Abstract
This paper studies the inhomogeneous defocusing coupled Schrödinger system
i
u
˙
j
+
Δ
u
j
=
|
x
|
-
ρ
(
∑
1
≤
k
≤
m
a
j
k
|
u
k
|
p
)
|
u
j
|
p
-
2
u
j
,
ρ
>
0
,
j
∈
[
1
,
m
]
.
i\dot{u}_{j}+\Delta u_{j}=\lvert x\rvert^{-\rho}\bigg{(}\sum_{1\leq k\leq m}a_%
{jk}\lvert u_{k}\rvert^{p}\biggr{)}\lvert u_{j}\rvert^{p-2}u_{j},\quad\rho>0,%
\,j\in[1,m].
The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of
f
∈
H
1
(
ℝ
N
)
{f\in H^{1}(\mathbb{R}^{N})}
such that
x
f
∈
L
2
(
ℝ
N
)
{xf\in L^{2}(\mathbb{R}^{N})}
. The present paper is a complement of the previous work by the first author and Ghanmi
[T. Saanouni and R. Ghanmi,
Inhomogeneous coupled non-linear Schrödinger systems,
J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption
x
u
0
∈
L
2
{xu_{0}\in L^{2}}
enables us to get the scattering in the mass-sub-critical regime
p
0
<
p
≤
2
-
ρ
N
+
1
{p_{0}<p\leq\frac{2-\rho}{N}+1}
, where
p
0
{p_{0}}
is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets
|
t
|
α
∥
u
(
t
)
∥
L
r
(
ℝ
N
)
≲
1
\lvert t\rvert^{\alpha}\lVert u(t)\rVert_{L^{r}(\mathbb{R}^{N})}\lesssim 1
for some
α
>
0
{\alpha>0}
and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of
e
i
⋅
Δ
u
0
{e^{i\cdot\Delta}u_{0}}
. This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms
p
0
<
p
<
2
-
ρ
N
-
2
+
1
{p_{0}<p<\frac{2-\rho}{N-2}+1}
. This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that
p
≥
2
{p\geq 2}
, which restricts the space dimensions to
N
≤
3
{N\leq 3}
. In a paper in progress, the authors treat the same problem in the complementary case
ρ
<
0
{\rho<0}
.