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Decay Results for a Viscoelastic Problem with Nonlinear Boundary Feedback and Logarithmic Source Term

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Abstract

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, \(g^{\prime }(t)\le -\xi (t) G(g(t))\). We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = sp and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature.

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Acknowledgments

The authors would like to express their profound gratitude to King Fahd University of Petroleum and Minerals (KFUPM) and University of Sharjah for their continuous support.

Funding

This work was funded by KFUPM under Project #SB191037.

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Authors and Affiliations

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

    Mohammad M. Al-Gharabli & Adel M. Al-Mahdi

  2. Department of Mathematics, University of Sharjah, P. O. Box, 27272, Sharjah, United Arab Emirates

    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. Decay Results for a Viscoelastic Problem with Nonlinear Boundary Feedback and Logarithmic Source Term. J Dyn Control Syst 28, 71–89 (2022). https://doi.org/10.1007/s10883-020-09522-1

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  • DOI: https://doi.org/10.1007/s10883-020-09522-1

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