I'm trying to understand how one can make precise statements about propagation speed for various (linear and nonlinear) PDEs (in particular, ones with infinite propagation speed) and what, if anything, these precise statements can be used to prove (which isn't trivial by other methods). An example set of questions I'd be interested in learning more about is the following:

Suppose we have a nice initial data $\varphi \in \mathcal{S}(\mathbb{R}^n)$ which is frequency supported in $|\xi| \leq N$. Let $u$ be the resulting (strong) solution to a linear or nonlinear Schrodinger equation with initial data $\varphi$. In what sense does $u$ propagate with speed at most $N$ in physical space (as predicted by the formula ''speed of propagation equals minus the gradient of the dispersion relation'' together with the initial frequency localization of $u$)? If, for example, $u$ at time zero (i.e., $\varphi$) has most of its $L^2$ norm in the set $|x| \leq M$, does most of its $L^2$ norm remain in the set $|x| \leq M + TN$ when $|t| \leq T$? What about $H^k$ norms (for integral $k \geq 1$, say)? How is this behavior different in the linear and nonlinear cases? What if we only assume that $\varphi$ has ''most'' of its frequency support in $|\xi| \leq N$ (as quantified by some weighted $L^2$ norm in frequency space)?

Tao mentions in *Nonlinear Dispersive Equations* that one can use the method of stationary phase to give precise meaning to statements about propagation speed, but I don't believe he gives any references. So, I'd be interested in any references which cover questions like the ones indicated above.