Some results on the global solvability for structurally damped models with a special nonlinearity - Duong, P. T. (Ukrains’kyi Matematychnyi Zhurnal, 2018)
The main purpose of this 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). The parameter δ (0, σ] describes the structural damping in the model varying from the exterior damping δ = 0 up to the visco-elastic type damping δ = σ. We will obtain 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.