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    Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system
February  2020, 13(2): 257-268. doi: 10.3934/dcdss.2020014

Decay in chemotaxis systems with a logistic term

Università di Cagliari, Dipartimento di Matematica e Informatica, Viale Merello 92, 09123 Cagliari, Italy

* Corresponding author: Monica Marras

Received  May 2017 Revised  May 2018 Published  January 2019

This paper is concerned with a general fully parabolic Keller-Segel system, defined in a convex bounded and smooth domain $Ω$ of $\mathbb{R}^N, $ for N∈{2, 3}, with coefficients depending on the chemical concentration, perturbed by a logistic source and endowed with homogeneous Neumann boundary conditions. For each space dimension, once a suitable energy function in terms of the solution is defined, we impose proper assumptions on the data and an exponential decay of such energies is established.

Citation: Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Decay in chemotaxis systems with a logistic term. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 257-268. doi: 10.3934/dcdss.2020014
References:
[1]

N. BellomoA. BelloquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel model of pattern formation in biological tissues, Math. Mod. Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[2]

T. Cieàlak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Diff. Eq., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[3]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1988.  Google Scholar

[4]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[5]

R. Kaiser and L. X. Xu, Nonlinear stability of the rotating Bé énard problem, the case Pr=1, Nonlin. Dffer. Equ. Appl., 5 (1998), 283-307.  doi: 10.1007/s000300050047.  Google Scholar

[6]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[7]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[8]

M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Dynamical System and Diff. Equ.(DCDS), supplement (2007), 704-712.   Google Scholar

[9]

M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete and Continuous Dynamical Systems, 32 (2012), 4001-4014.  doi: 10.3934/dcds.2012.32.4001.  Google Scholar

[10]

M. MarrasS. Vernier-Piro and G. Viglialoro, Estimates from below of blow-up time in a parabolic system with gradient term, International Journal of Pure and Applied Mathematics, 93 (2014), 297-306.   Google Scholar

[11]

M. MarrasS. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Mathematical Methods in the Applied Sciences, 39 (2016), 2787-2798.  doi: 10.1002/mma.3728.  Google Scholar

[12]

J. D. Murray, Mathematical Biology. I: An Introduction, Springer, New York, 2002.  Google Scholar

[13]

J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Springer, New York, 2003.  Google Scholar

[14]

L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model, Studies in Applied Mathematics, 123 (2009), 337-360.  doi: 10.1111/j.1467-9590.2009.00457.x.  Google Scholar

[15]

Y. Tao and S. Vernier Piro, Explicit lower bound for blow-up time in a fully parabolic chemotaxis system with nonlinear cross-diffusion, Journal of Mathematical Analysis and Applications, 436 (2016), 16-28.  doi: 10.1016/j.jmaa.2015.11.048.  Google Scholar

[16]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[17]

G. Viglialoro, Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Differential Integral Equations, 29 (2016), 359-376.   Google Scholar

[18]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[19]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxissystem with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.  Google Scholar

[20]

G. Viglialoro and T. Woolley, Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3023-3045.  doi: 10.3934/dcdsb.2017199.  Google Scholar

[21]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Part. Diff. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[22]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[23]

M. Winkler, Finite-time blow-up in higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

show all references

References:
[1]

N. BellomoA. BelloquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel model of pattern formation in biological tissues, Math. Mod. Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar

[2]

T. Cieàlak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Diff. Eq., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[3]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1988.  Google Scholar

[4]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[5]

R. Kaiser and L. X. Xu, Nonlinear stability of the rotating Bé énard problem, the case Pr=1, Nonlin. Dffer. Equ. Appl., 5 (1998), 283-307.  doi: 10.1007/s000300050047.  Google Scholar

[6]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[7]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[8]

M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Dynamical System and Diff. Equ.(DCDS), supplement (2007), 704-712.   Google Scholar

[9]

M. Marras and S. Vernier Piro, Blow-up phenomena in reaction-diffusion systems, Discrete and Continuous Dynamical Systems, 32 (2012), 4001-4014.  doi: 10.3934/dcds.2012.32.4001.  Google Scholar

[10]

M. MarrasS. Vernier-Piro and G. Viglialoro, Estimates from below of blow-up time in a parabolic system with gradient term, International Journal of Pure and Applied Mathematics, 93 (2014), 297-306.   Google Scholar

[11]

M. MarrasS. Vernier-Piro and G. Viglialoro, Blow-up phenomena in chemotaxis systems with a source term, Mathematical Methods in the Applied Sciences, 39 (2016), 2787-2798.  doi: 10.1002/mma.3728.  Google Scholar

[12]

J. D. Murray, Mathematical Biology. I: An Introduction, Springer, New York, 2002.  Google Scholar

[13]

J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Springer, New York, 2003.  Google Scholar

[14]

L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model, Studies in Applied Mathematics, 123 (2009), 337-360.  doi: 10.1111/j.1467-9590.2009.00457.x.  Google Scholar

[15]

Y. Tao and S. Vernier Piro, Explicit lower bound for blow-up time in a fully parabolic chemotaxis system with nonlinear cross-diffusion, Journal of Mathematical Analysis and Applications, 436 (2016), 16-28.  doi: 10.1016/j.jmaa.2015.11.048.  Google Scholar

[16]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[17]

G. Viglialoro, Blow-up time of a Keller-Segel-type system with Neumann and Robin boundary conditions, Differential Integral Equations, 29 (2016), 359-376.   Google Scholar

[18]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.  Google Scholar

[19]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxissystem with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.  Google Scholar

[20]

G. Viglialoro and T. Woolley, Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3023-3045.  doi: 10.3934/dcdsb.2017199.  Google Scholar

[21]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Part. Diff. Equ., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[22]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[23]

M. Winkler, Finite-time blow-up in higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

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