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

Abstract

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_{textrm{crit}}(n,gamma ):=1+frac{4}{n+2gamma } for some gamma in (0,frac{n}{2}) and low dimensions in the framework of Sobolev spaces of negative order. Precisely, global (in time) existence of small data Sobolev solutions of lower regularity is proved for p>p_{textrm{crit}}(n,gamma ), and blow-up of weak solutions in finite time even for small data if 1<p<p_{textrm{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.