On the Asymptotics of Some Strongly Damped Beam Equations with Structural Damping

Abstract

<p style='text-indent:20px;'>The Fourier transform, <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{F} $\end{document}</tex-math></inline-formula>, on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M3">\begin{document}$ (N \geq 1) $\end{document}</tex-math></inline-formula> transforms the Cauchy problem for a strongly damped beam equation with structural damping <inline-formula><tex-math id="M4">\begin{document}$ u_{tt} - \Delta u_t + \alpha (\Delta^2)u - \Delta u = 0, \ \alpha \geq 0 $\end{document}</tex-math></inline-formula>, to an ordinary differential equation in time. With <inline-formula><tex-math id="M5">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> being the weak solution of the problem given by the Fourier transform, the goal of the paper is to determine the asymptotic expansion of the squared <inline-formula><tex-math id="M6">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M7">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula>. With suitable, additional assumptions on the initial data <inline-formula><tex-math id="M9">\begin{document}$ u(0, x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ u_t(0, x) $\end{document}</tex-math></inline-formula>, we establish the behavior of the squared <inline-formula><tex-math id="M11">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M12">\begin{document}$ u(t, x) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M13">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula>.</p>