Journal of Applied Nonlinear Dynamics
Lower Bounds of FiniteTime BlowUp of Solutions to a TwoSpecies KellerSegel Chemotaxis Model
Journal of Applied Nonlinear Dynamics 10(1) (2021) 8193  DOI:10.5890/JAND.2021.03.005
G. Sathishkumar$^1$, L. Shangerganesh$^2$, S. Karthikeyan$^1$
$^1$ Department of Mathematics, Periyar University, Salem, 636 011, India
normalsize
$^2$ Department of Applied Sciences, National Institute of Technology, Goa, 403 401, India
Download Full Text PDF
Abstract
In this paper, we investigate the blowup phenomena of nonnegative solutions of a twospecies KellerSegel chemotaxis model with LotkaVolterra competitive source terms. We estimate the lower bounds for the blowup time of solutions of the model under the Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n, n\geq 1$. The firstorder differential inequality technique is applied to determine the results in various space dimensions by using different auxiliary functions.
References

[1] 
Keller, E.F. and Segel, A. (1970), Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26, 399415.


[2] 
Gao, H., Fu, S., and Mohammed, H. (2018), Existence of global solution to a twospecies KellerSegel chemotaxis model, International Journal of Biomathematics, 11, 1850036, (17 pages).


[3] 
Issa, T.B. and Shen, W. (2018), Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments, Journal of Dynamics and Differential Equations, 31, 23052338.


[4] 
Kurganov, A. and LukacovaMedvidova, M. (2014), Numerical study of twospecies chemotaxis models, Discrete and Continuous Dynamical Systems, 19, 131152.


[5] 
Li, Y. and Li, Y. (2014), Finitetime blowup in higher dimensional fullyparabolic chemotaxis system for two species, Nonlinear Analysis: Theory, Methods $\&$ Applications, 109, 7284.


[6] 
Li, Y. (2015), Global bounded solutions and their asymptotic properties under small initial data condition in a twodimensional chemotaxis system for two species, Journal of Mathematical Analysis and Applications, 429, 12911304.


[7] 
Lin, K., Mub, C., and Zhong, H. (2018), A new approach toward stabilization in a twospecies chemotaxis model with logistic source, Computers $\&$ Mathematics with Applications, 75, 837849.


[8] 
Mizukami, M. (2017), Boundedness and asymptotic stability in a twospecies chemotaxis competition model with signaldependent sensitivity, Discrete and Continuous Dynamical Systems, 22, 23012319.


[9] 
Stinner, C., Tello, J.I., and Winkler, M. (2014), Competitive exclusion in a twospecies chemotaxis model, Journal of Mathematical Biology, 68, 16071626.


[10] 
Tello, J.I. and Winkler, M. (2012), Stabilization in a twospecies chemotaxis system with a logistic source, Nonlinearity, 25, 14131425.


[11] 
Zhang, Q. and Li, Y. (2014), Global existence and asymptotic properties of the solution to a twospecies chemotaxis system, Journal of Mathematical Analysis and Applications, 418, 4763.


[12] 
Zhao, J., Mu, C., Wang, L., and Zhou, D. (2018), Blowup and bounded solutions in a twospecies chemotaxis system in twodimensional domains, Acta Applicandae Mathematicae, 153, 197220.


[13] 
An, X. and Song, X. (2017), The lower bound for the blowup time of the solution to a quasilinear parabolic problem, Applied Mathematics Letters, 69, 8286.


[14] 
Bao, A. and Song, X. (2014), Bounds for the blowup time of the solutions to quasilinear parabolic problems. Zeitschrift f$\ddot{u$r Angewandte Mathematik und Physik}, 65, 115123.


[15] 
Bhuvaneswari, V., Shangerganesh, L., and Balachandran, K. (2015), Global existence and blowup of solutions of quasilinear chemotaxis system, Mathematical Methods in the Applied Sciences, 38, 37383746.


[16] 
Chen, S. and Yu, D. (2007), Global existence and blowup solutions for quasilinear parabolic equations, Journal of Mathematical Analysis and Applications, 335, 151167.


[17] 
Horstmann, D. and Winkler, M. (2005), Boundedness vs. blowup in a chemotaxis system, Journal of Differential Equations, 215, 52107.


[18] 
Marras, M., Vernier Piro, S., and Viglialoro, G. (2014), Lower bounds for blowup time in a parabolic problem with a gradient term under various boundary conditions, Kodai Mathematical Journal, 37, 532543.


[19] 
Marras, M., Vernier Piro, S., and Viglialoro, G. (2016), Blowup phenomena in chemotaxis systems with a source term, Mathematical Methods in the Applied Sciences, 39, 27872798.


[20] 
Marras, M. and Viglialoro, G. (2016), Blowup time of a general KellerSegel system with source and damping terms, Comptes Rendus de LAcademie Bulgare des Sciences, 6, 687696.


[21] 
Payne, L.E. and Schaefer, P.W. (2006), Lower bounds for blowup time in parabolic problems under Neumann conditions, Applicable Analysis, 85, 13011311.


[22] 
Payne, L.E., Philippin, G.A., and VernierPiro, S. (2010), Blowup phenonena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis: Theory, Methods $\&$ Applications, 73, 971978.


[23] 
Payne, L.E. and Song, J.C. (2012), Lower bounds for blowup in a model of chemotaxis, Journal of Mathematical Analysis and Applications, 385, 672676.


[24] 
Sathishkumar, G., Shangerganesh, L., and Karthikeyan, S. (2019), Lower bounds for the finitetime blowup of solutions of a cancer invasion model, Electronic Journal of Qualitative Theory of Differential Equations, 12, 113.


[25] 
Shangerganesh, L., Nyamoradi, N., Sathishkumar, G., and Karthikeyan, S. (2019), Finitetime blowup of solutions to a cancer invasion mathematical model with haptotaxis effects, Computers \& Mathematics with Applications, 77, 22422254.


[26] 
Bai, X. and Winkler, M. (2016), Equilibration in a fully parabolic twospecies chemotaxis system with competitive kinetics, Indiana University Mathematics Journal, 65, 553583.


[27] 
Biler, P., Espejo, E.E., and Gurra, I. (2013), Blowup in higher dimensional two species chemotactic systems, Communications on Pure and Applied Analysis, 12, 8998.


[28] 
Black, T. and Lankeit, J. (2016), On the weakly competitive case in a twospecies chemotaxis model, IMA Journal of Applied Mathematics, 81, 860876.


[29] 
Conca, C., Espejo, E.E., and Vilches, K. (2011), Remarks on the blowup and global existence for a two species chemotactic KellerSegel system in $\mathbb{R}^2$, European Journal of Applied Mathematics, 22, 553580.


[30] 
Espejo, E.E., Stevens, A., and Vel\{a}zquez, J.J.L. (2009), Simultaneous finite time blowup in a twospecies model for chemotaxis, Analysis. International Mathematical Journal of Analysis and its Applications, 29, 317338.


[31] 
Espejo, E.E., Vilches, K., and Conca, C. (2013), Sharp condition for blowup and global existence in a two species chemotactic KellerSegel system in $\mathbb{R}^2$, European Journal of Applied Mathematics, 24, 297313.


[32] 
Fujie, K. and Senba, T. (2019), Blowup of solutions to a twochemical substances chemotaxis system in the critical dimension, Journal of Differential Equations, 266, 942976.


[33] 
Wang, L., Mu, C., Hu, X., and Zheng, P. (2018), Boundedness and asymptotic stability of solutions to a twospecies chemotaxis system with consumption of chemoattractant, Journal of Differential Equations, 264(5), 33693401.


[34] 
Wang, Q., Yang, J., and Zhang, L. (2017), Timeperiodic and stable patterns of a twocompetingspecies KellerSegel chemotaxis model: Effect of cellular growth, American Institute of Mathematical Sciences, 22, 35473574.


[35] 
Wang, Q., Zhang, L., Yang, J., and Hu, J. (2015), Global existence and steady states of a two competing species KellerSegel chemotaxis model, Kinetic and Related Models, 8, 777807.
