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Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems

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Abstract

In this paper, we consider the nonlinear abstract equation

$$\begin{aligned} u_{tt}+Au-\int _{0}^{t}g(t-s)Au(s)\mathrm{d}s+h(u_{t})=j(u) \end{aligned}$$

subject to a competing effect of viscoelastic and frictional dampings. With very general assumptions on the behavior of g at infinity and the behavior of h near 0, we establish explicit and optimal energy decay result. To the best of our knowledge, this is the first time we have such combination of generality and optimality in one explicit formula for the energy decay rates of this system.

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Acknowledgements

This work was supported by MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah.

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Correspondence to Muhammad I. Mustafa.

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Mustafa, M.I. Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems. Z. Angew. Math. Phys. 72, 67 (2021). https://doi.org/10.1007/s00033-021-01498-7

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