<inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> estimates for oscillating integrals and their applications to semi-linear models with <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-evolution like structural damping

Abstract

The present paper is a continuation of our recent paper [4]. We will consider the following Cauchy problem for semi-linear structurally damped σ-evolution models:Our aim is to study two main models including σ-evolution models with structural damping δ ∈ ( σ 2 , σ) and those with visco-elastic damping δ = σ. Here the function f (u, ut) stands for power nonlinearities |u| p and |ut| p with a given number p > 1. We are interested in investigating 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 by assuming additional L m regularity for the initial data, with q ∈ (1, ∞) and m ∈ [1, q).

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

“…Meanwhile, for the case δ = σ this phenomenon is no longer true, that is, some kind of wave structure appears and oscillations come into play from the asymptotic profile of solutions to (4). Furthermore, compared with the regularity of the initial data we can see that a smoothing effect appears for some derivatives of solutions to (4) with respect to the time variable (see [7]) in the latter case. This brings some benefits in treament of the corresponding semi-linear equations.…”

“…One of the main goals of this paper is to give a positive answer to this question. More recently, the authors in [6,7] have studied the following Cauchy problem for damped σ-evolution equations (see also [3,4,8]):…”