Let $u_0, u_1 \in C^\infty( \mathbb{R},\mathbb{R}^3)$. Consider the cubic defocusing NLW $$(\ast)\begin{cases} \Box u= |u|^2 u \\ (u,\partial_t u) \restriction_{t=0} = (u_0,u_1) ,\end{cases}$$ where $\Box = -\partial^2_{tt} + \Delta$ such that the energy is positive.
Now I want to show that there is a classical $C^\infty$ global solution.
My attempt: Fix $(t,x)$, and consider initial data $\psi_{1,x,t},\psi_{1,x,t}\in C_c^\infty$ such that $(\psi_{0,x,t},\psi_{1,x,t})\restriction_{B(x,\text{radius} = t+1)} = (u_0,u_1)\restriction_{B(x,\text{radius} = t+1}$. This problem has a unique global smooth solution, $\psi_{x,t}$. Do this for all $(t,x)$ and set $u(t,x) = \psi_{x,t}(t,x)$. By uniqueness of $(*)$ for $C_c^\infty$ data, this defines a smooth function solving $\Box u = |u|^2u$. Due to finite speed of propagation of $(*)$ for $C_c^\infty$ data, $u(t,x)$ depends only on the values of the initial data on the ball with radius $1+t$ around $x$, where, by construction, the initial data are the same. This gives the result.
Is still proof correct? Did I miss an important point?
Edit: I just figured out that this is Exercise 3.38 of Tao's book on nonlinear dispersive equations.