TITLE: Why is uniqueness important for PDEs?
QUESTION [30 upvotes]: Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, but I can't see why so much time is spent on uniqueness. Why do we care whether a solution is unique or not as long as we know that there is a solution? Is uniqueness just shown for the sake of it, ie. the sake of completeness, or is there some deeper reason why it's considered important to show uniqueness?

REPLY [4 votes]: Suppose you show that $$y=e^x \rightarrow \left(\frac{dy}{dx} = y \wedge y(0) = 1\right).$$
Have you solved the IVP? In my opinion, no: its not solved until you've also proved that 
$$\left(\frac{dy}{dx} = y \wedge y(0) = 1\right) \rightarrow y=e^x.$$
That's what uniqueness gives you; it lets you derive the second formula from the first, via this principle of logic that I still don't have a name for.