A new critical exponent for semi-linear damped wave equations with the initial data from Sobolev spaces of negative order

In this paper, we would like to study the critical exponent for semi-linear damped wave equations with power nonlinearity and the initial data belonging to Sobolev spaces of negative order ${\dot{H}}_m^{-\gamma }$. Precisely, we obtain a new critical exponent $$p_{\mathrm{c}}(m,\gamma ,n):=1+\frac{2m}{n+m\gamma }$$ for $m\in (1,2],\gamma \in [0,\frac{n(m-1)}{m})$ by proving the global (in time) existence of small data Sobolev solutions when $p\ge p_{\mathrm{c}}(m,\gamma ,n)$ and the blow-up result for weak solutions in finite time even for small data if $1<p<p_{\mathrm{c}}(m,\gamma ,n)$. In addition, a novelty of this paper is that the critical value $p=p_{\mathrm{c}}(m,\gamma ,n)$ belongs to the global existence range. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.