Abstract

The main purpose of the paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem utt+(−Δ)σu+(−Δ)δut=|ut|p,u(0,x)=u0(x),ut(0,x)=u1(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∈(σ2,σ) and in the exceptional case δ = 0.