-
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
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 |
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. Google Scholar |
| [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. Google Scholar |
| [3] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II,, Jahresber. Deutsch. Math.-Verein, 106 (2004), 51. Google Scholar |
| [4] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. Google Scholar |
| [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. Google Scholar |
| [6] |
E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. Google Scholar |
| [7] |
M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems,, Discrete Contin. Dyn. Syst., (2007), 704. Google Scholar |
| [8] |
M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions,, Num. Funct. Anal. Optim., 32 (2011). Google Scholar |
| [9] |
M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system,, Discrete Contin. Dyn. Syst., (2011), 1025. Google Scholar |
| [10] |
M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems,, Discrete Contin. Dyn. Syst., 32 (2012), 4001. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [17] |
Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria,, J. Math. Anal. Appl., 381 (2011), 521. Google Scholar |
| [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. Google Scholar |
| [19] |
G. Viglialoro, On the blow-up time of a parabolic system with damping terms,, C. R. Acad. Bulg. Sci., 67 (2014), 1223. Google Scholar |
| [20] |
M. Winkler, Finite time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. Google Scholar |
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. Google Scholar |
| [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. Google Scholar |
| [3] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II,, Jahresber. Deutsch. Math.-Verein, 106 (2004), 51. Google Scholar |
| [4] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. Google Scholar |
| [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. Google Scholar |
| [6] |
E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. Google Scholar |
| [7] |
M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems,, Discrete Contin. Dyn. Syst., (2007), 704. Google Scholar |
| [8] |
M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions,, Num. Funct. Anal. Optim., 32 (2011). Google Scholar |
| [9] |
M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system,, Discrete Contin. Dyn. Syst., (2011), 1025. Google Scholar |
| [10] |
M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems,, Discrete Contin. Dyn. Syst., 32 (2012), 4001. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [17] |
Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria,, J. Math. Anal. Appl., 381 (2011), 521. Google Scholar |
| [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. Google Scholar |
| [19] |
G. Viglialoro, On the blow-up time of a parabolic system with damping terms,, C. R. Acad. Bulg. Sci., 67 (2014), 1223. Google Scholar |
| [20] |
M. Winkler, Finite time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. Google Scholar |
| [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:
Tools
Metrics
Other articles
by authors
[Back to Top]