For example, a wikipedia (simplified) article (and there are many more).
The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that:
1.A solution exists
2.The solution is unique
3.The solution's behavior changes continuously with the initial conditions.
There are too many opinions and debates on (this) debate.Org site that are vague, and this (may be) the reason nothing truly gets answered. Not to mention scientific falsifiability / testability.