Abstract
In this paper we study the existence as well as uniform decay rates
of the energy associated with the nonlinear damped Schrödinger equation,
i
u
t
+
Δ
u
+
|
u
|
α
u
-
g
(
u
t
)
=
0
in
Ω
×
(
0
,
∞
)
,
iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty),
subject to Dirichlet boundary conditions, where
Ω
⊂
ℝ
n
{\Omega\subset\mathbb{R}^{n}}
,
n
≤
3
{n\leq 3}
, is a bounded domain with smooth boundary
∂
Ω
=
Γ
{\partial\Omega=\Gamma}
and
α
=
2
,
3
{\alpha=2,3}
. Our goal is to consider a different approach than the one used in [B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface,
Math. Z. 254 2006, 4, 729–749], so instead than using the properties of pseudo-differential operators introduced by cited authors, we consider a nonlinear damping, so that we remark that no
growth assumptions on
g
(
z
)
{g(z)}
are made near the origin.
Stabilization of Semilinear PDEs, and Uniform Decay under Discretization
