Well-posedness and Stability for Nonlinear Schroedinger Equations with Dynamic/Wentzell Boundary Conditions

Lefler, Christopher, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Lasiecka, Irena, Department of Mathematics, University of Virginia

Semilinear Schroedinger equations with cubic nonlinear part on bounded domains subject to a Wentzell (dynamic) boundary condition are studied in dimensions 2 and 3. We prove well-posedness on the Sobolev space H^2 as well as exponentially stable. The former result relies on first proving well-posedness of the linear model through treating the problem as a Wentzell problem, to which semigroup methods are applied. Obtaining well-posedness of the nonlinear model requires reformulating the problem as having a dynamic boundary condition, to which a fixed point argument is applied. Global well-posedness of the nonlinear model can only be achieved in 2 dimensions. In 3 dimensions we are able to prove global existence of weak solutions on the Sobolev space H^1 by the Galerkin approach, but we are not able to obtain uniqueness or continuous dependence on the initial data. Exponential stability of the linear model has been established previously in the literature. We adapt techniques from the linear model to achieve exponential stability of the nonlinear model.

PHD (Doctor of Philosophy)
mathematics, partial differential equations, nonlinear Schroedinger equation, well-posedness, stability, Wentzell boundary conditions, dynamic boundary conditions
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