Well-posedness and sharp uniform decay rates at the L2(Ω)-level of the Schrodinger equation with nonlinear boundary dissipation

We prove that the Schrodinger equation defined on a bounded open domain of R-n and subject to a certain attractive, nonlinear, dissipative boundary feedback is ( semigroup) well-posed on L-2(Omega) for any n = 1, 2, 3,..., and, moreover, stable on L-2(Omega) for n = 2, 3, with sharp ( optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L-2(Omega)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely - at the outset - on a far general result of interest in its own right: an energy estimate at the L-2(Omega)-level for a fully general Schrodinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/ micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H-1(Omega)- level energy estimate to the L-2(Omega)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear ( interior and) boundary dissipation.