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


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.


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

Mathematics Subject Classification (2010)

35K50 35K57 92D25 


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© 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

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