kdvbseg.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 01, (41–58). March 2010

Korteweg-de Vries-Burgers Equation on a Segment

Elena I. Kaikina

Instituto de Matemáticas,

UNAM Campus Morelia, AP 61-3 (Xangari),

Morelia CP 58089, Michoacán, MEXICO

email : ekaikina@matmor.unam.mx

and

Leonardo Guardado-Zavala, Hector F. Ruiz-Paredes,

S. Juarez Zirate

Posgrado Electrica, Instituto Tecnológico de Morelia,

CP 58120, Morelia, Michoacán, MÉXICO

emails: guardado@ps.itm.mx

hruiz@sirio.tsemor.mx

sjzirate@matmor.unam.mx

ABSTRACT

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers
equation on the interval (0, 1)





ut + uux − uxx + uxxx = 0, t > 0, x ∈ (0, 1)

u(x, 0) = u0(x), x ∈ (0, 1)

u(0, t) = u(1, t) = ux(1, t) = 0, t > 0.

(0.1)

We prove that if the initial data u0 ∈ L
2, then there exists a unique solution u ∈ C

(
[0, ∞) ; L2

)
∪

C
(
(0, ∞) ; H1

)
of the initial-boundary value problem (0.1). We also obtain the large time

asymptotic of solution uniformly with respect to x ∈ (0, 1) as t → ∞.



42 E.I. Kaikina et. al. CUBO
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RESUMEN

Estudiamos el siguiente problema de valor inicial en la frontera para la ecuación de Korteweg-
de Vries-Burgers en el intervalo (0, 1)





ut + uux − uxx + uxxx = 0, t > 0, x ∈ (0, 1)

u(x, 0) = u0(x), x ∈ (0, 1)

u(0, t) = u(1, t) = ux(1, t) = 0, t > 0.

(0.1)

Provamos que si el dato inicial u0 ∈ L
2, entonces existe una única solución u ∈ C

(
[0, ∞) ; L2

)
∪

C
(
(0, ∞) ; H1

)
del problema de valor inicial en la frontera (0.1). También obtenemos com-

portamiento asintótico de la solución con respecto a x ∈ (0, 1) cuando t → ∞.

Key words and phrases: Dissipative Nonlinear Evolution Equation, Large Time Asymptotics,

Korteweg-de Vries-Burgers equation.

Math. Subj. Class.: 35Q35.

1 Introduction

We study the global existence and large time asymptotic behavior of solutions to the initial-

boundary value problem for the Korteweg-de Vries-Burgers equation in the interval (0, 1)





ut + uux − uxx + uxxx = 0, t > 0, x ∈ (0, 1)
u(x, 0) = u0(x), x ∈ (0, 1)

u(0, t) = u(1, t) = ux(1, t) = 0, t > 0.

(1.1)

The Korteweg-de Vries-Burgers equation (1.1) is a simple universal model equation which

appears as the first approximation in the description of the dispersive-dissipative nonlinear waves,

and has many applications in various fields of Physics, Biology and Engineering. In the case of the

Cauchy problem some estimates for the time decay rates of solutions to the Korteweg-de Vries-

Burgers type equations were found in papers [3], [4], [5] and the large time asymptotics of solutions

was obtained in [6], [11], [12]. In the case of the boundary value problem on half-line the large

time asymptotics of solutions were studied in papers [2], [7], [8],[9], [10]. As far as we know large

time asymptotic behavior for solutions of the initial-boundary value problem for the Korteweg-de

Vries-Burgers equation (1.1) on the interval was not studied previously. In this paper we consider

(1.1) in the case of the initial data belonging to L2. We note here that we do not assume the

smallness condition on the data. In the case of large initial data it is more difficult than that of

small data to obtain exact representation of large time asymptotics of solutions and there are a

few results (see, e.g. [13]). Another difficulty in the study of the boundary value problem for the

Korteweg-de Vries-Burgers equation (1.1) is that the linear operator −∂2x + ∂3x is not self-adjoint



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Korteweg-de Vries-Burgers Equation ... 43

and we can not apply the Fourier method when we take the boundary value into account. To

avoid this difficulty we apply the Laplace transformation with respect to space variable to derive

the Green function of the resulting equation. For obtaining Lp -estimates of the Green function

we use the method of papers [8] and [9].

To state the results of the present paper precisely we give some notations. Let us denote

H1 =
{
ϕ ∈ L2 (0, 1) ; ‖ϕ‖

H1
= ‖ϕ‖

L2
+ ‖ϕx‖L2 < ∞

}
.

We introduce the function Λ(x) ∈ L∞(0, 1)

Λ(x) = −△̃(−ξ0, 1 − x)
△̃′(−ξ0, 1)

, △̃(ξ, y) =
3∑

j=1

e−φj yφ′j (ξ) ,

△̃′(ξ, 1) =
3∑

j=1

e−φj y
(
φ
′′
j (ξ) −

(
φ
′
j (ξ)

)2)
,

where

△̃(ξ, y) =
3∑

j=1

e−φj yφ′j (ξ) ,

△̃′(ξ, y) =
3∑

j=1

e−φj y
(
φ′′j (ξ) −

(
φ′j (ξ)

)2)
.

Here φl(ξ) are the roots of the characteristic equation −p2 + p3 + ξ = 0, such that Re φl(ξ) > 0,
l = 1, 2, and Re φ3(ξ) < 0, for all

ξ ∈ D0 =
{

ξ ∈ C : Re ξ ≥ 0, ξ /∈
[
0,

4

27

]}
,

and ξ0 ∈ C , Reξ0 > 0 is the first root of the equation
3∑

j=1

e−φj φ
′
j (−ξ) = 0. (1.2)

By the same letter C we denote different positive constants if it does not make confusion.

We state the main result of this paper.

Theorem 1.1. Suppose that the initial data u0 ∈ L2. Then there exists a unique solution of (1.1)

u ∈ C
(
[0, ∞) ; L2

)
∪ C

(
(0, ∞) ; H1

)

such that the solution has the following asymptotics

u(x, t) = AΛ(x)e−ξ0t + O
(
e−(ξ0+δ)t

)

for t → ∞ uniformly with respect to x ∈ (0, 1) , where δ > 0 is a constant satisfying the condition
such that |ξ0| + δ < |ξ1| , where ξ1 is the second root of (1.2), the constant A is defined by



44 E.I. Kaikina et. al. CUBO
12, 1 (2010)

A =

∫ 1

0

u0(y)△̃(−ξ0, y)dy +
∫ ∞

0

eξ0τ dτ

∫ 1

0

u(y, τ )uy(y, τ )△̃(−ξ0, y)dy.

Remark 1.1. By virtue of the numerical computations via program Maple we have ξ0 ≈ 70 and

ξ1 ≈ 200.

We organize our paper as follows. In Section 2 we solve the linear initial-boundary value

problem corresponding to (1.1). In Section 3 we prove the local existence of solutions to (1.1).

Section 4 is devoted to the proof of global existence of solutions to (1.1) for the case of small initial

data. We prove Theorem 1.1 in Section 5 by using the time decay estimates of solutions obtained

in Section 4.

2 Linear Problem

We consider the following linear initial-boundary value problem






ut − uxx + uxxx = f, t > 0, x ∈ (0, 1) ,
u(x, 0) = u0(x), x ∈ (0, 1) ,

u(0, t) = u(1, t) = ux(1, t) = 0, t > 0.

(2.1)

We define for x ∈ (0, 1)

△̃(ξ, x) =
3∑

j=1

e−φj xφ′j (ξ) , (2.2)

where φl(ξ) are the roots of the characteristic equation −p2 + p3 + ξ = 0, such that Re φl(ξ) > 0,
l = 1, 2, and Re φ3(ξ) < 0, for all

ξ ∈ D0 =
{

ξ ∈ C : Re ξ ≥ 0, ξ /∈
[
0,

4

27

]}
,

Denote by

Gf = Θ(x)
(∫ x

0

f (y)F1(x, y, t)dy +

∫ 1

x

f (y)F2(x, y, t)dy

)
,

where

Θ(x) =

{
1, x ∈ [0, 1]
0, x /∈ [0, 1] ,

F1(x, y, t) = −
1

2πi

∫ i∞

−i∞

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

and

F2(x, y, t) = −
1

2πi

∫ i∞

−i∞

dξeξt
△̃(ξ, 1 − x)△̃(ξ, y) − △̃(ξ, y − x)△̃(ξ, 1)

△̃(ξ, 1)
.



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Korteweg-de Vries-Burgers Equation ... 45

Proposition 2.1. Let the initial data u0 ∈ L2 and f ∈ L2 Then a solution u of (2.1) has the
following representation

u(x, t) = Gu0 +
∫ t

0

G(t − τ )f (τ )dτ .

Proof. To derive an integral representation for solutions of the problem (2.1) we suppose that there

exists a solution u(x, t) of problem (2.1), which is continued by zero outside of x < 0, x > 1

u(x, t) = 0 for all x /∈ [0, 1] ,
∂jxu(0, t) = lim

x→0+
∂jxu(x, t), j = 0, 1, 2

∂jxu(1, t) = lim
x→1−

∂jxu(x, t), j = 0, 1, 2.

Also we suppose that f = 0. We denote operator

P [φ(p, t)] = − 1
2πi

∫ i∞

−i∞

e(q−p) − 1
q − p

φ(q, t)dq.

We have for the Laplace transform

L {Ku} = P



K(p)û + p2
2∑

j=1

∂j−1x u(0, t) − e−p∂j−1x u(a, t)
pj

− p3
3∑

j=1

∂j−1x u(0, t) − e−p∂j−1x u(a, t)
pj



 .

Since L {u} is analytic for all p ∈ C we have

L {u} = û(p, t) = P [û(p, t)] . (2.3)

Applying the Laplace transform with respect to x to problem (2.1) we obtain





P [ût + K(p)û(p, t) + B1(p, t) − e−pB2(p, t)] = 0, t > 0, x > 0,
û(p, 0) = û0(p),

u(0, t) = u(1, t) = ux(0, t) = 0, t > 0,

(2.4)

where

B1(p, t) = p
2

2∑

j=1

∂j−1x u(0, t)

pj
− p3

3∑

j=1

∂j−1x u(0, t)

pj
, (2.5)

B2(p, t) = p
2

2∑

j=1

∂j−1x u(a, t)

pj
− p3

2∑

j=1

∂j−1x u(a, t)

pj
.

We rewrite (2.4) in the form

ût + K(p)û(p, t) + B1(p, t) − e−pB2(p, t) = Φ(p, t), (2.6)



46 E.I. Kaikina et. al. CUBO
12, 1 (2010)

where some function Φ(p, t) is analytic for all p ∈ C ,

|Φ(p, t)| ≤ C(t) 1 + |e
−p|

|p|
, |p| > 1 (2.7)

and

P [Φ(p, t)] = 0. (2.8)

Now we prove that under this conditions Φ(p, t) ≡ 0.To find Φ(p, t) we introduce functions

Ω1(z, t) =
1

2πi

∫ i∞

−i∞

1

q − z
Φ(q, t)dq,

Ω2(z, t) =
e−z

2πi

∫ i∞

−i∞

eq

q − z
Φ(q, t)dq.

Since Φ(p, t) satisfies Holder condition the functions Ω1(z, ξ), Ω2(z, ξ) are analytic in Re z 6= 0.
Denote by

Ω+1,2(p, t) = lim
z→p,Re z<0

Ω1,2(z, t),

and

Ω−1,2(p, t) = lim
z→p,Re z>0

Ω1,2(z, t).

for Re p = 0.Since function Φ(p, t) is analytic for all p ∈ C from estimate (2.7) we have

Ω−2 (p, ξ) = Ω
+
1 (p, ξ) = 0.

Another hand by Sokhotsky-Plemeli formula we get

Ω−2 (p, ξ) = V P
e−p

2πi

∫ i∞

−i∞

1

q − p Φ(p, t)dq −
1

2
Φ(p, t)

Ω+1 (p, ξ) = V P
1

2πi

∫ i∞

−i∞

1

q − p
Φ(p, t)dq +

1

2
Φ(p, t).

and therefore for Re p = 0

Ω−2 (p, ξ) − Ω
+
1 (p, ξ) = P [Φ(p, t)] − Φ(p, t) = 0.

Thus for Re p = 0

Φ(p, t) = P [Φ(p, t)] = 0

and therefore due to analycity Φ(p, t) ≡ 0 for all p ∈ C .

Applying the Laplace transformation to problem (2.6) with respect to time variable we write

Lt {û(p, t)} = ̂̂u(p, ξ) as

̂̂u(p, ξ) =
1

K(p) + ξ

(
û0(p) − B̂1(p, ξ) + e−pB̂2(p, ξ)

)
(2.9)



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Korteweg-de Vries-Burgers Equation ... 47

for p ∈ C, K(p) = −p2 + p3.

Here functions B̂1(p, ξ), B̂2(p, ξ) are the Laplace transforms of B1(p, t), B2(p, t) with respect

to time.

In order to get the integral formula for solution , we need to know the functions B̂1(p, ξ),

B̂2(p, ξ) . We will find its using the analytic condition of function ̂̂u for p ∈ C and Re ξ > 0

̂̂u(p, ξ) = P
{
̂̂u(p, ξ)

}
. (2.10)

Via (2.9) we rewrite (2.10) in the form

1

K(p) + ξ

(
û0(p) − B̂1(p, ξ) + e−pB̂2(p, ξ)

)
(2.11)

=
1

2πi

∫ i∞

−i∞

e(q−p) − 1
q − p

1

K(q) + ξ

(
û0(q) − B̂1(q, ξ) + e−pB̂2(q, ξ)

)
.

Let φl(ξ) are the roots of the characteristic equation −p2 + p3 + ξ = 0, such that Re φl(ξ) > 0,
l = 1, 2, and Re φ3(ξ) < 0, for all ξ ∈ D0 =

{
ξ ∈ C : Re ξ ≥ 0, ξ /∈

[
0, 4

27

]}
. Note that the functions

φl(ξ) are analytic in the domain
{
ξ ∈ C : ξ /∈

(
−∞, 4

27

]}
. We represent p2 = ξ

1−p
for |p| < 1 and

p3 = −ξ
1− 1

p

for |p| > 1, hence we get the asymptotics

φ1(ξ) =

{ √
ξ + O(|ξ|), ξ → 0, Im ξ > 0, 1 + O(|ξ|), ξ → 0, Im ξ < 0,

ei
π
3

3
√

ξ + O (1) , |ξ| → ∞,
(2.12)

φ2(ξ) =

{
1 + O(|ξ|), ξ → 0, Im ξ > 0,

√
ξ + O(|ξ|), ξ → 0, Im ξ < 0,

e−i
π
3

3
√

ξ + O (1) , |ξ| → ∞,
(2.13)

and

φ3(ξ) =

{
−
√

ξ + O(|ξ|), |ξ| → 0,
− 3
√

ξ + O (1) , |ξ| → ∞,
(2.14)

for all ξ ∈ C : ξ /∈
(
−∞, 4

27

]
(by

√
ξ and 3

√
ξ we denote the main value of the analytic function, i.e.√

1 =
3
√

1 = 1.)

By Cauchy Theorem we have for all p ∈ C

1

2πi

∫ i∞

−i∞

e(q−p) − 1
q − p

1

K(q) + ξ

(
û0(q) − B̂1(q, ξ) + e−pB̂2(q, ξ)

)

=
1

K(p) + ξ

(
û0(p) − B̂1(p, ξ) + e−pB̂2(p, ξ)

)

+
e(φ3(ξ)−p)

φ3 − p
φ
′
3(ξ)

(
−û0(φ3) + B̂1(φ3, ξ) − e−φ3 B̂2(φ3, ξ)

)

−
2∑

j=1

1

φj − p
φ
′
j (ξ)

(
û0(φj ) − B̂1(φj , ξ) + e−φj B̂2(φj , ξ)

)
.



48 E.I. Kaikina et. al. CUBO
12, 1 (2010)

Using (2.11) we get 



B̂2(φ3, ξ) = e
φ

3

(
−û0(φ3) + B̂1(φ3, ξ)

)

B̂1(φj , ξ) = û0(φj ) + e
−φj B̂2(φj , ξ), j = 1, 2.

(2.15)

So we need to put in the initial-boundary value problem one boundary data in the point x = 0

and two boundary data in the point x = 1.

Let, for example, u(0, t) = ux(1, t) = u(1, t) = 0. Thus from system (2.16) we get

{
−∂xxû(a, ξ) = eφ3(ξ)a (−û0(φ3) + ∂xû(0, ξ)(1 − φ3) − ∂xxû(0, ξ))

∂xû(0, ξ)(1 − φj ) − ∂xxû(0, ξ) = û0(φj ) − e−φj a∂xxû(a, ξ), j = 1, 2,
(2.16)

which is equal to



e−φ1(ξ) 1 − φ1 −1
e−φ2(ξ) 1 − φ2 −1
e−φ3(ξ) 1 − φ3 −1







∂xxû(1, ξ)

∂xû(0, ξ)

∂xxû(0, ξ)


 =




û0(φ1)

û0(φ2)

û0(φ3)


 . (2.17)

Denote the determinant of this system by △(φ1,φ2,φ3), then it has a form

△(φ1,φ2,φ3) =

∣∣∣∣∣∣∣∣

1 1 1

e−φ1(ξ) e−φ2(ξ) eφ3(ξ)

φ1(ξ)e
φ

1
(ξ) φ2(ξ)e

φ
2
(ξ) φ3(ξ)e

φ
3
(ξ)

∣∣∣∣∣∣∣∣

= e−φ1 (φ2 − φ3) + e−φ2 (φ3 − φ1) + e−φ3 (φ1 − φ2).

in the domain ξ ∈ D0.

Since
∑3

j=1 φj = 1 and

φ
′
1(ξ) = −

1

(φ1 − φ2)(φ1 − φ3)
; φ

′
2(ξ) = −

1

(φ2 − φ1)(φ2 − φ3)
;

φ′3(ξ) = −
1

(φ3 − φ1)(φ3 − φ2)
(2.18)

we can rewrite △(φ1,φ2,φ3) as

△(φ1,φ2,φ3) = V (ξ)
3∑

j=1

e−φj (ξ)φ′j (ξ). (2.19)

where

V (ξ) = (φ1 − φ2)(φ2 − φ3)(φ3 − φ1).

Since V (ξ) 6= 0 and Re φl(ξ) > 0, l = 1, 2, Re φ3(ξ) < 0 in domain ξ ∈ D0 we easily get for
|ξ| ≫ 1, ξ ∈ D0

△(φ1,φ2,φ3) 6= 0.



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Korteweg-de Vries-Burgers Equation ... 49

By numeric computations we can check that △(φ1,φ2,φ3) 6= 0 for all |ξ| ≤ C,
ξ ∈ D0 =

{
ξ ∈ C : Re ξ ≥ 0, ξ /∈

[
0, 4

27

]}
. Therefore there exists a unique solution of the system

(2.17) 


∂xxû(1, ξ)

∂xû(0, ξ)

∂xxû(0, ξ)


 =

∫ 1

0

dyu0(y)




e−φ1(ξ) 1 − φ1 −1
e−φ2(ξ) 1 − φ2 −1
e−φ3(ξ) 1 − φ3 −1




−1 


e−φ1y

e−φ2y

e−φ3y


 . (2.20)

Since u(0, t) = ux(1, t) = u(1, t) = 0 by (2.9) and (2.5) we have

̂̂u(p, ξ) =
1

K(p) + ξ
(û0(p) + (p − 1)ûx(0, ξ)) + ûxx(0, ξ) − e−paûxx(a, ξ)). (2.21)

Taking inverse Laplace transform with respect to variable p we get

û(x, ξ) = Θ(x)

∫ 1

0

dyu0(y)
1

2πi

∫ +i∞

−i∞

dpepx
1

K(p) + ξ

×(e−py + (p − 1)ûx(0, ξ)) + ûxx(0, ξ) − e−pûxx(a, ξ)).

By the method of residues we see that for ξ ∈ D0

û(x, ξ) = Θ(x)

∫ x

0

dyu0(y)F̂1(x, y, ξ) +

∫ 1

x

dyu0(y)F̂2(x, y, ξ), (2.22)

where

F1(x, y, ξ) = −φ′3eφ3(ξ)x
(
e−yφ3(ξ) + (φ3 − 1)ûx(0, ξ) + ûxx(0, ξ)

)
(2.23)

−
2∑

j=1

φ
′
j (ξ)e

−φj (1−x)ûxx(1, ξ)

and

F2(x, y, ξ) = −φ′3eφ3(ξ)x ((φ3 − 1)ûx(0, ξ) + ûxx(0, ξ)) (2.24)

−
2∑

j=1

φ
′
j (ξ)

(
e−φj (1−x)ûxx(1, ξ) − e−φj (ξ)(x−y)

)
.

Denote △̃(ξ, y) =
∑3

j=1 e
−φj yφ′j (ξ) .

Substituting (2.20) into (2.23) and (2.24) and using (2.18) we get

F̂1(x, y, ξ) = −
△̃(ξ, 1 − x)△̃(ξ, y)

△̃(ξ, 1)
,

F̂2(x, y, ξ) =
△̃(ξ, y − x)△̃(ξ, 1) − △̃(ξ, 1 − x)△̃(ξ, y)

△̃(ξ, 1)
.



50 E.I. Kaikina et. al. CUBO
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Since φ′l(ξ) = O
(
|ξ|−

1

2

)
, l = 1, 2, 3 for |ξ| < 1, ξ ∈ D0, we have

△̃(ξ, 1 − x)△̃(ξ, y)
△̃(ξ, 1)

=

(∑3
j=1 e

−φj (ξ)(1−x)φ′j

)(∑3
j=1 e

−φj (ξ)yφ′j

)

∑3
j=1 e

−φj (ξ)φ′j
= O

(
|ξ|−

1

2

)
(2.25)

and

△̃(ξ, y − x) = O
(
|ξ|−

1

2

)
(2.26)

for |ξ| < 1, ξ ∈ D0. Due to the fact that Re φl(ξ) > 0, l = 1, 2, Re φ3(ξ) < 0 for |ξ| > 1, ξ ∈ D0, we
obtain for |ξ| > 1

1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)

=
e−φ3(1−x+y)

(
φ′3
)2 (

1 +
∑2

i=1 O
(
e(−φj +φ3)(1−x)

φ
′

j

φ′
3

))

e−φ3 φ′3

(
1 +

∑2
j=1 O

(
e(−φj +φ3)

φ′
j

φ′
3

))

×



1 +
2∑

j=1

O

(
e(−φj +φ3)y

φ′j

φ′3

)



= eφ3(x−y)φ
′
3

(
1 +

2∑

i=1

O
(
e(−φj +φ3)y

)
+

2∑

i=1

O
(
e(−φj +φ3)(1−x)

))
. (2.27)

Therefore taking the asymptotics (2.12)-(2.14) into account we find that

F̂1(x, y, ξ) = O
(
ξ−

2

3 e−C
3
√

|ξ|(x−y)
)

(2.28)

for ξ ∈ D0, |ξ| > 1, x > y. Also from (2.27) we get

F̂2(x, y, ξ) = △̃(ξ, y − x) −
△̃(ξ, 1 − x)△̃(ξ, y)

△̃(ξ, 1)
=

2∑

j=1

(
φ′j e

−φj (y−x)

+O
(
φ
′
3e

− Re φj y+Re φ3x
)

+ O
(
φ
′
3e

− Re φj (1−x)+Re φ3(1−y)
))

= O
(
ξ−

2

3 e−C
3
√

|ξ|(y−x)
)

(2.29)

for ξ ∈ D0, |ξ| > 1, y > x. Thus there exist the inverse Laplace transforms for the functions
F̂1(x, y, ξ) and F̂2(x, y, ξ). Taking the inverse Laplace transformation of (2.23) and (2.24) we obtain

F1(x, y, t) = L−1
{
F̂1(x, y, ξ)

}
= − 1

2πi

∫

Γ0

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

and

F2(x, y, t) = L−1
{
F̂1(x, y, ξ)

}
= − 1

2πi

∫

Γ0

dξeξt
△̃(ξ, 1 − x)△̃(ξ, y) − △̃(ξ, y − x)△̃(ξ, 1)

△̃(ξ, 1)
,



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Korteweg-de Vries-Burgers Equation ... 51

where Γ0 = ∂D0, i.e.

Γ0 = (−i∞, −i0) ∪
[
−i0, 4

27
− i0

]
∪
[

4

27
+ i0, i0

]
∪ (i0, i∞) .

Since functions F̂l(x, y, ξ) are symmetric with respect to φ1,φ2,φ3,using the relations φ1 (ξ) =

φ2
(
ξ
)
, φ2 (ξ) = φ1

(
ξ
)
, φ3 (ξ) = φ3

(
ξ
)

for all ξ ∈ D0 we can change the contour of integration Γ0
to the imaginary axis (−i∞, i∞) to get

F1(x, y, t) = −
1

2πi

∫ i∞

−i∞

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

and

F2(x, y, t) = −
1

2πi

∫ i∞

−i∞

dξeξt
△̃(ξ, 1 − x)△̃(ξ, y) − △̃(ξ, y − x)△̃(ξ, 1)

△̃(ξ, 1)
.

Therefore taking inverse Laplace transform of (2.22) with respect to ξ we obtain

u(x, t) = Gu0.

Thus by Duhamel principle Proposition is proved.

Lemma 2.2. We have the asymptotics for large time

Fj (x, y, t) = −e−ξ0tΛ (x) △̃(−ξ0, y) + O
(
e−(ξ0+δ)t

)
(2.30)

and estimates

|∂nx Fj (x, y, t)| ≤ Ce−ξ0t {t}
−α |x − y|2α−1−n (2.31)

for x, y ∈ (0, 1) , x 6= y, t > 0, where α ∈
[
0, n+1

2

]
, n = 0, 1, j = 1, 2.

Proof. We consider a curve in the complex left-half plane Re ξ < 0 such that Re φ1 (ξ) = 0, it is

defined by the equation (iy)
2 − (iy)3 = ξ with y = Im φ1 (ξ) . Therefore there exists a contour

C0 =
{
ξ ∈ C, Re ξ < 0 : Re ξ = O

(
|ξ|

2

3

)}

such that

Re φl(ξ) > 0, l = 1, 2, Re φ3(ξ) < 0 for all ξ ∈ C0.

We also consider a contour

C1 = (−ξ0 − δ − i∞, −ξ0 − δ − i0) ∪ (−ξ0 − δ − i0, −i0)
∪ (i0, −ξ0 − δ + i0) ∪ (−ξ0 − δ + i0, −ξ0 − δ + i∞) .

Denote

C0 ∩ C1 = {z1, z2} , Im z1 > 0, Im z2 < 0, Re zl = −ξ0 − δ, l = 1, 2.



52 E.I. Kaikina et. al. CUBO
12, 1 (2010)

We now define a contour C = C2 ∪ C3, where

C2 = {ξ ∈ C1 : Im z2 ≤ Im ξ ≤ Im z1} .

C3 = {ξ ∈ C0 : Im ξ > Im z1 or Im ξ < Im z2} .

Note that the asymptotic formulas (2.12)-(2.14) are valid on the contour C. In view of them we
have (2.25)-(2.29) for ξ ∈ C. Therefore changing the contour of integration to C we obtain

F1(x, y, τ ) = −
1

2πi

∫

ξ∈C2

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

− 1
2πi

∫

ξ∈C3

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ. (2.32)

Since △̃(x + i0, q) = △̃(x − i0, q) we get

− 1
2πi

∫ −i0

−ξ
0
−δ−i0

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

−
1

2πi

∫ −ξ
0
−δ+i0

+i0

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

= −e−ξ0t △̃(−ξ0, 1 − x)△̃(−ξ0, y)
△̃′(−ξ0, 1)

. (2.33)

Also by (2.25) we have

1

2πi

∫ z1

−ξ
0
−δ+i0

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

+
1

2πi

∫ −ξ
0
−δ−i0

z2

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

= O
(
e−(ξ0+δ)t

)
. (2.34)

Taking into account (2.28) we get

∣∣∣∣∣
− 1

2πi

∫

ξ∈C3

eξt
1

△̃(ξ, 1)
△̃(ξ, 1 − x)△̃(ξ, y)dξ

∣∣∣∣∣

< Ce−(ξ0+δ)t
∫

ξ∈C3

e−Ct|ξ|
2

3 +t(ξ
0
+δ)−C|x−y||ξ|

1

3 |ξ|−
2

3 dξ

< Ce−(ξ0+δ)tt−α |x − y|2α−1 (2.35)

since C |ξ|
2

3 − (ξ0 + δ) ≥ 0 for ξ ∈ C3, where α ∈
[
0, 1

2

]
. By (2.33)-(2.35) we have from (2.32)

F1(x, y, t) = −e−ξ0t
△̃ (−ξ0, 1 − x) △̃ (−ξ0, y)

△̃′ (−ξ0, 1)
+ O

(
e−(ξ0+δ)t

)



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Korteweg-de Vries-Burgers Equation ... 53

for x, y > 0, t ≥ 1, and moreover

|F1(x, y, t)| ≤ Ce−ξ0t
(
1 + {t}−α |x − y|2α−1

)

for all x, y ∈ (0, 1) , x 6= y, t > 0, where α ∈
[
0, 1

2

]
. Thus the result of the lemma is true for the

case n = 0. Consider the case n = 1. In view of the asymptotic formulas (2.12)-(2.14) we get

∂x△̃(ξ, 1 − x)△̃(ξ, y)
△̃(ξ, 1)

=

(∑3
j=1 e

−φj (ξ)(1−x)
(
φ′j φj

))(∑3
j=1 e

−φj (ξ)yφ′j

)

∑3
j=1 e

−φj (ξ)φ′j
= O(1) (2.36)

and

∂x△̃(ξ, y − x) = O(1) (2.37)
for |ξ| < 1, ξ ∈ C2 and in the same argument as in the proof of the estimate (2.25) we get

∂x△̃(ξ, 1 − x)△̃(ξ, y)
△̃(ξ, 1)

= e−φ3(y−x)φ3φ
′
3

(
1 + O

(
e−C

3
√

|ξ|y
)

+ O
(
e−C

3
√

|ξ|(1−x)
))

(2.38)

and

∂x△̃(ξ, y − x) = e−φ3(y−x)φ3φ
′
3

(
1 + O

(
e−C

3
√

|ξ|y
)

+ O
(
e−C

3
√

|ξ|(1−x)
))

(2.39)

for all |ξ| > 1, ξ ∈ C3. Hence by the similar way to (2.33)-(2.35) we get

|∂xF1(x, y, t)| ≤ e−ξ0t
∣∣∣∣∣
△̃(−ξ0, y)∂x△̃(−ξ0, 1 − x)

△̃′(−ξ0, 1)

∣∣∣∣∣
+ Ce−ξ0t

+Ce−(ξ0+δ)t
∫

ξ∈C3

e−Ct|ξ|
2

3 +t(ξ
0
+δ)−C|x−y||ξ|

1

3 |ξ|−
1

3 dξ

≤ e−ξ0t
(
C + C {t}−α |x − y|2α−2

)

for all x, y ∈ (0, 1) , x 6= y, t > 0, where α ∈ [0, 1] .The function F2(x, y, t) is considered in the same
way. Lemma 2.2 is proved.

Now we prove the local existence for the linear problem (2.1).

Theorem 2.3. Let the initial data u0 ∈ L2 and f ∈ L2. Then for any T > 0 there exists a unique
solution u ∈ C

(
[0, T ] ; L2

)
∪ C

(
(0, T ] ; H1

)
of the linear initial-boundary value problem (2.1) such

that

sup
t∈(0,T ]

tα ‖∂nx u (t)‖L2 ≤ Cλ

provided that

λ = ‖u0‖L2 + T
1−β sup

t∈(0,T ]

tβ ‖f (t)‖
L2

< ∞,

where n = 0, 1, α ∈
(

n
2
, 1
)
, β ∈ (0, 1) .



54 E.I. Kaikina et. al. CUBO
12, 1 (2010)

Proof. ¿From Proposition and estimates of Lemma 2.2 we have

‖∂nx u (t)‖L2 ≤ t
−α

∥∥∥∥
∫ 1

0

u0(y) |x − y|2α−1−n dy
∥∥∥∥
L2

+

∥∥∥∥
∫ t

0

τ −α (t − τ )−β dτ
∫ 1

0

(t − τ )β f (y, t − τ ) |x − y|2α−1−n dy
∥∥∥∥
L2

≤ Ct−α
(
‖u0(y)‖L2 + T

1−β sup
t∈(0,T ]

tβ ‖f (·, t)‖
L2

)

for n = 0, 1. So we have the estimate of the theorem. Theorem 2.3 is proved.

3 Local Existence for the Nonlinear Problem

In this section we prove the following result.

Theorem 3.1. Suppose that the initial data u0(x) ∈ L2. Then there exists a unique solution

u(x, t) ∈ C ([0, T ] ; L∞) ∪ C
(
(0, T ] ; H1

)

where T > 0 depends on ‖u0‖L2 .

Proof. We prove the local existence of solutions by the contraction mapping principle. Let u(x, t)

be a solution of the following linear problem




ut + N(w) − uxx + uxxx = 0, t > 0, x ∈ (0, 1) ,
u(x, 0) = u0(x), x ∈ (0, 1) ,

u(0, t) = u(1, t) = ux(1, t) = 0, t > 0,

(3.1)

where N(w) = wwx, w is taken from the closed ball

H
1
ρ =

{
w ∈ C

(
(0, T ] ; H1

)
; sup
t∈(0,T ]

1∑

n=0

tαn‖∂nx w‖L2 ≤ ρ
}

,

where αn ∈
(

n
2
, 1
)

and satisfies the boundary condition w(0, t) = w(1, t) = wx(1, t) = 0. The

initial-value problem (3.1) defines a mapping M by u = M(w) and we will show that M is the

contraction mapping from H1ρ into itself for a sufficiently small T > 0. Since w ∈ H1ρ, using the
estimate ‖w‖

L∞
≤ 2 ‖w‖

L2
‖wx‖L2 , we have

sup
t∈(0,T ]

tβ ‖N(w)(t)‖
L2

≤ C sup
t∈(0,T ]

tβ ‖w(t)‖
1

2

L2
‖wx(t)‖

3

2

L2
≤ Cρ2,

where β = α0+3α1
2

< 1. Via Theorem 2.3 problem (3.1) has a unique solution u(x, t) ∈ C
(
(0, T ] ; H1

)
,

such that

sup
t∈(0,T ]

tαn ‖∂nx u‖L2 ≤ Cλ,



CUBO
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Korteweg-de Vries-Burgers Equation ... 55

where

λ = ‖u0‖L2 + T
1−β sup

t∈(0,T ]

tβ ‖N(w)(t)‖
L2

.

Therefore we obtain the estimate

sup
t∈(0,T ]

1∑

n=0

tαn‖∂nx u‖L2 ≤ C‖u0‖L2 + CT 1−βρ2 ≤ ρ (3.2)

if T = (2Cρ)
− 1

1−β =
(
4C2 ‖u0‖L2

)− 1
1−β and C ‖u0‖L2 ≤

ρ
2
. Thus the mapping M transforms the

closed ball H1ρ with a center at the origin and a radius ρ into itself. Analogously we can estimate

the difference

sup
t∈(0,T ]

1∑

n=0

tαn ‖∂nx (u − ũ) ‖L2 ≤
1

2
sup

t∈(0,T ]

1∑

n=0

tαn ‖∂nx (w − w̃) ‖L2

for sufficiently small T > 0. Therefore the mapping M is a contraction mapping in H1ρ and there

exists a unique solution u(x, t) ∈ C
(
(0, T ] ; H1

)
of the initial-value problem (1.1). Theorem 3.1 is

proved.

Remark 3.1. By virtue of estimate (3.2) we see that if the initial data u0 are small, i.e. the norm

‖u0‖L2 ≤ ε, where ε > 0 is sufficiently small, then there exists T ≥ 1, such that there exists a
unique solution u, which is also small: supt∈(0,T ]

∑1
n=0 t

αn ‖∂nx u‖L2 < Cε.

4 Large Time Asymptotics

In this section we give sufficient conditions for global existence of solutions to the initial-boundary

value problem (1.1) with small initial data

‖u0‖L2 < ε1, (4.1)

where ε1 > 0 is sufficiently small.

Theorem 4.1. Suppose that the initial data u0 ∈ L2 and satisfy (4.1). Then there exists a unique
solution u of (1.1) such that

u ∈ C
(
[0, ∞) ; L2

)
∪ C

(
(0, ∞) ; H1

)
.

Furthermore the solution has the following asymptotics

u(x, t) = AΛ(x)e−ξ0t + O
(
e−(ξ0+δ)t

)
(4.2)

for t → ∞ uniformly with respect to x ∈ (0, 1) , where ξ0 > 0, δ > 0, the constant A, the function
Λ(x) ∈ L∞ are defined below in the proof.



56 E.I. Kaikina et. al. CUBO
12, 1 (2010)

Proof. According to Theorem 3.1 and Remark 3.1 we see that after some time T ≥ 1 the solution
is small in the norm H1. Therefore we can only consider the initial-boundary problem (1.1) for

t ≥ 1 with small initial data u (x, 1) such that ‖u (·, 1)‖
H1

≤ ε2, where ε2 > 0 is sufficiently small.
Let us prove that

eξ0t ‖u(t)‖
H1

< ε (4.3)

for all t ≥ 1, with some small ε > 0. By the contradiction we can find a maximal interval [1, T1]
such that the estimate

eξ0t ‖u(t)‖
H1

≤ ε (4.4)

is true for all t ∈ [1, T1] and estimate (4.3) is violated at time t = T1. From (4.4) and estimates
(2.31) of Lemma 2.2 we obtain for n = 0, 1

∥∥∥u(n)x (·, t)
∥∥∥
L2

≤ Ce−ξ0t ‖u (·, 1)‖
L2

+

∫ t

1

dτ

∫ x

0

|uuy(y, τ )| ‖∂nx F1 (·, y, t − τ )‖L∞ dy

+

∫ t

1

dτ

∫ 1

x

|uuy(y, τ )| ‖∂nx F2 (·, y, t − τ )‖L∞ dy

≤ Cε2e−ξ0t + Ce−ξ0t
∫ t

1

eξ0τ ‖u(τ )‖
1

2

L2
‖ux(τ )‖

3

2

L2
{t − τ}−

3

4 dτ

≤ Cε2e−ξ0t + Cε2e−ξ0t
∫ t

1

e−ξ0τ {t − τ}−
3

4 dτ ≤ C
(
ε2 + ε

2
)
e−ξ0t

for t ∈ [1, T1] . The contradiction obtained proves (4.3). Now using the estimate (4.3) and Lemma
2.2 we prove that the solution has asymptotics (4.2) for t → ∞ uniformly with respect to x > 0,
where

Λ(x) = −
△̃(−ξ0, 1 − x)
△̃′(−ξ0, 1)

,

A =

∫ 1

0

u0(y)△̃(−ξ0, y)dy +
∫ ∞

0

eξ0τ dτ

∫ 1

0

u(y, τ )uy(y, τ )△̃(−ξ0, y)dy.

Indeed, due to asymptotics (2.30) of Lemma 2.2 we have

u(x, t) = AΛ(x)e−ξ0t + R(x, t), (4.5)

where in view of (4.3) we have

|R(x, t)| ≤ e−(ξ0+δ)t ‖u (·, 1)‖
H1

+ C

∫ t

1

e−(ξ0+δ)(t−τ )dτ

∫ 1

0

|uuy| dy

+e−ξ0t |Λ (x)|
∫ ∞

t

eξ0τ dτ

∫ 1

0

|uuy|
∣∣∣△̃(−ξ0, y)

∣∣∣dy

= O
(
e−(ξ0+δ)t

)

for all t ≥ 1, where δ > 0. Theorem 4.1 is proved.



CUBO
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Korteweg-de Vries-Burgers Equation ... 57

5 Large Initial Data

We consider the initial-boundary value problem (1.1) with any initial data ‖u0‖L2 ≤ C. Multiplying
equation (1.1) by u and integrating with respect to x ∈ (0, 1) we get

d

dt
‖u‖2

L2
+ 2

∫ 1

0

(
u2ux − uuxx + uuxxx

)
dx = 0.

We have ∫ 1

0

u2uxdx =
1

3
u3
∣∣∣∣
1

0

= 0,

∫ 1

0

uuxxdx = uux|10 −
∫ 1

0

u2xdx = −
∫ 1

0

u2xdx,

∫ 1

0

uuxxxdx = uuxx|10 −
1

2
u2x
∣∣1
0

=
1

2
u2x (0, t)

in view of the boundary data, hence

d

dt
‖u‖2

L2
+ 2

∫ 1

0

u2xdx + u
2
x (0, t) = 0.

Integration with respect to t > 0 yields

‖u (t)‖
L2

+ 2

∫ t

0

‖ux (τ )‖2L2 dτ ≤ ‖u0‖L2

for all t ∈ (0, ∞) . We see that the norm ‖u (t)‖
L2

≤ ‖u0‖L2 for all t ≥ 0. By the standard
continuation process via Theorem 3.1 we obtain that there exists a unique global solution u ∈
C
(
(0, ∞) ; H1

)
since the existence time T depends only on ‖u0‖L2 . Moreover we see that for

any ε > 0 there exists a time T > 0 such that ‖ux (T )‖2L2 < ε. By the inequality u2 (x, T ) =
2
∫ x
0

uuydy ≤ 2 ‖u‖L2 ‖ux‖L2 we see that the norm ‖u (T )‖L∞ , and hence the norm ‖u (T )‖L2 , are
also small by the estimate ‖u (T )‖

L2
≤ ‖u (T )‖

L∞
. Then we consider the initial-boundary value

problem (1.1) for t ≥ T and apply Theorem 4.1 whence the result of Theorem 1.1 follows. Theorem
1.1 is proved.

Acknowledgments. The work of One of the authors (E.I.K.) is partially supported by

CONACYT.

Received: September, 2008. Revised: October, 2009.

References

[1] Aikawa, H. and Hayashi, N., Holomorphic solutions of semilinear heat equations, Complex

Variables, 16 (1991), pp. 115–125.



58 E.I. Kaikina et. al. CUBO
12, 1 (2010)

[2] Aikawa, H., Hayashi, N. and Saitoh, S., The Bergman space on a sector and the heat

equation, Complex Variables, 15 (1990), pp. 27–36.

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