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 $$p_{\rm c}(m,\gamma,n): = 1 + \frac{2m}{n+m\gamma}$$ for $m \in (1, 2], \,\gamma \in \big[0, \frac{n(m-1)}{m}\big)$ by proving the global (in time) existence of small data Sobolev solutions when $p \geq p_{\rm c}(m,\gamma,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(m,\gamma,n)$. In addition, a novelty of this paper is that the critical value $p= p_{\rm c}(m,\gamma,n)$ belongs to the global existence range. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.