New general decay result for a system of viscoelastic wave equations with past history

Abstract

<p style='text-indent:20px;'>This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels <inline-formula><tex-math id="M1">\begin{document}$ g_i : [0, +\infty) \rightarrow (0, +\infty) $\end{document}</tex-math></inline-formula> satisfying <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ H_i $\end{document}</tex-math></inline-formula> are functions satisfying some specific properties. Under this very general assumption on the behavior of <inline-formula><tex-math id="M4">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity, we establish a relation between the decay rate of the solutions and the growth of <inline-formula><tex-math id="M5">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.