Defocusing semilinear wave equation ($u_{tt}-\Delta u+u|u{|}^{p-1}=0$)

The defocusing power-type wave equation has the conserved nonnegative energy

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.

For a radial function $u(t,r)$ in three dimensions, setting $w=ru$ removes the radial first derivative because $r\Delta u=w_{rr}$. The defocusing power equation becomeson the half-line.

For $q=w_t+w_r$ and a radial weight $\psi$,