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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)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

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

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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.

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