Critical exponent for semilinear damped wave equations with weighted nonlinear terms and data from Sobolev spaces of negative order

In this paper, we would like to study the critical exponent for semilinear damped wave equations with the nonlinearity terms of Coulomb-type singularities $|x|^{-\alpha} |u(t,x)|^p$ and the initial data belonging to Sobolev spaces of negative order $\dot{H}^{-\beta}$. Precisely, we obtain a critical exponent $$p_{\rm c}(\alpha,\beta,n): = 1 + \frac{4-2\alpha}{n+2\beta} $$ for $1 \leq n \leq 4$ and $ 0 \leq \alpha, \beta < n/2,$ by proving the global (in time) existence of small data solutions when $p \geq p_{\rm c}(\alpha,\beta,n)$ and the blow-up result for weak solutions in finite time even for small data if $1 < p < p_{\rm c}(\alpha,\beta,n)$. Furthermore, we are going to provide lifespan estimates for solutions when a blow-up phenomenon occurs.