# Bounding the solution of a wave equation in 3 dimensions

Let $u:{\mathbb{R}^ + } \times {\mathbb{R}^3} \to \mathbb{R}$ be a solution of the Cauchy problem

$\left\{ \begin{gathered} {u_{tt}} - \Delta u = 0 \\ u\left( {0,x} \right) = {u_0}\left( x \right) \\ {u_t}\left( {0,x} \right) = {u_1}\left( x \right) \\ \end{gathered} \right.$

Assuming that ${u_0}$ and ${u_1}$ are smooth and compactly supported, show that

$\left( {\exists C > 0} \right)\left( {\forall t > 0} \right)\left( {\forall x \in {\mathbb{R}^3}} \right)\left| {u\left( {t,x} \right)} \right| \leqslant \frac{C}{t}$.

Edit: Kirchoff's formula in 3 dimensions gives

$u\left( {t,x} \right) = \frac{1}{{4{t^2}\pi }}\int_{S\left( {x,t} \right)} {t{u_1}\left( y \right) + {u_0}\left( y \right) + \nabla {u_0}\left( y \right)\left( {y - x} \right)d{S_y}}$

which would directly give only $\left| {u\left( {t,x} \right)} \right| \leqslant Ct$ which isn't good enough.

Source: Evans, partial differential equations, problem 18 on page 89

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