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A new general decay result for abstract evolution equation with time-dependent nonlinear dissipation

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Abstract

The following abstract problem:

$$\begin{aligned} u_{tt}(t)+A u(t)-\int _{0}^{t}g(t-s)A u(s)ds+\eta (t)h(u_t)=\nabla F (u(t)) \end{aligned}$$

of second order evolution equation with time-dependent dissipation is considered. We prove, under a very general and wide class of kernel, a decay result. We establish optimal explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our result improves and generalizes many other results in the literature.

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Acknowledgements

The authors thank KFUPM for its continuous support. This work is funded by KFUPM under Project Number SB181039.

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Authors and Affiliations

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

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

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

    Mohammad Kafini

Authors
  1. Adel M. Al-Mahdi
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  2. Mohammad M. Al-Gharabli
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  3. Mohammad Kafini
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Correspondence to Mohammad Kafini.

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Al-Mahdi, A.M., Al-Gharabli, M.M. & Kafini, M. A new general decay result for abstract evolution equation with time-dependent nonlinear dissipation. Ann Univ Ferrara 65, 201–230 (2019). https://doi.org/10.1007/s11565-019-00325-2

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  • DOI: https://doi.org/10.1007/s11565-019-00325-2

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