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 $\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ or $\mathrm{\mid }(x,t){\mathrm{\mid }}^{-\alpha }$. In particular, we show that space-time weights $\mathrm{\mid }(x,t){\mathrm{\mid }}^{-\alpha }$ admit stronger singularities and require weaker regularity assumptions on the initial data compared to purely spatial weights $\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$.