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

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  • 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.
    Mathematics Subject Classification: Primary: 35K55, 35B44. Secondary: 92C17.


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  • [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.


    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


    D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II, Jahresber. Deutsch. Math.-Verein, 106 (2004), 51-69


    D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215, (1) (2005), 52-107


    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


    E. F. Keller and A. Segel A, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415


    M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Discrete Contin. Dyn. Syst., Suppl. (2007), 704-712


    M. Marras, Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Num. Funct. Anal. Optim., 32 (2011), 453- 468


    M. Marras and S. Vernier Piro, Bounds for blow-up time in nonlinear parabolic system, Discrete Contin. Dyn. Syst., Suppl. (2011), 1025-1031


    M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete Contin. Dyn. Syst., 32, (11) (2012), 4001-4014


    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


    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


    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


    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


    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


    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


    Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529


    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.


    G. Viglialoro, On the blow-up time of a parabolic system with damping terms, C. R. Acad. Bulg. Sci., 67, (2014), 1223-1232


    M. Winkler, Finite time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100, (5) (2013), 748-767

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