Unique continuation

A question arising by analogy from the theory of unique continuation in elliptic equations, and also in control theory, is the following: if u is a solution to a nonlinear Schrodinger equation, and u(t_0) and u(t_1) is specified on a domain D at two different times t_0, t_1, does this uniquely specify the solution everywhere at all other intermediate times?

• For the 1D cubic NLS, with D equal to a half-line, and u assumed to vanish on D, this is in Zg1997.
• For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in Bo1997b.
• For D the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in KnPoVe2003
• For D in a half-plane, and allowing potentials in various Lebesgue spaces, this is in IonKn-p
• A local unique continuation theorem (asserting that a non-zero solution cannot vanish on an open set) is in Isk1993