Compact supports of initial data implies compact supports for all $t$ in semi-linear wave equation $-\frac{\partial^2}{\partial t^2}u+\Delta u=u^3$.

Question

Problem: Let $u(t,x,y)$ be a smooth real function defined on $\mathbb R \times \mathbb R^2$ where $t \in\mathbb R$ and $(x,y) \in\mathbb R^2$. We assume that it solves the following semi-linear wave equation:$$-\frac{\partial^2}{\partial t^2}u+\Delta u=u^3$$ If the supports of the initial data $u(0,x)$ and $\frac{\partial u}{\partial t}(0,x)$ are compact, prove that, for all $t_0\in \mathbb R$, the supports of $u(t_0, x)$ and $\frac{\partial u}{\partial t} (t_0, x)$ are compact.$$$$ We know that for a general wave equation, the solution travels with finite speed. Since the initial data has compact support, from an intuition this problem is obvious. But I'm a starter in PDE and I don't know how to explicitly and strictly write this intuition out into a formal solution. Are there any brief way to express this intuition out for this problem? Thanks!