ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Lower Bounds of Finite-Time Blow-Up of Solutions to a Two-Species Keller-Segel Chemotaxis Model

Journal of Applied Nonlinear Dynamics 10(1) (2021) 81--93 | 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

Abstract

In this paper, we investigate the blow-up phenomena of non-negative solutions of a two-species Keller-Segel chemotaxis model with Lotka-Volterra competitive source terms. We estimate the lower bounds for the blow-up time of solutions of the model under the Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^n, n\geq 1$. The first-order differential inequality technique is applied to determine the results in various space dimensions by using different auxiliary functions.

References

1.  [1] Keller, E.F. and Segel, A. (1970), Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26, 399-415.
2.  [2] Gao, H., Fu, S., and Mohammed, H. (2018), Existence of global solution to a two-species Keller-Segel chemotaxis model, International Journal of Biomathematics, 11, 1850036, (17 pages).
3.  [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, 2305-2338.
4.  [4] Kurganov, A. and Lukacova-Medvidova, M. (2014), Numerical study of two-species chemotaxis models, Discrete and Continuous Dynamical Systems, 19, 131-152.
5.  [5] Li, Y. and Li, Y. (2014), Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Analysis: Theory, Methods $\&$ Applications, 109, 72-84.
6.  [6] Li, Y. (2015), Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species, Journal of Mathematical Analysis and Applications, 429, 1291-1304.
7.  [7] Lin, K., Mub, C., and Zhong, H. (2018), A new approach toward stabilization in a two-species chemotaxis model with logistic source, Computers $\&$ Mathematics with Applications, 75, 837-849.
8.  [8] Mizukami, M. (2017), Boundedness and asymptotic stability in a two-species chemotaxis competition model with signal-dependent sensitivity, Discrete and Continuous Dynamical Systems, 22, 2301-2319.
9.  [9] Stinner, C., Tello, J.I., and Winkler, M. (2014), Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68, 1607-1626.
10.  [10] Tello, J.I. and Winkler, M. (2012), Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25, 1413-1425.
11.  [11] Zhang, Q. and Li, Y. (2014), Global existence and asymptotic properties of the solution to a two-species chemotaxis system, Journal of Mathematical Analysis and Applications, 418, 47-63.
12.  [12] Zhao, J., Mu, C., Wang, L., and Zhou, D. (2018), Blow-up and bounded solutions in a two-species chemotaxis system in two-dimensional domains, Acta Applicandae Mathematicae, 153, 197-220.
13.  [13] An, X. and Song, X. (2017), The lower bound for the blow-up time of the solution to a quasi-linear parabolic problem, Applied Mathematics Letters, 69, 82-86.
14.  [14] Bao, A. and Song, X. (2014), Bounds for the blow-up time of the solutions to quasi-linear parabolic problems. Zeitschrift f$\ddot{u$r Angewandte Mathematik und Physik}, 65, 115-123.
15.  [15] Bhuvaneswari, V., Shangerganesh, L., and Balachandran, K. (2015), Global existence and blow-up of solutions of quasilinear chemotaxis system, Mathematical Methods in the Applied Sciences, 38, 3738-3746.
16.  [16] Chen, S. and Yu, D. (2007), Global existence and blow-up solutions for quasilinear parabolic equations, Journal of Mathematical Analysis and Applications, 335, 151-167.
17.  [17] Horstmann, D. and Winkler, M. (2005), Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215, 52-107.
18.  [18] Marras, M., Vernier Piro, S., and Viglialoro, G. (2014), Lower bounds for blow-up time in a parabolic problem with a gradient term under various boundary conditions, Kodai Mathematical Journal, 37, 532-543.
19.  [19] Marras, M., Vernier Piro, S., and Viglialoro, G. (2016), Blow-up phenomena in chemotaxis systems with a source term, Mathematical Methods in the Applied Sciences, 39, 2787-2798.
20.  [20] Marras, M. and Viglialoro, G. (2016), Blow-up time of a general Keller-Segel system with source and damping terms, Comptes Rendus de LAcademie Bulgare des Sciences, 6, 687-696.
21.  [21] Payne, L.E. and Schaefer, P.W. (2006), Lower bounds for blow-up time in parabolic problems under Neumann conditions, Applicable Analysis, 85, 1301-1311.
22.  [22] Payne, L.E., Philippin, G.A., and Vernier-Piro, S. (2010), Blow-up phenonena for a semilinear heat equation with nonlinear boundary condition II, Nonlinear Analysis: Theory, Methods $\&$ Applications, 73, 971-978.
23.  [23] Payne, L.E. and Song, J.C. (2012), Lower bounds for blow-up in a model of chemotaxis, Journal of Mathematical Analysis and Applications, 385, 672-676.
24.  [24] Sathishkumar, G., Shangerganesh, L., and Karthikeyan, S. (2019), Lower bounds for the finite-time blow-up of solutions of a cancer invasion model, Electronic Journal of Qualitative Theory of Differential Equations, 12, 1-13.
25.  [25] Shangerganesh, L., Nyamoradi, N., Sathishkumar, G., and Karthikeyan, S. (2019), Finite-time blow-up of solutions to a cancer invasion mathematical model with haptotaxis effects, Computers \& Mathematics with Applications, 77, 2242-2254.
26.  [26] Bai, X. and Winkler, M. (2016), Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana University Mathematics Journal, 65, 553-583.
27.  [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, 89-98.
28.  [28] Black, T. and Lankeit, J. (2016), On the weakly competitive case in a two-species chemotaxis model, IMA Journal of Applied Mathematics, 81, 860-876.
29.  [29] Conca, C., Espejo, E.E., and Vilches, K. (2011), Remarks on the blow-up and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, European Journal of Applied Mathematics, 22, 553-580.
30.  [30] Espejo, E.E., Stevens, A., and Vel\{a}zquez, J.J.L. (2009), Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis. International Mathematical Journal of Analysis and its Applications, 29, 317-338.
31.  [31] Espejo, E.E., Vilches, K., and Conca, C. (2013), Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, European Journal of Applied Mathematics, 24, 297-313.
32.  [32] Fujie, K. and Senba, T. (2019), Blow-up of solutions to a two-chemical substances chemotaxis system in the critical dimension, Journal of Differential Equations, 266, 942-976.
33.  [33] Wang, L., Mu, C., Hu, X., and Zheng, P. (2018), Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, Journal of Differential Equations, 264(5), 3369-3401.
34.  [34] Wang, Q., Yang, J., and Zhang, L. (2017), Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth, American Institute of Mathematical Sciences, 22, 3547-3574.
35.  [35] Wang, Q., Zhang, L., Yang, J., and Hu, J. (2015), Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinetic and Related Models, 8, 777-807.