[17]	Julián López-Gómez, Pavol Quittner. Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 169-186. doi: 10.3934/dcds.2006.14.169
[18]	Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[19]	Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489
[20]	Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

Impact Factor:

American Institute of Mathematical Sciences