A general decay result for the Cauchy problem of plate equations with memory

Abstract

In this paper, we investigate the general decay rate of the solutions for a class of plate equations with memory term in the whole space \begin{document}$ \mathbb{R}^n $\end{document}, \begin{document}$ n\geq 1 $\end{document}, given by

\begin{document}$ \begin{equation*} u_{tt}+\Delta^2 u+ u+ \int_0^t g(t-s)A u(s)ds = 0, \end{equation*} $\end{document}

with \begin{document}$ A = \Delta $\end{document} or \begin{document}$ A = -Id $\end{document}. We use the energy method in the Fourier space to establish several general decay results which improve many recent results in the literature. We also present two illustrative examples by the end.