# Regularity Analysis for Two Coupled Second Order Evolution Equations

## TL;DR

This paper analyzes the regularity properties of the semigroup generated by a system of two coupled second order evolution equations with indirect damping, providing a complete characterization of when the system's solutions are analytic or Gevrey regular.

## Contribution

It derives the asymptotic eigenvalue expressions and partitions the parameter space to classify the semigroup's regularity, offering a sharp and comprehensive analysis.

## Key findings

- Semigroup exhibits analyticity in certain parameter regions.
- Semigroup exhibits Gevrey class regularity in other regions.
- Complete characterization of regularity based on eigenvalue asymptotics.

## Abstract

We investigate the regularity of the strongly continuous semigroup associated with a system of two coupled second order evolution equations with indirect damping, whose stability was recently studied by Hao et al. By deriving the asymptotic expression of the eigenvalues the generator, we partition the parameter space into several disjoint regions, where the semigroup exhibits either analyticity or Gevrey class regularity. Together with the estimate of the resolvent of the generator on the imaginary axis, we give a complete and sharp regularity characterization for this system.