# Study of semi-linear $\sigma$-evolution equations with frictional and visco-elastic damping

## TL;DR

This paper investigates semi-linear $\sigma$-evolution equations with dual damping mechanisms, analyzing their long-term behavior, asymptotic expansions, and critical exponents, providing new insights into their diffusion phenomena and solution existence.

## Contribution

It introduces a comprehensive analysis of semi-linear $\sigma$-evolution equations with frictional and visco-elastic damping, including higher order asymptotics, diffusion phenomena, and critical exponent determination.

## Key findings

- Global existence of small data solutions for $\sigma \\ge 1$
- Detailed asymptotic expansions of solutions
- Identification of critical exponents for integer $\sigma$

## Abstract

In this article, we study semi-linear $\sigma$-evolution equations with double damping including frictional and visco-elastic damping for any $\sigma\ge 1$. We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the $L^p-L^q$ framework, with $1\le p\le q\le \infty$, to the corresponding linear equations. By assuming additional $L^{m}$ regularity on the initial data, with $m\in [1,2)$, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of the global obtained solutions as well to semi-linear equations. Moreover, we also determine the so-called critical exponent when $\sigma$ is integers.