Asymptotic profiles for a wave equation with parameter dependent logarithmic damping

TL;DR

This paper analyzes the long-term behavior of solutions to a wave equation with weak logarithmic damping, providing asymptotic profiles and optimal energy estimates in the whole space.

Contribution

It introduces a detailed analysis of a wave equation with parameter-dependent logarithmic damping, focusing on asymptotic profiles and energy decay in the L^{2} sense.

Findings

Asymptotic profiles of solutions are characterized.

Optimal decay estimates for total energy are established.

Hypergeometric functions are utilized in the analysis.

Abstract

We study a nonlocal wave equation with logarithmic damping which is rather weak in the low frequency zone as compared with frequently studied strong damping case. We consider the Cauchy problem for this model in the whole space and we study the asymptotic profile and optimal estimates of the solutions and the total energy as time goes to infinity in L^{2}-sense. In that case some results on hypergeometric functions are useful.