Some results on the global solvability for structurally damped models with a special nonlinearity

Abstract

The main purpose of this paper is to prove the global (in time) existence of solution for the semilinear Cauchy problem $$u_{tt} + (-\Delta )^{\sigma} u + (-\Delta )^{\delta} u_t = |u_t|^p,\; u(0, x) = u_0(x),\; u_t(0, x) = u_1(x)$$. The parameter $\delta \in (0, \sigma]$ describes the structural damping in the model varying from the exterior damping $\delta = 0$ up to the visco-elastic type damping $\delta = \sigma$. 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 $\delta \in \Bigl(\cfrac{\sigma}{2} , \sigma \Bigr)$, and in the exceptional case $\delta = 0$.