On uniqueness properties of solutions of the generalized fourth-order Schrödinger equations

Abstract

In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schrödinger equations in any dimension d of the following forms, i∂tu+∑j=1d∂xj4u=Vt,xu,andi∂tu+∑j=1d∂xj4u+Fu,u‾=0. We show that a linear solution u with fast enough decay in certain Sobolev spaces at two different times has to be trivial. Consequently, if the difference between two nonlinear solutions u 1 and u 2 decays sufficiently fast at two different times, it implies that u1≡u2 .