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.
The Cauchy-Kowalevski theorem gives local existence of analytic solutions of analytic PDEs: https://en.wikipedia.org/wiki/Cauchy%E2%80%93Kowalevski_theorem Of course, a linear constant coefficient PDE is analytic, but this theorem is far more general and allows nonlinear PDEs as well.
In general, if you allow the coefficients to be non-constant smooth functions (not necessarily analytic), then it is possible to construct PDEs of this type for which there are no smooth solutions. There are many papers along these lines, one is "An example of a smooth linear PDE without solution" by Hans Lewy.