Abstract

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models: with , and . Here the function stands for the power nonlinearities and with a given number . We are interested in investigating 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 regularity on the initial data, we use and estimates with and , to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on spaces.