= Morawetz estimate for the defocusing wave equation
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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> is
$$
\int_0^\infty\int_{\mathbb R^3}\frac{|u(t,x)|^{p+1}}{|x|}\,dx\,dt\leq C_pE.
$$
The 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.<Nonlinear dispersive equations: local and global analysis by Terence Tao>[section3.5, Morawetz inequality for the wave equation, printed p161]{ref}
Use the radial multiplier $u_r+u/r$ in the <Morawetz identity for the defocusing wave equation>. Its action is bounded by $\|u_t\|_2(\|\nabla u\|_2+\|u/r\|_2)\leq CE$ 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.
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