On the critical exponent and sharp lifespan estimates for semilinear damped wave equations with data from Sobolev spaces of negative order

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 $p=p_{mathrm{crit}}(n,gamma):=1+frac{4}{n+2gamma}$ for some $gammain(0,frac{n}{2})$ 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.