An application of L1 estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models

We study the following Cauchy problems for semi-linear structurally damped sigma-evolution models: 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) with sigma >= 1, mu > 0 and delta is an element of (0, sigma/2). Here the function f(u,u(t)) stands for the power nonlinearities vertical bar u vertical bar(p) and vertical bar u(t)vertical bar(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 Faa di Bruno's formula. By assuming additional L-m regularity on the initial data, we use (L-m boolean AND L-q) - L-q and L-q - L-q estimates with q is an element of (1, infinity)) and m is an element of [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. (C) 2019 Elsevier Inc. All rights reserved.