I am studying some results about existence and uniqueness of solutions to some PDEs and many times Banach fixed point Theorem is used. I saw that the ideia is to consider the integral formulation and use the Theorem to prove existence and uniqueness in a ball but I can't understant how the uniquess extends to all the space.
Any one can explain how to get uniquess in all the space and how existence and uniquess of solution to the integral formulation garantees the same for the PDE?
Any help is appreciated.