Consider the heat equation $u_t = \Delta u$ on $\mathbb{R}^n$ with initial data $u(0, x) = f(x)$. Suppose $f$ is smooth and compactly supported. Do we necessarily have that $u$ has non-compact support for all $t > 0$?
What about the same question for the Schrödinger equation $u_t = i\Delta u$ with smooth compactly supported $f$?
The solution $u(t,x)$ of the heat equation is analytic in $x$ as a function of $x$ for any $t>0$, so it cannot have compact support. This is independent of the smoothies or size of the initial data.
For the Schrödinger equation, the same is true if the initial data decays fast enough at $\infty$, in particular,if it has compact support.