General and optimal decay for a quasilinear parabolic viscoelastic system

In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying \begin{document}$ g'(t) \leq - \xi(t) H(g(t)) $\end{document} , where \begin{document}$ H $\end{document} is an increasing, convex function and \begin{document}$ \xi $\end{document} is a nonincreasing function. Precisely, we establish a general and optimal decay result for a large class of relaxation functions which improves and generalizes several stability results in the literature. In particular, our result extends an earlier one in the literature, namely, the case of the polynomial rates when \begin{document}$ H(t) = t^p, \ t\geq 0, \forall p>1 $\end{document} , instead the parameter \begin{document}$ p \in [1, \frac{3}{2}[ $\end{document} .