Consider the Cauchy problem to the free wave equation
$\begin{align} u_{tt}-\Delta u=0,\newline u(0,x)=\phi(x), \newline u_t(0,x)=\psi(x) \end{align} \quad \text{with } t \geq 0, x \in \mathbb{R}^n$
where $\phi, \, \psi \in \mathcal{S}(\mathbb{R}^{n})$. Using Fourier transform, I found a solution to this problem in terms of Fourier multipliers. How can I show that this solution is unique?
Any help is appreciated.