A dissipative logarithmic type evolution of second order in time

TL;DR

This paper introduces a second-order evolution model with non-local logarithmic damping in Euclidean space, analyzing its long-term behavior and decay rates of solutions in the $L^2$ sense.

Contribution

It presents a novel logarithmic-type second-order model with non-local damping and provides a spectral approach to study its asymptotic behavior and decay properties.

Findings

Proved asymptotic profile of solutions as time approaches infinity.

Established optimal decay rates for solutions in $L^2$-sense.

Analyzed the spectral properties related to the damping mechanism.

Abstract

In this paper, we introduce a logarithmic-type second-order model with a non-local logarithmic damping mechanism in $R^N$. We present a motivation with a spectral approach to consider the equation, we consider the Cauchy problem associated with the model. More precisely, we study the asymptotic behavior of solutions as $t$ goes to infinity in $L^2$-sense; namely, we prove results on the asymptotic profile and optimal decay of solutions as time goes to infinity in $L^2$-sense.