Strongly damped wave equations with mass-like terms of the logarithmic-Laplacian

TL;DR

This paper analyzes the long-term behavior of solutions to strongly damped wave equations with logarithmic mass-like terms, revealing how the parameter  influences diffusive or singular growth properties in different dimensions.

Contribution

It extends previous work by deriving the leading asymptotic term of solutions for a range of  values, providing detailed growth and decay characterizations in L^2-norm.

Findings

Small  leads to diffusive behavior in solutions.

Large  causes growth rates indicating singularity.

The asymptotic analysis varies with spatial dimension n.

Abstract

We consider strongly damped wave equations with logarithmic mass-like terms with a parameter $\theta \in (0; 1]$. This research is a part of a series of wave equations that was initiated by Char\~ao-Ikehata [6], Char\~ao-D'Abbicco-Ikehata considered in [5] depending on a parameter $\theta \in (1/2,1)$ and Piske- Char\~ao-Ikehata [26] for small parameter $\theta \in (0,1/2)$. We derive a leading term (as time goes to infinity) of the solution, and by using it, a growth and a decay property of the solution itself can be precisely studied in terms of L^2-norm. An interesting aspect appears in the case of n = 1, roughly speaking, a small $\theta$ produces a diffusive property, and a large $\theta$ gives a kind of singularity, expressed by growth rates.