On Uniqueness for Supercritical Nonlinear Wave and Schrodinger Equations (original) (raw)

Uniqueness properties of solutions to Schrödinger equations

Bulletin of the American Mathematical Society, 2012

For a proof of this estimate, see [25], [7]. The SUCP of Carleman-Müller follows easily from (1.2) (see [37] for instance). In the late 1950s and 1960s there was a great deal of activity on the subject of SUCP and the closely related uniqueness in the Cauchy problem, some highlights being [1] and [8], respectively, both of which use the method of Carleman estimates. These results and methods have had a multitude of applications to many areas of analysis, including to nonlinear problems. (For a recent example, see [38] for an application to energy critical nonlinear wave equations). In connection with the Carleman-Müller SUCP a natural question is, How fast is a solution u allowed to vanish before it must vanish identically? By considering n = 2, u(x 1 , x 2) = (x 1 + ix 2) N , we see that to make sense of the question, a normalization is required; for instance, sup |x|<3/4

Unique continuation properties of the nonlinear Schrödinger equation

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1997

Consider the unique continuation problem for the nonlinear Schrödinger (NLS) equationBy using the inverse scattering transform and some results from the Hardy function theory, we prove that if u ∈ C(R; H1(R)) is a solution of the NLS equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the NLS equation, then u vanishes identically if it vanishes on two horizontal half lines in the x–t space. This implies that the solution u must vanish everywhere if it vanishes in an open subset in the x–t space.