A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

Abstract

We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in \begin{document}$ {{\bf R}}^{n} $\end{document}, and study the asymptotic profile and optimal decay rates of solutions as \begin{document}$ t \to \infty $\end{document} in \begin{document}$ L^{2} $\end{document}-sense. The operator \begin{document}$ L $\end{document} considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [7]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.