Suppose $u \in C_{x, t}^2([0, 1] \times [0, \infty))$, and $u$ is solution of following equation: $$ \begin{cases}u_{tt} = u_{xx} \\ u|_{\partial[0, 1]} = 0\end{cases}$$
If $u \not\equiv 0 $, could it be that for some fixed $x_0 \in [0, 1],$ $u(x_0, t) = 0$ for any $t \in [0, \infty]$?
Answer: (by @Paul) Yes, $u(x, t) = \sin(2\pi t)\sin(2\pi x)$.
Editted question: Could it be that for some fixed interval $[a, b] \subset [0, 1]$, $u(x, t) = 0$, for any $x \in [a, b],$ and $t \in [0, \infty]$?
Sure. $u= sin ( 2\pi t) sin( 2\pi x)$ is just such a solution.