On Morawetz estimates for the elastic wave equation

We establish Morawetz-type estimates for solutions to the elastic wave equation with singular weights of the form $|x|^{-\alpha}$ or $|(x,t)|^{-\alpha}$. In particular, we show that space-time weights $|(x,t)|^{-\alpha}$ admit stronger singularities and require weaker regularity assumptions on the initial data compared to purely spatial weights $|x|^{-\alpha}$.