Advertisement

Acta Applicandae Mathematicae

, Volume 160, Issue 1, pp 1–20 | Cite as

Extended Global Asymptotic Stability Conditions for a Generalized Reaction–Diffusion System

  • Salem Abdelmalek
  • Samir BendoukhaEmail author
  • Belgacem Rebiai
  • Mokhtar Kirane
Article

Abstract

In this paper, we consider the general reaction–diffusion system proposed in Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017) as a generalization of the original Lengyel–Epstein model developed for the revolutionary Turing-type CIMA reaction. We establish sufficient conditions for the global existence of solutions. We also follow the footsteps of Lisena (Appl. Math. Comput. 249:67–75, 2014) and other similar studies to extend previous results regarding the local and global asymptotic stability of the system. In the local PDE sense, more relaxed conditions are achieved compared to Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017). Also, new extended results are achieved for the global existence, which when applied to the Lengyel–Epstein system, provide weaker conditions than those of Lisena (Appl. Math. Comput. 249:67–75, 2014). Numerical examples are used to affirm the findings and benchmark them against previous results.

Keywords

Reaction diffusion equations Lengyel–Epstein system Global existence Global asymptotic stability 

Mathematics Subject Classification (2010)

35K50 35K57 92D25 

References

  1. 1.
    Abdelmalek, S., Bendoukha, S.: On the global asymptotic stability of solutions to a generalized Lengyel–Epstein system. Nonlinear Anal., Real World Appl. 35, 397–413 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abdelmalek, S., Bendoukha, S., Rebiai, B.: On the stability and nonexistence of Turing patterns for the generalised Lengyel–Epstein model. Math. Methods Appl. Sci. 40, 1–11 (2017).  https://doi.org/10.1002/mma.4457 CrossRefzbMATHGoogle Scholar
  3. 3.
    Ahmad, S., Rao, M.: Theory of Ordinary Differential Equations. Affiliated East–West Press Private Limited, Delhi (1999) Google Scholar
  4. 4.
    Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, San Diego (1985) zbMATHGoogle Scholar
  5. 5.
    Casten, R., Holland, C.J.: Stability properties of solutions to systems of reaction–diffusion equations. SIAM J. Appl. Math. 33, 353–364 (1977) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conway, E., Hoff, D., Smoller, J.: Large time behavior of solutions of systems of nonlinear reaction–diffusion equations. SIAM J. Appl. Math. 35, 1–16 (1978) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Kepper, P., Boissonade, J., Epstein, I.: Chlorite–iodide reaction: a versatile system for the study of nonlinear dynamical behavior. J. Phys. Chem. 94, 6525–6536 (1990) CrossRefGoogle Scholar
  8. 8.
    Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice–Hall, Englewood Cliffs (1964) zbMATHGoogle Scholar
  9. 9.
    Lengyel, I., Epstein, I.R.: Modeling of Turing structures in the chlorite–iodide–malonic acid–starch reaction system. Science 251, 650–652 (1991) CrossRefGoogle Scholar
  10. 10.
    Lengyel, I., Epstein, I.R.: A chemical approach to designing Turing patterns in reaction–diffusion system. Proc. Natl. Acad. Sci. USA 89, 3977–3979 (1992) CrossRefzbMATHGoogle Scholar
  11. 11.
    Lisena, B.: On the global dynamics of the Lengyel–Epstein system. Appl. Math. Comput. 249, 67–75 (2014) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ni, W.M., Tang, M.: Turing patterns in the Lengyel–Epstein system for the CIMA reaction. Trans. Am. Math. Soc. 357, 3953–3969 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Turing, A.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser 237(641), 37–72 (1952) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Weinberger, H.F.: Invariant sets for weakly coupled parabolic and elliptic systems. Rend. Mat. 8, 295–310 (1975) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mathematics and Computer Science DepartmentTebessa UniversityTebessaAlgeria
  2. 2.Department of Electrical Engineering, College of EngineeringTaibah UniversityYanbuSaudi Arabia
  3. 3.LaSIE, Faculté des Sciences, Pole Sciences et TechnologiesUniversité de La RochelleLa Rochelle CedexFrance
  4. 4.NAAM Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  5. 5.RUDN UniversityMoscowRussia

Personalised recommendations