# General Energy Decay Rates for Some Viscoelastic Problems

General Energy Decay Rates for Some Viscoelastic Problems. PhD thesis, King Fahd University of Petroleum and Minerals.

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The work in this dissertation is concerned with the longtime behavior of some viscoelastic problems. Precisely, we will consider four problems, where the relaxation function satisfies a new relation of the form \begin{equation*} g^{\prime}(t) \leq - \xi(t) g^{p}(t),\hspace{0.5cm}\forall t\geq0, \end{equation*} where $1\leq p<\frac{3}{2}$ and $\xi: \mathrm{I\!R}_{+} \longrightarrow (0, +\infty)$ is a nonincreasing differentiable function.\\ Using the multiplier method, we could establish some new explicit decay results depending on $p$, $\xi$, and other parameters in the problem such as the behavior of the feedback function and/or the degree of the nonlinearity of the frictional damping when it is present in the equation.\\ Our work generalized many results in the literature such as [27, 31, 36] and improved others. Particularly, it gaves a better rate of decay than that of [29]. In addition, our results answered partially the question in [2], namely, looking for decay rates induced by relaxation functions satisfying \begin{equation*} g^{\prime}(t) \leq - \xi(t) H(g(t)),\hspace{0.5cm}\forall t\geq0, \end{equation*} for functions $\xi$ and $H$.