I am trying to solve this problem:
Let $u$ solve the wave equation $u_{tt}-\Delta{u}=0$ in $\mathbb{R}^n\times[0,\infty)$ and $h:\mathbb{R}^n\rightarrow\mathbb{R}$ be a harmonic with $h(x)=u(x,0)$ and $u_t(x,0)=0$. Show that $u$ is time independent.
I have:
Since $u$ solves the wave equation $u_{tt}-\Delta{u}=0$. At the point $(x,0)$ we have $0-\Delta{h}=0$. Because $h$ is harmonic, $h$ also solves the wave equation.
From here I am not sure how to continue, I am not even sure if this is correct. I would appreciate some hints to know how to proceed.