failure of uniqueness

Failure of uniqueness is the unsettling situation where an initial value problem has a solution, but more than one — sometimes infinitely many curves all pass through the same starting point obeying the same rule. It clashes with the intuition that a starting state plus a law of motion should determine the future, and that intuition is usually right; but it is not a theorem unless you add a hypothesis. When the hypothesis is missing, the future genuinely forks.

The famous example is y' = y^(2/3) with y(0) = 0. One solution is the flat line y = 0, sitting at the origin forever. But another is y = (x/3)^3, which also passes through (0, 0) and satisfies the equation. Worse, you can wait any amount of time T and then peel off: stay on y = 0 until x = T, then follow y = ((x - T)/3)^3. Every choice of T gives a different valid solution, so there are infinitely many. The mechanism is that f(y) = y^(2/3) has an infinite slope at y = 0 — its derivative (2/3) y^(-1/3) is unbounded there — so no Lipschitz constant exists at the origin, and solutions are free to branch.

The lesson is exactly what the Lipschitz condition is for: it is the precise property that forbids this branching. A bounded slope in y means two solutions cannot peel apart from a shared point — Gronwall's inequality turns 'bounded spreading rate' into 'no spreading at all from a common start.' So non-uniqueness is not a paradox or a flaw in the theory; it is the theory honestly telling you that a square-root-style flatness at the equilibrium lets trajectories rest there for a while and then leave at any moment, and only extra smoothness rules that out.