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

In this paper, we derive suitable optimal \(L^p-L^q\) $L^p-L^q$ decay estimates, \(1\le p\le 2\le q\le \infty \) $1\le p\le 2\le q\le \infty$ , for the solutions to the \(\sigma \) $\sigma$ -evolution equation, \(\sigma >1\) $\sigma >1$ , with scale-invariant time-dependent damping and power nonlinearity  \(|u|^p\) ${|u|}^p$ , \(\begin{aligned} u_{tt}+(-\Delta )^\sigma u + \frac{\mu }{1+t} u_t= |u|^{p}, \quad t\ge 0, \quad x\in {{\mathbb {R}}}^n, \end{aligned}\) $$\begin{array}{c}\hfill u_{\mathit{tt}}+{(-\mathrm{\Delta })}^{\sigma }u+\frac{\mu }{1+t}{u}_t={|u|}^p,t\ge 0,x\in {\mathbb{R}}^n,\end{array}$$ where  \(\mu >0\) $\mu >0$ ,   \(p>1\) $p>1$ . The critical exponent \(p=p_c\) $p=p_c$ for the global (in time) existence of small data solutions to the Cauchy problem is related to the long time behavior of solutions, which changes accordingly \(\mu \in (0, 1)\) $\mu \in (0,1)$ or \(\mu >1\) $\mu >1$ . Under the assumption of small initial data in \(L^m({{\mathbb {R}}}^n)\cap L^2({{\mathbb {R}}}^n), m=1,2\) $L^m({\mathbb{R}}^n)\cap L^2({\mathbb{R}}^n),m=1,2$ , we find the critical exponent at low space dimension n with respect to \(\sigma \) $\sigma$ , namely, \(\begin{aligned} p_c= \max \left\{ {{\bar{p}}}(\gamma _{m}), {{\bar{p}}} (\gamma _{m}+\mu -1) \right\} , \quad \gamma _{m}{\mathrm {\,:=\,}}\frac{n}{m\sigma }, \quad \mu >1-\gamma _m, \end{aligned}\) $$\begin{array}{c}\hfill p_c=max\left\{\overline{p},({\gamma }_m),,,\overline{p},({\gamma }_m+\mu -1)\right\},{\gamma }_m:=\frac{n}{m\sigma },\mu >1-{\gamma }_m,\end{array}$$ where \( {{\bar{p}}}(\gamma ){\mathrm {\,:=\,}}1+ \frac{2}{\gamma }\) $\overline{p}(\gamma ):=1+\frac{2}{\gamma }$ is the well known Fujita exponent. Hence, \(p_c={{\bar{p}}}(\gamma _{m})\) $p_c=\overline{p}({\gamma }_m)$ if \(\mu >1\) $\mu >1$ , whereas \(p_c={{\bar{p}}} (\gamma _{m}+\mu -1)\) $p_c=\overline{p}({\gamma }_m+\mu -1)$ is a shift of Fujita type exponent if \(\mu \in (0, 1)\) $\mu \in (0,1)$ .