$L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\sigma$-evolution like structural damping

TL;DR

This paper extends previous work by deriving $L^1$ estimates for oscillating integrals and applies these results to analyze global existence of small data solutions in semi-linear structurally damped $\sigma$-evolution models with power nonlinearities.

Contribution

It introduces new $L^1$ estimates for oscillating integrals and applies them to establish global existence results for semi-linear $\sigma$-evolution equations with structural damping.

Findings

Established $L^1$ estimates for oscillating integrals.

Proved global existence of small data solutions for certain damping models.

Analyzed models with power nonlinearities in $L^q$ and $L^m$ spaces.

Abstract

The present paper is a continuation of our recent paper \cite{DaoReissig}. We will consider the following Cauchy problems for semi-linear structurally damped $\sigma$-evolution models: \begin{equation*} u_{tt}+ (-\Delta)^\sigma u+ \mu (-\Delta)^\delta u_t = f(u,u_t),\, u(0,x)= u_0(x),\, u_t(0,x)=u_1(x) \end{equation*} with $\sigma \ge 1$, $\mu>0$ and $\delta \in (\frac{\sigma}{2},\sigma]$. Our aim is to study two main models including $\sigma$-evolution models with structural damping $\delta \in (\frac{\sigma}{2},\sigma)$ and those with visco-elastic damping $\delta=\sigma$. Here the function $f(u,u_t)$ stands for power nonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. We are interested in investigating the global (in time) existence of small data solutions to the above semi-linear models from suitable spaces basing on $L^q$ space by assuming additional $L^{m}$ regularity on the initial data, with $q\in (1,\infty)$ and $m\in [1,q)$.