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

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}+ (-Δ)^σu+ μ(-Δ)^δu_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 $m\in [1,q)$.

arXiv admin note: substantial text overlap with arXiv:1808.02706