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In this paper, we derive suitable optimal $L^p-L^q$ decay estimates, $1\leqp\leq 2\leq q\leq \infty$, for the solutions to the $\sigma$-evolutionequation, $\sigma>1$, with scale-invariant time-dependent damping and powernonlinearity~$|u|^p$, \[ u_{tt}+(-\Delta)^\sigma u + \frac{\mu}{1+t} u_t=|u|^{p}, \] where $\mu>0$, $p>1$. The critical exponent $p=p_c$ for the global(in time) existence of small data solutions to the Cauchy problem is related tothe long time behavior of solutions, which changes accordingly $\mu \in (0, 1)$or $\mu>1$. Under the assumption of small initial data in $L^1\cap L^2$, wefind the critical exponent \[ p_c=1+ \max\left\{\frac{2\sigma}{[n-\sigma+\sigma\mu]_+}, \frac{2\sigma}{n} \right\}=\begin{cases} 1+ \frac{2\sigma}{[n-\sigma+\sigma\mu]_+}, \quad \mu \in (0,1)\\ 1+ \frac{2\sigma}{n}, \quad \mu>1. \end{cases} \]  For $\mu>1$ it is well known as Fujita type exponent, whereas for $\mu \in(0, 1)$ one can read it as a shift of Kato exponent.

The move from Fujita to Kato type exponent for a class of semilinear evolution equations with time-dependent damping