If you are given just the values of the function on the line x=0, then isn't the solution determined uniquely by analytic continuation, so that the differential equation doesn't even matter?

Also, for hyperbolic equations, the boundary conditions along with solving the differential equation gives you the solution in a region bounded by the characteristics and the boundary. However, once you have the solution in this region, don't you also have the solution everywhere, by analytic continuation? For example, you just extend the solution by power series about a point in your region into the new region?