Morawetz estimate for the defocusing wave equation
= Morawetz estimate for the defocusing wave equation
{c}
For a finite-energy defocusing power-wave solution in three dimensions,
$$
\int_0^\infty\int_{\mathbb R^3}\frac{|u(t,x)|^{p+1}}{|x|}\,dx\,dt
\lesssim E(u(0)).
$$
Hardy's inequality bounds the Morawetz action by the conserved energy.