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

TL;DR

This paper introduces a new logarithmic-Laplacian plate equation with nonlocal damping, analyzing the long-term behavior and decay rates of solutions, and classifies solutions based on initial data regularity into diffusion-like, wave-like, or mixed types.

Contribution

It presents a novel logarithmic-Laplacian plate model with nonlocal damping and studies its asymptotic profiles and decay rates, extending previous work on wave equation dissipation.

Findings

Solutions exhibit diffusion-like, wave-like, or mixed behavior depending on initial data.

The paper establishes optimal decay rates for solutions in the L^2 sense.

Asymptotic profiles are classified based on initial data regularity.

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 the whole space, and study the asymptotic profile and optimal decay rates of solutions as time goes to infinity in L^{2}-sense. The operator L considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charao-Ikehata in 2020. 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.