A dissiptive logarithmic type evolution equation: asymptotic profile and optimal estimates

TL;DR

This paper introduces a new wave-like equation with a weakly effective nonlocal logarithmic damping mechanism, analyzing its long-term behavior and decay rates in the whole space.

Contribution

It presents a novel logarithmic damping model for wave equations and studies its asymptotic profile and decay rates, extending understanding of weak damping effects.

Findings

Derived the asymptotic profile of solutions.

Established optimal decay rates in L^2 sense.

Compared effectiveness of logarithmic damping with fractional damping.

Abstract

We introduce a new model of the logarithmic type of wave-like equation with a nonlocal logarithmic damping mechanism, which is rather weakly effective as compared with frequently studied fractional damping cases. We consider the Cauchy problem for this new model in the whole space, and study the asymptotic profile and optimal decay and/or blowup rates of solutions as time goes to infinity in L^{2}-sense. The operator L considered in this paper was used to dissipate the solutions of the wave equation in the paper studied by Charao-Ikehata in 2020, and in the low frequency parameters the principal part of the equation and the damping term is rather weakly effective than those of well-studied power type operators.