# Proof that solutions of the wave equation with smooth initial data have to be smoot

Is it possible to show that the $n$-dimensional initial value problem of the wave equation \begin{align*} u_{tt}(x,t)-\Delta u(x,t)&=0\qquad \mbox{in } \mathbb{R}^n \times (0,\infty)\\ u(x,0)&=g(x)\qquad \mbox{on } \mathbb{R}^n\\ u_t(x,0)&=h(x)\qquad \mbox{on } \mathbb{R}^n\\ \end{align*} with $g,h\in C^\infty$ does lead to $u\in C^\infty$ solutions without explicitly solving it or using the fact that the system ist hyperbolic.

In other words is there a theorem like: For every linear PDE with constant coefficients $C^\infty$ initial data does lead to $C^\infty$ solutions.

• From what I have found the theorem is always stated in the general form with analytic coefficients and analytic initial data. Is it possible to reduce the initial data to $C^\infty$ if i have constant coefficients and achieve $C^\infty$ solutions? – lecovee Jan 28 '17 at 11:50