On semilinear damped wave equations with initial data in homogeneous Sobolev spaces

In this paper, we study semilinear damped equations $u_{tt}+u_t-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^p$ with the initial data in $({\dot{H}}^{-\gamma }\cap H^s)\times ({\dot{H}}^{-\gamma }\cap L^2)$. Chen-Reissig (2023) studied the case $0<\gamma <\frac{n}{2}$ and showed that the exponent $p_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=1+\frac{4}{n+2\gamma }$ of $p$ distinguishes the time global existence and the blow-up of solution. In this paper, we discuss the case $\gamma \ge \frac{n}{2}$.