Suppose there is a spacetime. If I have two functions $u_1$, $u_2$ satisfying one wave equation, and there is a subset $A$ such that $u_1,u_2$ coincide in $A$, then do we have that $u_1,u_2$ coincide in the domain of dependence of $A$? If $A$ is a neighbourhood of a Cauchy surface, then this is true, and I wonder what if $A$ is arbitrary?
We can assume necessary smoothness.
Thanks in advance.