Before revision: These finite-duration solutions can't be analytical functions on the whole real line, and because they will being non-Lipschitz functions at their ending time, they don't stand uniqueness of solutions of Lipschitz differential equations.
After revision: These finite-duration solutions can't be analytical functions on the whole real line, and because they are non-Lipschitz functions at their ending time, they are not unique solutions of Lipschitz differential equations.