Asymptotic profiles for damped plate equations with rotational inertia terms

TL;DR

This paper investigates the long-term behavior of solutions to damped plate equations with rotational inertia, deriving asymptotic profiles in L^2 sense for initial data of varying regularity, including cases with regularity-loss.

Contribution

It provides a detailed analysis of asymptotic profiles for damped plate equations with rotational inertia, especially addressing the regularity-loss phenomena in low regularity initial data.

Findings

Asymptotic profiles are derived for high and low regularity initial data.

The regularity-loss structure of solutions is characterized.

The Fourier splitting method and explicit solution expressions are employed.

Abstract

We consider the Cauchy problem for plate equations with rotational inertia and frictional damping terms. We will derive asymptotic profiles of the solution in L^2-sense as time goes to infinity in the case when the initial data have high and low regularity, respectively. Especially, in the low regularity case of the initial data one encounters the regularity-loss structure of the solutions, and the analysis is more delicate. We employ the so-called Fourier splitting method combined with the explicit expression of the solutions (high frequency estimates) and the method due to Ikehata (low frequency estimates).