The critical exponent for nonlinear damped $\sigma$-evolution equations

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.