Asymptotic Stability for a Viscoelastic Equation with Nonlinear Damping and Very General Type of Relaxation Functions

  • Farida Belhannache
  • Mohammad M. AlgharabliEmail author
  • Salim A. Messaoudi


In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and a relaxation function satisfying g′(t) ≤ −ξ(t)G(g(t)). Using the Galaerkin method, we establish the existence of the solution and prove an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This work generalizes and improves earlier results in the literature. In particular, those of Messaoudi (2016) and Mustafa (Math Methods Appl Sci. 2017;V41:192–204).


Viscoelasticity Optimal decay Relaxation functions Convexity 

Mathematics Subject Classification (2010)

35B35 35L55 75D05 74D10 93D20 



The authors thank an anonymous referee for his/her very careful reading and valuable suggestions. This work was funded by KFUPM under Project #IN161006.

Funding information

This work was funded by KFUPM under Project #IN161006.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsMohammed Seddik Ben Yahia University-JijelJijelAlgeria
  2. 2.The Preparatory Year ProgramKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  3. 3.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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