We will consider the coupled system of damped $\sigma$-equation \begin{equation*}\begin{cases} \partial_t^2U + (-\Delta)^\sigma U + D \partial_t U = F(U), & t>0, \ x\in\R^n, \\ (U,\partial_tU)(0,x)=(0,\Phi(x)), & x\in\R^n, \end{cases} \end{equation*} where $\sigma>1$,~$U=(u_1,u_2)\tm$,~$\Phi=(\varphi_1,\varphi_2)\tm$ and \begin{equation*} D U = \begin{pmatrix} a_1(-\Delta)^{\frac{\theta_1}2} & 0 \\ 0 & a_2 (-\Delta)^{\frac{\theta_2}2} \end{pmatrix},\quad \theta_1,\theta_2\in[\sigma,2\sigma], \ a_1,a_2\geq0, \end{equation*} is a damping term in the noneffective case. We will analyze the various scenarios of existence of global-in-time solution that arise due to the semilinear term $F(U)=(f_1(U),f_2(U))\tm$, in the case of a general semilinear coupling \[ F(U)=\begin{pmatrix} c_{11}|u_1|^{p_{11}} + c_{12}|u_2|^{p_{12}}\\ c_{21}|u_1|^{p_{21}} + c_{22}|u_2|^{p_{22}} \end{pmatrix}, \] % or in the case of weak coupling of derivative type \[ F(U)=\begin{pmatrix} |\partial_t u_2|^{p_{2}}\\ |\partial_t u_1|^{p_{1}} \end{pmatrix}. \] Sharp $L^p-L^q$ estimates for the corresponding linear scalar equation and a suitable application of the test function method are crucial for determining the so-called critical curves for the existence of small data solutions.\\ Based on a joint work with Marcello D'Abbicco (University of Bari).