Electronic Journal of Differential Equations, Vol. 2007(2007), No. 06, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
STABILIZATION OF SOLUTIONS TO HIGHER-ORDER
NONLINEAR SCHRÖDINGER EQUATION WITH LOCALIZED
DAMPING
ELENI BISOGNIN, VANILDE BISOGNIN, OCTAVIO PAULO VERA VILLAGRÁN
Abstract. We study the stabilization of solutions to higher-order nonlinear
Schrödinger equations in a bounded interval under the effect of a localized
damping mechanism. We use multiplier techniques to obtain exponential decay
in time of the solutions of the linear and nonlinear equations.
1. Introduction
In this work we consider the initial-value problem of the higher-order nonlinear
Schrödinger equation with localized damping
iut + αuxx + iβuxxx + |u|2 u + ia(x)u = 0
(1.1)
u(x, 0) = u0 (x)
(1.2)
where 0 < x < L and t > 0, and with boundary conditions
u(0, t) = u(L, t) = 0
ux (L, t) = 0
for all t > 0
for all t > 0
(1.3)
(1.4)
with α, β ∈ R, β 6= 0, u = u(x, t) a complex valued function, a = a(x) a nonnegative
everywhere function such that a(x) ∈ C ∞ ((0, L)) and a(x) ≥ a0 > 0. Equation
(1.1) is a particular case of the equation
iut + ωuxx + iβuxxx + γ|u|2 u + iδ|u|2 ux + iu2 ux = 0 x, t ∈ R
u(x, 0) = u0 (x)
(1.5)
where ω, β, γ, δ are real numbers and β 6= 0. This equation was first proposed by
Hasegawa and Kodama [9] as a model for the propagation of a signal in a optic fiber
(see also [11]). The equation (1.5) can be reduced to other well known equations.
For instance, setting ω = 1, β = δ = = 0 in (1.5) we have the semilinear
Schrödinger equation,
iut + uxx + γ|u|2 u = 0.
(1.6)
2000 Mathematics Subject Classification. 35K60, 93C20.
Key words and phrases. Higher order nonlinear Schrödinger equation; stabilization;
localized damping.
c
2007
Texas State University - San Marcos.
Submitted August 2, 2006. Published January 2, 2007.
1
2
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
If we let β = γ = 0 and ω = 1 in (1.5) we obtain the nonlinear Schrödinger equation
iut + uxx + iδ|u|2 ux + iu2 ux = 0.
(1.7)
Letting α = γ = = 0 in (1.5) arises is the complex modified Korteweg-de Vries
(KdV) equation
iut + iβuxxx + iδ|u|2 ux = 0.
(1.8)
The initial value problem for the equations (1.6), (1.7) and (1.8) has been extensively studied in the last few years, see for instance [3, 6, 10, 27] and references
therein. In 1992, Laurey [14] considered the equation (1.5) and proved local wellposedness of the initial value problem associated for data in H s (R), s > 3/4, and
global well-posedness in H s (R), s ≥ 1. In 1997, Staffilani [30] established local
well-posedness for data in H s (R), s ≥ 1/4 in (1.5) improving Laurey’s result. A
similar result was given in [4, 5] with w(t), β(t) real functions. Recently, Sepúlveda
and Vera [28] showed that C ∞ solutions u(x, t) are obtained for all t > 0 if the
initial data u0 (x) decays faster than polynomially on R+ = {x ∈ R : x > 0} and
has a certain initial Sobolev regularity. In [2] Bisognin and Vera considered the
equation (1.5) with δ = = 0 and proved the unique continuation property.
This paper concerns the exponential stabilization of the solution of (1.1) when
the damping a = a(x) is effective only on a subset of the interval (0, L). This
problem was extensively studied in the context of wave equations, see Dafermos [7],
Haraux [8], Slemrod [29], Zuazua [36] and Nakao [17]. The same problem has been
also studied for the KdV equation. Here we can mention the works of Komornik,
Russell and Zhang [13]. Using a different damping mechanism they obtained the
exponential decay with periodic boundary conditions. In [20], Menzala, Vasconcellos and Zuazua studied the nonlinear KdV equation inspired in the work of Rosier
[23]. They studied the stabilization of solutions for the KdV equation in a bounded
interval under the effect of a localized damping mechanism. Using compactness
arguments, the smoothing effect of the KdV equation on the line and the unique
continuation results, the authors deduced the exponential decay in time of the solutions of the linear equation and a local uniform stabilization result of the solutions
of the nonlinear equation when the localized damping is active simultaneously only
in a neighborhood of both extremes x = 0, x = L. The same result was obtained
by the KdV coupled system by Menzala, Bisognin and Bisognin [21]. The main
result of this paper says that the total energy E(t) associated to (1.1) decays exponentially as t → +∞, for bounded sets of initial data. In order to prove the result
we use multipliers together with compactness arguments and smoothing properties
proved by M. Sepúveda and Vera [28] and the Unique Continuation Principle valid
for this problem, see Bisognin and Vera [2].
This paper is organized as follows: In section two, we study the existence of
global solution to the linear and nonlinear problem. In section three, we study
the stabilization result of the problem. First we prove the exponential decay in
the linear problem and of at end we prove the stabilization of the solution of the
nonlinear problem. The notation that we use in this article is standard and can be
found in Temam [32].
2. Stabilization of solutions of the linear problem
In this section we are interested in proving the global existence and uniqueness
of the solution and the exponential decay of the solution of the linear problem
EJDE-2007/06
STABILIZATION OF SOLUTIONS
3
associated to (1.1)-(1.4). We consider the problem
iut + αuxx + iβuxxx + ia(x)u = 0
for all x ∈ I
u(x, 0) = u0 (x),
u(0, t) = u(L, t) = 0,
ux (L, t) = 0,
for all t > 0
for all t > 0
(2.1)
(2.2)
(2.3)
(2.4)
where I = (0, L), a ∈ C ∞ (I), a(x) ≥ a0 > 0 is assumed to be nonnegative everywhere in an open non empty proper subset ω of I and we will prove the global
existence of the problem (2.1)-(2.4).
We consider a ≡ 0 and the operator A = −β∂x3 + iα∂x2 with domain
D(A) = {v ∈ H 3 (I) : v(0) = v(L) = 0, vx (L) = 0} ⊆ L2 (I)
Lemma 2.1. Let a ≡ 0 and β > 0. Then, the operator A is the infinitesimal
generator of a strongly continuous semigroup {S(t)}t≥0 of contractions in L2 (I).
Proof. It is easy to prove that A is closed. Let us to prove that A is dissipative.
Integration by parts give us
Z L
Z L
β 2
2
|vx |2 dx.
(Av, v)L2 (I) =
(−βvxxx + iαvxx )v dx = − |vx (0, t)| − iα
2
0
0
Hence,
β
Re(Av, v)L2 (I) = − |vx2 (0, t)|2 ≤ 0,
2
where A is dissipative. On the other hand, the adjoint of the operator A is given
by
H ∗ v = βvxxx − iαvxx
with domain
D(H ∗ ) = {v ∈ H 3 (I) : v(0, t) = v(L, t) = 0, vx (0, t) = 0} ⊆ L2 (I).
A similar calculation shows that
Z L
Z L
β 2
2
∗
(H v, v)L2 (I) =
|vx |2 dx.
(βvxxx − iαvxx )v dx = − |vx (0, t)| + iα
2
0
0
Hence
β
Re(Av, v)L2 (I) = − |vx2 (0, t)|2 ≤ 0.
2
The conclusion of Lemma 2.1 follows from the Stone Theorem [19] of semigroup
theory.
The above discussion proves the following result.
Theorem 2.2. Let u0 ∈ D(A), a ≡ 0 and β > 0. Then, there exists a unique
function u such that u ∈ C(0, +∞ : H 3 (I)) ∩ C 1 (0, +∞ : L2 (I)) which solves
(2.1)-(2.4).
The well-posedness of system (2.1)-(2.4), when a 6≡ 0 can be handled in a similar
way by considering the term a(x)u as a linear perturbation of the case a ≡ 0.
Now, we will prove the exponential decay of the total energy E(t) associated to
(2.1)-(2.4) under suitable assumptions on the open subset ω of I. We denote by
4
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
{S(t)}t≥0 the semigroup of contractions associated with A, and by H the Banach
space C([0, T ] : L2 (I)) ∩ L2 (0, T : H 1 (I)) with the norm
kvkH = sup kv(·, t)kL2 (I) +
T
Z
[0,T ]
kv(·, t)k2H 1 (I)
1/2
(2.5)
0
Theorem 2.3. Consider the solution of the problem (2.1)-(2.4). Then there exist
c > 0 and µ > 0 such that
ku(·, t)k2L2 (I) ≤ cku0 k2L2 (I) e−µt
(2.6)
for all t ≥ 0 and u0 ∈ L2 (I).
For the proof of the above theorem we need the following result.
Lemma 2.4. Let |α| < 3β. Then
(1) The map u0 ∈ L2 (I) → S(t)u0 ∈ H is continuous.
(2) For u0 ∈ L2 (I), ∂x u(0, ·) makes sense in L2 (I) and
T
Z
|u0 |2 dx ≤
0
1
T
Z
T
1
kux (0, ·)kL2 (0,T ) ≤ √ ku0 kL2 (I)
β
Z L
Z T
Z
|u|2 dx dt + β
|ux (0, t)|2 dt + 2
0
0
0
(2.7)
T
0
Z
L
a(x)|u|2 dx.
0
(2.8)
Proof. (1) For u0 ∈ L2 (I), let u = S(t)u0 be the mild solution of (2.1)-(2.4). By
Theorem 2.2, u ∈ C(0, T : L2 (I)) and
kukC(0,T :L2 (I)) ≤ ku0 kL2 (I) .
(2.9)
To see that u ∈ L2 (0, T : H 1 (I)) we first assume that u0 ∈ D(A). Let ξ = ξ(x, t) ∈
C ∞ ([0, L] × [0, T ]). Multiplying the equation (2.1) by ξu we have
iξuut + αξuuxx + iβξuuxxx + iξa(x)|u|2 = 0.
(2.10)
Applying the conjugate, we have
−iξuut + αξuuxx − iβξuuxxx − iξa(x)|u|2 = 0.
(2.11)
Subtracting (2.10) and (2.11) and integrating over x ∈ (0, L), we have
i
d
dt
L
Z
ξ|u|2 dx − i
0
Z
L
ξt |u|2 dx + iβ
L
Z
ξuuxx dx − α
+α
0
L
Z
ξuuxxx dx + iβ
0
0
Z
Z
L
ξuuxxx dx
0
Z
ξuuxx dx + 2i
0
L
L
2
ξa(x)|u| dx = 0.
0
(2.12)
EJDE-2007/06
STABILIZATION OF SOLUTIONS
5
Each term in the above equation is treated separately, integrating by parts and
using the boundary conditions we obtain
L
Z
L
Z
ξuuxxx dx =
0
Z
0
0
0
ξux uxx dx
0
L
ξx |ux | dx −
ξux uxx dx
0
L
Z
0
Z L
L
Z
ξuuxx dx = −
ξ|ux |2 dx
ξx uux dx −
0
Z L
ξuuxx dx = −
0
Z L
ξx uux dx −
0
L
Z
2
0
L
L
Z
ξx |ux |2 dx +
0
+ ξ(0, t)|ux (0, t)|2
Z L
Z
L
ξxx uux dx +
ξuuxxx dx =
Z
L
Z
ξxx uux dx + 2
0
ξ|ux |2 dx.
0
Replacing these expression in (2.12) and performing straightforward calculations,
i
L
Z
d
dt
ξ|u|2 dx − i
L
Z
0
ξt |u|2 dx + iβ
L
Z
0
ξxx (|u|2 )x dx + 3iβ
0
L
ξx |ux |2 dx
0
L
Z
+ iβξ(0, t)|ux (0, t)|2 − 2iα Im
Z
Z
ξx uux dx + 2i
L
ξa(x)|u|2 dx = 0
0
0
and
d
dt
L
Z
ξ|u|2 dx −
Z
0
L
ξt |u|2 dx + β
L
Z
0
ξxx (|u|2 )x dx + 3β
Z
0
Z
2
ξx |ux |2 dx
0
L
+ βξ(0, t)|ux (0, t)| − 2α Im
L
2
ξx uux dx + 2
0
(2.13)
L
Z
ξa(x)|u| dx = 0.
0
Let ξ(x, t) ≡ x, then in (2.13), we obtain
L
Z
d
dt
x|u|2 dx + 3β
0
L
Z
|ux |2 dx − 2α Im
L
Z
L
Z
xa(x)|u|2 dx = 0.
uux dx + 2
0
0
0
Hence,
d
dt
Z
L
x|u|2 dx + 3β
0
Z
L
|ux |2 dx + 2
0
Z
L
xa(x)|u|2 dx
0
L
Z
L
|u|2 dx + |α|
uux dx ≤ |α|
= 2α Im
Z
0
0
(2.14)
L
Z
|ux |2 dx
0
then
Z
d
dt
L
Z
2
x|u| dx + (3β − |α|)
0
L
Z
2
|ux | dx + 2
0
L
2
Z
xa(x)|u| dx ≤ |α|
0
L
|u|2 dx
0
and
d
dt
Z
L
2
Z
x|u| dx+(3β −|α|)
0
L
2
Z
|ux | dx+2
0
0
L
xa(x)|u|2 dx ≤ |α|kuk2L2 (0,L) . (2.15)
6
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
Integrating (2.15) over t ∈ [0, T ] we have
Z
Z TZ L
Z L
|ux |2 dx dt + 2
x|u|2 dx + (3β − |α|)
L
Z
x|u0 |2 dx + |α|
≤
Z
L
Z
xa(x)|u|2 dx dt
0
0
0
0
0
T
T
kuk2L2 (0,L) dt
0
0
≤ Lku0 k2L2 (0,L) + T |α|ku0 k2L2 (0,L)
= [L + T |α|]ku0 k2L2 (0,L)
Hence,
Z
L
x|u|2 dx + (3β − |α|)kux k2L2 (0,T :L2 (0,L)) + 2
T
Z
0
L
Z
xa(x)|u|2 dx dt
0
0
≤ [L + T |α|]ku0 k2L2 (0,L) .
Using that a(x) ≥ a0 > 0, we obtain
Z L
x|u|2 dx + (3β − |α|)kux k2L2 (0,T :L2 (0,L)) ≤ [L + T |α|]ku0 k2L2 (0,L) .
(2.16)
0
In particular, using that |α| < 3β,
(3β − |α|)kux k2L2 (0,T :L2 (0,L)) ≤ [L + T |α|]ku0 k2L2 (0,L) .
and
1
[L + T |α|] ku0 k2L2 (0,L) .
(2.17)
(3β − |α|)
By the density of D(A) in L2 (I) the result extends to arbitrary u0 ∈ L2 (I).
We remark that: (a) The estimate (2.16) gives a smoothing effect. (b) In (2.14)
using Young’s estimate and assuming that β > 0 we have
Z L
Z
Z L
|α|2 L 2
2α Im
uux dx ≤
|u| dx + 2β
|ux |2 dx.
2β 0
0
0
kux k2L2 (0,T :L2 (0,L)) ≤
Then, in we obtain (2.15),
Z
Z L
Z L
d L
|α|2
2
2
kuk2L2 (0,L)
x|u| dx + β
|ux | dx + 2
xa(x)|u|2 dx ≤
dt 0
2β
0
0
and the assumption that |α| < 3β can be removed.
(2) We also assume u0 ∈ D(A) and taking ξ(x, t) = 1 in (2.13), we have
Z
Z L
d L 2
|u| dx + β|ux (0, t)|2 + 2
a(x)|u|2 dx = 0.
(2.18)
dt 0
0
Hence, integrating (2.18) over t ∈ [0, T ] we have
Z L
Z T
Z TZ L
Z
2
2
2
|u| dx + β
|ux (0, t)| dt + 2
a(x)|u| dx dt =
0
0
0
Using a(x) ≥ a0 > 0, we obtain
Z L
Z
|u|2 dx + β
0
Z
T
2
Z
|ux (0, t)| dt ≤
β
0
0
|u0 |2 dx
0
T
|ux (0, t)|2 dt ≤
Z
L
|u0 |2 dx
0
0
L
2
Z
|u0 | dx −
0
L
L
2
Z
|u| dx ≤
0
0
L
|u0 |2 dx;
EJDE-2007/06
STABILIZATION OF SOLUTIONS
7
therefore, (2.7) is proved.
On the other hand, taking ξ(x, t) = T − t in (2.13) we have
Z L
Z
d L
|u|2 dx
(T − t)|u|2 dx +
dt 0
0
Z L
(T − t)a(x)|u|2 dx = 0.
+ β(T − t)|ux (0, t)|2 + 2
(2.19)
0
Integrating (2.19) over t ∈ [0, T ] we have
Z L
Z TZ L
−T
|u0 |2 dx +
|u|2 dx dt
Z
0
T
0
0
(T − t)|ux (0, t)|2 dt + 2
+β
0
T
Z
0
Z
L
(T − t)a(x)|u|2 dx dt = 0.
0
Then
Z L
T
|u0 |2 dx
0
T
Z
L
Z
|u|2 dx dt + β
=
0
0
T
Z
|u|2 dx dt + β
≤
0
Z
(T − t)|ux (0, t)|2 dt + 2
0
L
Z
T
Z
0
T
Z
T |ux (0, t)|2 dt + 2
Z
0
and
Z
L
1
|u0 | dx ≤
T
T
Z
L
Z
0
L
Z
0
2
0
T
0
Z
T
L
(T − t)a(x)|u|2 dx dt
0
T a(x)|u|2 dx dt
T
|u| dx dt + β
0
Z
0
Z
2
T
|ux (0, t)|2 dt
0
Z
+2
(2.20)
L
2
a(x)|u| dx dt.
0
0
Equation (2.20) holds trivially for any u0 ∈ L2 (0, L).
Proof. ][Proof of Theorem 2.3] To show the result,from (2.20), it suffices to prove
Z Z
Z T
Z TZ L
1 T L 2
|u| dx dt ≤ β
|ux (0, t)|2 dt + 2
a(x)|u|2 dx dt.
(2.21)
T 0 0
0
0
0
Let us argue by contradiction. Suppose that (2.21) is not valid. Then, there will
exist a sequence of solutions {un } of (2.1)-(2.4) such that
kun k2L2 (0,T :L2 (I))
= +∞.
R
R R
n→∞ β T |u0 (0, t)|2 dt + 2 T L a(x)|u |2 dx dt
n
n
0
0
0
lim
Let
λn = kun kL2 (0,T :L2 (I))
and vn (x, t) =
un (x, t)
.
λn
We have that vn solves the (2.1)-(2.4) problem with initial data vn (x, 0) =
Furthermore,
Z
β
0
T
|vn0 (0, t)|2 dt + 2
Z
0
kvn kL2 (0,T :L2 (I)) = 1,
Z L
a(x)|vn |2 dx dt → 0,
un (x,0)
λn .
(2.22)
T
0
as
n → ∞.
(2.23)
8
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
In view of (2.20), it follows that vn (x, 0) is bounded in L2 (I). Thus
kvn (·, t)kL2 (I) ≤ c,
for all 0 ≤ t ≤ T.
According to (2.7)
kvn kL2 (0,T :H 1 (I)) ≤ c(T )kvn (·, t)kL2 (I) ≤ constant, for all n ∈ N.
(2.24)
Estimates (2.24) and (2.22) tell us that
i(vn )t = −α(vn )xx − iβ(vn )xxx − ia(x)vn
is bounded in L2 (0, T : H −2 (I)). Since the embedding H 1 (I) ,→ L2 (I) is compact
it follows that (vn ) is relatively compact in L2 (0, T : L2 (I)). By extracting a
subsequence we may deduce that
vn * v
weakly in L2 (0, T : H −2 (I)),
vn → v
strongly in L2 (0, T : L2 (I)).
Since
kvn kL2 (0,T :L2 (I)) = 1
(2.25)
kvkL2 (0,T :L2 (I)) = 1.
(2.26)
it follows that
By the weak lower semicontinuity, we have
Z T
Z
0 = lim inf β
|vn0 (0, t)|2 dt + 2
n→∞
Z
≥β
0
T
0
T
|v 0 (0, t)|2 dt + 2
Z
0
0
T
Z
L
Z
a(x)|vn |2 dx dt
0
L
a(x)|v|2 dx dt
0
which guarantees that a(x)v ≡ 0, and in particular, v ≡ 0 in ω × (0, T ). On the
other hand, the limit v satisfies
ivt + αvxx + iβvxxx + ia(x)v = 0.
Using Holmgren’s Uniqueness Theorem (see [22]) we deduce that v ≡ 0 in I ×(0, T ).
This contradicts (2.22). Consequently, (2.21) has to be true. On the other hand,
we have
Z L
d
ku(·, t)k2L2 (I) + 2
a(x)|u|2 dx ≤ −β|ux (0, t)|2 ≤ 0
dt
0
and
Z T
Z TZ L
2
2
2
ku(·, T )kL2 (I) = ku0 kL2 (I) − β
|ux (0, t)| dt − 2
a(x)|u|2 dx dt
0
0
0
which together with (2.20) give us the inequality
(1 + c)ku(·, T )k2L2 (I)
≤ (1 + c) ku0 k2L2 (I) − β
Z
T
Z
|ux (0, t)|2 dt − 2
0
≤ cku0 k2L2 (I) − β
Z
0
≤ cku0 k2L2 (I) .
T
|ux (0, t)|2 dt + 2
Z
0
T
Z
0
0
L
T
Z
L
a(x)|u|2 dx dt
0
a(x)|u|2 dx dt
EJDE-2007/06
STABILIZATION OF SOLUTIONS
9
Consequently,
c
<1
1. + c
Therefore, by a semigroup property, the conclusion of the Theorem follows.
ku(·, T )k2L2 (I) ≤ µku0 k2L2 (I)
with µ =
3. Stabilization of the solution of the non-linear problem
In this section we prove the existence of a global solution (and uniqueness) and
the exponential decay of the solution of the nonlinear problem (1.1)-(1.4). The
proof of the result needs of the Unique Continuation Principle since we are dealing
with a nonlinear equation.
Theorem 3.1 (local existence and uniqueness). Let |α| < 3β and u0 ∈ L2 (I).
Then, there exist T0 > 0 and a unique function u ∈ L∞ (0, T0 : L2 (I)) ∩ L2 (0, T0 :
H01 (I)) that satisfies (1.1)-(1.4).
Proof. Let T > 0 and consider the set of functions X(T ) = {u : u ∈ L2 (0, T :
H01 (I))} with the norm
Z T
1/2
kukX(T ) =
ku(s)k2H 1 (I) ds
.
0
∞
We define the map P : X(T ) → L (0, +∞ : L2 (I)) given by
Z t
P (u)(t) = S(t)u0 +
S(t − τ )g(u)(τ ) dτ
(3.1)
0
where g(u) = |u|2 u. In order to prove local existence (and uniqueness) it is sufficient
to prove that P maps X(T ) into itself continuously and it is a contraction for
T > 0 sufficiently small. According to the results of section two it follows that the
semigroup of contractions {S(t)}t≥0 corresponding to the linear system satisfies the
following properties:
kS(t)ukL2 ≤ ku0 kL2
(3.2)
kS(t)u0 kL2 (0,T :H01 (I)) ≤ c(T )ku0 kL2
(3.3)
for all T > 0, where c(T ) =
hand, the function
|α|T +L
3β−|α| .
It follows that S(t)u0 ∈ X(T ). On the other
Z
t
S(t − s)g(u)
J(t) =
0
is a solution to the problem
iJt + αJxx + iβJxxx + ia(x)J = F,
where F = −g(u). We can follow the same idea due to L. Rosier [23] (Proposition
4.1) to prove that J ∈ X(T ) and F → J which maps L2 (0, T : L2 (I)) to X(T ) is
continuous. Furthermore, the map that associates to each u in L2 (0, T : H01 (I)) the
element g(u) in L1 (0, T : L2 (I)) is also continuous. Consequently, P maps X(T )
into X(T ) continuously. Now, let us prove that P is a contraction in a suitable ball
of X(T ) provided that T > 0 is chosen sufficiently small. Let u and v be elements
of X(T ), then
Z t
P (u) − P (v) = −
S(t − τ )[g(u) − g(v)] dτ.
0
10
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
Direct calculation, (3.1), (3.3) and Holder’s inequality yield
kP (u)(t) − P (v)(t)k2H 1 (I)
Z
2
t
S(t − τ )[g(u) − g(v)] dτ H 1 (I)
=
0
Z
t
2
≤
kS(t − τ )[g(u) − g(v)]kH 1 (I) dτ
0
1/2 i2
h Z t 1/2 Z t
≤
dτ
kS(t − τ )[g(u) − g(v)]k2H 1 (I) dτ
0
0
Z t Z t
≤
dτ
kS(t − τ )[g(u) − g(v)]k2H 1 (I) dτ
0
0
Z t
≤ M 2T
kg(u) − g(v)k2L2 (I) dτ
0
≤T
|α|T + L 2
3β − |α|
T
Z
kg(u) − g(v)k2L2 (I) dt.
0
Hence,
kP (u)(t) − P (v)(t)k2L∞ (0,T :H 1 (I))
2 Z T
|α|T + L
kg(u) − g(v)k2L2 (I) dt.
≤T
3β − |α|
0
(3.4)
On the other hand, using that ||u| − |v|| ≤ |u − v|, we have
kg(u) − g(v)kL2 (I)
= k|u|2 u − |v|2 vkL2 (I)
= k|u|2 (u − v) + (|u|2 − |v|2 )vkL2 (I)
= k|u|2 (u − v) + (|u| + |v|)(|u| − |v|)vkL2 (I)
≤ k|u|2 (u − v)kL2 (I) + k(|u| + |v|)(|u| − |v|)vkL2 (I)
≤ kuk2L∞ (I) ku − vkL2 (I) + kukL∞ (I) + kvkL∞ (I) kvkL∞ (I) ku − vkL2 (I)
≤ kuk2L∞ (I) ku − vkL2 (I) + kukL∞ (I) kvkL∞ (I) + kvk2L∞ (I) ku − vkL2 (I)
3 ≤
kuk2L∞ (I) + kvk2L∞ (I) ku − vkL2 (I) .
2
Thus
kg(u) − g(v)k2L2 (I) ≤
2
3 kuk2L∞ (I) + kvk2L∞ (I) ku − vk2L2 (I) .
2
(3.5)
Therefore, from (3.4), we have
kP (u)(t) − P (v)(t)k2L∞ (0,T :H 1 (I))
Z
2
|α|T + L 2 t
≤ cT
kuk2H 1 (I) + kvk2H 1 (I) ku − vk2L2 (I) dt.
3β − |α|
0
Then
kP (u)(t) − P (v)(t)k2X(T ) ≤ cT
2
|α|T + L 2
kuk2X(T ) + kvk2X(T ) ku − vk2X(T )
3β − |α|
EJDE-2007/06
STABILIZATION OF SOLUTIONS
11
and
kP (u)(t) − P (v)(t)kX(T ) ≤ cT 1/2
|α|T + L kuk2X(T ) + kvk2X(T ) ku − vkX(T ) .
3β − |α|
This shows that P is a contraction in the ball BR = {u ∈ X(T ) : kukX(T ) ≤ R}
with
|α|T + L 2
2cT 1/2
R < 1.
(3.6)
3β − |α|
Therefore, the proof will be complete if we show that for a suitable choice of R
and T satisfying (3.6), the map P maps BR into itself. Putting all the previous
estimates together, we have
Z T
2
kP (u)k2H 1 (I) dt.
kP (u)kX(T ) =
0
From (3.5),
kP (u)k2X(T ) ≤ cT
|α|T + L 2
|α|T + L 2
ku0 k6X(T ) ≤ cT
R6
3β − |α|
3β − |α|
for all u ∈ BR . Choosing R = ku0 kL2 (I) from (3.7) we deduce that
|α|T + L kP (u)k2X(T ) ≤ cT 1/2
ku0 k2L2 (I) ku0 kL2 (I) .
3β − |α|
Let us choose T > 0 sufficiently small and such that
|α|T + L cT 1/2
ku0 k2L2 (I) < 1.
3β − |α|
Hence, P map BR into itself.
(3.7)
(3.8)
Theorem 3.2 (Global existence and uniqueness). Let |α| < 3β and u0 ∈ L2 (I).
Then, there exists a unique function u ∈ L∞ (0, T : L2 (I)) ∩ L2 (0, T : H01 (I)) that
satisfies the problem (1.1)-(1.4).
Proof. It follows from Theorem 3.1 that we can extend the solution u to the maximal
interval of existence 0 ≤ t < Tmax . We need to prove that Tmax = +∞. Let T > 0
such that 0 < T < Tmax and let us get bounds for the solution u in the interval
0 ≤ t < T . Due to Theorem 3.1 we know that the solution u belongs to X(T ) and
satisfies
Z t
u(t) = S(t)u0 +
S(t − τ )g(u(τ )) dτ.
0
It follows that
−|u|2 u ∈ L2 (0, T : L2 (I)).
The global existence is an immediate consequence of the a priori estimate obtained
by multiplying the equation in (1.1)-(1.4) by u. In fact
iuut + iβuuxxx + αuuxx + |u|4 + ia(x)|u|2 = 0.
(3.9)
Applying conjugate in (3.9) we have
−iuut − iβuuxxx + αuuxx + |u|4 − ia(x)|u|2 = 0.
(3.10)
Subtracting (3.9) with (3.10), integrating over x ∈ [0, L] and using boundary conditions we obtain
Z
Z L
d L 2
2
|u| dx + β|ux (0, t)| +
a(x)|u|2 dx = 0.
(3.11)
dt 0
0
12
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
Integrating over t ∈ [0, T ]
Z L
Z T
Z
|u|2 dx + β
|ux (0, t)|2 dt +
0
0
T
EJDE-2007/06
L
Z
a(x)|u|2 dx dt = ku0 k2L2 (I) .
(3.12)
0
0
Therefore, u ∈ L∞ (0, T : L2 (I)) for any 0 < T < Tmax .
Now, we multiply the equation in (1.1)-(1.4) by xu
ixuut + iβxuuxxx + αxuuxx + x|u|4 + ixa(x)|u|2 = 0.
(3.13)
Applying conjugate in (3.9) we have
−ixuut − iβxuuxxx + αxuuxx + x|u|4 − ixa(x)|u|2 = 0.
(3.14)
Subtracting (3.13) with (3.14), integrating over x ∈ [0, L] we obtain
Z
Z L
Z L
d L
2
x|u| dx + iβ
xuuxxx dx + iβ
xuxxx dx
i
dt 0
0
0
Z L
Z L
Z L
α
xuuxx dx − α
xuuxx dx + 2
x|u|4 dx = 0.
0
0
0
Performing similar calculations as in section two, we obtain
Z
Z L
d L
2
x|u| dx + 3iβ
|ux |2 dx + iβ|ux (0, t)|2
i
dt 0
0
Z L
Z L
− 2iα Im
uux dx + 2i
xa(x)|u|2 dx = 0
0
0
then
d
dt
L
Z
2
Z
L
x|u| dx + 3β
0
|ux |2 dx + β|ux (0, t)|2
0
Z
− 2α Im
L
Z
uux dx + 2
0
(3.15)
L
2
xa(x)|u| dx = 0.
0
Performing straightforward calculation we obtain
Z
Z L
Z L
d L
x|u|2 dx + (3β − |α|)
|ux |2 dx + β|ux (0, t)|2 + 2
xa(x)|u|2 dx
dt 0
0
0
(3.16)
Z L
≤ |α|
|u|2 dx.
0
Therefore, integrating over t ∈ [0, T ] we have
" Z Z
#
Z TZ L
Z L
T
L
1
|ux |2 dx dt ≤
|α|
|u|2 dx dt +
|u0 |2 dx
(3β − |α|)
0
0
0
0
0
(3.17)
1
ku0 k2L2 (I)
≤
(3β − |α|)
and u ∈ L2 (0, T : H01 (I)), for any 0 < T < Tmax . Estimates (3.11) and (3.17) allow
us to conclude that Tmax = +∞. Thus, the global existence follows. Uniqueness
can be shown in the standard way using Gronwall’s inequality.
Now, we have to prove the exponential decay of the solutions of the nonlinear
problem (1.1)-(1.4). The proof of the result needs that the Unique Continuation
Principle(UCP) holds because we are dealing with a nonlinear equation. The next
EJDE-2007/06
STABILIZATION OF SOLUTIONS
13
Theorem contains the result of (UCP) for the problem (1.1)-(1.4). The proof will
be given later.
Theorem 3.3. Assume that the set ω contains two sets of the form (0, δ) and
(L − δ, L) for some δ > 0. Let u ∈ L∞ (0, T : L2 (I))) ∩ L2 (0, T : H 1 (I))) be the
global solution of the problem
iut + iβuxxx + αuxx + δ|u|2 u + ia(x)u = 0
u(0, t) = u(L, t),
t ∈ (0, T )
(3.18)
(3.19)
t ∈ (0, T )
(3.20)
in ω × (0, T )
(3.21)
ux (L, t) = 0,
u(x, t) ≡ 0
in I × (0, T )
with ≥ 0 and T > 0, then necessarily u ≡ 0 in I × (0, T ).
For the moment, let us assume that ω satisfies the (UCP). Then we have the
following result.
Theorem 3.4. Let |α| < 3β, a = a(x) a non-negative function, a ∈ C ∞ (I) such
that a(x) ≥ a0 > 0 is assumed to be nonnegative everywhere in an open non empty
proper subset ω. Let u be the global solution of the problem (1.1)-(1.4). Then, for
any L > 0 and R > 0 there exist positive constants c > 0 and µ > 0 such that
E(t) ≤ cku0 k2L2 (I) e−µt
for any t ≥ 0 and any solution of (1.1)-(1.4) with u0 ∈ L2 (I) such that ku0 kL2 (I) ≤
R.
Proof. We proceed as in he proof of Theorem 2.3. From (3.11) we have
Z L
Z TZ L
2
2
|u| dx + β|ux (0, t)| + 2
a(x)|u|2 dx dt = ku0 k2L2 (I) .
0
0
(3.22)
0
Next, we multiply the equation in (1.1)-(1.4) by (T − t)u,
i(T −t)uut +iβ(T −t)uuxxx +α(T −t)uuxx +(T −t)|u|4 +i(T −t)a(x)|u|2 = 0. (3.23)
Applying conjugate we have
−i(T − t)uut − iβ(T − t)uuxxx + α(T − t)uuxx + (T − t)|u|4 − i(T − t)a(x)|u|2 = 0.
(3.24)
Subtracting (3.23) with (3.24) and performing straightforward calculations we obtain
Z
Z L
Z L
d L
(T − t)|u|2 dx +
|u|2 dx + β(T − t)|ux (0, t)|2 + 2
(T − t)a(x)|u|2 dx = 0.
dt 0
0
0
(3.25)
Integrating over t ∈ [0, T ], we have
Z L
Z TZ L
Z T
(T − t)|ux (0, t)|2 dt
−T
|u0 |2 dx +
|u|2 dx dt + β
Z
0
T
0
Z
+2
0
0
0
L
(T − t)a(x)|u|2 dx dt = 0.
0
14
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
and
L
Z
|u0 |2 dx =
0
1
T
T
Z
Z
|u|2 dx dt +
0
T Z L
0
2
+
T
L
Z
β
T
Z
T
(T − t)|ux (0, t)|2 dt
0
(T − t)a(x)|u|2 dx dt.
0
0
Consequently,
Z L
Z Z
Z T
Z TZ L
1 T L 2
2
2
|u0 | dx ≤
|u| dx dt + β
a(x)|u|2 dx dt.
|ux (0, t)| dt + 2
T 0 0
0
0
0
0
(3.26)
To show the result it is sufficient to prove
Z TZ L
Z T
Z TZ L
|u|2 dx dt ≤ c β
|ux (0, t)|2 dt + 2
a(x)|u|2 dx dt
(3.27)
0
0
0
0
0
for some positive constant c independent of the solution u.
Let us argue by contradiction. Suppose that (3.27) is not true. Then, there will
exist a sequence of solutions un of (1.1)-(1.4) such that
kun k2L2 (0,T :L2 (I)))
lim R
= +∞.
R R
n→∞ β T |un (0, t)|2 dt + 2 T L a(x)|un |2 dx dt
x
0
0
0
Let λn = kun kL2 (0,T :L2 (I)) and v n (x, t) =
satisfies
un
λn .
For each n ∈ N the function v n
i(v n )t + α(v n )xx + iβ(v n )xxx + (λn )2 |v n |2 v n + ia(x)v n = 0 in I × (0, T ) (3.28)
vxn (L, t) = 0,
n
for all t > 0
n
v (0, t) = v (L, t) = 0,
(3.29)
for all t > 0
(3.30)
n
v n (x, 0) =
u (x, 0)
,
λn
for all x ∈ I.
(3.31)
We have
Z
β
T
|unx (0, t)|2 dt + 2
0
Z
0
kv n kL2 (0,T :L2 (I)) = 1,
T Z L
a(x)|un |2 dx dt → 0
(3.32)
as n → +∞.
(3.33)
0
n
In view of (3.26) it follows that v (x, 0) is bounded in L2 (I). Thus,
kv n (·, t)kL2 (I) ≤ c for all
0 ≤ t ≤ T.
(3.34)
According to (2.17)
kv n kL2 (0,T :H 1 (I)) ≤ c(T )kv n (·, t)kL2 (I) ,
n 2 n
2
∀n ∈ N.
(3.35)
1
On the other hand, |v | v belongs to L (0, T : L (I)) and
k|v n |2 v n kL2 (0,T :L1 (I)) ≤ kv n k2L∞ (0,T :L2 (I)) kv n kL2 (0,T :H 1 (I))
(3.36)
and by (3.35) we obtain a constant c > 0 such that
k|v n |2 v n kL2 (0,T :L1 (I)) ≤ c.
n
n
(3.37)
Since (λ ) is a bounded sequence, because ku (·, 0)kL2 (I) ≤ R, it follows by (3.18)(3.21), (3.35) and (3.37) that
(v n )t = −α(v n )xx − iβ(v n )xxx − (λn )2 |v n |2 v n − ia(x)v n
EJDE-2007/06
STABILIZATION OF SOLUTIONS
15
is bounded in L2 (0, T : H −2 (I)). Since the embedding H 1 (I) ,→ L2 (I) is compact
it follows that (v n ) is relatively compact in L2 (0, T : L2 (I)). By extracting a
subsequence we can deduce that
vn * v
weakly in L2 (0, T : H −2 (I)),
vn → v
strongly in L2 (0, T : L2 (I)).
Since kvn kL2 (0,T :L2 (I)) = 1, then
kvkL2 (0,T :L2 (I)) = 1.
By lower semicontinuity, we have
Z
Z T
|vxn (0, t)|2 dt + 2
0 = lim inf β
n→∞
Z
≥β
T
(3.38)
Z
0
0
T
|vx (0, t)|2 dt + 2
Z
0
T
0
Z
L
a(x)|v n |2 dx dt
0
L
a(x)|v|2 dx dt
0
which guarantees that av ≡ 0, and in particular v ≡ 0 in ω × (0, T ). We now
distinguish the following two situations:
(1) There exists a subsequence of (λn ) also denoted by (λn ) such that
λn → 0
as n → ∞.
In this case, the limit v satisfies the linear problem
ivt + iβvxxx + αvxx + δ|v|2 v + ia(x)v = 0
v(0, t) = v(L, t)
vx (L, t) = 0
v(x, t) = 0
for all t ∈ (0, T )
for all t ∈ (0, T )
in ω × (0, T )
Then, by Holmgren’s uniqueness Theorem (see [22]), v ≡ 0 in I × (0, T ) and this
contradicts (3.38).
(2) There exists a subsequence of (λn ) also denoted by (λn ) and λ > 0 such that
λn → λ. In this case, the limit function v solves (3.29)-(3.31) and, by the (UCP)
assumed to hold for the subset ω, we have that v ≡ 0 in I × (0, T ), and again, in
this case, we have a contradiction.
In the cases (1) and (2) we have a contradiction. Hence, (3.27) holds and the
proof is complete.
Proof of Theorem 3.3. From Theorem 3.2 we obtain that if u0 ∈ L2 (I) then
u ∈ L∞ (0, T : L2 (I)) ∩ L2 (0, T : H01 (I))
and ut ∈ L2 (0, T : H0−2 (I)). Consequently, we know that u is weakly continuous
from [0, T ] into L2 (I). According to the structure of ω, u ≡ 0 in {(0, δ) × (L −
δ, L)} × (0, T ). Let us define the extended function
(
u(x, t) if (x, t) ∈ (δ, L − δ) × (0, T )
u(x, t) =
0
if (x, t) ∈ {R − (δ, L − δ)} × (0, T ).
Then, u satisfies
iut + αuxx + iβuxxx + λ|u|2 u + ia(x)u = 0 in R × (0, T )
u(x, 0) = φ(x)
in R.
(3.39)
(3.40)
16
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
and
(
u0 (x) if x ∈ (δ, L − δ)
φ(x) =
0
if x ∈ {R − (δ, L − δ)}.
If we consider v(x, t) = u(x + t, t), then v solves
ivt + αvxx + iβvxxx + λ|v|2 v + ia(x)v = 0
u(x, 0) = φ(x)
in R × (0, T )
in R.
(3.41)
(3.42)
Since φ has compact support and belongs to L2 (R), we have
Z
φ2 (x)e2bx dx < ∞, ∀b > 0.
(3.43)
R
Thus, by regularizing properties (see [28]), v ∈ C ∞ (I × (0, T )). Therefore, v is
smooth as well, and applying the unique continuation result we have that v ≡ 0
and u ≡ 0, x ∈ I, t ∈ (0, T ).
Remark about the hypothesis |α| < 3β. We consider the Gauge transformation
u(x, t) = eid2 x+id3 t v(x − d1 t, t) ≡ eθ v(η, ξ)
and θ = id2 x + id3 t, η = x − d1 t, ξ = t. Then
ut = id3 eθ v − d1 eθ vη + eθ vξ ,
ux = id2 eθ v + eθ vη ,
uxx = −d22 eθ v + 2id2 eθ vη + eθ vηη ,
uxxx = −id32 eθ v − 3d22 eθ vη + 3id2 eθ vηη + eθ vηηη .
Replacing in (1.5), we have
− d3 eθ v − id1 eθ vη + ieθ vξ − ωd22 eθ v + 2iωd2 eθ vη + ωeθ vηη
βd33 eθ v − 3iβd22 eθ vη − 3βd2 eθ vηη + iβeθ vηηη + γ|v|2 eθ v
− δd2 |v|2 eθ v + iδ|v|2 eθ vη + d2 eθ v 2 v + ieθ v 2 vη = 0
and
ivξ + (ω − 3βd2 )vηη + iβvηηη + (2iωd2 − 3iβd22 − id1 + iδ|v|2 + iv 2 )vη
(βd32 − ωd22 − d3 + γ|v|2 − δd2 |v|2 )v + d2 v 2 v = 0 .
Then
d1 =
ω2
,
3β
d2 =
ω
,
3β
d3 =
−2ω 3
.
27β 2
This way in (1.5) we obtain
ivξ + iβvηηη + i(δ|v|2 + v 2 )vη + γ −
ωδ 2
δ 2
|v| v +
v v = 0,
3β
3β
but v 2 v = vvv = |v|2 v, then using the Gauge transformation we have the equivalent
problem to (1.5)
δ
ωδ 2
ivξ + iβvηηη + iδ|v|2 vη + iv 2 vη + γ +
−
|v| v = 0 η, ξ ∈ R
3β
3β
(3.44)
ω
v(η, 0) = e−i 3β η u0 (η).
EJDE-2007/06
STABILIZATION OF SOLUTIONS
Here, rescaling the equation, we take β = 1.
δ
ωδ 2
ivt + ivxxx + iδ|v|2 vx + iv 2 vx + γ +
−
|v| v = 0 x, t ∈ R
3
3
ω
v(x, 0) = e−i 3 x u0 (x).
17
(3.45)
The above Gauge’s transformation is a bicontinuous map from Lp ([0, T ] : H s (R)
to itself, as far as 0 < T < ∞. With this, the imposed assumption |ω| < 3β can be
removed.
Acknowledgments. The authors would like to express his gratitude to the anonymous referee for his/her valuable suggestions pointing out important references
[15, 18, 24, 25] which were missing in the first version of this work.
The first two authors were supported by Fundação de Amparo à pesquisa do Estado do Rio Grande do Sul. FAPERGS. The third author was supported by Proyectos de Investigación Internos 061008 1/R Universidad del Bı́o-Bı́o and Fundação de
Amparo à pesquisa do Estado do Rio Grande do Sul. FAPERGS.
References
[1] C. Bardos, G. Lebeau and J. Rauch. Sharp sufficient condition for the observation, control
and stabilzatio of waves from the boundary, SIAM, J. Control Optim., 30(1992) 1024-1065.
[2] V. Bisognin and O. Vera. On the unique continuation property for the higher order nonlinear
Schrödinger equation with constant coefficients, Turk. J. Math. 30(2006) 1-38.
[3] J. Bona and R. Scott. Solutions of the Korteweg-de Vries equation in fractional order Sobolev
space, Duke Math. J., 43(1976) 87-99.
[4] X. Carvajal. Local well-posedness for a higher order nonlinear Schrödinger equation in
Sobolev space of negative indices, EJDE, 13(2004) 1-10.
[5] X. Carvajal and F. Linares. A higher order nonlinear Schrödinger equation with variable
coefficients, Differential and Integral Equations, 16(2003) 1111 - 1130.
[6] A. Cohen. Solutions of the Korteweg-de Vries equation from irregular data, Duke Math.,
45(1991) 149-181.
[7] C. Dafermos. On the existence and asymptotic stability, Arch. Rational Mech. Anal., 22(1968)
241-271.
[8] A. Haraux. Stabilization of trayectories for some weakly damped hyperbolic equations, J. Diff.
Eq., 59, 2(1985) 145-154.
[9] A. Hasegawa and Y. Kodama. Nonlinear pulse propagation in a monomode dielectric guide,
IEEE. J. Quant. Elect., 23(1987) 510-524.
[10] T. Kato. On the Cauchy problem for the ( generalized ) Korteweg - de Vries equations, Adv.
in Math. Suppl. Studies, Studies in Appl. Math., 8(1983) 93-128.
[11] Y. Kodama. Optical solitons in a monomode fiber, J. Phys. Stat., 39(1985) 596.614.
[12] V. Komornik. Exact controllability and stabilization, the multiplier method, R. A. M. 36,
John Wiley-Mason, 1994.
[13] V. Komornik, D. L. Russel and B-Y. Zhang. Stabilization de l’équation de Korteweg-de Vries,
C. R. Acad. Sci. Paris, Serie I, Math., 312(1991) 841-843.
[14] C. Laurey. Le probléme de Cauchy pour une équation de Schrödinger non-linéaire de ordre
3, C. R. Acad. Sci. Paris, 315(1992) 165-168.
[15] F. Linares and A. F. Paxoto. On the exponential decay of the critical generalized Korteweg-de
Vries equation with localized damping, to appear in the proceeding of the AMS.
[16] J. L. Lions. Exact controllability, stabilization and perturbation for distributed system, SIAM
Rev., 30(1988) 1-68.
[17] M. Nakao. Decay of solutions of the wave equation with localized nonlinear dissipation, Math.
Ann., 305(1996) 403-417.
[18] A. F. Paxoto. Unique continuation and decay for the Korteweg- de Vries equation with localized damping, ESAIM: COCV 11 (3)(2005) 473-486.
[19] A. Pazy, Semigroups of linear operators and applications to partial differential equations,
Springer-Verlag, New York, 1983.
18
E. BISOGNIN, V. BISOGNIN, O. P. VERA V.
EJDE-2007/06
[20] G. Perla Menzala, C. F. Vasconcellos and E. Zuazua. Stabilization of the Korteweg-de Vries
with localized damping, Quarterly of Applied Math., LX, 1(2002) 111-129.
[21] G. Perla Menzala, E. Bisognin and V. Bisognin. Exponential stabilization of a coupled system
of Korteweg-de Vries equations with localized damping, Adv. in Diff. eqns., 8(2003) 442-469.
[22] M. Renardy and R. C. Rogers. An introduction to partial differential equations, Text in
Applied Mathematics, 13, 1992, Springer-Verlag.
[23] L. Rosier. Exact Boundary controllability for the Korteweg-de Vries equation on a bounded
domain, ESAIM Contrl, Optimization and Calculus of Variations, 12(1997) 33-55.
[24] L. Rosier. Control of the surface of a fluid by a wavemaker, ESAIM: COCV 10 (3)(2006)
927-956.
[25] L. Rosier and B.-Y. Zhang. Global stabilization of the generalized Korteweg-de Vries equation
posed on a finite domain, SIAM J. Contr. Optim. 45 (3)(2006) 927-956.
[26] D. L. Russell and B-Y. Zhang. Exact controllability and stabilization of the Korteweg-de Vries
equations, Trans. AMS, 348(1996) 3643-3672.
[27] J. C Saut and R. Temam. Remark on the Korteweg - de Vries equation, Israel J. Math.,
24(1976) 78-87.
[28] M. Sepúlveda and O. Vera. Gain in regularity for the higher order nonlinear Schrödinger
equation with constant coefficients. Submitted.
[29] M. Slemrod. Weak asymptotc decay via a ”Relaxed Invariant Principle” for a wave equation
with nonlinear, non monotone damping, Proc. Royal Soc. Edim., 113(1989) 87-58.
[30] G. Staffilani. On the generalized Korteweg-de Vries type equation, Diff. and Int. Eqns., Vol.
10, 4(1997) 777-796.
[31] D. Tataru. Carleman estimates and unique continuatin for the Schrödinger equation, Differential and Integral Equation, 8, 4(1995) 901-905.
[32] R. Temam. ”Navier-Stokes equations”, studies in Mathematics and its Applicaons, 2, NorthHolland, 1977.
[33] O. Vera. Gain of regularity for a Korteweg-de Vries-Kawahara type equation, EJDE, Vol
2004, 71(2004) 1-24.
[34] B-Y. Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM, J. Math Anal.
23(1991) 55-71.
[35] B-Y. Zhang, Exact boundary controllability for the Korteweg-de Vries equation, SIAM, J.
Control Optim., 37(1999) 548-565.
[36] E. Zuazua. Exponential decay for the semilinear wave equation with locally distributed damping, Comm P. D. E. 15(l)(1990) 205-235.
Eleni Bisognin
Centro Universitário Franciscano-UNIFRA, Rua dos Andrade 1614, CEP: 97010-032,
Santa Maria, R. S., Brasil
E-mail address: eleni@unifra.br
Vanilde Bisognin
Centro Universitário Franciscano-UNIFRA, Rua dos Andrade 1614, CEP: 97010-032,
Santa Maria, R. S., Brasil
E-mail address: vanilde@unifra.br
Octavio Paulo Vera Villagrán
Departamento de Matemática, Universidad del Bı́o-Bı́o, Collao 1202, Casilla 5-C, Concepción, Chile
E-mail address: overa@ubiobio.cl octavipaulov@yahoo.com