Definition (well-posed problem)

The initial value problem

,

is said to be well-posed if:

(1) a unique solution exists,

(2) for any , there exists a positive constant such that when

and , a unique solution to the problem

,

exists with

for all in .

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In other words, not only needs there to be a unique solution, but when the problem is perturbed slightly, another unique solution must exists which is close to the first. This is a fundamental requirement when trying to address this problem numerically, where errors due to number representation and round-off means we're often solving a slightly perturbed problem.

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