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

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

For $u_{tt}-\Delta u+u^3=0$ in three spatial dimensions, conserved positive wave energy $E$ controls $\Vert \nabla u{\Vert }_2$. The Sobolev inequality gives $\Vert u^2\nabla u{\Vert }_2\lesssim \Vert \nabla u{\Vert }_2^2\Vert D^{2}u{\Vert }_2\lesssim E\Vert D^{2}u{\Vert }_2$. Differentiated wave energy estimates and the Gronwall inequality then give $\Vert u(t){\Vert }_{{\dot{H}}^2}+\Vert u_t(t){\Vert }_{{\dot{H}}^1}\le CD{e}^{CEt}$, where $D$ bounds the corresponding initial Sobolev norms.

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.

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.

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.

In three dimensions set $w=ru$ for a radial function satisfying $\square u=F$. If the source has support in $s\le r\le s+1$ and $|F(s,r)|\le (1+s{)}^{-2}$, the Duhamel principle bounds its contribution to $w(t,t+a)$ by $\frac{1}{2}\log (1+t)$ for $1/2\le a\le 1$. Its backward characteristic interval meets the source in a set of length at most one, and $r|F|\le (1+s{)}^{-1}$. Division by $r\simeq 1+t$ gives $|u|\lesssim \log (2+t)/(1+t)$ for compactly supported initial data.

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