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: 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∈(σ2,σ] . Our aim is to study two main models including σ -evolution models with structural damping δ∈(σ2,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 Lq spaces by assuming additional Lm regularity for the initial data, with q∈(1,∞) and m∈[1,q) .