Morawetz identity for the defocusing wave equation

Multiplying a defocusing wave equation by $\nabla \psi \cdot \nabla u+(\Delta \psi )u/2$ and integrating by parts gives a weighted spacetime identity. For a radial solution and radial weight, its coercive terms are

In three dimensions, $\Delta |x|=2/|x|$ away from the origin and $-\Delta (1/|x|)=4\pi {\delta }_0$. Hence $-{\Delta }^2|x|=8\pi {\delta }_0$ as a distribution.

For a finite-energy defocusing power-wave solution in three dimensions,Hardy's inequality bounds the Morawetz action by the conserved energy.