International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 2 (2015), 96-103
http://www.etamaths.com

ON THE STABILIZATION OF THE LINEAR KAWAHARA

EQUATION WITH PERIODIC BOUNDARY CONDITIONS

PATRICIA N. DA SILVA AND CARLOS F. VASCONCELLOS∗

Abstract. We study the stabilization of global solutions of the linear Kawa-

hara equation (K) with periodic boundary conditions under the effect of a

localized damping mechanism. The Kawahara equation is a model for small
amplitude long waves. Using separation of variables, the Ingham inequality,

multiplier techniques and compactness arguments we prove the exponential

decay of the solutions of the (K) model.

1. Introduction

In this paper we study the stabilization of global solutions of the linear Kawahara
equation (K) with periodic boundary conditions under the effect of a localized
damping mechanism, that is, we consider the following problem:



ut + βux + κuxxx + ηuxxxxx + a(x)u = 0 x ∈ (0, 2π), t > 0
u(0, t) = u(2π,t), t > 0

ux(0, t) = ux(2π,t), t > 0

uxx(0, t) = uxx(2π,t), t > 0

uxxx(0, t) = uxxx(2π,t), t > 0

uxxxx(0, t) = uxxxx(2π,t), t > 0

u(x, 0) = u0(x), x ∈ (0, 2π)

(1.1)

The parameter η is a negative real number, κ 6= 0, β is a real number and a ∈
L∞(0, 2π), a ≥ 0 a.e. in (0, 2π) and we assume that a(x) ≥ a0 > 0 a.e. in an open
subinterval ω of (0, 2π), where the damping is effectively acting .
In the Kawahara equation

(1.2) ut + ux + κuxxx + ηuxxxxx + uux = 0,

the conservative dispersive effect is represented by the term (κuxxx + ηuxxxxx).
This equation is a model for plasma wave, capilarity-gravity water waves and other
dispersive phenomena when the cubic KdV-type equation is weak. Kawahara [10]
pointed out that it happens when the coefficient of the third order derivative in
the KdV equation becomes very small or even zero. It is then necessary to take
into account the higher order effect of dispersion in order to balance the nonlinear
effect. Kakutani and Ono [9] showed that for a critical value of angle between the
magneto-acoustic wave in a cold collision-free plasma and the external magnetic

2010 Mathematics Subject Classification. 35Q35, 35B40, 35Q53.
Key words and phrases. exponential decay; periodic boundary; Kawahara equation.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

96



STABILIZATION OF THE LINEAR KAWAHARA EQUATION 97

field, the third order derivative term in the KdV equation vanishes and may be
replaced by the fifth order derivative term. Following this idea, Kawahara [10]
studied a generalized nonlinear dispersive equation which has a form of the KdV
equation with an additional fifth order derivative term. This equation has also
been obtainded by Hasimoto [8] for the shallow wave near critical values of surface
tension. More precisely, in this work Hasimoto found these critical values when the
Bond number is near to one third.
While analyzing the evolution of solutions of the water wave-problem, Schneider and
Wayne [19] also showed that the coefficient of the third order dispersive term in
nondimensionalized statements of the KdV equation vanishes when the Bond num-
ber is equal to one third. The Bond number is proportional to the strength of the
surface tension and in the KdV equation it is related to the leading order dispersive
effects in the water-waves problem. With its disappearance, the resulting equation
is just Burger’s equation whose solutions typically form shocks in finite time. Thus,
if we wish to model interesting behavior in the water-wave problem it is necessary
to include higher order terms. That is, it is necessary to consider the Kawahara
equation. In any case, the inclusion of the fifth order derivative term takes into
account the comparative magnitude of the coefficients of the third and fifth power
terms in the linearized dispersion relation.
Berloff and Howard [3] presented the Kawahara equation as the purely dispersive
form of the following nonlinear partial differential equation

ut + u
rux + auxx + buxxx + cuxxxx + duxxxxx = 0.

The above equation describes the evolution of long waves in various problems in fluid
dynamics. The Kawahara equation corresponds to the choice a = c = 0 and r = 1
and describes water waves with surface tension. Bridges and Derks [6] presented
the Kawahara equation – or fifth-order KdV-type equation – as a particular case
of the general form

(1.3) ut + κuxxx + ηuxxxxx =
∂

∂x
f(u,ux,uxx)

where u(x,t) is a scalar real valued function, κ and η 6= 0 are real parameters
and f(u,ux,uxx) is some smooth function. The form (1.2) occurs most often in
applications and corresponds to the choice of f in (1.3) with the form f(u,ux,uxx) =

−u
2

2
.

As noted by Kawahara [10], we may assume without loss of generality that η < 0
in (1.2). In fact, if we introduce the following simple transformations

u →−u, x →−x and t → t

we can obtain an equation of the form of equation (1.2) in which κ and η are
replaced, respectively, by −κ and −η.
Hagarus et al. pointed out that the Kawahara equation

(1.4) ut = uxxxxx −εuxxx + uux
in which ε is a real parameter models water waves in the long-wave regime for
moderate values of surface tension, Weber numbers close to 1/3; and that for such
Weber numbers the usual description of long water waves via the Korteweg-de Vries
(KdV) equation fails since the cubic term in the linear dispersion relation vanishes
and fifth order dispersion becomes relevant at leading order, ω(k) = k5 + εk3.



98 SILVA AND VASCONCELLOS

Positive (resp. negative) values of the parameter ε in (1.4) correspond to Weber
numbers larger (resp. smaller) than 1/3.
Dispersive problems have been object of intensive research (see, for instance, the
classical paper of Benjamin-Bona-Mahoni [2], Biagioni-Linares [4], Bona-Chen [5],
Menzala et al. [15], Rosier [16], and references therein). Recently global stabi-
lization of the generalized KdV system have been obtained by Rosier-Zhang [17]
and Linares-Pazoto[12] with critical exponents. For the stabilization of global so-
lutions of the Kawahara under the effect of a localized damping mechanism, see
Vasconcellos and Silva [20, 21].
For controllability problems involving dispersive systems, we can consider the works
of Russel-Zhang [18] and Laurent et al. [12] about the KdV system; the paper by
Linares-Ortega [14], where the Benjamin-Ono equation has been analyzed and the
paper of Zhang and Zhao [22] for the Kawahara equation.
The total energy associated with the (1.1) system is defined by

E(t) =
1

2

∫ 2π
0

|u(x,t)|2dx =
1

2
‖u(t)‖2.

Using the above boundary conditions we prove that

dE

dt
=
η

2
|uxx(0, t)|2 −

∫ 2π
0

a(x)|u(x,t)|2dx ≤ 0, ∀t > 0.

So, E(t) is a nonincreasing function of time. This paper is devoted to analyze the
following questions: Does the energy E(t) → 0 as t → ∞? Is it possible to find a
rate of decay of the energy?
Then, we can state our main result:

Theorem 1.1. There exist C > 0 and γ > 0 such that the energy E(t) associated
to the problem (1.1) satisfies

E(t) ≤ Ce−γt‖u0‖2L2(0,2π)
for all u0 ∈ L2(0, 2π).

To prove the above theorem we need some generalizations of Ingham inequality (see
for instance [1],[7] and [11]), multiplier techniques and compactness arguments.
We organize this work as follows.
In Section 2, we present some auxiliary lemmas, useful to demonstrate our main
result. In Section 3, we prove Theorem 1.1 and in Section 4, we present our final
remarks.

2. Auxiliary Lemmas

Lemma 2.1. Consider the problem:


vt + βvx + κvxxx + ηvxxxxx = 0 x ∈ (0, 2π), t > 0
v(0, t) = v(2π,t), t > 0

vx(0, t) = vx(2π,t), t > 0

vxx(0, t) = vxx(2π,t), t > 0

vxxx(0, t) = vxxx(2π,t), t > 0

vxxxx(0, t) = vxxxx(2π,t), t > 0

v(x, 0) = u0(x), x ∈ (0, 2π)

(2.5)



STABILIZATION OF THE LINEAR KAWAHARA EQUATION 99

The parameter η is a negative real number, κ 6= 0 and β is a real number. Then,
for T > 0, there exists a constant C1 = C1(T) > 0 such that

‖u0‖2L2(0,2π) ≤ C1
∫ T
0

∫
ω

|v(x,t)|2dxdt,

where ω is an open subinterval of (0, 2π).

Proof. We assume a solution v of the system (2.5) can be written as v(x,t) =
X(x)T(t). Then

XT ′ + βTX′ + κTX′′′ + ηTX′′′′′ = 0

that is
T ′

T
= −

βX′ + κX′′′ + ηX′′′′′

X
= λ

for some constant λ. Thus, we obtain


βX′ + κX′′′ + ηX′′′′′ + λX = 0 x ∈ (0, 2π),
X(0) = X(2π),

X′(0) = X′(2π),

X′′(0) = X′′(2π),

X′′′(0) = X′′′(2π),

X′′′′(0) = X′′′′(2π),

(2.6)

and

(2.7) T ′ −λT = 0
To solve (2.6), we use the characteristic equation

ηr5 + κr3 + βr + λ = 0.

We can show that the eigenvalues λ are pure imaginary numbers. Notice that for
each k ∈ Z, the function

φk(x) =
1
√

2π
eikx

is an eigenfunction of (2.6) associated with the eigenvalue

λk = (−ηk5 + κk3 −βk)i.
Furthermore, for any l ∈ Z, let

ml = #{k ∈ Z, λk = λl}.
Then, ml ≤ 5 for any l and in particular m(l) = 1, if |l| is large enough. Moreover,
(2.8) lim

|k|→∞
|λk −λk+1| = ∞.

We have

(2.9) X(x) = Cke
ikx, k ∈ Z.

Then, by (2.7) and (2.9), it follows that

(2.10) v(x,t) =
∑
k∈Z

cke
i(kx+σkt), σk = −ηk5 + κk3 −βk

where
u0(x) =

∑
k∈Z

cke
ikx.



100 SILVA AND VASCONCELLOS

As pointed out by, Jaffard and Micu [7],

lim sup
n
|λn+1 −λn| >

2π

T

gives a sufficient condition for the validity of an Ingham type inequality For each
T , since we have (2.8), from an Ingham inequality (see for instance Theorem 3.5
in Baiocchi, Komornik and Loreti [1] for Ingham inequalities for sequences with
repeated eigenvalues and with weak gap conditions.), it follows that there exists a
constant C = C(T) > 0 such that

(2.11) ‖u0‖2L2(0,2π) =
∑
k∈Z
|ck|2 ≤ C(T)

∫ T
0

∣∣∣∣∣∑
k∈Z

cke
iσkt

∣∣∣∣∣
2

dt

Therefore, using (2.11) and the Fubini Theorem, we have∫ T
0

∫
ω

|v(x,t)|2dxdt =
∫
ω

∫ T
0

∣∣∣∣∣∑
k∈Z

cke
ikxeiσkt

∣∣∣∣∣
2

dtdx

≥
1

C(T)

∫
ω

∑
k∈Z

∣∣ckeikx∣∣2 dx = 1
C(T)

∫
ω

∑
k∈Z
|ck|

2
dx

=
l(ω)

C(T)

∑
k∈Z
|ck|

2
=

l(ω)

C(T)
‖u0‖2L2(0,2π)·

�

Here, we denote by l(ω) the length of subset ω

Lemma 2.2. Let w be a solution of the following problem:


wt + βwx + κwxxx + ηwxxxxx = −a(x)u(x,t) x ∈ (0, 2π), t > 0
w(0, t) = w(2π,t), t > 0

wx(0, t) = vw(2π,t), t > 0

wxx(0, t) = wxx(2π,t), t > 0

wxxx(0, t) = wxxx(2π,t), t > 0

wxxxx(0, t) = wxxxx(2π,t), t > 0

w(x, 0) = 0, x ∈ (0, 2π)

(2.12)

where a = χω, ω ⊂ (0, 2π) and u is the solution of (1.1). The parameter η is a
negative real number, κ 6= 0 and β is a real number. Then, for T > 0, there exists
a constant C2 = C2(T) > 0 such that

‖w(t)‖2L2(0,2π) ≤ C2
∫ T
0

∫
ω

|u(x,t)|2dxdt.

Proof. If we multiply the equation (2.12) by w, integrate in (0, 2π) and use the
periodic boundary conditions, we have

1

2

d

dt
‖w(t)‖2L2(0,2π) = −

∫
ω

u(x,t)w(x,t)dx, t > 0.

Thus
d

dt
‖w(t)‖2L2(0,2π) ≤

∫
ω

|u(x,t)|2dx +
∫ 2π
0

|w(x,t)|2dx.



STABILIZATION OF THE LINEAR KAWAHARA EQUATION 101

Now, if γ(t) = ‖w(t)‖2
L2(0,2π)

, we obtain{
γ′(t) ≤ g(t) + γ(t)
γ(0) = 0

where g(t) =
∫
ω
|u(x,t)|2dx. Hence, by Gronwall inequality, there exists a constant

C2 = C2(T) > 0, such that

γ(t) ≤ C2(T)
∫ T
0

g(t)dt, t ∈ (0,T)

and the Lemma follows. �

Lemma 2.3. For each T > 0, there exists a constant C3 = C3(T) > 0 such that

1

2
‖u0‖2L2(0,2π) ≤ C3

∫ T
0

∫
ω

|u(x,t)|2dxdt,

where u is the solution of (1.1).

Proof. Let v and w be respectively the solutions of the problems (2.5) and (2.12).
So we have u = v + w (or v = u−w). Now using Lemmas 2.1 and 2.2, we obtain

‖u0‖2L2(0,2π) ≤ C1
∫ T
0

∫
ω

|v(x,t)|2dxdt

≤ 2C1

[∫ T
0

∫
ω

|u(x,t)|2dxdt +
∫ T
0

∫
ω

|w(x,t)|2dxdt

]

≤ 2C1

[∫ T
0

∫
ω

|u(x,t)|2dxdt + C2T
∫ T
0

∫
ω

|u(x,t)|2dxdt

]

= 2C1(1 + C2T)

∫ T
0

∫
ω

|u(x,t)|2dxdt.

The inequality stated in the lemma holds with C3 = C1(1 + C2T). �

3. Proof of Theorem 1.1

Now, we are able to prove Theorem 1.1. In fact, if we multiply the equation in
system (1.1) by u and integrate in (0, 2π), we have

1

2

d

dt
‖u(t)‖2L2(0,2π) = −

∫
ω

|u(x,t)|2dxdt ≤ 0

So, E(t) = 1
2
‖u(t)‖2

L2(0,2π)
is a decreasing function of time and moreover

E(T) −E(0) = −
∫ T
0

∫
ω

|u(x,t)|2dxdt.

Thus

(1 + C3)E(T) = −C3
∫ T
0

∫
ω

|u(x,t)|2dxdt + C3E(0) + E(T).

Since

E(T) ≤ E(0) =
1

2
‖u0‖2L2(0,2π),

it follows, by Lemma 2.3, that:

(1 + C3)E(T) ≤ C3E(0).



102 SILVA AND VASCONCELLOS

Therefore

E(T) ≤
C3

1 + C3
E(0), T > 0.

Finally, we use the semigroup property to obtain Theorem 1.1.

Remark 3.1. In the Lemma 2.2 and in the Theorem 1.1, we can consider a ∈
L∞(0, 2π), a ≥ 0 a.e. in (0, 2π) and assume that a(x) ≥ a0 > 0 a.e. in an open
subinterval ω of (0, 2π) and the proofs follow in the same way.

4. Final Remarks

We can observe that, if we consider the parameter β = 0 in the system (1.1) the
Theorem 1.1 follows similarly.
Now, we will make some comments concerning the exact controllability for Kawa-
hara system:
In the linear case, boundary exact controllability is proved, using HUM method
and multipliers techniques, by Vasconcellos-Silva [20].
In the nonlinear case, internal exact controllability can be found in Zhang-Zhao
[22], where was considered periodic domain with an internal control acting on an
arbitrary small nonempty subdomain of [0, 2π]. Aided by the Bourgain smoothing
property of the Kawahara equation on a periodic domain, it was showed that the
system is locally exactly controllable.
We believe that it is possible to show the boundary exact controllability for linear
Kawahara system in periodic domain
That is, we consider the following problem:
Given u0 and uT in L

2(0,L), find hj ∈ L2(0,L), j = 0, 1, 2, 3, 4 such that the
solution of the bellow system:



ut + βux + κuxxx + ηuxxxxx = 0 x ∈ (0, 2π), t > 0
u(0, t) −u(2π,t) = h0, t > 0
ux(0, t) −ux(2π,t) = h1, t > 0
uxx(0, t) −uxx(2π,t) = h2, t > 0
uxxx(0, t) −uxxx(2π,t) = h3, t > 0
uxxxx(0, t) −uxxxx(2π,t) = h4, t > 0
u(., 0) = u0

(4.13)

satisfies u(·,T) = uT .
As proved by Rosier in [16] for linear KdV system, we could use HUM method and
generalizations of Ingham inequalities to obtain the above problem.

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STABILIZATION OF THE LINEAR KAWAHARA EQUATION 103

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Instituto de Matemática e Estat́ıstica (IME)- UERJ, R. São Francisco Xavier, 524,

Sala 6016, Bloco D - CEP 20550-013, Rio de Janeiro, RJ, Brasil

∗Corresponding author