After having done some research, I could not find a reference for the following. Suppose I have a problem of the following type, on $(t,r) \in \mathbb{R} \times \mathbb{R}^2$: $$ \begin{array}{ll} \square u = \partial_t^2u - \Delta u = f(\cdot, u, Du) \\ u(0) = u_0, \qquad \partial_t u (0) = u_1. \end{array} $$ $f$ is assumed to be smooth here. My questions are:
Given smooth and compactly supported initial data $(u_0, u_1)$, is it true that there is a local-in-time existence result for this equation, such that the resulting solution is itself smooth?
Given smooth initial data $(u_0, u_1)$, with $\|u_0 \|_{H^1} + \| \partial_t u\|_{L^2} < C$ is it true that there is a local-in-time existence result for this equation, such that the resulting solution is itself smooth?
In particular, I think these results (if any) should rely on an iteration argument. I also know that, in general, $L^\infty$ bounds on the first derivatives combined with energy estimates give higher regularity. I would be really interested in understanding how (and if) the two (iteration + derivative bounds) fit together. Any help would be appreciated.