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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

Blow-up and Lower Bounds of Solutions to a Two-Species Keller-Segel Chemotaxis Model in $mathbb{R}^2$

Journal of Applied Nonlinear Dynamics 15(1) (2026) 97--109 | DOI:10.5890/JAND.2026.03.006

G. Sathishkumar$^{1}$, L. Shangerganesh$^2$, S. Karthikeyan$^3$, J.J. Nieto$^4$

$^1$ Department of Mathematics, Faculty of Science and Humanities, SRM Institute of Science and Technology, Chennai Ramapuram, Tamil Nadu, 600 089, India

$^2$ Department of Applied Sciences, National Institute of Technology Goa, Cuncolim, Goa, 403 703, India

$^3$ Department of Mathematics, Periyar University, Salem, Tamil Nadu, 636 011, India

$^4$ CITMAga, Departamento de Estat'istica, An'alise Matem'atica e Optimizaci'on, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

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Abstract

This paper investigates the blow-up phenomena of non-negative solutions of a two-species Keller-Segel chemotaxis model with the Lotka-Volterra competitive source terms under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^2$ with smooth boundary. We establish the results for the finite time blow-up of solutions when $\frac{\chi_1}{\alpha}=\frac{\chi_2}{\beta}$ for some positive constants $\alpha$ and $\beta$. The concavity method determines the main result in a two-dimensional space domain with a suitable auxiliary function. Also, the lower bounds for the finite time blow-up of solutions using the differential inequality techniques are estimated.

Acknowledgments

The authors declare that they have no conflict of interest.

References

  1. [1]  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.
  2. [2]  Anbu, A., Natesan, B.B., Lingeshwaran, S., and Kallumgal, D. (2023), Blow-up phenomena for a sixth-order partial differential equation with a general nonlinearity, Journal of Dynamical and Control Systems, 29, 1653-1667.
  3. [3]  Jaiswala, A. and Tyagi, J. (2024), Finite time blow-up in a parabolic-elliptic Keller–Segel system with flux dependent chemotactic coefficient, Nonlinear Analysis: Real World Applications, 75, 103985.
  4. [4]  Biler, P., Boritchev, A., and Brandolese, L. (2023), Large global solutions of the parabolic-parabolic Keller–Segel system in higher dimensions, Journal of Differential Equations, 344, 891-914.
  5. [5]  Beltrán-Larrotta, C.M., Rueda-Gómez, D.A., and Villamizar-Roa, É.J. (2023), On a chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics: Theoretical and numerical analysis, Applied Numerical Mathematics, 184, 77-100.
  6. [6]  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.
  7. [7]  Biler, P., Espejo Arenas, E.E., and Guerra, I. (2013), Blow-up in higher dimensional two-species chemotactic systems, Communications on Pure and Applied Analysis, 12, 89-98.
  8. [8]  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.
  9. [9]  Conca, C., Espejo Arenas, 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.
  10. [10]  Ding, J. and Pang, W. (2023), Blow-up behavior for a degenerate parabolic system subject to Neumann boundary conditions, Applicable Analysis, 102, 3795-3811.
  11. [11]  Espejo Arenas, E.E., Stevens, A., and Velázquez, J.J.L. (2009), Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29, 317-338.
  12. [12]  Espejo Arenas, 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.
  13. [13]  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.
  14. [14]  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.
  15. [15]  Gilbarg, D. and Trudinger, N.S. (2001), Elliptic Partial Differential Equations of Second Order, Springer, New York.
  16. [16]  Guillén-González, F., Rodríguez-Bellido, M.A., and Rueda-Gómez, D.A. (2022), Comparison of two finite element schemes for a chemo-repulsion system with quadratic production, Applied Numerical Mathematics, 173, 193-210.
  17. [17]  Han, Y. (2022), Blow-up phenomena for a reaction diffusion equation with special diffusion process, Applicable Analysis, 101, 1971-1983.
  18. [18]  Horstmann, D. and Winkler, M. (2005), Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215, 52-107.
  19. [19]  Itô, S. (1992), Diffusion Equations, Translations of Mathematical Monographs, Vol. 114, American Mathematical Society, Providence, Rhode Island.
  20. [20]  Li, Y. and Li, Y. (2014), Finite time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Analysis, 109, 72-84.
  21. [21]  Lin, K., Mu, 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.
  22. [22]  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.
  23. [23]  Mizukami, M., Tanaka, Y., and Yokota, T. (2022), Can chemotactic effects lead to blow-up or not in two-species chemotaxis-competition models?, Zeitschrift für Angewandte Mathematik und Physik, 73, 239.
  24. [24]  Murray, J.D. (1993), Mathematical Biology, 2nd ed., Springer, Berlin.
  25. [25]  Nagai, T. (2001), Blow-up of non-radial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, Journal of Inequalities and Applications, 6, 37-55.
  26. [26]  Naito, Y. (2021), Blow-up criteria for the classical Keller–Segel model of chemotaxis in higher dimensions, Journal of Differential Equations, 297, 144-174.
  27. [27]  Nieto, J.J. (2022), Fractional Euler numbers and generalized proportional fractional logistic differential equation, Fractional Calculus and Applied Analysis, 25, 876-886.
  28. [28]  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.
  29. [29]  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.
  30. [30]  Rahmoune, A. (2022), Bounds for blow-up time in a nonlinear generalized heat equation, Applicable Analysis, 101, 1871-1879.
  31. [31]  Ren, G. (2020), Boundedness and stabilization in a two-species chemotaxis system with logistic source, Zeitschrift für Angewandte Mathematik und Physik, 71, 177.
  32. [32]  Sathishkumar, G., Shangerganesh, L., and Karthikeyan, S. (2018), Lower bounds of finite-time blow-up of solutions to a two-species Keller–Segel chemotaxis model, Journal of Applied Nonlinear Dynamics, 7, 55-67.
  33. [33]  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.
  34. [34]  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.
  35. [35]  Tanaka, Y. (2023), Finite-time blow-up in a two-species chemotaxis-competition model with degenerate diffusion, Acta Applicandae Mathematicae, 186, 13.
  36. [36]  Tao, Y. and Winkler, M. (2014), Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant, Journal of Differential Equations, 257, 784-815.
  37. [37]  Tello, J.I. and Winkler, M. (2012), Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25, 1413-1425.
  38. [38]  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.
  39. [39]  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, Discrete and Continuous Dynamical Systems Series B, 22, 3547-3574.
  40. [40]  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, 3369-3401.
  41. [41]  Winkler, M. (2010), Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35, 1516-1537.
  42. [42]  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.

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