Morawetz identity for the defocusing wave equation
= Morawetz identity for the defocusing wave equation
{c}
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
$$
\psi''|\nabla u|^2-\frac14(\Delta^2\psi)u^2
+\frac{p-1}{2(p+1)}(\Delta\psi)|u|^{p+1}.
$$