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General and Optimal Decay Result for a Viscoelastic Problem with Nonlinear Boundary Feedback

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Abstract

In this paper, we consider a viscoleastic equation with a nonlinear feedback localized on a part of the boundary and a relaxation function satisfying g′(t) ≤−ξ(t)G(g(t)). We establish an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our results are obtained without imposing any restrictive growth assumption on the damping term. This work generalizes and improves earlier results in the literature, in particular those of Messaoudi (Topological Methods in Nonlinear Analysis 51(2):413–427, 2018), Messaoudi and Mustafa (Nonlinear Analysis: Theory Methods & Applications 72(9–10):3602–3611, 2010), Mustafa (Mathematical Methods in the Applied Sciences 41(1): 192–204, 2018) and Wu (Zeitschrift für angewandte Mathematik und Physik 63(1):65–106, 2012).

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Acknowledgments

The authors thank KFUPM for its continuous support. This work was funded by KFUPM under Project #IN161006.

Funding

This work was funded by KFUPM under Project #IN161006.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Mohammad M. Al-Gharabli

  2. The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Adel M. Al-Mahdi & Salim A. Messaoudi

Authors
  1. Mohammad M. Al-Gharabli
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  2. Adel M. Al-Mahdi
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  3. Salim A. Messaoudi
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Correspondence to Mohammad M. Al-Gharabli.

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Al-Gharabli, M.M., Al-Mahdi, A.M. & Messaoudi, S.A. General and Optimal Decay Result for a Viscoelastic Problem with Nonlinear Boundary Feedback. J Dyn Control Syst 25, 551–572 (2019). https://doi.org/10.1007/s10883-018-9422-y

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  • DOI: https://doi.org/10.1007/s10883-018-9422-y

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