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

Abstract

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models:utt+(−Δ)σu+μ(−Δ)δut=f(u,ut),u(0,x)=u0(x),ut(0,x)=u1(x) with σ≥1, μ>0 and δ∈(0,σ2). Here the function f(u,ut) stands for the power nonlinearities |u|p and |ut|p with a given number p>1. We are interested in investigating L1 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 Lm regularity on the initial data, we use (Lm∩Lq)−Lq and Lq−Lq estimates with q∈(1,∞) and m∈[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 Lq spaces.