The main purpose of the paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem u tt + − Δ σ u + − Δ δ u t = u t p , u 0 x = u 0 x , u t 0 x = u 1 x . $$ {u}_{tt}+{\left(-\Delta \right)}^{\sigma }u+{\left(-\Delta \right)}^{\delta }{u}_t={\left|{u}_t\right|}^p,\kern0.5em u\left(0,x\right)={u}_0(x),\kern0.5em {u}_t\left(0,x\right)={u}_1(x). $$
The parameter δ ∈ (0, σ] describes the structural damping in the model varying from the exterior damping δ = 0 to the viscoelastic type damping δ = σ. We determine the admissible sets of the parameter p for the global solvability of this semilinear Cauchy problem with arbitrary small initial data u0, u1 in the hyperbolic-like case δ ∈ σ 2 σ $$ \delta \in \left(\frac{\sigma }{2},\sigma \right) $$ and in the exceptional case δ = 0.