Some $L^1$-$L^1$ estimates for solutions to visco-elastic damped $\sigma$-evolution models

TL;DR

This paper establishes $L^1-L^1$ estimates for solutions to a class of visco-elastic damped $\sigma$-evolution equations with $\sigma>1$ across all space dimensions, enhancing understanding of their decay properties.

Contribution

It provides the first $L^1-L^1$ estimates for solutions to visco-elastic damped $\sigma$-evolution models with $\sigma>1$, extending previous results to all space dimensions.

Findings

Derived $L^1-L^1$ decay estimates for solutions.

Applicable to all space dimensions $n \\ge 1$.

Improves understanding of damping effects in $\sigma$-evolution models.

Abstract

This note is to conclude $L^1-L^1$ estimates for solutions to the following Cauchy problem for visco-elastic damped $\sigma$-evolution models: \begin{equation} \begin{cases} u_{tt}+ (-\Delta)^\sigma u+ (-\Delta)^\sigma u_t = 0, &\quad x\in \mathbb{R}^n,\, t \ge 0, \\ u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x), &\quad x\in \mathbb{R}^n, \label{pt1.1} \end{cases} \end{equation} where $\sigma> 1$, in all space dimensions $n\ge 1$.