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

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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.

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Authors and Affiliations

  1. Mathematics and Computer Science Department, Tebessa University, Tebessa, 12062, Algeria

    Salem Abdelmalek & Belgacem Rebiai

  2. Department of Electrical Engineering, College of Engineering, Taibah University, Yanbu, Saudi Arabia

    Samir Bendoukha

  3. LaSIE, Faculté des Sciences, Pole Sciences et Technologies, Université de La Rochelle, Avenue M. Crepeau, 17042, La Rochelle Cedex, France

    Mokhtar Kirane

  4. NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Mokhtar Kirane

  5. RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russia

    Mokhtar Kirane

Authors
  1. Salem Abdelmalek
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  2. Samir Bendoukha
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  3. Belgacem Rebiai
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  4. Mokhtar Kirane
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Corresponding author

Correspondence to Samir Bendoukha.

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Abdelmalek, S., Bendoukha, S., Rebiai, B. et al. Extended Global Asymptotic Stability Conditions for a Generalized Reaction–Diffusion System. Acta Appl Math 160, 1–20 (2019). https://doi.org/10.1007/s10440-018-0191-0

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  • DOI: https://doi.org/10.1007/s10440-018-0191-0

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