Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 66, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

INVISCID LIMIT OF LINEARLY DAMPED AND FORCED

NONLINEAR SCHRÖDINGER EQUATIONS

NIKOLAOS GIALELIS

Abstract. We approximate a solution of the nonlinear Schrödinger Cauchy

problem by solutions of the linearly damped and driven nonlinear Schrödinger

Cauchy problems in any open subset of Rn and, for the case n = 1, we provide
an estimate of the convergence rate. In doing so, we extract a sufficient relation

between the external force and the constant of damping.

1. Introduction

In this work we are interested in the n-dimensional linearly damped, driven
nonlinear Schrödinger equation (LDDNLS), with the common case of pure power
nonlinearity, i.e.

iut + ∆u+ λ|u|αu+ iγu = f, ∀(t, x) ∈ [0, T ]× U, (1.1)

where λ ∈ R∗ and α > 0, γ > 0 and u = u(t, x; γ), f = f(t, x; γ) are complex-valued
functions for t ∈ [0, T ] with T > 0 and x ∈ U with U ⊆ Rn being an arbitrary
open set. γ is the constant of zero order dissipation and f an external excitation.
The goal is to show, under certain conditions, that (1.1) can be considered as a
perturbation of the associated nonlinear Schrödinger equation (NLS), i.e.

ivt + ∆v + λ|v|αv = 0, ∀(t, x) ∈ [0, T ]× U. (1.2)

NLS models with gain and loss effects have found applications to many physi-
cal fields such as nonlinear optics and fluid mechanics (see [3] and the references
therein). The use of damping and forcing effects for (1.2) is not a novelty for physi-
cists (see e.g. [6] and [20]). On the other hand, some cases of (1.1) have already
been studied, concerning the solvability and the long time behavior of solutions and
their attractors of Cauchy problems (see [2, 13, 14, 15, 16, 17, 18, 24]). Compar-
isons between the two equations have also been made (see [12] about some blowup
issues). Even though these two equations seem quite similar, they share important
differences. In particular, many of the symmetries of (1.2) do not hold for (1.1),
such as the known scaling symmetry, the Galilean invariance and the time reversal
symmetry (see [22]). To the author’s best knowledge, some questions of “inviscid
limit” type for these equations still remain unasked. In [5], (1.1) arises from a
perturbation study of the sine-Gordon equation and in [26] it is shown that (1.2)

2010 Mathematics Subject Classification. 35Q55, 35B20, 35A01.
Key words and phrases. Nonlinear Schrödinger equation; inviscid limit; linear damping;

forcing term.
c©2020 Texas State University.

Submitted November 12, 2018. Published June 29, 2020.

1



2 N. GIALELIS EJDE-2020/66

is the inviscid limit of complex Ginzburg-Landau equation. However, it is natural
for us to expect that (1.1) could be a perturbation of (1.2) and this viewpoint is
the scope of this study.

Here, we extract a sufficient relation between f and γ of the form ‖f‖ = O(γ),
as γ ↘ 0 (see (6.1)), to obtain two approximation results in Section 6. First (see
Proposition 6.1 and Corollary 6.2), we approximate a solution (or the solution in
case of uniqueness) v of the NLS initial-boundary value problem

ivt + ∆v + λ|v|αv = 0, ∀(t, x) ∈ (0, T ]× U
v = v0, on {t = 0} × U
v = 0, on [0, T ]× ∂U,

(1.3)

by a sequence {um}∞m=1 of solutions of the LDDNLS initial-boundary value prob-
lems of the form

iut + ∆u+ λ|u|αu+ iγu = f, ∀(t, x) ∈ (0, T ]× U
u = u0, on {t = 0} × U
u = 0, on [0, T ]× ∂U,

(1.4)

as γm ↘ 0, fm → 0 and u0m → v0. Second (see Proposition 6.3), we estimate the
rate of this approximation for certain cases. We note that the convergences above
will be rigorously interpreted.

In proving the above results, we first show, in Sections 4 and 5, the existence of
a bounded solution of (1.4), which satisfies a certain estimate (see Theorems 4.1,
4.2 and 5.1). The aforementioned sufficient condition ‖f‖ = O(γ), as γ ↘ 0, comes
naturally from that estimate. We emphasize that the technique we use differs from
the classic one of “regularized nonlinearities” presented in [9] and this is also a third
goal that we reach with the present work.

We note that, since our main interest lies in inviscid limit results, we deal with the
defocusing and the subcritical focusing case, as well as the critical focusing case with
sufficiently small initial datum (see (4.1)), where the analysis for the extraction of
energy estimates is not that extended in comparison with the supercritical focusing
case for sufficiently small initial datum. Hence, we exclude this case, not bacause
of inefficiency of our approach, but to keep the work as compact as possible and
stay focused on our main result.

2. Notation

We denote by ∗ ∨ ? := max{∗, ?} and by B%(x) ⊂ Rn the open ball of radius
% > 0 centered at x. If p, r ∈ [1,∞] and k,m ∈ N0, then we write

| · |m,r,U := ‖ · ‖Wm,r(U), | · |−m,U := ‖ · ‖H−m(U)

| · |k,p,T ;m,r,U := ‖ · ‖Wk,p(0,T ;Wm,r(U)), | · |k,p,T ;−m,U := ‖ · ‖Wk,p(0,T ;H−m(U)).

We omit p =∞, T =∞ and U = Rn from the notation.
For m ∈ N0 and U , we consider that the space Hm(U) ≡ Wm,2(U) is equipped

with the inner product (∗, ?)Hm(U) → C defined as

(u, v)Hm(U) :=
∑

0≤|α|≤m

∫
U

(Dα
wu)(Dα

wv)dx, ∀u, v ∈ Hm(U).

When m = 0, we simply write (∗, ?) := (∗, ?)H0(U) ≡ (∗, ?)L2(U).



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 3

Let F(U1;C) be a function space over U1 ⊂ U2 ⊆ Rn and f ∈ F(U1). We denote
by EU2f its extension by zero in U2 \ U1 and EU2F(U1) := {EU2f

∣∣f ∈ F(U1)}. We
omit U2 = Rn from these notations. Moreover, if g ∈ F(U2), we denote by RU1

g
and RU1

F(U2) the restriction of g in U1 and the set of these restricted functions,
respectively.

We write C and c for any non-negative constant factor and exponent, respec-
tively. These constants may be explicitly calculated in terms of known quantities
and may change from line to line and also within a certain line in a given computa-
tion. We also employ the letter K for any increasing function K : [0,∞)m → [0,∞),

as well as K̃ : [0,∞)2 × (0,∞)→ [0,∞), such that

(1) K̃(·, ·, z0) is increasing, for fixed z0 > 0 and also

(2) there exists K such that K̃(x,O(z), z)→ K(x0), as (x, z)→ (x0, 0).

When U appears as subscript in an element, it denotes that this depends on it,
while its absence designates independence. If u : [0, T ]×U → C, with u(t, ·) ∈ F(U)
for each t ∈ [0, T ], then, following the notation of, e.g., [11] and [23], we associate
with u the mapping u : [0, T ] → F(U ;C), defined by [u(t)](x) := u(t, x), for every
x ∈ U and t ∈ [0, T ].

3. Preliminaries

Lemma 3.1. Let u, v ∈ Lα+2(U). Then∫
U

|u|α+1|v|dx ≤ |u|α+1
0,α+2,U |v|0,α+2,U , (3.1)

||u|αu− |v|αv|0,α+2
α+1 ,U

≤ C(|u|c0,α+2,U + |v|c0,α+2,U )|u− v|0,α+2,U . (3.2)

Proof. The first inequality follows from (7.4) for p = α+2
α+1 and q = α + 2. As for

the second one, we apply (7.2), (7.4) for p = α+ 1 and q = α+1
α and (7.1). �

Next, we set

α ∈

{
(0,∞), if n = 1, 2

(0, 4
n−2 ], otherwise.

(3.3)

In view of (3.1) and the scaling invariant embedding H1
0 (U) ↪→ Lα+2(U) (notice

that U is assumed to be just an open set and then see Remark 7.5, we define

g : H1
0 (U)→ L

α+2
α+1 (U) ↪→ H−1(U) to be the nonlinear and bounded operator such

that

〈g(u;α), v〉 := λ

∫
U

|u|αu vdx, for v ∈ H1
0 (U).

Next, we recall the following well establish result.

Lemma 3.2. For every f ∈ H−1(U) there exists {fj}nj=0 ⊂ L
2(U) such that

〈f, v〉 =

∫
U

vf0 +

n∑
j=1

(∂jv)fjdx, ∀v ∈ H1
0 (U)

and, in particular, we have

(v, f) = 〈f, v〉, ∀v ∈ H1
0 (U), ∀f ∈ L2(U).



4 N. GIALELIS EJDE-2020/66

Proof. The first result follows from a direct application the complex version of
Riesz-Fréchet representation theorem (see [8, Proposition 11.27]). The second is a
direct consequence of the first one. �

Now, for the above operator we have the following estimate.

Proposition 3.3. Let u, v ∈ H1
0 (U). Then

|g(u)− g(v)|0,α+2
α+1 ,U

≤ K
(
|u|1,2,U , |v|1,2,U

)
|u− v|0,α+2,U . (3.4)

The proof of the above proposition is a direct application of (3.2) and the scaling
invariant embedding H1

0 (U) ↪→ Lα+2(U).

We further define N [·, ·],Nγ [·, ·] : (H1
0 (U))

2 → C to be the forms which are
associated with the operators ∆+g and ∆+g+iγI, respectively, such thatN [u, v] :=
〈∆u, v〉 + 〈g(u), v〉 and Nγ [u, v] := 〈∆u, v〉 + 〈g(u), v〉 + iγ〈u, v〉, for every u, v ∈
H1

0 (U).
We then restate problems (1.3) and (1.4) as Cauchy ones: for f : [0, T ]→ L2(U),

we seek solutions v,u ∈ L∞(0, T ;H1
0 (U)) ∩W 1,∞(0, T ;H−1(U)) of

〈iv′, u〉+N [v, u] = 0, ∀u ∈ H1
0 (U), a.e. in [0, T ]

v(0) = v0.
(3.5)

and
〈iu′, v〉+Nγ [u, v] = 〈f , v〉, ∀v ∈ H1

0 (U), a.e. in [0, T ]

u(0) = u0.
(3.6)

Also, we provide an estimate for the forms N and Nγ .

Proposition 3.4. Let u, v ∈ H1
0 (U). Then

|N [u, v]|+ |Nγ [u, v]| ≤ K
(
|u|1,2,U , |v|1,2,U

)
. (3.7)

The proof of the above proposition is and application of (7.4) (p = p = 2), (3.1)
and the scaling invariant embedding H1

0 (U) ↪→ Lα+2(U). Some useful results also
follow.

Lemma 3.5. Let α be as in (3.3) and u ∈ H1
0 (U). Then

|u|α+2
0,α+2,U ≤ C|Du|

nα
2

0,2,U |u|
4−nα

2 +α

0,2,U . (3.8)

If, in addition, n = 2 and τ ∈ (1,∞), then

|u|2τ0,2τ,U ≤ C|Du|
2(τ−1)
0,2,U |u|

2
0,2,U . (3.9)

Proof. The first inequality is direct from Theorem 7.4 (and Remark 7.5) for p =
α + 2, r = q = 2, j = 0, m = 1 and θ = nα

2(α+2) . As for the second one we set

α = 2(τ − 1) in (3.8). �

Remark 3.6. If

α ∈

{
(0,∞), if n = 1, 2

(0, 4
n−2 ), otherwise,

(3.10)

then the exponent of the term |u|0,2,U in (3.8) is strictly positive and hence that

term does not vanish. Moreover, an estimate of the constant in (3.9) is

C ≤ (4π)
(1−τ)

τ τ , (3.11)

for an elegant proof of which we refer to [21] and the references therein.



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 5

Lemma 3.7. Let α ∈ (0, 4/n), ε > 0 and u ∈ H1
0 (U). Then

|u|α+2
0,α+2,U ≤ ε|Du|

2
0,2,U + C|u|c0,2,U . (3.12)

The above lemma is an application of (7.3) for p = 4
nα and q = 4

4−nα into (3.8).

Proposition 3.8. (i) Let H be a Hilbert space, as well as {uk}∞k=1 ⊂ L∞(0, T ;H)
and u : [0, T ]→ H with uk(t) ⇀ u(t) in H, for a.e. t ∈ [0, T ]. If ‖uk‖L∞(0,T ;H) ≤
C uniformly for all k ∈ N∗, then u ∈ L∞(0, T ;H) with ‖u‖L∞(0,T ;H) ≤ C, where

C is the same in both inequalities.
(ii) Let F be a Banach space with the Radon-Nikodym property with respect

to the Lebesgue measure in (0, T,B([0, T ])) and {uk}∞k=1 ∪ {u} ⊂ L∞(0, T ;F∗)
with uk

∗
⇀ u in L∞(0, T ;F∗) (That is, uk

∗
⇀ u in σ(L∞(0, T ;F∗), L1(0, T ;F)).

Note that L∞(0, T ;F∗) ∼= (L1(0, T ;F))∗ (see, e.g., [10, Theorem 1, §IV.1].) If
‖uk‖L∞(0,T ;F∗) ≤ C uniformly for all k ∈ N∗, then ‖u‖L∞(0,T ;F∗) ≤ C, where C is

the same in both inequalities.

Proof. (i) We derive that ‖u(t)‖H ≤ C, for a.e. t ∈ [0, T ], from the (sequentially)
weak lower semi-continuity of the norm. The result follows directly.

(ii) Let v ∈ F be such that ‖v‖F ≤ 1 and set v : [0, T ]→ F the constant function
with v(t) := v, for all t ∈ [0, T ]. We have∫ s+h

s

〈uk,v〉dt ≤ Ch,

] for every s ∈ (0, T ) and every sufficiently small h > 0. Letting k → ∞, dividing
both parts by h and then letting h→ 0, we obtain 〈u(s), v〉 ≤ C, for every s ∈ (0, T ).
Since v arbitrary, the proof is complete. �

Proposition 3.9. Let U1 ⊂ U2 ⊆ Rn, m ∈ N0 and {uk}∞k=1 ∪ {u} ⊂ Hm(U2) such
that uk ⇀ u in Hm(U2). Then RU1uk ⇀ RU1u in Hm(U1). The analogous result
for Lp, with p ∈ (1,∞), instead of Hm also holds.

Proof. We show the first result and in analogous fashion we obtain the second one.
Let v ∈ C∞c (U1), then we have

(RU1uk −RU1u, v)Hm(U1) =

m∑
|β|=0

∫
U1

Dβ(RU1uk −RU1u)Dβvdx

=

m∑
|β|=0

∫
U2

Dβ(uk − u)DβEU2
vdx

= (uk − u, EU2
v)Hm(U2) → 0,

hence, the result follows from a denseness argument. �

Proposition 3.10. Let {um}∞m=1∪{u} ⊂ H1(U) such that um ⇀ u in H1(U) and
um ⇀ u in L2(U). Then Dum ⇀ Du in L2(U).

Proof. Let v ∈ C∞c (U). Then

(Dum −Du, v) = (um − u, v)H1(U) − (um − u, v)→ 0,

hence, the result follows from a denseness argument. �



6 N. GIALELIS EJDE-2020/66

4. LDDNLS Cauchy problem in bounded open sets

In this section we assume U ⊂ Rn is bounded.

Theorem 4.1. Let α be as in (3.10), f ∈W 1,∞(0, T ;L2(U)) and u0 ∈ H1
0 (U). If

λ < 0, or

λ > 0 and α ∈ (0,
4

n
), or

λ > 0, α =
4

n
and |u0|0,2,U ∨

1

γ
|f |0,T ;0,2,U < λ−1/α|R|0,2,

(4.1)

where R as in Theorem 7.6, then there exist a solution u ∈ L∞(0, T ;H1
0 (U)) ∩

W 1,∞(0, T ;H−1(U)) of (3.6), such that

|u|0,T ;1,2,U + |u′|0,T ;−1,U ≤ K̃ := K̃(|u0|1,2,U , |f |1,T ;0,2,U , γ). (4.2)

Proof. Step 1. We use the standard Faedo-Galerkin method. It holds true that
H1

0 (U) ↪→↪→ L2(U) (see Remark 7.5), hence there exists a countable subset of
H1

0 (U) ∩ C∞(U), which is an orthogonal basis of L2(U), e.g., the complete set of
eigenfunctions for the operator −∆ in H1

0 (U) (This specific subset is an orthogonal
basis of both H1

0 (U) and L2(U)). Let {wk}∞k=1 ⊂ H1
0 (U) ∩ C∞(U) be that basis,

appropriately normalized so that {wk}∞k=1 be an orthonormal basis of L2(U). Fixing

any m ∈ N∗, we define dm : Jm → Cm, with dm(t) := [d1
m(t), . . . , dmm(t)]

T
, to be

the unique, absolutely continuous, maximal solution (i.e. Jm with 0 ∈ Jm is the
maximal interval on which the solution is defined) of the initial-value problem

dm
′(t) = Fm(t,dm(t)), ∀t ∈ J∗m

dm(0) = [(u0, w1), . . . , (u0, wm)]
T
,

where Fm ∈ C([0, T ]
2m+1

;Cm) with

F km(t, dm(t)) := iNγ [

m∑
l=1

dlm(t)wl, wk]− i(wk, f(t)), ∀k = 1, . . . ,m.

Now, we define um : Jm → H1
0 (U) ∩ C∞(U), with

um(t) :=

m∑
k=1

dkm(t)wk.

It is then trivial to verify that

〈iu′m, wk〉+Nγ [um, wk] = 〈f , wk〉, (4.3)

everywhere in Jm and for all k ∈ {1, . . . ,m}. Note that u0m := um(0, ·) = um(0)→
u0 in L2(U) and |u0m|0,2,U ≤ |u0|0,2,U . Furthermore, |u0m|1,2,U ≤ |u0|1,2,U . Indeed,

we can argue as in Step 3. of the proof of [11, Theorem 2, Section 6.5] to deduce
|Du0m|0,2,U ≤ |Du0|0,2,U . Moreover, we set f0 := f(0), since f ∈ C([0, T ];L2(U)).

Step 2. We multiply the variational equation (4.3) by dkm(t), sum for k = 1, . . . ,m
and take imaginary parts of both sides to find

d

dt
|um|20,2,U + 2γ|um|20,2,U ≤ 2|(f ,um)|,



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 7

hence, from (7.3) for ε = γ/2 (p = q = 2),

d

dt
|um|20,2,U + γ|um|20,2,U ≤

1

γ
|f |20,2,U ≤

1

γ
|f |20,T ;0,2,U ,

which implies the estimate

|um|0,2,U ≤ |u0|0,2,U ∨
1

γ
|f |0,T ;0,2,U , ∀t ∈ [0, T ], (4.4)

therefore, since m ∈ N∗ is arbitrary, Jm ≡ [0, T ], for all m ∈ N∗ and

|um|0,2,U ≤ K̃, ∀t ∈ [0, T ], ∀m ∈ N∗. (4.5)

Step 3α. We multiply the variational equation (4.3) by dkm
′
(t) + γdkm(t), sum for

k = 1, . . . ,m and take real parts of both sides to find

d

dt
J [um, f ]+γJ [um, f ]+

γ

2
|Dum|20,2,U−

γλ(α+ 1)

α+ 2
|um|α+2

0,α+2,U = Re(f ′,um), (4.6)

where

J [v, g] :=
1

2
|Dv|20,2,U −

λ

α+ 2
|v|α+2

0,α+2,U + Re(g, v), ∀v ∈ H1
0 (U), g ∈ L2(U).

Note that J [u0m, f0] ≤ K(|u0|1,2,U , |f |0,T ;0,2,U ). To show that

|Dum|0,2,U ≤ K̃, ∀m ∈ N∗, (4.7)

we consider the following cases.

(i) Since γ
2 |Dum|20,2,U −

γλ(α+1)
α+2 |um|

α+2
0,α+2,U ≥ 0, from (7.4) (p = q = 2) and (4.5)

we obtain

d

dt
J [um, f ] + γJ [um, f ] ≤ |um|0,2,Ω|f

′|0,2,U ≤ K̃|f
′|0,T ;0,2,U ,

which implies

J [um, f ] ≤ J [u0m, f0] ∨ 1

γ
K̃|f ′|0,T ;0,2,U .

Hence
1

2
|Dum|20,2,U ≤ K̃|f |0,T ;0,2,U + J [u0m, f0] ∨ 1

γ
K̃|f ′|0,T ;0,2,U ,

therefore we obtain (4.7).
(ii) Using (3.12) for ε = α+2

2λ(α+1) to estimate the last term on the left-hand side of

(4.6), we have

d

dt
J [um, f ] + γJ [um, f ] ≤ K̃(γ + |f ′|0,T ;0,2,U ),

which implies

J [um, f ] ≤ J [u0m, f0] ∨ K̃(1 +
1

γ
|f ′|0,T ;0,2,U ).

Therefore, applying again (3.12) for ε = ε̃(α+2)
λ and some ε̃ ∈ (0, 1/2), we obtain

1

2
|Dum|20,2,U ≤ K̃(1 + |f |0,T ;0,2,U ) + J [u0m, f0] ∨ K̃(1 +

1

γ
K̃|f ′|0,T ;0,2,U ),

hence (4.7) follows.



8 N. GIALELIS EJDE-2020/66

Using (7.6) for Ccr to estimate the last term on the left-hand side of (4.6), as well
as (4.4), we have

d

dt
J [um, f ] + γJ [um, f ] ≤ K̃(γ + |f ′|0,T ;0,2,U ),

since 1
2 −

λ
α+2Ccr(|u0|0,2,U ∨

1
γ |f |0,T ;0,2,U )

α
> 0. (4.7) then follows.

Step 3β. From (4.5) and (4.7) we conclude that {um}∞m=1 is uniformly bounded
in L∞(0, T ;H1

0 (U)), with

|um|0,T ;1,2,U ≤ K̃, ∀m ∈ N∗. (4.8)

Notice that we avoid to use the Poincaré inequality along with (4.7) for the above
bound.

Step 4. We fix an arbitrary v ∈ H1
0 (U) with |v|1,2,U ≤ 1 and write v = Pv ⊕

(I −P)v, where P is the projection in span{wk}mk=1. Since u′m ∈ span{wk}mk=1 and
N [h, g] linear for g, from the variational equation (4.3) we obtain that

〈iu′m, v〉 = −Nγ [um,Pv] + 〈f ,Pv〉.

Applying (3.7) we derive |〈iu′m, v〉| ≤ K̃+ |f |0,T ;0,2,U . Hence {u′m}
∞
m=1 is uniformly

bounded in L∞(0, T ;H−1(U)), with

|u′m|0,T ;−1,U ≤ K̃, ∀m ∈ N∗. (4.9)

Step 5α. From (4.8), (4.9), [9, Theorem 1.3.14 i)] and Proposition 3.8 (i), there
exist a subsequence {uml}

∞
l=1 ⊆ {um}

∞
m=1 and a function u ∈ L∞(0, T ;H1

0 (U)) ∩
W 1,∞(0, T ;H−1(U)), such that

uml(t) ⇀ u(t) in H1
0 (U), (4.10)

for every t ∈ [0, T ] and |u|0,T ;1,2,U ≤ K̃.

Step 5β. H−1(U) is separable since H1
0 (U) is separable, hence by the Dunford-

Pettis theorem (see [10, Theorem 1, §III.3]) we have

L∞(0, T ;H−1(U)) ∼= (L1(0, T ;H1
0 (U)))

∗
.

From the the above, (4.9), the Banach-Alaoglu-Bourbaki theorem (see [8, Theorem
3.16]) and Proposition 3.8 (ii), there exist a subsequence of {uml

}∞l=1, which we still
denote as such and a function h ∈ L∞(0, T ;H−1(U)), such that

u′ml
∗
⇀ h in L∞(0, T ;H−1(U)) and |h|0,T ;−1,U ≤ K̃. (4.11)

From the convergence in (4.10), [23, Lemma 1.1, Chapter 3], along with the Leibniz
rule, we can derive that∫ T

0

〈u′ml , ψv〉dt→
∫ T

0

〈u′, ψv〉dt, forallψ ∈ C1
c ([0, T ]), v ∈ H1

0 (U),

hence h ≡ u′.

Step 6α. Since U is bounded, H1
0 (U) ↪→↪→ L2(U) ↪→ H−1(U). Hence, from

(4.8), (4.9) and the Aubin-Lions-Simon lemma (see [7, Theorem II.5.16]), there
exist a subsequence of {uml}

∞
l=1, which we still denote as such and a function

y ∈ C([0, T ];L2(U)), such that

uml → y in C([0, T ];L2(U)). (4.12)



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 9

From the convergence in (4.10), we deduce that y ≡ u.

Step 6β. From (4.8), (4.12), (3.8) and Remark 3.6 we have

uml → u in C([0, T ];Lα+2(U)). (4.13)

Step 6γ. From (3.4), (4.8), the bound in (4.10), (4.12) and (4.13) we obtain

g(uml)→ g(u) in C([0, T ];L
α+2
α+1 (U)). (4.14)

Step 7α. Let now ψ ∈ C∞c ([0, T ]) and fix N ∈ N∗. We choose ml such that

N ≤ ml and v ∈ span{wk}Nk=1, hence, by the linearity of the inner product, we
obtain from (4.3) that∫ T

0

〈iu′ml , ψv〉+Nγ [uml , ψv]dt =

∫ T

0

〈f , ψv〉dt.

In view of Proposition 3.10, we then pass to the weak, ∗-weak and strong limits
(since ψv ∈ L1(0, T ;H1

0 (U))), to obtain∫ T

0

〈iu′, ψv〉+Nγ [u, ψv]dt =

∫ T

0

〈f , ψv〉dt.

Since ψ is arbitrary, u satisfies the variational equation in (3.6) for every v ∈
span{wk}Nk=1. By the linear and continuous dependence on v, we obtain the desired
result, after letting N →∞.

Step 7β. Finally, u satisfies the initial condition, i.e. u(0) ≡ u0, which follows
from (4.12) for t = 0 combined with um(0)→ u0 in L2(U) from Step 1. �

We can also get the following well-known result, by slightly modifying, in an
evident way, the above proof.

Theorem 4.2. Let α be as in (3.10) and v0 ∈ H1
0 (U). If

λ < 0, or

λ > 0 and α ∈ (0,
4

n
), or

λ > 0, α =
4

n
and |v0|0,2,U < λ−1/α|R|0,2,

(4.15)

where R as in Theorem 7.6, then there exist a solution v ∈ L∞(0, T ;H1
0 (U)) ∩

W 1,∞(0, T ;H−1(U)) of (3.5), such that

|v|0,T ;1,2,U + |v′|0,T ;−1,U ≤ K(|v0|1,2,U ). (4.16)

5. LDDNLS Cauchy problem in unbounded sets

In this section, we assume that U ⊆ Rn is unbounded. The concept behind the
proof of the following result is that of [4, Theorem 1.3].

Theorem 5.1. Let U ⊆ Rn be unbounded, α be as in (3.10), f ∈W 1,∞(0, T ;L2(U))
and u0 ∈ H1(U). Then the conclusions of Theorem 4.1 and Theorem 4.2 still hold.



10 N. GIALELIS EJDE-2020/66

Proof. We deal with the extension of Theorem 4.1 for unbounded sets. The second
result follows similarly.

Step 1. Since U open, we fix an arbitrary B%(x0) ⊂ U . Let u0k := RUηku0, for
all k ∈ N∗, where {ηk}∞k=1 as in Appendix 8. Hence, for all k ∈ N∗, we have

|u0k|0,2,U ≤ |u0|0,2,U and |u0k|1,2,U ≤ C|u0|1,2,U . (5.1)

From the first inequality in (5.1), the required bound of |u0|0,2,U for the critical

focusing case iii) in (4.1) remains the same, as in the corresponding case of bounded
open sets. We also notice that

u0k = 0, in Bak(x0)
T ∩ U,

hence, by fixing a δ = δ(%, a1) such that δ < a1−% and by setting Bk := Bak+δ(x0)∩
U , for every k ∈ N∗, we obtain that {RBku0k}

∞
k=1 ⊂ H1

0 (Bk) (see also [8, Lemma
9.5]). Moreover,

u0k → u0 in L2(U). (5.2)

Indeed,

|u0k − u0|0,2,U = |(ηk − 1)u0|0,2,U ≤ |u0|0,2,Bak−1
(x0)T∩U → 0.

Step 2α. Fixing any k ∈ N∗, we consider (3.6) in U = Bk, where we take RBku0k

as our initial datum. and we set uk ∈ L∞(0, T ;H1
0 (Bk))∩W 1,∞(0, T ;H−1(Bk)) to

be a solution that Theorem 4.1 provides. From its proof, it follows that there exist
a sequence {ukm}

∞
m=1 of absolutely continuous functions from [0, T ] to H1

0 (Bk) ∩
C∞(Bk), such that

|ukm|0,T ;1,2,Bk
+ |ukm

′|0,T ;−1,Bk
≤ K̃(|u0k|1,2,Bk , |f |1,T ;0,2,Bk

, γ), ∀m ∈ N∗. (5.3)

and
ukm(t) ⇀ uk(t) in H1

0 (Bk), for every t ∈ [0, T ],

ukm
′ ∗
⇀ uk

′
in L∞(0, T ;H−1(Bk)).

(5.4)

From (5.1) and (5.3) we deduce that

|ukm|0,T ;1,2,Bk
+ |ukm

′|0,T ;−1,Bk
≤ K̃, ∀m ∈ N∗. (5.5)

Step 2β. From the fact that the local regularity of the eigenfunctions at the
boundary depends on the local smoothness of the boundary and also that ∂Bk\∂U ∈
C∞, we obtain that ukm(t) and ukm

′
(t) are smooth on ∂Bk \ ∂U for every t ∈ [0, T ],

with

R∂Bk\∂Uu
k
m = R∂Bk\∂Uu

k
m

′
= 0, ∀m ∈ N∗.

Therefore, the extensions by zero vkm := EUukm, for all m ∈ N∗, are continuous in

∂Bk \ ∂U and thus {vkm}∞m=1 and {vkm
′}∞m=1 are sequences of functions mapping to

H1
0 (U). Evidently,

|vkm|0,T ;1,2,U = |ukm|0,T ;1,2,Bk
and |vkm

′|0,T ;−1,U = |ukm
′|0,T ;−1,Bk

,

hence, from (5.5), we obtain

|vkm|0,T ;1,2,U + |vkm
′|0,T ;−1,U ≤ K̃, ∀m ∈ N∗.



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 11

Step 2γ. Dealing as in Step 4 of the proof of Theorem 4.1, there exist a subsequence
{vkml}

∞
l=1
⊆ {vkm}

∞
m=1 and a function vk ∈ L∞(0, T ;H1

0 (U))∩W 1,∞(0, T ;H−1(U)),
such that

vkml(t) ⇀ vk(t) in H1
0 (U), for every t ∈ [0, T ],

vkml
′ ∗
⇀ vk

′
in L∞(0, T ;H−1(U)),

|vk|0,T ;1,2,U + |vk′|0,T ;−1,U ≤ K̃.

(5.6)

Since k ∈ N∗ is arbitrary, {vk}∞k=1 ⊂ L∞(0, T ;H1
0 (U)) ∩W 1,∞(0, T ;H−1(U)) and

the above estimate is satisfied for each k ∈ N∗.
Step 3α. Dealing again as before, there exist a subsequence {vkl}∞l=1 ⊆ {vk}

∞
k=1

and a function u ∈ L∞(0, T ;H1
0 (U)) ∩W 1,∞(0, T ;H−1(U)), such that

vkl(t) ⇀ u(t) in H1
0 (U), for every t ∈ [0, T ],

vkl
′ ∗
⇀ u′ in L∞(0, T ;H−1(U)),

|u|0,T ;1,2,U + |u′|0,T ;−1,U ≤ K̃.

(5.7)

Step 3β. From (3.4), (3.8), Remark 3.6, the estimate in (5.6) and [9, Lemma

3.3.6] we deduce that {g(vkl)}∞l=1 is bounded in C0, 12 ([0, T ];L
α+2
α+1 (U)). Hence, from

Proposition 1.1.2 in the same book, there exist a subsequence of {vkl}∞l=1, which

we still denote as such, and a function y ∈ C([0, T ];L
α+2
α+1 (U)), such that

g(vkl(t)) ⇀ y(t) in L
α+2
α+1 (U), for every t ∈ [0, T ]. (5.8)

Step 4α. Let Ω be any bounded ⊂ U , such that H1(Ω) ↪→↪→ L2(Ω), e.gȧ ball. For
k ∈ N∗ big enough so that Ω ⊆ Bk, we have

〈vk, EUv〉 = (uk, EBkv), 〈g(vk), EUv〉 = 〈g(uk), EBkv〉, 〈vk′, EUv〉 = 〈uk′, EBkv〉,
(5.9)

for every v ∈ C∞c (Ω). Indeed, for the first equality, from (5.6) we obtain∫
U

vkmlEUvdx→
∫
U

vkEUvdx ,

and from (5.4) we obtain∫
U

vkmlEUvdx =

∫
Bk

RBkvkmlEBkvdx→
∫
Bk

ukEBkvdx.

The second equality follows similarly. The third equality follows from the first one
and Lem1.1, Ch3, in [23]. Now, since uk is a solution of (3.6) in Bk,

〈iuk′, EBkv〉+Nγ [uk, EBkv] = 〈f , EBkv〉, ∀v ∈ C∞c (Ω), a.e. in [0, T ],

hence, from (5.9),

〈ivk′, EUv〉+Nγ [vk, EUv] = 〈f , EUv〉, ∀v ∈ C∞c (Ω), a.e. in [0, T ]. (5.10)

Step 4β. From the first convergence in (5.7), the weak lower semi-continuity of the
H1-norm and the aforementioned compact embedding, we obtain that there exist
a subsequence of {vkl}∞l=1, which we still denote as such, for which we have

vkl(t)→ u(t) in L2(Ω), for every t ∈ [0, T ]. (5.11)



12 N. GIALELIS EJDE-2020/66

We set k = kl in (5.10) and we pass to the limit l → ∞. From (5.7), (5.8), (5.11)
and Proposition 3.10, we deduce that∫ T

0

(〈iu′, EUv〉+ 〈∆u, EUv〉+ 〈y, EUv〉+ iγ〈u, EUv〉)ψdt =

∫ T

0

〈f , EUv〉ψdt,

for every v ∈ C∞c (Ω) and ψ ∈ C∞c ([0, T ]), hence

〈iu′, EUv〉+ 〈∆u, EUv〉+ 〈y, EUv〉+ iγ〈u, EUv〉 = 〈f , EUv〉, (5.12)

for all v ∈ C∞c (Ω), a.e. in [0, T ].

Step 4γ. From (5.8) and [restr]Proposition 3.9 we have

g(RΩv
kl(t)) = RΩg(vkl(t)) ⇀ RΩy(t) in L

α+2
α+1 (U), for every t ∈ [0, T ]. (5.13)

On the other hand, from (5.11) and Proposition 3.9,

RΩv
kl(t)→ RΩu(t) in L2(Ω), for every t ∈ [0, T ].

From (3.4), (3.8), Remark 3.6 and the latter convergence we obtain

g(RΩv
kl(t))→ g(RΩu(t)) = RΩg(u(t)) in L

α+2
α+1 (U), for every t ∈ [0, T ]. (5.14)

From (5.13) and (5.14) we derive RΩg(u) ≡ RΩy and so (5.12) gets the form

i〈u′, EUv〉+Nγ [u, EUv] = 〈f , EUv〉, ∀v ∈ C∞c (Ω), a.e. in [0, T ].

Since Ω is arbitrary, u satisfies the variational equation in (3.6).

Step 5. As far as the initial condition is concerned, we fix an arbitrary t0 ∈ (0, T ].
Let v ∈ H1

0 (U) be arbitrary and φ ∈ C1([0, T ]) such that φ(0) 6= 0 and φ(t0) = 0.
We then have from [23, Lemma 1.1, Chapter 3], along with the Leibniz rule, that∫ t0

0

〈vkm
′
, φv〉dt = −

∫ t0

0

〈vkm, φ′v〉dt− 〈vkm(0), φ(0)v〉,∫ t0

0

〈u′, φv〉dt = −
∫ t0

0

〈u, φ′v〉dt− 〈u(0), φ(0)v〉.
(5.15)

Moreover, 〈vkm(0), φ(0)v〉 = 〈ukm(0), φ(0)RBkv〉, hence, by setting m = ml and
letting l→ 0, we obtain∫ t0

0

〈vk′, φv〉dt = −
∫ t0

0

〈vk, φ′v〉dt− 〈RBku0k, φ(0)RBkv〉.

Since 〈RBku0k, φ(0)RBkv〉 = 〈u0k, φ(0)v〉, we set k = kl and we pass to the limit
l→∞, applying (5.2), to obtain∫ t0

0

〈u′, φv〉dt = −
∫ t0

0

〈u, φ′v〉dt− 〈u0, φ(0)v〉. (5.16)

From the second equation in (5.15) and (5.16), we conclude that u(0) = u0. �

6. NLS as limit case γ → 0 of LDDNLS

Here we consider {u0m}
∞
m=1 ∪ {v0} ⊂ H1

0 (U), {fm}∞m=1 ⊂ W 1,∞(0, T ;L2(U))
and {γm}∞m=1 ⊂ (0,∞) with γm ↘ 0, such that

|fm|1,T ;0,2,U = O(γm), as m→∞,
u0m → v0, in H1

0 (U).
(6.1)



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 13

Proposition 6.1. For every v0 and {(u0m, fm, γm)}∞m=1 as above, as well as ev-
ery corresponding sequence {um}∞m=1 of solutions of (3.6), which Theorem 4.1
or 5.1 provides, there exist a subsequence {uml}

∞
l=1 ⊆ {um}

∞
m=1 and a solution

v ∈ L∞(0, T ;H1
0 (U)) ∩W 1,∞(0, T ;H−1(U)) of (3.5), such that

uml(t) ⇀ v(t) in H1
0 (U), for every t ∈ [0, T ],

u′ml
∗
⇀ v′ in L∞(0, T ;H−1(U)),

|uml |0,T ;1,2,U + |uml ′|0,T ;−1,U + |v|0,T ;1,2,U + |v′|0,T ;−1,U ≤ K(|v0|1,2,U ),

for all m ∈ N∗.

Proof. In view of the From the above proofs, it is sufficient to show that
{|um|0,T ;1,2,U + |um′|0,T ;−1,U}

∞
m=1

is bounded. Indeed, it is direct from the limit

property of K̃ that

|um|0,T ;1,2,U + |um′|0,T ;−1,U ≤ K(|v0|1,2,U ), ∀m ∈ N∗. �

Before we proceed to the next result, we make a short, needed note about the
uniqueness of solutions of the problems (3.5) and (3.6). It is easy to see that
uniqueness results for (3.6) follow exactly as for (3.5). In particular (see [9]), for
the case n = 1 as well as for n = 2, α ∈ (0, 2], we obtain uniqueness in every
open U ⊆ Rn, from the embedding H1

0 (U) ↪→ L∞(U) and Trudinger’s inequality
respectively. One can also utilize (3.9) and (3.11) instead of Trudinger’s inequality
(see also the proof of point (ii) in Proposition 6.3 below. As for the case U =
Rn, uniqueness follows for all n ∈ N∗ from the dispersive properties (see also the
Strichartz estimates) of every solution.

Corollary 6.2. If the solutions of (3.5) and (3.6) are unique, then, for every
v0 and {(u0m, fm, γm)}∞m=1 as above, the corresponding sequence {um}∞m=1 of so-
lutions of (3.6) converges to the corresponding solution v ∈ L∞(0, T ;H1

0 (U)) ∩
W 1,∞(0, T ;H−1(U)) of (3.5), in the sense that

um(t) ⇀ v(t) in H1
0 (U), for every t ∈ [0, T ],

u′m
∗
⇀ v′ in L∞(0, T ;H−1(U))

|um|0,T ;1,2,U + |um′|0,T ;−1,U ≤ K(|v0|1,2,U ), ∀m ∈ N∗.

Proof. From Proposition 6.1 and uniqueness, we have that, for every such v0 and
{(u0m, fm, γm)}∞m=1, there exists a subsequence {uml}

∞
l=1 ⊆ {um}

∞
m=1 such that

uml(t) ⇀ v(t) in H1
0 (U), for every t ∈ [0, T ],

u′ml
∗
⇀ v′ in L∞(0, T ;H−1(U)).

(6.2)

Seeking a contradiction, we assume that a sequence {um}∞m=1 does not converge to
v in the above sense, e.g. there exists t0 ∈ [0, T ] such that

um(t0) 6⇀ v(t0) in H1
0 (U).

The second case follows similarly. Then there exist ε > 0, v0 ∈ H1
0 (U) and a

subsequence of {um}∞m=1, that we still denote as such, for which we have

|(um(t0), v0)H1
0 (U) − (v(t0), v0)H1

0 (U)| ≥ ε, ∀m ∈ N∗,

which is a contradiction to (6.2). The estimate follows from the limit property of

K̃. �



14 N. GIALELIS EJDE-2020/66

Next, we extract some estimates for the rate of the above convergence. We note
that they involve the uniqueness cases, even though we do not make use of this
property in the process.

Proposition 6.3. For every convergent sequence {um}∞m=1 of solutions of (3.6)
to a solution v of (3.5), as in Proposition 6.1 or Corollary 6.2, we set wm :=
um − v, for all m ∈ N∗. If n = 1, then there exist C11 = C11(|v0|1,2,U ), C12 =

C12(|v0|1,2,U , |fm|1,T ;0,2,U , γm) with C12 = O(γ2
m), as m→∞, such that

|wm|20,2,U ≤ |u0m − v0|20,2,Ue
C11t + C12(1− eC11t), ∀t ∈ [0, T ], (6.3)

for every m ∈ N∗. In particular, if |u0m − v0|0,2,U = O(γm), as m→∞, then

|wm|0,T ;0,2,U = O(γm), as m→∞.

Proof. Let m ∈ N∗. Then

iw′m + ∆wm + g(um)− g(v) + iγmum
H−1(U)

= fm, a.e. in [0, T ]. (6.4)

Applying (7.2) and dealing as usual we obtain

d

dt
|wm|20,2,U ≤ C

∫
U

|wm|2(|um|α + |v|α)dx+ |wm|20,2,U

+ Cγ2
m|um|

2
0,2,U + C|fm|21,T ;0,2,U ,

a.e. in [0, T ]. From the embedding H1
0 (U) ↪→ L∞(U) we obtain (6.3) with

C11 = 1 +K1(|v0|1,2,U ) and C12 =
C

C11

(K2(|v0|1,2,U )γ2
m + |fm|21,T ;0,2,U ),

for increasing, non-negative K1 and K2. �

7. Useful inequalities

We first mention two elementary inequalities.

Theorem 7.1. Let p > 0, α ≥ 0 and z1, z2 ∈ C. Then

|z1 + z2|p ≤ C(|z1|p + |z2|p), (7.1)

||z1|αz1 − |z2|αz2| ≤ C|z1 − z2|(|z1|α + |z2|α). (7.2)

We also mention the Young inequality with constant ε and the Hölder inequality.

Theorem 7.2. Let a, b ∈ [0,∞) and p, q ∈ (1,∞), such that 1
p + 1

q = 1. Then

ab ≤ εap + Cbq, ∀ε > 0, where C =
1

(εp)
q
p q
. (7.3)

Theorem 7.3. Let p, q ∈ [1,∞], such that 1
p + 1

q = 1, u ∈ Lp(U) and v ∈ Lq(U).

Then ∫
U

|uv|dx ≤ |u|0,p,U |v|0,q,U . (7.4)

The following result is a version of the Gagliardo-Nirenberg interpolation in-
equality (see [9]).



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 15

Theorem 7.4. Let q, r ∈ [1,∞] and j,m ∈ N0 such that j < m. Then∑
|β|=j

|Dβu|0,p ≤ C
( ∑
|β|=m

|Dβu|0,r
)θ
|u|1−θ0,q , ∀u ∈ Cmc (Rn), (7.5)

where
1

p
=
j

n
+ θ
(1

r
− m

n

)
+ (1− θ)1

q
, ∀θ ∈ [

j

m
, 1],

where C is a constant depending only on n, m, j, q, r and θ.
There is an exception: If r > 1 and m − j − n

r ∈ N0, then (7.5) holds only for

all θ ∈ [ jm , 1).

Remark 7.5. The following Sobolev embeddings are true (see [8])

Wm,p(Rn) ↪→ Lq(Rn), where
1

q
=

1

p
− m

n
with mp < n,

Wm,p(Rn) ↪→ Lq(Rn), where q ∈ [p,∞) with mp = n,

Wm,p(Rn) ↪→ L∞(Rn), with mp > n.

It is then easy to see that the following embeddings

Wm,p
0 (U) ↪→ Lq(U), where

1

q
=

1

p
− m

n
with mp < n,

Wm,p
0 (U) ↪→ Lq(U), where q ∈ [p,∞) with mp = n,

Wm,p
0 (U) ↪→ L∞(U), with mp > n

are also true for every U ⊆ Rn. These embeddings are, additionally, scaling
invariant, since, for every inequality of the corresponding embedding, we have
CU = CRn = C for every U ⊆ Rn. Indeed, we only have to notice that

ECmc (U) ⊂ Cmc (Rn) and |Dβu|0,p,U = |DβEu|0,p,

for every u ∈ Cmc (U), every multi-index β such that 0 ≤ |β| ≤ m, and every
p ∈ [1,∞] (see also [1]). Using the above arguments, we see that Theorem 7.4 is also
true for every u ∈Wm,p

0 (U) and also (7.5) is scaling invariant in the aforementioned
space.

We note that the embeddings

Wm,p(U) ↪→ Lq(U), where
1

q
=

1

p
− m

n
with mp < n,

Wm,p(U) ↪→ Lq(U), where q ∈ [p,∞) with mp = n,

Wm,p(U) ↪→ L∞(U), with mp > n,

are true for appropriate choices of U ⊆ Rn. Possible such choices are: (i) Rn+, (ii)
any U that satisfies the cone condition, (iii) any bounded U with a locally Lipschitz
boundary, (iv) any Lipschitz domain, etc. (see [8, 1, 19] for definitions and more
examples/counterexamples). Evidently, these embeddings and the corresponding
inequalities depend on the choice of U . Moreover, for the above special cases of
U ⊆ Rn, the (compact) Rellich-Kondrachov embeddings

W 1,p(U) ↪→↪→ Lq(U), where q ∈ [1, p∗) and
1

q∗
=

1

p
− 1

n
with p < n,

W 1,p(U) ↪→↪→ Lq(U), where q ∈ [p,∞) with p = n,

W 1,p(U) ↪→↪→ C(U), with p > n,



16 N. GIALELIS EJDE-2020/66

are true if, in addition, U is bounded. On the contrary, if we replace W 1,p(U)

with W 1,p
0 (U), there is no restriction on the choice of U , except for being bounded.

The latter follows from the fact that we only need the aforementioned continuous
embeddings and the boundedness of U , in order to prove the compact ones.

In particular, applying the above remark, we modify a well-known result from
[25] (see also [9, 22]).

Theorem 7.6. Let α = 4
n and R ∈ H1(Rn) be the spherically symmetric, positive

ground state of the elliptic equation −∆R + R = |R|αR, in H−1(Rn). Then, the
best constant C in

|u|α+2
0,α+2,U ≤ C|Du|

2
0,2,U |u|

α
0,2,U , ∀u ∈ H1

0 (U), for any open U ⊆ Rn (7.6)

is C = Ccr := α+2
2|R|α0,2

.

8. Cut-off functions

If δ > 0, we set U δ ⊃ U for

Uδ := U ∪ ∪x∈∂UB(x, δ).

Proposition 8.1. Let U and δ > 0. Then there exists φ ∈ C∞c (Rn; [0, 1]) such that

(1) supp (φ) ⊆ U δ,
(2) φ ≡ 1 in U , and
(3) ‖∇kφ‖L∞(Rn) ≤

Ck
δk

, for every k ∈ N0 (C0 = 1).

Proof. We consider φ = ϕδ ∗ χU , i.e.

φ(x) =

∫
Rn
ϕδ(x− y)χU (y)dy =

∫
B(x,δ)

ϕδ(x− y)χU (y)dy, ∀x ∈ Rn,

where ϕδ stands for the standard mollifier with supp (ϕ) ⊆ B(0, δ) and also χU
for the characteristic function of U . It is well known that φ ∈ C∞(Rn) with
Dαφ = Dαϕδ ∗χU , for every α ∈ Nn0 with |α| ≥ 1. If x ∈ U , then B(x, δ) ⊂ U , thus

φ(x) =

∫
B(x,δ)

ϕδ(x− y)dy = 1, ∀x ∈ U.

Similarly we can obtain φ(x) ∈ [0, 1] for every x ∈ Rn, since the same is true for

χU . If x ∈ U δ
c
, then B(x, δ) ∩ U = ∅, thus φ(x) = 0 for every such x and so

supp (φ) ⊆ U δ. Lastly, from the Faá di Bruno formula, we have

|Dαφ(x)| ≤
∫
Rn
|Dαϕδ(x− y)||χU (y)|dy ≤ ‖∇|α|ϕδ‖L1(Rn) ≤

C|α|

δ|α|
, ∀α ∈ Nn0 . �

If B%(x0) ⊂ Rn fixed and {ak}∞k=1 ⊂ R+ increasing, such that ak > % for all
k ∈ N∗ and ak ↗∞, we can obtain {ηk}∞k=1 ⊂ C∞c (Rn) such that

ηk(x) =

{
1, x ∈ Bak−1

(x0)

0, x ∈ Bak(x0)
T
,
∀k ∈ N∗ \ {1} and η1(x) =

{
1, x ∈ B%(x0)

0, x ∈ Ba1(x0)
T
.

In view of the above result, if, in addition, ak+1 − ak = a1 − % = C uniformly
for all k ∈ N∗ (i.e. C is independent of k), then |Dβηk|0,∞ ≤ Cm, for some



EJDE-2020/66 LINEARLY DAMPED AND FORCED NLS EQUATIONS 17

{Cm}∞m=0 ⊂ R+, uniformly for all k ∈ N∗ and every multi-index β such that
|β| = m. In particular, C0 = 1. In fact, if f ∈ C∞(R) with

f(t) :=

{
e−1/t, t > 0

0, t ≤ 0,

then we can directly construct such a sequence as follows

ηk(x;x0, ak−1, ak) :=
f(ak − |x− x0|)

f(|x− x0| − ak−1) + f(ak − |x− x0|)
,

for all x ∈ Rn and all k ∈ N∗ \ {1}, and

η1(x;B%(x0), a1) :=
f(a1 − |x− x0|)

f(|x− x0| − %) + f(a1 − |x− x0|)
, ∀x ∈ Rn.

Acknowledgments. The author wishes to thank Prof. I. G. Stratis for fruitful dis-
cussions. This research is co-financed by Greece and the European Union (European
Social Fund- ESF) through the Operational Programme Human Resources Devel-
opment, Education and Lifelong Learning in the context of the project Strengthen-
ing Human Resources Research Potential via Doctorate Research (MIS-5000432),
implemented by the State Scholarships Foundation (IKY).

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Nikolaos Gialelis

Department of Mathematics, National and Kapodistrian University of Athens, Panepis-

timioupolis, GR-157 84, Athens, Greece
Email address: ngialelis@math.uoa.gr


	1. Introduction
	2. Notation
	3. Preliminaries
	4. LDDNLS Cauchy problem in bounded open sets
	5. LDDNLS Cauchy problem in unbounded sets
	6. NLS as limit case 0 of LDDNLS
	7. Useful inequalities
	8. Cut-off functions
	Acknowledgments

	References