Decay characterization of solutions to semi-linear structurally damped $\sigma$-evolution equations with time-dependent damping

In this paper, we study the Cauchy problem to the linear damped $\sigma$-evolution equation with time-dependent damping in the effective cases \begin{equation*} u_{t t}+(-\Delta)^\sigma u+b(t)(-\Delta)^\delta u_t=0, \end{equation*} and investigate the decay rates of the solution and its derivatives that are expressed in terms of the decay character of the initial data $u_0(x)=u(0, x)$ and $u_1(x)=u_t(0, x)$. We are interested also in the existence and decay rate of the global in time solution with small data for the corresponding semi-linear problem with the nonlinear term of power type $||D|^\gamma u|^p$. The blow-up results for solutions to the semi-linear problem in the case $\gamma=0$ are presented to show the sharpness of the exponent $p$.