  In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leq
p\leq q\leq \infty$, for the solutions to the $\sigma$-evolution equation,
$\sigma>1$, with structural damping and power nonlinearity $|u|^{1+\alpha}$ or
$|u_t|^{1+\alpha}$, \[ u_{tt}+(-\Delta)^\sigma u +(-\Delta)^\theta
u_t=\begin{cases} |u|^{1+\alpha}, \\ |u_t|^{1+\alpha}, \end{cases}\] where
$t\geq0$ and $x\in\mathbb{R}^n$. Using these estimates, we can solve the
problem of finding the critical exponents for the two nonlinear problems above
in the so-called non-effective case, $\theta\in(\sigma/2,\sigma]$. This latter
is more difficult than the effective case $\theta\in[0,\sigma/2)$, since the
asymptotic profile of the solution involves a diffusive component and an
oscillating one. The novel idea in this paper consists in treating separately
the two components to neglect the loss of decay rate created by the interplay
of the two components. We deal with the oscillating component, by localizing
the low frequencies, where oscillations appear, in the extended phase space.
This strategy allows us to recover a quasi-scaling property which replaces the
lack of homogeneity of the equation.
