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, Differential Equations and Applications, AIMS Proceedings, 2015, (Madrid, 2014), 409-417. [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 [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 [4] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215, (1) (2005), 52-107 [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 [6] E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415 [7] M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Discrete Contin. Dyn. Syst., Suppl. (2007), 704-712 [8] M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim., 32 (2011), 453- 468 [9] M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system, Discrete Contin. Dyn. Syst., Suppl. (2011), 1025-1031 [10] M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete Contin. Dyn. Syst., 32, (11) (2012), 4001-4014 [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 [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 [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 [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 [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 [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 [17] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529 [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. [19] G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67, (2014), 1223-1232 [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

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. [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 [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 [4] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215, (1) (2005), 52-107 [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 [6] E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415 [7] M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Discrete Contin. Dyn. Syst., Suppl. (2007), 704-712 [8] M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim., 32 (2011), 453- 468 [9] M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system, Discrete Contin. Dyn. Syst., Suppl. (2011), 1025-1031 [10] M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete Contin. Dyn. Syst., 32, (11) (2012), 4001-4014 [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 [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 [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 [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 [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 [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 [17] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529 [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. [19] G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67, (2014), 1223-1232 [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
 [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] Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022075 [3] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [4] Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569 [5] 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 [6] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [7] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 [8] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [9] Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671 [10] Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683 [11] Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194 [12] Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134 [13] Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013 [14] Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096 [15] Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 [16] 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 [17] Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11 [18] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [19] María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic and Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024 [20] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

Impact Factor: