Uniform stabilization of the fourth order Schrödinger equation

We study both boundary and internal stabilization problems for the fourth order Schrödinger equation in a smooth bounded domain Ω of RnRn. We first consider the boundary stabilization problem. By introducing suitable dissipative boundary conditions, we prove that the solution decays exponentially in an appropriate energy space. In the internal stabilization problem, by assuming that the damping term is effective on a neighborhood of a part of the boundary, we prove the exponential decay of the L2(Ω)L2(Ω)-energy of the solution. Both results are established by using multiplier techniques and compactness/uniqueness arguments.