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 $(\dot{H}^{-\gamma}\cap H^s)\times(\dot{H}^{-\gamma}\cap L^2)$ with the dimension $n\le n$.

Chen-Reissig(2023) studied the case $0<\gamma\le\min\{\frac{n}{2}, (-n+\sqrt{n^2+16n})/4\}$ and showed that the exponent $p_{\mathrm{crit}}=1+4/(n+2\gamma)$ of $p$ distinguishes the time global existence and the blow-up of solution.

In this paper, we discuss the case $\gamma\ge\min\{\frac{n}{2}, (-n+\sqrt{n^2+16n})/4\}$ and show that the critical exponent is not $1+4/(n+2\gamma)$ but $1+\frac{2}{n}$ for $n=1,2$, and $(n+\sqrt{n^2+16n})/(2n)$ for $3\le n\le 6$.