Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains

Well-posedness and stability for Schrödinger equations with infinite memory

“…Moreover, the uniqueness of solution is proved when d ∈ {1, 2}. Similar results were obtained in [1] and [2] in d-dimensional Riemannian manifolds and nonlinear local damping ic(x)g(y) (instead of icy) but with b = 0, where g is a given function satisfying some properties. The authors of [7] considered in Riemannian manifolds two more general forms than (1.3) by taking f (|y|) and (−∆) k (c(x)y) instead of b|y| p and cy, respectively, where f is a given function satisfying some properties and k ∈ {0, 1 2 }.…”

Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

“…In fact, the smoothing effect is a natural property for the whole R n or non-trapping exterior domains as considered in the works of Constantin and Saut [14] or Burq, Gérard and Tzvetkov [10]. It has been considered previously in unbounded domains as powerful tool in order to achieve the exponential stability, see, for instance, the works [8] and [12]. On the other hand, for compact manifolds the smoothing effect has been introduced firstly by Aloui [1,2], by forcing it precisely by means of the stronger damping iαb(x)(−∆) 1 2 b(x) above mentioned where the the region ω such that b(x) > 0 satisfies the wellknown geometric control condition (GCG, in short).…”

Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold

“…where C is a positive constant. Therefore the main result follows from semigroup property (see [4]).…”

“…Stabilization. Let's start the main section of this work with a technical Lemma whose the similar proof can be found in [6] and [4]. In fact we only join our ideas.…”