Global existence for semi-linear structurally damped σ-evolution models

The main purpose of this paper is to study the global existence of small data solutions for semi-linear structurally damped σ-evolution models of the formutt+(−Δ)σu+μ(−Δ)δut=f(|D|au,ut),u(0,x)=u0(x),ut(0,x)=u1(x) with σ≥1σ≥1, μ>0μ>0 and δ=σ2. This is a family of structurally damped σ  -evolution models interpolating between models with exterior damping δ=0δ=0 and those with visco-elastic type damping δ=σδ=σ. The function f(|D|au,ut)f(|D|au,ut) represents power non-linearities ||D|au|p||D|au|p for a∈[0,σ)a∈[0,σ) or |ut|p|ut|p. Our goal is to propose a Fujita type exponent diving the admissible range of powers p   into those allowing global existence of small data solutions (stability of zero solution) and those producing a blow-up behavior even for small data. On the one hand we use new results from harmonic analysis for fractional Gagliardo–Nirenberg inequality or for superposition operators (see Appendix A), on the other hand our approach bases on Lp−LqLp−Lq estimates not necessarily on the conjugate line for solutions to the corresponding linear models assuming additional L1L1 regularity for the data. The linear models we have here in mind arevtt+(−Δ)σv+μ(−Δ)δvt=0,v(0,x)=v0(x),vt(0,x)=v1(x) with σ≥1σ≥1, μ>0μ>0 and δ∈(0,σ]δ∈(0,σ].