If you have a PDE that models physical phenomena, it is fundamental that:
- there must exist a solution for every physically valid initial condition, otherwise it means that the equation does not describe certain cases of reality
- the solution must be unique, otherwise how are we to choose between the multiple solutions?
Unlike for ordinary differential equations which have the Picard–Lindelöf theorem, the existence and uniqueness of solution is not well solved for PDEs.
For example, Navier-Stokes existence and smoothness was one of the Millennium Prize Problems.