  We study semilinear damped wave equations with power nonlinearity $|u|^p$ and
initial data belonging to Sobolev spaces of negative order $\dot{H}^{-\gamma}$.
In the present paper, we obtain a new critical exponent
$p=p_{\mathrm{crit}}(n,\gamma):=1+\frac{4}{n+2\gamma}$ for some
$\gamma\in(0,\frac{n}{2})$ and low dimensions in the framework of Soblev spaces
of negative order. Precisely, global (in time) existence of small data Sobolev
solutions of lower regularity is proved for $p>p_{\mathrm{crit}}(n,\gamma)$,
and blow-up of weak solutions in finite time even for small data if
$1<p<p_{\mathrm{crit}}(n,\gamma)$. Furthermore, in order to more accurately
describe the blow-up time, we investigate sharp upper bound and lower bound
estimates for the lifespan in the subcritical case.
