An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $\sigma$-evolution models

Abstract

We study 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 $\delta \in (0,\frac{\sigma }{2})$. Here the function $f(u,u_t)$ stands for the power nonlinearities $|u{|}^p$ and $|u_t{|}^p$ with a given number $p>1$. We are interested in investigating $L^1$ estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faà di Bruno's formula. By assuming additional $L^m$ regularity on the initial data, we use $(L^m\cap L^q)-L^q$ and $L^q-L^q$ estimates with $q\in (1,\infty )$ and $m\in [1,q)$, to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on $L^q$ spaces.