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 $δ \in (0, σ]$ 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 $δ \in \Bigl(\cfrac{σ}{2} , σ \Bigr)$, and in the exceptional case $δ = 0$.