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

In this paper, we study semilinear damped equations $u_{tt}+u_t-\Delta u=|u|^p$ 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 $p_{\mathrm{crit}}=1+\frac{4}{n+2\gamma}$ of $p$ distinguishes the time global existence and the blow-up of solution. In this paper, we discuss the case $\gamma\ge\frac{n}{2}$.