Well-posedness and uniform stability for nonlinear Schrödinger equations with dynamic/wentzell boundary conditions


In this paper, the Shrödinger equation with a defocusing nonlinear term and dynamic boundary conditions defined on a smooth boundary of a bounded domain Ω ⊂ ℝN, N = 2, 3 is considered. Local well-posedness of strong H2 solutions is also established. In the case, N = 2 local solutions are shown to be global, and existence of weak H1 solutions in dimensions N = 2, N= 3 is shown. The energy corresponding to a weak solution is shown to satisfy uniform decay rates under appropriate monotonicity conditions imposed on the nonlinear terms appearing in the dynamic boundary conditions. The proof of well-posedness is critically based on converting the equation into a Wentzell boundary value problem associated with Shrödinger dynamics. The analysis of this later problem with nonhomegenous boundary data allows us to build a theory suitable for the treatment of the dynamic boundary conditions.

Publication Title

Indiana University Mathematics Journal