Morawetz estimate for the defocusing wave equation

Let $u$ be a global sufficiently regular finite-energy solution of $u_{tt}-\Delta u+|u{|}^{p-1}u=0$ on ${\mathbb{R}}^3$, with $p>1$, and let $E$ be its conserved nonnegative wave energy. Then the Morawetz estimate for the defocusing wave equation isThe constant depends on $p$ and the spatial dimension, not the upper endpoint of the time interval. A finite-time version holds on every interval of smooth existence. A global spacetime bound does not itself assert existence of a global solution.^[ref]

Use the radial multiplier $u_r+u/r$ in the Morawetz identity for the defocusing wave equation. Its action is bounded by $\Vert u_t{\Vert }_2(\Vert \nabla u{\Vert }_2+\Vert u/r{\Vert }_2)\le CE$ using the Hardy inequality in Euclidean space. The radial weight has nonnegative angular terms and $-{\Delta }^2|x|=8\pi {\delta }_0$; the remaining nonlinear term has positive coefficient proportional to $(p-1)/(p+1)$. Integration in time bounds its spacetime integral by endpoint actions, and monotone convergence gives the infinite-time statement. The estimate limits how long nonlinear wave energy can remain concentrated near a fixed location.