# Thresholds for low regularity solutions to wave equations with structural damping

## TL;DR

This paper investigates the long-term behavior of solutions to a wave equation with structural damping, identifying new thresholds that determine whether diffusion or wave-like properties dominate in low regularity scenarios.

## Contribution

It introduces new thresholds that distinguish between diffusion-dominated and wave-dominated behaviors in low regularity solutions of damped wave equations.

## Key findings

- Identified thresholds for diffusion versus wave dominance.
- Extended previous research on regularity-loss dissipative wave equations.
- Clarified the influence of initial data regularity on solution behavior.

## Abstract

We study the asymptotic behavior of solutions to wave equations with a structural damping term \[ u_{tt}-\Delta u+\Delta^2 u_t=0, \qquad u(0,x)=u_0(x), \,\,\, u_t(0,x)=u_1(x), \] in the whole space. New thresholds are reported in this paper that indicate which of the diffusion wave property and the non-diffusive structure dominates in low regularity cases. We develop to that end the previous author's research in 2019 where they have proposed a threshold that expresses whether the parabolic-like property or the wave-like property strongly appears in the solution to some regularity-loss type dissipative wave equation.