The Korteweg–de Vries–Kawahara equation in a bounded domain and some numerical results

Applied Mathematics and Computation 190 (2007) 912–936

www.elsevier.com/locate/amc

The Korteweg–de Vries–Kawahara equationin a bounded domain and some numerical results

Juan Carlos Ceballos a, Mauricio Sepulveda b,*, Octavio Paulo Vera Villagran a

a Departamento de Matematica, Universidad del Bıo-Bıo, Collao 1202, Casilla 5-C, Concepcion, Chileb Departamento de Ingenierıa Matematica, Facultad de Ciencias Fısicas y Matematicas,

Universidad de Concepcion, Casilla 160-C, Concepcion, Chile

Abstract

We are concerned with the initial-boundary problem associated to the Korteweg–de Vries–Kawahara perturbed by adispersive term which appears in several fluids dynamics problems. We obtain local smoothing effects that are uniform withrespect to the size of the interval. We also propose a simple finite different scheme for the problem and prove its uncon-ditional stability. Finally we give some numerical examples.� 2007 Elsevier Inc. All rights reserved.

Keywords: Evolution equations; Gain in regularity; Sobolev space; Numerical methods

1. Introduction

We consider the non linear problem for the Korteweg–de Vries–Kawahara (KdV–K) equation

0096-3

doi:10

* CoE-m

yahoo

ut þ guxxxxx þ uxxx þ uux þ ux ¼ 0; x 2 ½0; L½; t 2 ½0; T ½; ð1:1Þuð0; tÞ ¼ g1ðtÞ; uxð0; tÞ ¼ g2ðtÞ; t 2 ½0; T ½; ð1:2ÞuðL; tÞ ¼ 0; uxðL; tÞ ¼ 0; uxxðL; tÞ ¼ 0; t 2 ½0; T ½; ð1:3Þuðx; 0Þ ¼ u0ðxÞ; ð1:4Þ

where u ¼ uðx; tÞ is a real-valued function, g 2 R. Eq. (1.1)–(1.4) is a version of the Benney–Lin equation

ut þ guxxxxx þ bðuxxxx þ uxxÞ þ uxxx þ uux ¼ 0 ð1:5Þ
with �1 < x < þ1, t 2 ½0; T �, T is an arbitrary positive time, b > 0 and g 2 R.
The above equation is a particular case from a Benney–Lin equation derived by Benney [1] and later by Lin[2] (see also [3,4,5] and references therein). It describes one dimensional evolutions of small but finite ampli-

003/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

.1016/j.amc.2007.01.107

rresponding author.ail addresses: [email protected] (J.C. Ceballos), [email protected] (M. Sepulveda), [email protected], octavipaulov@

.com (O.P. Vera Villagran).

mailto:[email protected]





J.C. Ceballos et al. / Applied Mathematics and Computation 190 (2007) 912–936 913

tude long waves in various problems in fluid dynamics. This also can be seen as an hybrid of the well knownfifth order Korteweg–de Vries (KdV) equation or Kawahara equation. In 1997, Biagioni and Linares [6] moti-vated by the results obtained by Bona et al. [7] showed that the initial value problem (1.5) is globally well-posed in H sðRÞ, s P 0. The initial value problem (1.5) has been studied in the last few years, see for instancefor comprehensive descriptions of results pertaining to the KdVK equation [6,8] and references therein.

Guided by experimental studied on water waves in channels [9,10,11], Bona and Winter [12,13] consideredthe Korteweg de Vries equation

ut þ uxxx þ uux þ ux ¼ 0 for x; t P 0; ð1:6Þuðx; 0Þ ¼ f ðxÞ for x P 0; ð1:7Þuð0; tÞ ¼ gðtÞ for t P 0 ð1:8Þ

and proved that such that a quarter plane problem is well posed. See [14,15,16] for theory involving nonlin-earities having more general form. Bona and Bryant [9] had studied the same quarter plane problem for theregularized long-wave equation

ut þ ux þ uux � uxxt ¼ 0 ð1:9Þ

and proved it to be well posed. In [17,18], Bona and Luo studied an initial and boundary-value problem fromthe nonlinear wave equation

ut þ uxxx þ ½P ðuÞ�x ¼ 0 ð1:10Þ
in the quarter plane fðx; tÞ : x P 0 and t P 0g with the initial data and boundary data specified at t ¼ 0 andon x ¼ 0, respectively. With suitable restrictions on P and with conditions imposed on the initial data andboundary data which are quite reasonable with regard to potential applications, the aforementioned initialboundary value problem for (1.10) is show to be well posed.
In this work, motivated by the results obtained by Colin and Gisclon [19] we show that the Korteweg deVries Kawahara equation has smoothing effects that are uniform with respect to the size of the interval and wealso to propose a simple finite different scheme for the problem and prove its stability This paper is organizedas follows: In Section 2 outlines briefly the notation and terminology to be used subsequently and presents astatement of the principal result. In Section 3 we consider the linear problem and we will find a priori esti-mates. In Section 4 we obtain estimates that are independent of L for the non-homogeneous linear system.In section 5 we prove the existence of a time T min depending only on kgkH1ð0;T Þ and ku0kL2ðð1þx2Þ dxÞ but noton L such that uL exists on ½0; T min� thanks to uniform (with respect to L) smoothing effects. In Section 6we present a finite different scheme for the initial boundary value problem (KdVK); we prove its stabilityand present some numerical experiments. The paper concludes with some commentary concerning aspectsnot covered in the present study. Our main result reads as follows

Theorem 1.1 (Existence and Uniqueness). Let g 6 0, w0 2 L2ðð1þ x2ÞdxÞ, g 2 H 1locðRþÞ and 0 < L < þ1.

Then there exists a unique weak maximal solution defined over ½0; T L� to ðRÞ. Moreover, there exists T min > 0independent of L, depending only on kw0kL2ð0;LÞ and kgkH1ð0;T Þ such that T L P T min. The solution w depends

continuously on w0 and g in the following sense: Let a sequence wn0 ! w0 in L2ðð1þ x2ÞdxÞ, let a sequence gn ! g

in H 1locðRþÞ and denote by wn the solution with data ðwn

0; gnÞ and T n

L its existence time. Then

lim infn!þ1

T nL P T L

and for all t < T L, wn exists on the interval ½0; T � if n is large enough and un ! u in HT.

2. Preliminaries

Let X a bounded domain in R. For any real p in the interval ½1;1�, LpðXÞ denotes the collection of real-valued Lebesgue measurable pth-power absolutely integrable functions defined on X. As usual, L1ðXÞ denotesthe essentially bounded real-valued functions defined on X. These spaces get their usual norms,

914 J.C. Ceballos et al. / Applied Mathematics and Computation 190 (2007) 912–936

kukLpðXÞ ¼Z

XjuðxÞjp dx

� �1=p

for 1 6 p < þ1

and

kukL1ðXÞ ¼ supx2X

essjuðxÞj:

For an arbitrary Banach space X, the associated norm will be denoted k � kX . The following spaces will inter-vene in the subsequent analysis. For any real p in the interval ½1;1� and �1 6 a < b 6 þ1 the notationLpða; b : X Þ denotes the Banach space of measurable functions u : ða; bÞ ! X whose norms are pth-power inte-grable (essentially bounded if p ¼ þ1). These spaces get their norms,

kukpLpða;b:X Þ ¼

Z b

akuð�; tÞkp

X dt for 1 6 p < þ1

and

kukL1ða;b:X Þ ¼ suppt2ða;bÞ

kuð�; tÞkX for p ¼ þ1:

In this paper, we assume that for 0 < L < þ1 the initial data u0 2 L2ð0; LÞ and that xu0 2 L2ð0; LÞ and weintroduce

ku0kL2ðð1þx2Þ dxÞ ¼Z L

0

u20ð1þ x2Þdx

� �1=2

:

We introduce the following spaces: E :¼ ff 2 L1ð0; T : L2ðð1þ x2ÞdxÞÞ;ffiffitp

f 2 L2ð0; T : L2ðð1þ x2ÞdxÞÞg.This space is endowed with the norm

kf kE ¼Z T

0

Z L

0

f 2ðx; tÞð1þ x2Þdx� �1=2

dt þZ T

0

Z L

0

t½f ðx; tÞ�2ð1þ x2Þdxdt� �1=2

¼Z T

0

kf kL2ðð1þx2Þ dxÞ dt þffiffitp

f��

L2ð0;T :L2ðð1þx2Þ dxÞÞ ¼ kf kL1ð0;T :L2ðð1þx2Þ dxÞÞ þffiffitp

f��

L2ð0;T :L2ðð1þx2Þ dxÞÞ:

Let T > 0

HT ¼ fu 2 Cð½0; T � : L2ðð1þ x2ÞdxÞÞ; ux 2 L2ð0; T : L2ðð1þ xÞdxÞÞffiffitp

ux 2 L2ð½0; T � : L2ðð1þ xÞdxÞÞ;ffiffitp

uxx 2 L2ð½0; T � : L2ð0; LÞÞg:

This space is endowed with the norm

kukH ¼ kukL1ð0;T :L2ðð1þx2Þ dxÞÞ þ kuxkL2ð0;T :L2ðð1þxÞ dxÞÞ þffiffitp

ux

�� L2ð0;T :L2ðð1þxÞ dxÞÞ þ

ffiffitp

uxx

�� L2ð0;T :L2ð0;LÞÞ:

Let X a bounded domain in R. If 1 6 p 6 þ1, and m P 0 is an integer, let W m;pðXÞ be the Sobolev space ofLpðXÞ-functions whose distributional derivatives up to order m also lie in LpðXÞ. The norm on W m;pðXÞ is

kukpW m;pðXÞ ¼

Xa6m

koaxukp

LpðXÞ:

The space C1ðXÞ ¼T

jP0CjðXÞ will be used, but its usual Frechet-space topology will not be needed. DðXÞ isthe subspace of C1ðXÞ of functions with compact support in X. Its dual space, D0ðXÞ, is the space of Schwartzdistributions on X. When p ¼ 2, W m;pðXÞ will be denoted by H mðXÞ. This is a Hilbert space, andH 0ðXÞ ¼ L2ðXÞ. The notation H1ðXÞ ¼

TjP0H jðXÞ will be used for the C1-functions on X, all of whose deriv-

atives lie in L2ðXÞ. Finally, H mlocðXÞ is the set of real-valued functions u defined on X such that, for each

u 2 DðXÞ, uu 2 H mðXÞ. This space is equipped with the weakest topology such that all of the mappingsu! uu, for u 2 DðXÞ, are continuous from H m

locðXÞ into H mðXÞ. With this topology, H mlocðXÞ is a Frechet

space. Let Rþ denote the positive real numbers ð0;1Þ. A simple but pertinent example of the localized Sobolevspace is H m

locðRþÞ. Interpreting the foregoing definitions in this special case, u 2 H mlocðRþÞ if and only if


u 2 H mð0; T Þ, for all finite T > 0. Moreover, un ! u in HmlocðRþÞ if and only if un ! u in H mð0; T Þ, for each

T > 0.We consider the inhomogeneous initial value problem

ouot ðtÞ ¼ AuðtÞ þ f ðtÞ; t > 0;

uð0Þ ¼ x;

�

where f : ½0; T � ! X and A is the infinitesimal generator of a C0 semigroup SðtÞ so that the correspondinghomogeneous equation, i. e., the equation with f � 0, has a unique solution for every initial value x 2 DðAÞ.

By c, a generic constant, not necessarily the same at each occasion, which depend in an increasing way onthe indicated quantities.

The following result is going to be used several times in the rest of this paper.

Lemma 2.1. We have the following inequalities:

kuk2L2ðð1þxÞ dxÞ 6 3kuk2

L2ðð1þx2Þ dxÞ; ð2:1Þ

kukL1ð0;T Þ 6 ðcþffiffiffiffiTpÞkukH1ð0;T Þ; ð2:2Þ

kukL1ð0;T :L2ðð1þx2Þ dxÞÞ 6ffiffiffiffiTpkukL2ð0;T :L2ðð1þx2Þ dxÞÞ ð2:3Þ

and, if u 2 H 1ð0; LÞ, uð0Þ ¼ 0 then

kukL1ð0;LÞ 6 2kukL2ð0;LÞkuxkL2ð0;LÞ: ð2:4Þ

3. Uniform estimates on the solutions to the linear homogeneous problem

We consider the linear problem for the Korteweg de Vries Kawahara equation

ut þ guxxxxx þ uxxx þ ux ¼ 0; x 2 ½0; L½; t 2 ½0; T ½; ð3:1Þuð0; tÞ ¼ 0; uxð0; tÞ ¼ 0; t 2 ½0; T ½; ð3:2ÞuðL; tÞ ¼ 0; uxðL; tÞ ¼ 0; uxxðL; tÞ ¼ 0; t 2 ½0; T ½; ð3:3Þuðx; 0Þ ¼ u0ðxÞ; ð3:4Þ

where u ¼ uðx; tÞ is a real-valued function, g 2 R.

Lemma 3.1. Let u0 2 L2ð0; LÞ and g < 0. Then there exists a continuous function t! cðtÞ such that

kukL1ð0;T :L2ð0;LÞÞ 6 cðT Þku0kL2ð0;LÞ; ð3:5Þkuxxð0; �ÞkL2ð0;T Þ 6 cðT Þku0kL2ð0;LÞ: ð3:6Þ

Proof. Multiplying (3.1) by u and integrating over x 2 ð0; LÞ we have

Z L

0

uut dxþ gZ L

0

uuxxxxx dxþZ L

0

uuxxx dxþZ L

0

uux dx ¼ 0: ð3:7Þ

Each term is treated separately integrating by parts
Z L
0

uut dx ¼ 1

2

d

dt

Z L

0

u2 dx; gZ L

0

uuxxxxx dx ¼ � 1

2gu2

xxð0; tÞ;Z L

0

uuxxx dx ¼ 0;

Z L

0

uux dx ¼ 0:

Replacing in (3.7) we have

1

2

d

dt

Z L

0

u2 dx� 1

2gu2

xxð0; tÞ ¼ 0


then

d

dtkuk2

L2ð0;LÞ � gu2xxð0; tÞ ¼ 0: ð3:8Þ

Integrating (3.8) in t 2 ð0; T Þ we have

kuk2L2ð0;LÞ � g

Z t

0

u2xxð0; sÞds ¼ ku0k2

L2ð0;LÞ: ð3:9Þ

Then, using g < 0, the Gronwall inequality and straightforward calculus we have

kuk2L2ð0;LÞ � gkuxxð0; �Þk2

L2ð0;T Þ ¼ cðT Þku0k2L2ð0;LÞ ð3:10Þ

and (3.5,3.6) follows. h


kuxkL2ð0;T :L2ð0;LÞÞ 6 cðT Þku0kL2ð0;LÞ; ð3:11ÞkuxxkL2ð0;T :L2ð0;LÞÞ 6 cðT Þku0kL2ð0;LÞ; ð3:12ÞkukL1ð0;T :L2ðð1þxÞ dxÞÞ 6 cðT Þku0kL2ð0;LÞ: ð3:13Þ

Remark 3.1. From (3.12) we obtain that if u0 2 L2ð0; LÞ then u 2 H 1ð0; LÞ. This means that we have a gain oftwo derivatives in regularity, while in the Korteweg–de Vries equation only one derivative is gained.

Proof. Multiplying Eq. (3.1) by xu and integrating over x 2 ð0; LÞ we have
Z L
0

xuut dxþ gZ L

0

xuuxxxxx dxþZ L

0

xuuxxx dxþZ L

0

xuux dx ¼ 0: ð3:14Þ

Each term in (3.14) is treated separately
Z L
0

xuut dx ¼ 1

2

d

dt

Z L

0

xu2 dx; gZ L

0

xuuxxxxx dx ¼ � 5

2gZ L

0

u2xx dx;Z L

0

xuuxxx dx ¼ 3

2

Z L

0

u2x dx;

Z L

0

xuux dx ¼ � 1

2

Z L

0

u2 dx:

Hence, replacing in (3.14) we have

d

dt

Z L

0

xu2 dx� 5gZ L

0

u2xx dxþ 3

Z L

0

u2x dx�

Z L

0

u2 dx ¼ 0

then

d

dt

Z L

0

xu2 dx� 5gZ L

0

u2xx dxþ 3

Z L

0

u2x dx ¼

Z L

0

u2 dx ¼ kuk2L2ð0;LÞ: ð3:15Þ

where using (3.5) we obtain

d

dt

Z L

0

xu2 dx� gZ L

0

u2xx dxþ 3

Z L

0

u2x dx 6 cðT Þku0k2

L2ð0;LÞ: ð3:16Þ

Integrating (3.16) in t 2 ð0; T Þ we have
Z L
0

xu2 dx� gZ t

0

Z L

0

u2xx dxdsþ 3

Z t

0

Z L

0

u2x dxds

6 cðT Þku0k2L2ð0;LÞ þ

Z L

0

xu20 dx 6 cðT Þku0k2

L2ð0;LÞ þ LZ L

0

u20 dx

¼ cðT Þku0k2L2ð0;LÞ þ Lku0k2

L2ð0;LÞ 6 cðT Þku0k2L2ð0;LÞ


then using that g < 0 and

kuk2L1ð0;T :L2ðx dxÞÞ � gkuxxk2

L2ð0;T :L2ð0;LÞÞ þ 3kuxk2L2ð0;T :L2ð0;LÞÞ 6 cðT Þku0k2

L2ð0;LÞ ð3:17Þ

we have that (3.11) and (3.12) follows. Moreover, adding (3.5) with the first term in (3.22), we obtain(3.13). h

Z L
0

u2x2 dx 6 cðT Þku0k2L2ð0;LÞ; ð3:18ÞZ T

0

Z L

0

u2xxdxdt 6 cðT Þku0k2

L2ð0;LÞ; ð3:19ÞZ T

0

Z L

0

u2xxxdxdt 6 cðT Þku0k2

L2ð0;LÞ; ð3:20Þ

kukL1ð0;T :L2ðð1þx2Þ dxÞÞ 6 cðT Þku0kL2ð0;LÞ; ð3:21ÞkuxkL2ð0;T :L2ðð1þxÞ dxÞÞ 6 cðT Þku0kL2ð0;LÞ: ð3:22Þ

Proof. Multiplying Eq. (3.1) by x2u and integrating over x 2 ð0; LÞ we have

Z L

0

x2uut dxþ gZ L

0

x2uuxxxxx dxþZ L

0

x2uuxxx dxþZ L

0

x2uux dx ¼ 0: ð3:23Þ

Each term is treated separately, integrating by parts
Z L
0

x2uut dx ¼ 1

2

d

dt

Z L

0

x2u2 dx; gZ L

0

x2uuxxxxx dx ¼ �5gZ L

0

xu2xx dx;Z L

0

x2uuxxx dx ¼ 3

Z L

0

xu2x dx;

Z L

0

x2uux dx ¼Z L

0

xu2 dx:

Hence, in (3.23) we have

1

2

d

dt

Z L

0

x2u2 dx� 5gZ L

0

xu2xx dxþ 3

Z L

0

xu2x dx�

Z L

0

xu2 dx ¼ 0

then

d

dt

Z L

0

x2u2 dx� 10gZ L

0

xu2xx dxþ 6

Z L

0

xu2x dx ¼ 2

Z L

0

xu2 dx 6 2LZ L

0

u2 dx

hence

d

dt

Z L

0

x2u2 dx� 10gZ L

0

xu2xx dxþ 6

Z L

0

xu2x dx 6 2L

Z L

0

u2 dx

then, using (3.5) we obtain

d

dt

Z L

0

x2u2 dx� 10gZ L

0

xu2xx dxþ 6

Z L

0

xu2x dx 6 2LcðT Þku0k2

L2ð0;LÞ: ð3:24Þ

Integrating (3.24) over t 2 ð0; T Þ we have
Z L
0

x2u2 dx� 10gZ t

0

Z L

0

xu2xx dxdsþ 6

Z t

0

Z L

0

xu2x dxds 6 2LcðT Þku0k2

L2ð0;LÞ þZ L

0

x2u20 dx

6 2LcðT Þku0k2L2ð0;LÞ þ L2

Z L

0

u20 dx 6 cðT Þku0k2

L2ð0;LÞ:


We obtain
Z L
0

x2u2 dx� 10gZ t

0

Z L

0

xu2xx dxdsþ 6

Z t

0

Z L

0

xu2x dxds 6 cðT Þku0k2

L2ð0;LÞ ð3:25Þ

and (3.18)–(3.20) follows. From (3.5) and (3.18) we obtain (3.21) and from (3.11) with (3.19) we have (3.22).The result follows. h

ffiffitp
uxxð0; �Þ��

L2ð0;T Þ 6 cðT Þku0kL2ð0;LÞ; ð3:26Þffiffitp

ux

�� L2ð0;T :L2ð0;T ÞÞ 6 cðT Þku0kL2ð0;LÞ; ð3:27ÞZ t

0

Z L

0

su2xxxdxds 6 cðT Þku0kL2ð0;LÞ; ð3:28Þffiffi

tp

u��

L1ð0;T :L2ð1þxÞ dxÞ 6 cðT Þku0kL2ð0;LÞ: ð3:29Þ

Proof. In (3.10) we have

d

dt

Z L

0

u2 dx� gu2xxð0; tÞ ¼ cðT Þku0kL2ð0;LÞ: ð3:30Þ

Multiplying (3.30) by t and integrating the resulting expression over t 2 ½0; T � we have
Z t
0

sd

ds

Z L

0

u2 dx� �

ds� gZ t

0

su2xxð0; sÞds 6 cðT Þku0kL2ð0;LÞ

then

tZ L

0

u2 dx� gZ t

0

su2xxð0; sÞds 6 cðT Þku0kL2ð0;LÞ þ kuk

2L2ð0;T :L2ð0;LÞÞ: ð3:31Þ

Using (3.5) we obtain
Z L
0

tu2 dx� gffiffitp

uxxð0; �Þ�� 2

L2ð0;T Þ 6 cðT Þku0k2L2ð0;LÞ ð3:32Þ

and (3.26) follows. Multiplying (3.16) by t and integrating the resulting expression over t 2 ½0; T � we have
Z t
0

sd

ds

Z L

0

xu2 dx� �

ds� gZ t

0

Z L

0

su2xx dxdsþ 3

Z t

0

Z L

0

su2x dxds 6 cðT Þku0k2

L2ð0;LÞ ð3:33Þ

hence

tZ L

0

xu2 dx� gffiffitp

uxx

�� 2

L2ð0;T :L2ð0;T ÞÞ þ 3ffiffitp

ux

�� 2

L2ð0;T :L2ð0;T ÞÞ 6 cðT Þku0k2L2ð0;LÞ ð3:34Þ

and (3.27) follows. Moreover, from (3.32) and (3.34) we obtain (3.29). h

Z T
0

Z L

0

su2xxdxdt 6 cðT Þku0kL2ð0;LÞ; ð3:35ÞZ T

0

Z L

0

su2xxxdxdt 6 cðT Þku0kL2ð0;LÞ; ð3:36Þffiffi

tp

ux

�� L2ð0;T :L2ð1þxÞ dxÞ 6 cðT Þku0kL2ð0;LÞ: ð3:37Þ


Proof. Multiplying (3.24) by t, and integrating by parts the first term we obtain

tZ L

0

x2u2 dx� 10gZ t

0

Z L

0

sxu2xx dxdsþ 6

Z t

0

Z L

0

sxu2x dxds 6 cðT Þku0k2

L2ð0;LÞ ð3:38Þ

and (3.35), (3.36) follows. From (3.27) and (3.35) we obtain (3.37). h

4. Non-homogeneous linear estimates

We consider the non-homogeneous linear problem for the Korteweg de Vries Kawahara equation

vt þ gvxxxxx þ vxxx þ vx ¼ f ðx; tÞ; x 2 ½0; L½; t 2 ½0; T ½; ð4:1Þvð0; tÞ ¼ 0; vxð0; tÞ ¼ 0; t 2 ½0; T ½; ð4:2ÞvðL; tÞ ¼ 0; vxðL; tÞ ¼ 0; vxxðL; tÞ ¼ 0; t 2 ½0; T ½; ð4:3Þvðx; 0Þ ¼ 0; ð4:4Þ

where v ¼ vðx; tÞ, g 2 R.

Lemma 4.1. Let f 2 E. Then there exists a continuous function t! cðtÞ such that

kvkL1ð0;T :L2ðð1þx2Þ dxÞÞ 6 cðT Þkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ: ð4:5Þ

Proof. Multiplying (4.1) by v, integrating over ð0; LÞ and performing similar calculus to (3.10) we have

d

dt

Z L

0

v2 dx� gvxxð0; tÞ 6 2

Z L

0

vf dx: ð4:6Þ

Multiplying (4.1) by x2v, integrating over ð0; LÞ and performing similar calculus to (3.30) we have

d

dt

Z L

0

x2v2 dx� 10gZ L

0

xv2xx dxþ 6

Z L

0

xv2x dx 6 2

Z L

0

x2vf dxþ 2LZ L

0

v2 dx: ð4:7Þ

Adding (4.6) with (4.7) we have

d

dt

Z L

0

v2ð1þ x2Þdx� 10gZ L

0

xv2xx dxþ 6

Z L

0

xv2x dx 6 2

Z L

0

vf ð1þ x2Þdxþ 2LZ L

0

v2 dx

6 2kvkL2ðð1þx2Þ dxÞkf kL2ðð1þx2Þ dxÞ þ 2LZ L

0

v2 dx: ð4:8Þ

Integrating over t 2 ð0; T Þ we obtain
Z L
0

v2ð1þ x2Þdx� 10gZ t

0

Z L

0

xv2xx dxdsþ 6

Z t

0

Z L

0

xv2x dxds

6 2

Z t

0

kvkL2ðð1þx2Þ dxÞkf kL2ðð1þx2Þ dxÞ dsþ 2LZ t

0

Z L

0

v2 dxds

6 2kvkL1ð0;T :L2ðð1þx2Þ dxÞÞkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ þ 2LZ t

0

Z L

0

v2 dxds:

Hence

kvkL1ð0;T :L2ðð1þx2Þ dxÞÞ � 10gZ t

0

Z L

0

xv2xx dxdsþ 6

Z t

0

Z L

0

xv2x dxds

6 2kvkL1ð0;T :L2ðð1þx2Þ dxÞÞkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ þ 2Lkvk2L2ð0;T :L2ð0;LÞÞ


thus using estimates as in Section 3; (3.5), (3.6), (3.12), (3.21) and straightforward calculus we obtain

kvkL1ð0;T :L2ðð1þx2Þ dxÞÞ � 10gZ t

0

Z L

0

xv2xx dxdsþ 6

Z t

0

Z L

0

xv2x dxds 6 cðT Þkf k2

L1ð0;T :L2ðð1þx2Þ dxÞÞ ð4:9Þ

and (4.5) follows. h


kvxkL2ð0;T :L2ðð1þxÞ dxÞÞ 6 ckf kL1ð0;T :L2ðð1þx2Þ dxÞÞ; ð4:10Þffiffitp

vx

�� L2ð0;T :L2ðð1þxÞ dxÞÞ 6 ckf kL1ð0;T :L2ðð1þx2Þ dxÞÞ: ð4:11Þ

Proof. From (4.1)–(4.4) we have (see [20] for the construction of this semigroup)

vðx; tÞ ¼Z t

0

Sðt � sÞf ðx; sÞds:

Differentiating in x-variable we have

vxðx; tÞ ¼Z t

0

oxSðt � sÞf ðx; sÞds; ð4:12Þ

applying k � kL2ðð1þxÞ dxÞ

kvxðx; tÞkL2ðð1þxÞ dxÞ 6

Z t

0

koxSðt � sÞf ðx; sÞkL2ðð1þxÞ dxÞ ds

multiplying by wðtÞ 2 L2ð0; T Þ

kvxðx; tÞkL2ðð1þxÞ dxÞwðtÞ 6Z t

0

koxSðt � sÞf ðx; sÞkL2ðð1þxÞ dxÞ ds� �

integrating over t 2 ð0; T Þ
Z T
0

kvxðx; tÞkL2ðð1þxÞ dxÞwðtÞdt 6Z T

0

Z t

0


wðtÞdt

applying j � j
Z T
0

kvxðx; tÞkL2ðð1þxÞ dxÞwðtÞdt

�� 6

Z T

0

Z t

0


jwðtÞjdt:

Using Fubini’s Theorem and the Cauchy–Schwartz inequality we have
Z T
0


�� 6

Z T

0

Z T

0

jwðtÞjkoxSðt � sÞf ðx; sÞkL2ðð1þxÞ dxÞ dt ds

6

Z T

0

Z T

0

jwðtÞj2 dt� �1=2 Z T

0

koxSðt � sÞf ðx; sÞk2L2ðð1þxÞ dxÞ dt

� �1=2

ds

6

Z T

0


0

Z T

0

koxSðt � sÞf ðx; sÞk2L2ðð1þxÞ dxÞ dt

� �1=2

ds

6 MZ T

0


0

Z T

0

kf ðx; sÞk2L2ðð1þxÞ dxÞ dt

� �1=2

ds

6 MZ T

0


0

kf ðx; sÞkL2ðð1þxÞ dxÞ

Z T

0

dt� �1=2

ds

6 MT 1=2

Z T

0


0

kf ðx; sÞkL2ðð1þxÞ dxÞ ds


6

ð2:1Þ ffiffiffi3p

MT 1=2

Z T

0


0

kf ðx; sÞkL2ðð1þx2Þ dxÞds

6 cZ T

0

jwðtÞj2 dt� �1=2

kf kL1ð0;T :L2ðð1þx2Þ dxÞÞ:

Therefore
Z T
0


�� 6 c

Z T

0

jwðtÞj2 dt� �1=2

kf kL1ð0;T :L2ðð1þx2Þ dxÞÞ: ð4:13Þ

If wðtÞ ¼ kvxðx; tÞkL2ðð1þxÞ dxÞ then
Z T
0

kvxðx; tÞk2L2ðð1þxÞ dxÞ dt

�� 6 c

Z T

0

kvxðx; tÞk2L2ðð1þxÞ dxÞ dt

� �1=2

kf kL1ð0;T :L2ðð1þx2Þ dxÞÞ

then

kvxðx; tÞk2L2ð0;T :L2ðð1þxÞ dxÞÞ 6 ckvxðx; tÞkL2ð0;T :L2ðð1þxÞ dxÞÞkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ:

Moreover, multiplying byffiffitp

and using similar calculus as in (4.10) we obtain (4.11). h

ffiffitp
vxx

�� L2ð0;T :L2ð0;LÞÞ 6 cðT Þkf kL1ð0;T :L2ð1þx2Þ dxÞ; ð4:14ÞZ T

0

Z L

0

tv2x dxdt 6 cðT Þkf kL1ð0;T :L2ð1þx2Þ dxÞ: ð4:15Þ

Proof. Multiplying (4.1) by xv and integrating over x 2 ½0; L� we have
Z L
0

xvvt dxþ gZ L

0

xvvxxxxx dxþZ L

0

vxxx dxþZ L

0

xvvx dx ¼Z L

0

xvf ðx; tÞdx

performing similar calculus and straightforward estimates as in (3.16) and using Lemma 2.1 (2.1), we have

d

dt

Z L

0

xv2 dx� gZ L

0

v2xx dxþ 3

Z L

0

v2x dx 6 cðT Þkf k2

L1ð0;T :L2ð1þx2Þ dxÞ: ð4:16Þ

Multiplying by t and using straightforward calculus we obtain

tZ L

0

xv2 dx� gkffiffitp

vxxk2L2ð0;T :L2ð0;LÞÞ þ 3

Z t

0

Z L

0

sv2x dxds 6 cðT Þkf k2

L1ð0;T :L2ð1þx2Þ dxÞ ð4:17Þ

the result follows. h


ffiffitp

v��

L1ð0;T :L2ðð1þx2Þ dxÞÞ 6 ckf kL1ð0;T :L2ðð1þx2Þ dxÞÞ; ð4:18ÞZ T

0

Z L

0

tv2xxdxdt 6 cðT Þkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ; ð4:19ÞZ T

0

Z L

0

tv2xxxdxdt 6 cðT Þkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ; ð4:20Þffiffi

tp

vx

�� L2ð0;T :L2ð1þxÞ dxÞ 6 cðT Þkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ; ð4:21Þ

kffiffitp

vxxkL2ð0;T :L2ð1þxÞ dxÞ 6 cðT Þkf kL1ð0;T :L2ðð1þx2Þ dxÞÞ: ð4:22Þ


Proof. From (4.8) we have

d

dt

Z L

0

v2ð1þ x2Þdx� 10gZ L

0

xv2xx dxþ 6

Z L

0

xv2x dx 6 2kvkL2ðð1þx2Þ dxÞkf kL2ðð1þx2Þ dxÞ þ 2L

Z L

0

v2 dx: ð4:23Þ

Multiplying by t, we perform similar calculus as in (4.9) and we integrate over t 2 ½0; T �
ffiffitp
v�� 2

L1ð0;T :L2ðð1þx2Þ dxÞÞ � 10gZ t

0

Z L

0

sxv2xx dxds

6

Z t

0

Z L

0

sxv2x dxds 6 cðT Þkf k2

L1ð0;T :L2ðð1þx2Þ dxÞÞ ð4:24Þ

and we obtain (4.18), (4.19) and (4.20). From (4.15) and (4.19) we have (4.21) and from (4.14) with (4.20) wehave (4.22). h

5. Nonlinear case

We now prove in this section a local existence result for the nonlinear system

ut þ guxxxxx þ uxxx þ uux þ ux ¼ 0; x 2 ½0; L½; t 2 ½0; T ½; ð5:1Þuð0; tÞ ¼ g0ðtÞ; uxð0; tÞ ¼ g1ðtÞ; t 2 ½0; T ½; ð5:2ÞuðL; tÞ ¼ 0; uxðL; tÞ ¼ 0; uxxðL; tÞ ¼ 0; t 2 ½0; T ½; ð5:3Þuðx; 0Þ ¼ u0ðxÞ; x 2 ½0; L�; ð5:4Þ

where u ¼ uðx; tÞ, g 2 R.Let ni be a smooth function defined over Rþ such that

nðLÞ ¼ nð1ÞðLÞ ¼ nð2ÞðLÞ ¼ 0 and nðkÞi ð0Þ ¼1; i ¼ k;

0; i 6¼ k:

�

Definition 5.1. A weak solution of (5.1)–(5.4) on ½0; T � is a function uðx; tÞ 2 HT such that

wðx; tÞ ¼ uðx; tÞ �X1

i¼0

niðxÞgiðtÞ ð5:5Þ

satisfy

wðx; tÞ ¼ SðtÞw0ðxÞ �X1

i¼0

Z t

0

½niosgi þ gnð5Þi gi þ nð3Þi gi þ nð1Þi gi þ ðwþ nigiÞðwx þ ½nð1Þi �giÞ�ds;

where w0ðxÞ ¼ u0ðxÞ �P1

i¼0niðxÞgið0Þ.

We consider the change of function (5.5) in ((5.1)–(5.4)). Hence this change of the function yields an equiv-alent problem, i.e., we transform the original problem into a problem with a Dirichlet boundary conditiongi ¼ 0 ði ¼ 0; 1Þ,

wt þ gwxxxxx þ wxxx þ wx ¼ �F ðw;wx; giÞ; x 2 ½0; L�; t 2 ½0; T ½; ð5:6Þwð0; tÞ ¼ 0; wxð0; tÞ ¼ 0; t 2 ½0; T ½; ð5:7ÞwðL; tÞ ¼ 0; wxðL; tÞ ¼ 0; wxxðL; tÞ ¼ 0; t 2 ½0; T ½; ð5:8Þ

wðx; 0Þ ¼ w0ðxÞ � u0ðxÞ � u0ðxÞ �X1

i¼0

niðxÞgið0Þ; x 2 ½0; L�; ð5:9Þ

where

F ðw;wx; gÞ �X1

i¼0

½niosgi þ gnð5Þi gi þ nð3Þi gi þ nð1Þi gi þ ðwþ nigiÞðwx þ nð1Þi giÞ�:


We write (5.6) as

wðx; tÞ ¼ SðtÞw0ðxÞ �Z t

0

Sðt � sÞF ðw;wx; giÞds; ð5:10Þ

where SðtÞ is the linear semigroup. We introduce the following functional C defined by

Cðw0; gi;wÞ ¼ SðtÞw0ðxÞ �Z t

0

Sðt � sÞF ðw;wx; giÞds:

Remark 5.1. We would like to construct a mapping C : HT ! HT with the following property: Givenn ¼ CðwÞ with kwkHT

6 R we have kCðwÞkHT¼ knkHT

6 R. In fact, this property tell us thatC : BRð0Þ ! BRð0Þ where BRð0Þ is a ball in the space HT . Then if we want to prove that there exists aunique solution u defined on HT , weak solution of ((5.1)–(5.4)), it is enough to apply the Banach’s fixed pointTheorem for u! Cðu0; gi; uÞ on BRð0Þ, (which is a complete metric space) which yields local existence anduniqueness.

Lemma 5.1. There exist a constant cðT Þ depending on T. independent of L such that for all u0 2 L2ð0; LÞ and

g < 0

kSðtÞu0kH 6 cðT Þku0kL2ð0;LÞ ð5:11Þ
and the map T ! cðT Þ is continuous.
Proof. Is a consequence of inequalities (3.21) and (3.22). h

For the non-homogeneous problem and using that vðx; tÞ ¼R t

0Sðt � sÞf ðx; sÞds, we have

Lemma 5.2. There exist a constant cðT Þ depending on T independent of L such that for all f 2 E and g < 0
Z t
0

Sðt � sÞf ðsÞds

��

��H

6 cðT Þkf kE ð5:12Þ

and the map T ! cðT Þ is continuous.

Proof. Is a consequence of inequalities (4.5), (4.10), (4.11) and (4.14). h

Lemma 5.3. For all u 2 H 1ð0; LÞ such that uð0Þ ¼ 0, we have
ffiffiffixp
u��

L1ð0;LÞ 6 5 kuk1=2

L2ð1þxÞkuxk1=2

L2ð1þxÞ þ kuxkL2ð1þxÞ

h i: ð5:13Þ

Theorem 5.1. Let g < 0. Suppose that w0; z0 2 L2ð0; T Þ, and gi and hi are in H 1locðRþÞ. Then there exists a con-

tinuous function t! cðtÞ such that for all T 2 ½0; T 0� we have

kCðw0; gi;wÞ � Cðz0; hi; zÞkH 6 cðT Þkw0 � z0kL2ð0;LÞ þ cðT ÞffiffiffiffiTp½kgi � hikH1ð0;T Þ þ kw� zkH þ kzkH�

þ cðT ÞffiffiffiffiTp

1þffiffiffiffiTp�

kw� zkHkwkH þ kw� zkHkwkHh i

ð5:14Þ

Proof. Let Cðw0; gi;wÞ ¼ SðtÞw0ðxÞ �R t

0Sðt � sÞF ðw;wx; giÞds then

Cðw0; gi;wÞ � Cðz0; hi; zÞ ¼ SðtÞðw0 � z0ÞðxÞ �Z t

0

Sðt � sÞ½F ðw;wx; giÞ � F ðz; zx; hiÞ�ds: ð5:15Þ

Let

KðsÞ ¼ F ðw;wx; giÞ � F ðz; zx; hiÞ¼ nin

ð1Þi ðg2

i � h2i Þ þ ½gnð5Þi þ nð3Þi þ nð1Þi �ðgi � hiÞ þ niosðgi � hiÞ

þ nð1Þi ðwgi � zhiÞ þ niðwxgi � zxhiÞ þ wwx � zzx

¼ K1 þ K2 þ K3;


where

K1ðsÞ ¼ ninð1Þi ðg2

i � h2i Þ þ ½gnð5Þi þ nð3Þi þ nð1Þi �ðgi � hiÞ þ niosðgi � hiÞ;

K2ðsÞ ¼ nð1Þi ðwgi � zhiÞ þ niðwxgi � zxhiÞ;K3ðsÞ ¼ wwx � zzx:

Hence in (5.15), using Lemmas 5.1 and 5.2 we have

kCðw0; gi;wÞ � Cðz0; hi; zÞkH6 cðT Þkw0 � z0kL2ð0;LÞ þ cðT ÞkKkE6 cðT Þkw0 � z0kL2ð0;LÞ þ cðT Þ kKkL1ð0;T :L2ðð1þx2Þ dxÞÞ þ k

ffiffitp

KkL2ð0;T :L2ðð1þx2Þ dxÞÞ

h i: ð5:16Þ

We estimate separately K1; K2; K3

kK1kL2ðð1þx2Þ dxÞ

6 kninð1Þi ðg2

i � h2i Þ þ ½gnð5Þi þ nð3Þi þ nð1Þi �ðgi � hiÞkL2ðð1þx2Þ dxÞ þ kniosðgi � hiÞkL2ðð1þx2Þ dxÞ

6 k½ninð1Þi ðgi þ hiÞ þ gnð5Þi þ nð3Þi þ nð1Þi �ðgi � hiÞkL2ðð1þx2Þ dxÞ þ kniðosgi � oshiÞkL2ðð1þx2Þ dxÞ

6 cð1þ jgiðtÞj þ jhiðtÞjÞjgiðtÞ � hiðtÞj þ cjotgiðtÞ � othiðtÞj¼ cð1þ jgiðtÞj þ jhiðtÞjÞjgiðtÞ � hiðtÞj þ cjg0iðtÞ � h0iðtÞj: ð5:17Þ

Integrating over t 2 ½0; T � we have

kK1kL1ð0;T :L2ðð1þx2Þ dxÞÞ

6 cZ t

0

ð1þ jgiðsÞj þ jhiðsÞjÞjgiðsÞ � hiðsÞjdsþ cZ t

0

jg0iðsÞ � h0iðsÞjds

6 cffiffiffiffiTpkgi � hikL2ð0;T Þ þ kgikL2ð0;T Þkgi � hikL2ð0;T Þ þ khikL2ð0;T Þkgi � hikL2ð0;T Þ þ c

ffiffiffiffiTpkg0i � h0ikL2ð0;T Þ

6 cffiffiffiffiTpkgi � hikH1ð0;T Þ þ kgikL2ð0;T Þkgi � hikH1ð0;T Þ þ khikL2ð0;T Þkgi � hikH1ð0;T Þ þ c

ffiffiffiffiTpkgi � hikH1ð0;T Þ:

Using that gi and hi are in H 1locðRþÞ we obtain

kK1kL1ð0;T :L2ðð1þx2Þ dxÞÞ 6 cffiffiffiffiTpkgi � hikH1ð0;T Þ: ð5:18Þ

Similarly, elevating square (5.17) we have

kK1k2L2ðð1þx2Þ dxÞ 6 c2½ð1þ jgiðtÞj þ jhiðtÞjÞjgiðtÞ � hiðtÞj þ jg0iðtÞ � h0iðtÞj�

2

¼ c2ð1þ jgiðtÞj þ jhiðtÞjÞ2jgiðtÞ � hiðtÞj2 þ c2jg0iðtÞ � h0iðtÞj2 þ 2c2ð1þ jgiðtÞj

þ jhiðtÞjÞjgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞj

¼ c2ð1þ jgiðtÞj2 þ jhiðtÞj2 þ 2jgiðtÞj þ 2jhiðtÞj þ 2jgiðtÞjjhiðtÞjÞjgiðtÞ � hiðtÞj2

þ c2jg0iðtÞ � h0iðtÞj2 þ 2c2jgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞj þ 2c2jgiðtÞjjgiðtÞ � hiðtÞjjg0iðtÞ

� h0iðtÞj þ 2c2jhiðtÞjjgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞj

¼ c2jgiðtÞ � hiðtÞj2 þ jgiðtÞj2jgiðtÞ � hiðtÞj2 þ jhiðtÞj2jgiðtÞ � hiðtÞj2 þ 2c2jgiðtÞjjgiðtÞ

� hiðtÞj2 þ 2jhiðtÞjjgiðtÞ � hiðtÞj2 þ 2jgiðtÞjjhiðtÞjjgiðtÞ � hiðtÞj2 þ c2jg0iðtÞ � h0iðtÞj2

þ 2c2jgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞj þ 2c2jgiðtÞjjgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞjþ 2c2jhiðtÞjjgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞj: ð5:19Þ


Multiplying (5.19) by t and integrating over t 2 ½0; T � we have
Z T
0

tkK1k2L2ðð1þx2Þ dxÞ dt 6 c2

Z T

0

tjgiðtÞ � hiðtÞj2 dt þZ T

0

tjgiðtÞj2jgiðtÞ � hiðtÞj2 dt þ

Z T

0

tjhiðtÞj2jgiðtÞ

� hiðtÞj2 dt þ 2c2

Z T

0

tjgiðtÞjjgiðtÞ � hiðtÞj2 dt þ 2

Z T

0

tjhiðtÞjjgiðtÞ

� hiðtÞj2 dt þ 2

Z T

0

tjgiðtÞjjhiðtÞjjgiðtÞ � hiðtÞj2 dt þ c2

Z T

0

tjg0iðtÞ

� h0iðtÞj2 dt þ 2c2

Z T

0

tjgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞjdt þ 2c2

Z T

0

tjgiðtÞjjgiðtÞ

� hiðtÞjjg0iðtÞ � h0iðtÞjdt þ 2c2

Z T

0

tjhiðtÞjjgiðtÞ � hiðtÞjjg0iðtÞ � h0iðtÞjdt

6 c2Tkgi � hik2L2ð0;T Þ þ T kgik

2L1ð0;T Þkgi � hik2

L2ð0;T Þ þ T khik2L1ð0;T Þkgi

� hik2L2ð0;T Þ þ 2c2TkgikL1ð0;T Þkgi � hik2

L2ð0;T Þ þ 2T khikL1ð0;T Þkgi � hik2L2ð0;T Þ

þ 2TkgikL1ð0;T ÞkhikL1ð0;T Þkgi � hik2L2ð0;T Þ þ c2Tkg0i � h0ik

2L2ð0;T Þ þ 2c2Tkgi

� hikL2ð0;T Þkg0i � h0ikL2ð0;T Þ þ 2c2T kgikL1ð0;T Þkgi � hikL2ð0;T Þkg0i � h0ikL2ð0;T Þ

þ 2c2TkhikL1ð0;T Þkgi � hikL2ð0;T Þkg0i � h0ikL2ð0;T Þ ð5:20Þ
hence ffiffi
tp

K1

�� 2

L2ð0;T :L2ðð1þx2Þ dxÞÞ 6 c2Tkgi � hik2L2ð0;T Þ þ T kgik

2L1ð0;T Þkgi � hik2

L2ð0;T Þ þ Tkhik2L1ð0;T Þkgi � hik2

L2ð0;T Þ

þ 2c2T kgikL1ð0;T Þkgi � hik2L2ð0;T Þ þ 2TkhikL1ð0;T Þkgi � hik2

L2ð0;T Þ

þ 2TkgikL1ð0;T ÞkhikL1ð0;T Þkgi � hik2L2ð0;T Þ þ c2Tkg0i � h0ik

2L2ð0;T Þ þ 2c2T kgi

� hikL2ð0;T Þkg0i � h0ikL2ð0;T Þ þ 2c2TkgikL1ð0;T Þkgi � hikL2ð0;T Þkg0i � h0ikL2ð0;T Þ

þ 2c2T khikL1ð0;T Þkgi � hikL2ð0;T Þkg0i � h0ikL2ð0;T Þ:

Using that H 1ð0; T Þ,!L1ð0; T Þ and straightforward calculus we have
ffiffitp
K1

�� L2ð0;T :L2ðð1þx2Þ dxÞÞ 6 c

ffiffiffiffiTpkgi � hikH1ð0;T Þ: ð5:21Þ

From (5.18) and (5.21) we obtain

kK1kE 6 cffiffiffiffiTpkgi � hikH1ð0;T Þ: ð5:22Þ

By other hand

kK2kL2ðð1þx2Þ dxÞ 6 knð1Þi ðw� zÞgikL2ðð1þx2Þ dxÞ þ kn

ð1Þi zðgi � hiÞkL2ðð1þx2Þ dxÞ

þ knð1Þi ðwx � zxÞgikL2ðð1þx2Þ dxÞ þ knð1Þi zxðgi � hiÞkL2ðð1þx2Þ dxÞ

6 cjgiðtÞjkw� zkL2ðð1þx2Þ dxÞ þ cjgiðtÞ � hiðtÞjkzkL2ðð1þx2Þ dxÞ þ cjgiðtÞjkwx

� zxkL2ðð1þx2Þ dxÞ þ cjgiðtÞ � hiðtÞjkzxkL2ðð1þx2Þ dxÞ

Integrating over t 2 ½0; T � and using the Holder inequality
Z T
0

kK2kL2ðð1þx2Þ dxÞ dt 6 cZ T

0

jgiðtÞjkw� zkL2ðð1þx2Þ dxÞ dt þ cZ T

0

jgiðtÞ � hiðtÞjkzkL2ðð1þx2Þ dxÞ dt

þ cZ T

0

jgiðtÞjkwx � zxkL2ðð1þx2Þ dxÞ dt þ cZ T

0

jgiðtÞ � hiðtÞjkzxkL2ðð1þx2Þ dxÞ dt

6 ckgikL2ð0;T Þkw� zkL2ð0;T :L2ðð1þx2Þ dxÞÞ þ ckgi � hikL2ð0;T ÞkzkL2ð0;T :L2ðð1þx2Þ dxÞÞ

þ ckgikL2ð0;T Þkwx � zxkL2ð0;T :L2ðð1þx2Þ dxÞÞ þ ckgi � hikL2ð0;T ÞkzxkL2ð0;T :L2ðð1þx2Þ dxÞÞ

6 ckgikH1ð0;T Þkw� zkL2ð0;T :L2ðð1þx2Þ dxÞÞ þ ckgi � hikH1ð0;T ÞkzkL2ð0;T :L2ðð1þx2Þ dxÞÞ

þ ckgikH1ð0;T Þkwx � zxkL2ð0;T :L2ðð1þx2Þ dxÞÞ þ ckgi � hikH1ð0;T ÞkzxkL2ð0;T :L2ðð1þx2Þ dxÞÞ


Using that gi and hi are in H 1locðRþÞ we obtain

kK2kL1ð0;T :L2ð1þx2ÞÞ 6 ckwx � zxkL2ð0;T :L2ðð1þx2Þ dxÞÞ þ ckzxkL2ð0;T :L2ðð1þx2Þ dxÞÞ 6 ckw� zkH þ ckzxkH: ð5:23Þ

The similar form, we obtain

kffiffitp

K2kL1ð0;T :L2ð1þx2ÞÞ 6 cffiffiffiffiTpðkw� zkH þ ckzxkHÞ: ð5:24Þ

From (5.22) and (5.24) we obtain

kK2kE 6 cffiffiffiffiTpðkw� zkH þ ckzxkHÞ: ð5:25Þ

We estimate the term K3 ¼ wwx � zzx ¼ ðw� zÞwx þ zðwx � zxÞ, then

kwwx � zzxkE ¼ kðw� zÞwx þ zðwx � zxÞkE 6 kðw� zÞwxkE þ kzðwx � zxÞkE
hence
kK3kE 6 kðw� zÞwxkE þ kzðwx � zxÞkE¼ kðw� zÞwxkL1ð0;T :L2ðð1þx2Þ dxÞÞ þ k

ffiffitpðw� zÞwxkL2ð0;T :L2ðð1þx2Þ dxÞÞ þ kzðwx � zxÞkL1ð0;T :L2ðð1þx2Þ dxÞÞ

þ kffiffitp

zðwx � zxÞkL2ð0;T :L2ðð1þx2Þ dxÞÞ

¼ kðw� zÞwxkL1ð0;T :L2ðð1þx2Þ dxÞÞ þ kzðwx � zxÞkL1ð0;T :L2ðð1þx2Þ dxÞÞ þ kffiffitpðw� zÞwxkL2ð0;T :L2ðð1þx2Þ dxÞÞ

þ kffiffitp

zðwx � zxÞkL2ð0;T :L2ðð1þx2Þ dxÞÞ

¼ K3;1 þ K3;2;

where

K3;1 ¼ kðw� zÞwxkL1ð0;T :L2ðð1þx2Þ dxÞÞ þ kzðwx � zxÞkL1ð0;T :L2ðð1þx2Þ dxÞÞ

and

K3;2 ¼ kffiffitpðw� zÞwxkL2ð0;T :L2ðð1þx2Þ dxÞÞ þ k

ffiffitp

zðwx � zxÞkL2ð0;T :L2ðð1þx2Þ dxÞÞ:

Using that: if a P 0 and b P 0 thenffiffiffiffiffiffiffiffiffiffiffiaþ bp

6ffiffiffiap þ

ffiffiffibp

we have

K3;1 ¼ kðw� zÞwxkL1ð0;T :L2ðð1þx2Þ dxÞÞ þ kzðwx � zxÞkL1ð0;T :L2ðð1þx2Þ dxÞÞ

¼Z T

0

kðw� zÞwxkL2ðð1þx2Þ dxÞdt þZ T

0

kðwx � zxÞzkL2ðð1þx2Þ dxÞ dt

¼Z T

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðw� zÞ2w2xð1þ x2Þdx

sdt þ

Z T

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðwx � zxÞ2z2ð1þ x2Þdx

sdt

¼Z T

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðw� zÞ2w2x dxþ

Z L

0

ðw� zÞ2w2xx2 dx

sdt

þZ T

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðwx � zxÞ2z2 dxþZ L

0

ðwx � zxÞ2z2x2 dx

sdt

6

Z T

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðw� zÞ2w2x dx

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðw� zÞ2w2xx2 dx

s24

35dt

þZ T

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðwx � zxÞ2z2 dx

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0


s24

35dt ð5:26Þ


but, using Lemma 2.1 (2.4) follows that
Z L
0

ðw� zÞ2wx dx 6 kw� zk2L1ð0;LÞ

Z L

0

w2x dx 6 kw� zk2

L1ð0;LÞ

Z L

0

w2xð1þ xÞdx

6 2kw� zkL2ðð1þxÞ dxÞkwx � zxkL2ðð1þxÞ dxÞkwxk2L2ðð1þxÞ dxÞ ð5:27Þ

and
Z L
0

ðw� zÞ2w2xx2 dx 6 kxðw� zÞ2kL1ð0;LÞ

Z L

0

xw2x dx 6

ffiffiffixpðw� zÞ

�� 2

L1ð0;LÞ

Z L

0

w2xð1þ xÞdx

then kxðw� zÞwxk2L2ð0;LÞ 6 k

ffiffiffixp ðw� zÞk2

L1ð0;LÞR L

0w2

xð1þ xÞdx. Hence using Lemma 5.3 we have

kxðw� zÞwxkL2ð0;LÞ 6ffiffiffixpðw� zÞ

�� L1ð0;LÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

w2xð1þ xÞdx

s

6 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikw� zkL2ðð1þxÞ dxÞ

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikwx � zxkL2ðð1þxÞ dxÞ

qþ kw� zkL2ðð1þxÞ dxÞ

h ikwxkL2ðð1þxÞ dxÞ: ð5:28Þ

This way, using (5.27) and (5.28) for the first term in (5.26) we obtain

Z T

0


0

ðw� zÞ2w2x dx

sþ


0

ðw� zÞ2w2xx2 dx

s24

35dt

6

Z T

0


0

ðw� zÞ2w2x dx

sdt þ

Z T

0


0

ðw� zÞ2w2xx2 dx

sdt

6

ffiffiffi2p Z T

0

kw� zk1=2

L2ðð1þxÞ dxÞkwx � zxk1=2

L2ðð1þxÞ dxÞkwxkL2ðð1þxÞ dxÞ dt

þ 5

Z T

0

kw� zk1=2


L2ðð1þxÞ dxÞ þ kw� zkL2ðð1þxÞ dxÞ

h ikwxkL2ðð1þxÞ dxÞ dt

6 cZ T

0

kw� zk1=2


L2ðð1þxÞ dxÞkwxkL2ðð1þxÞ dxÞ dt

þ 5

Z T

0

kw� zkL2ðð1þxÞ dxÞkwxkL2ðð1þxÞ dxÞ dt

6 ckw� zkHZ T

0

kwx � zxk1=2

L2ðð1þxÞ dxÞkwxkL2ðð1þxÞ dxÞ dt þ 5kw� zkHZ T

0

kwxkL2ðð1þxÞ dxÞ dt

6 cffiffiffiffiTpþ T 1=4

� kw� zkHkwkH:

Therefore

Z T

0


0

ðw� zÞ2w2x dx

sþ


0

ðw� zÞ2w2xx2 dx

s24

35dt 6 c

ffiffiffiffiTpþ T 1=4

� kw� zkHkwkH: ð5:29Þ

Using the similar technique for the second term we obtain

Z T

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0

ðwx � zxÞ2z2 dx

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ L

0


s24

35dt 6 c

ffiffiffiffiTpþ T 1=4

� kw� zkHkwkH: ð5:30Þ

This way from (5.29) and (5.30) we have

K3;1 6 cðffiffiffiffiTpþ T 1=4Þkw� zkHkwkH: ð5:31Þ


Now we estimate the term K3;2. Using Lemma 5.4, we estimate the following term:

kxðw� zÞwxkL2ð0;LÞ ¼ffiffiffixpðw� zÞ

ffiffiffixp

wx

�� L2ð0;LÞ 6

ffiffiffixpðw� zÞ

�� L1ð0;LÞ

ffiffiffixp

wx

�� L2ð0;LÞ



qþ kw� zkL2ðð1þxÞ dxÞ

� ffiffiffixp

wx

�� L2ð0;LÞ



q ffiffiffixp

wx

�� L2ð0;LÞ

þ 5kw� zkL2ðð1þxÞ dxÞffiffiffixp

wx

�� L2ð0;LÞ



qkwxkL2ðð1þxÞ dxÞ

þ 5kw� zkL2ðð1þxÞ dxÞkwxkL2ðð1þxÞ dxÞ; ð5:32Þ

hence from (5.32)

kffiffitp

xðw� zÞwxkL2ð0;T :L2ð0;LÞÞ ¼ kffiffitp ffiffiffi

xpðw� zÞ

ffiffiffixp

wxkL2ð0;T :L2ð0;LÞÞ

6 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikw� zkL1ð0;T :L2ðð1þxÞ dxÞÞ

qkffiffitp

wxkL1ð0;T :L2ð1þxÞ dxÞ

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikwx � zxkL2ð0;T :L2ðð1þxÞ dxÞÞ

q ffiffiffiffiTpþ 5kw

� zkL1ð0;T :L2ðð1þxÞ dxÞÞkffiffitp

wxkL2ð0;T :L2ð1þxÞ dxÞ

6 ckw� zk1=2H kwkH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikwx � zxkL2ð0;T :L2ðð1þxÞ dxÞÞ

q ffiffiffiffiTpþ ckw� zkH

ffiffiffiffiTpkwkH

6 cffiffiffiffiTpkw� zkHkwkH; 80 6 t 6 T : ð5:33Þ

This way from (5.33), the first term in K3;2 is estimated by

kffiffitpðw� zÞwxkL2ð0;T :L2ðð1þx2Þ dxÞÞ 6 c

ffiffiffiffiTpkw� zkHkwkH ð5:34Þ

and the similar form we estimate the second term in K3;2.

kffiffitpðwx � zxÞzkL2ð0;T :L2ðð1þx2Þ dxÞÞ 6 c

ffiffiffiffiTpkw� zkHkwkH ð5:35Þ

then

kK3kE 6 cffiffiffiffiTpkw� zkHkwkH: ð5:36Þ

Therefore replacing in (5.16) the terms in (5.22), (5.25), (5.36) we have

kCðw0; gi;wÞ � Cðz0; hi; zÞkH 6 cðT Þkw0 � z0kL2ð0;LÞ þ cðT Þ kK1kE þ kK2kE þ kK3kE½ �

6 cðT Þkw0 � z0kL2ð0;LÞ þ cðT ÞffiffiffiffiTp½kgi � hikH1ð0;T Þ þ kw� zkH þ kzkH�



kw� zkHkwkH þ kw� zkHkwkHh i

; ð5:37Þ

where we obtain

kCðw0; gi;wÞ � Cðz0; hi; zÞkH6 cðT Þkw0 � z0kL2ð0;LÞ þ cðT Þ

ffiffiffiffiTpkgi � hikH1ð0;T Þ þ kw� zkH þ kzkHh i

þ cðT ÞffiffiffiffiTp½ð1þ

ffiffiffiffiTpÞkw� zkHkwkH þ kw� zkHkwkH� ð5:38Þ

The result follows. h

Theorem 5.2. Let g < 0. Let gi 2 H 1locðRþÞ. Then there exists a time T 1 2�0; T 0� such that the map

C : BRð0Þ ! BRð0Þ with u! Cðw0; gi;wÞ maps the ball BRð0Þ into itself.


Proof. Using (5.14) with z0 ¼ 0, hi ¼ 0, and z ¼ 0 we have

kCðw0; gi;wÞkH 6 cðT Þkw0kL2ð0;LÞ þ cðT ÞffiffiffiffiTpkgikH1ð0;T Þ þ kwkHh i



kwk2H þ kwk

2H

h i: ð5:39Þ

Let

R2¼ cðT 0Þkw0kL2ð0;LÞ þ cðT 0Þ

ffiffiffiffiffiT 0

pkgikH1ð0;T Þ ð5:40Þ

then, if w 2 BRð0Þ we have

kCðw0; gi;wÞkH 6R2þ cðT Þ

ffiffiffiffiTp


R2 þ 2Rh i

: ð5:41Þ

We choose T such that

cðT ÞffiffiffiffiTp½ð1þ

ffiffiffiffiTpÞR2 þ 2R� 6 R

2ð5:42Þ

hence kCukH 6 R and the Theorem follows. h

Theorem 5.3. Let g < 0. Assume that gi 2 H 1locðRþÞ. Then, there exists a time T 2 2�0; T 1� such that the applica-

tion w! Cðw0; gi;wÞ is a contraction over ðBRð0Þ; k � kHÞ.

Proof. Using (5.14) with z0 ¼ w0, hi ¼ gi, we have

kCðw0; gi;wÞ � Cðw0; gi; zÞkH6 cðT Þ

ffiffiffiffiTp½kw� zkH þ kwkH� þ cðT Þ

ffiffiffiffiTp½ð1þ

ffiffiffiffiTpÞkw� zkHkwkH þ kw� zkHkwkH�: ð5:43Þ

If w; z 2 BRð0Þ we obtain

kCðw0; gi;wÞ � Cðw0; gi; zÞkH 6 cðT ÞffiffiffiffiTp½kw� zkH þ R� þ cðT Þ

ffiffiffiffiTp½ð1þ

ffiffiffiffiTpÞkw� zkHRþ kw� zkHR�

6 cðT ÞffiffiffiffiTp½½2þ 2R� þ

ffiffiffiffiTp

R�kw� zkH ð5:44Þ
such that, if T is small enough namely
cðT ÞffiffiffiffiTp½½2þ 2R� þ

ffiffiffiffiTp

R� 6 1

then the application w! Cðw0; gi;wÞ is a contraction over ðBRð0Þ; k � kHÞ. h

Theorem 5.4. If g < 0, there exists a unique w defined in HT , weak solution of (1.1)–(1.4), (1.4).

Proof. To apply the Banach’s fixed point Theorem for ðw0; g;wÞ ! Cðw0; g;wÞ on BRð0Þ, which is a completemetric space and yields local existence and uniqueness. h

Theorem 5.5. If g < 0. Then the solution w depends continuously on w022ðð1þ x2ÞdxÞ and gi 2 H 1locðRþÞ.

Proof. From (5.14) for small times, one gets

kw� zkH 6 cðT Þkw0 � z0kL2ð0;LÞ þ cðT ÞffiffiffiffiTp½kgi � hikH1ð0;T Þ þ kw� zkH þ kzkH�

þ cðT ÞffiffiffiffiTp½ð1þ

ffiffiffiffiTpÞkw� zkHkwkH þ kw� zkHkwkH�: ð5:45Þ

Then if w0 ! z0 in L2ðð1þ x2ÞdxÞ and if gi ! hi in H 1ð0; T Þ one gets that w! z in HT . The proof follows. h

These results were obtained locally in time. But, since the time interval where this result holds depends onlyon kw0kL2ð0;LÞ and kgikH1ð0;T Þ, it can be extended as long the solution exists. Indeed we obtain

Theorem 5.6 (Existence and Uniqueness). Let g < 0, u0 2 L2ðð1þ x2ÞdxÞ, gi 2 H1locðRþÞ for i ¼ 0; 1, and

0 < L < þ1. Then there exists a unique weak maximal solution defined over ½0; T L� for (1.1)–(1.4), (1.4).Moreover, there exists T min > 0 independent of L, depending only on ku0kL2ð0;LÞ and kgikH1ð0;T Þ such that


T L P T min. The solution u depends continuously on u0 and gi in the following sense: Let a sequence un0 ! u0 in

L2ðð1þ x2ÞdxÞ, let a sequence gni ! gi in H 1

locðRþÞ and denote by un the solution with data ðun0; g

ni Þ and T n

L its

existence time. Then

lim infn!þ1

T nL P T L ð5:46Þ

and for all t < T L, un exists on the interval ½0; T � if n is large enough and un ! u in HT .

6. Numerical methods

We consider finite differences based on unconditionally stable schemes similar to the described in [19] forthe KdV equations.

Description of the scheme. We note by vni the approximate value of uðiDx; nDtÞ, solution of the nonlinear

problem (1.1)–(1.4), where Dx is the space-step, and Dt is the time-step, for i ¼ 0; . . . ;N , and n ¼ 0; . . . ;M .Define the discrete space

X N ¼ fu ¼ ðu0; u1; . . . ; uN Þ 2 RNþ1ju0 ¼ u1 ¼ 0 and uN ¼ uN�1 ¼ uN�2 ¼ 0g
and ðDþuÞi ¼
uiþ1�ui

Dx and ðD�uÞi ¼ui�ui�1

Dx the classical difference operators. In order to obtain a positive matrixwe have to choose a particular discretization. The numerical scheme for the nonlinear problem ((1.1)–(1.4))reads as follows:

vnþ1 � vn

Dtþ Avnþ1 þ a

2D� vn½ �2 ¼ 0; ð6:1Þ

where A ¼ gDþDþDþD�D� þ DþDþD� þ 12ðDþ þ D�Þ, with a ¼ 1 for the nonlinear case, and a ¼ 0 for the lin-

ear case.

Remark 6.1. We remark that A is well defined as a linear application X N ! RNþ1, in the sense that we do notneed additional point on the outside of ½0; L� to compute Au. Thus A is represented by a 6-diagonal positivedefined matrix of ðN þ 1Þ � ðN þ 1Þ:

Au ¼

c1 d1 e1 f1

b2 c2 d2 e2. .

.0

a3 b3 c3. .

. . .. . .

.

. .. . .

. . .. . .

. . ..

fn�3

. .. . .

. . .. . .

.en�2

0 an�1 bn�1 cn�1 dn�1

an bn cn

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

; ð6:2Þ

where ai ¼ � gDx5, for i ¼ 3; . . . ; n, fi ¼ g

Dx5, for i ¼ 1; . . . ; n� 3, and

bi ¼5 g

Dx5 � 1Dx3 � 1

2Dx ; for i ¼ 2; . . . ; n� 1;

3 gDx5 � 1

Dx3 � 12Dx ; for i ¼ n;

(

ci ¼�9 g

Dx5 þ 3Dx3 ; for i ¼ 1; n� 1;

�10 gDx5 þ 3

Dx3 ; for i ¼ 2; . . . ; n� 2;

�3 gDx5 þ 1

Dx3 ; for i ¼ n;

8><>:

di ¼10 g

Dx5 � 3Dx3 þ 1

2Dx ; for i ¼ 1; . . . ; n� 2;

5 gDx5 � 2

Dx3 þ 12Dx ; for i ¼ n� 1;

(

ei ¼�5 g

Dx5 þ 1Dx3 ; for i ¼ 1; . . . ; n� 3;

�4 gDx5 þ 1

Dx3 ; for i ¼ n� 2:

(


We consider the linear operators Dþ and D� as matrices of size ðN þ 1Þ � ðN þ 1Þ and we note the followinginternal product ðz;wÞ ¼

PN1¼0ziwi and ðz;wÞx ¼ ðz; xwÞ ¼

PN1¼0iDxziwi, and the norms in RNþ1 : jzj ¼

ffiffiffiffiffiffiffiffiffiffiðz; zÞ

pand jzjx ¼

ffiffiffiffiffiffiffiffiffiffiffiffiðz; zÞx

p. Then, we have the following lemma :

Lemma 6.1. For all z;w 2 RNþ1, we have

ðDþz;wÞ ¼ zN wN � z0w0 � ðz;D�wÞ; ð6:3Þ

ðDþz; zÞ ¼ 1

2

z2N

Dx� z2

0

Dx� DxjDþzj2

�; ð6:4Þ

ðDþz;wÞx ¼ NzN wN � ðz;D�wÞx þ Dxðz;D�wÞ � ðz;wÞ; ð6:5Þ

ðDþz; zÞx ¼1

2

Nz2N

Dx� DxjDþzj2x � jzj

2

�: ð6:6Þ

Proof. Eqs. (6.3) and (6.5) are result of summing by parts. Eq. (6.4) is result of usingða� bÞa ¼ 1

2ða2 � b2Þ þ 1

2ða� bÞ2: with zi ¼ a and ziþ1 ¼ b, and summing over i ¼ 0; . . . ;N . The last equality

(6.6) is result of the same identity with zi ¼ a and ziþ1 ¼ b, multiplying by iDx and summing overi ¼ 0; . . . ;N . h

In order to obtain estimates for the solution of the numerical scheme for the linear case, we have the fol-lowing lemmas describing the quadratic forms associated to the different matrices.

Lemma 6.2. For all u 2 X N , we have

1

2ððDþ þ D�Þu; uÞ ¼ 0; ð6:7Þ

ðDþDþD�u; uÞ ¼ Dx2jDþD�uj2; ð6:8Þ

ðDþDþDþD�D�u; uÞ ¼ � 1

2Dx½D�D�u�20 �

Dx2jDþD�D�uj2: ð6:9Þ

Remark 6.2. Since u 2 X N , the first term in the right-hand side of (6.9) is given by 1Dx ½D

�D�u�20 ¼ ðDxÞu22.

Proof. The matrix 12ðDþ þ D�Þ is clearly antisymmetric and we have (6.7). Using (6.4) with z ¼ D�u we obtain

(6.8), and using the same identity with z ¼ D�D�u we obtain (6.9). h

Corollary 6.1. If g 6 0, then I þ DtA is positive definite, and for any un 2 X N there exists a unique solution unþ1

of (6.1).

Proof. From Lemma 6.2 we have for all u 2 X N with u 6¼ 0,

ððI þ DtAÞu; uÞP juj2 þ DtDx2jDþD�uj2 � gDxDt

2jDþD�D�uj2 � gDt

2DxD�D�u½ �20 > 0; ð6:10Þ

when g 6 0. h

The following estimate shows that the numerical scheme (6.1) with a ¼ 0 is l2-stable and unconditionallystable.

Proposition 6.1. Let g 6 0. For any vn 2 X N satisfying the linear scheme (6.1) with a ¼ 0, there exists CðT Þ > 0such that jvnj 6 jv0j.

Proof. Multiplying the numerical scheme (6.1) by vnþ1 we obtain

jvnþ1j2 þ DtðAvnþ1; vnþ1Þ ¼ ðvnþ1; vnÞ ð6:11Þ
and then, using the same identity of the proof of the Lemma 6.1 with a ¼ vkþ1 and b ¼ vk and summing fork ¼ 0; . . . ; n� 1 we have


jvnj2 þXn�1

k¼0

jvkþ1 � vkj2 þ 2DtXn

k¼1

ðAvk; vkÞ ¼ jv0j2:

From (6.10) this last equality becomes

jvnj2 þ DtXn�1

k¼0

Dtvkþ1 � vk

Dt

��2

þ DxXn

k¼1

DtjDþD�vkj2 � gDxXn

k¼1

DtjDþD�D�vkj2

� gDx

Xn

k¼1

Dt½D�D�vk�20 6 jv0j2: � ð6:12Þ

In order to obtain the unconditional stability for the nonlinear version of the scheme, we will find a discreteestimate that is equivalent to that of (3.28) (see Proposition 3.1). Let us denote by x the sequence xi ¼ iDx. Wehave:

Lemma 6.3. For all u 2 X N , we have

1

2ððDþ þ D�Þu; xuÞ ¼ 1

4Dx2jDþuj2 � 1

2juj2;

ðDþDþD�u; xuÞ ¼ 3

2jD�uj2 þ Dx

2jDþD�uj2x �

Dx2

2jDþD�uj2;

ðDþDþDþD�D�u; xuÞ ¼ � 5

2jD�D�uj2 � Dx

2jDþD�D�uj2x

þ Dx2jDþD�D�uj2 � 1

2D�D�u½ �20;

where ðxuÞi ¼ iDxui.

Proof. Using (6.3), (6.4) and (6.5) we have

ððDþ þ D�Þu; xuÞ ¼ ðD�u; uÞx � ðu;D�uÞx þ Dxðu;D�uÞ � juj2 ¼ �DxðDþu; uÞ � juj2 ¼ Dx2

2jDþuj2 � juj2

and then we have the first identity of the Lemma. Following the same idea and applying the identities of Lem-ma 6.1 is easy to prove the rest of the identities. h

Proposition 6.2. Let g 6 0. For any vn 2 X N satisfying the linear scheme (6.1) with a ¼ 0, there exists CðT Þ > 0such that jvnjx 6 CðT Þjv0jx and

Xn

k¼1

DtjD�vkj2 !1

2

6 CðT Þjv0j;

Xn

k¼1

DtjDþD�vkj2 !1

2

6 CðT Þjv0j; if g < 0:

Proof. We multiply the numerical scheme (6.1) with a ¼ 0 by xvnþ1. Then, applying Lemma 6.3, and the sameidentity of the proof of Lemma 6.1 with a ¼ ffiffiffi

xp

vkþ1 and b ¼ ffiffiffixp

vk, we deduce

jvnj2x þDtXn�1

k¼0

Dtvkþ1 � vk

Dt

��2

x

þ 3þDx2

2

�Xn

k¼1

DtjD�vkj2 þDxXn

k¼1

DtjDþD�vkj2x � ð5gþDx2ÞXn

k¼1

DtjDþD�vkj2

� gDxXn

k¼1

DtjDþD�D�vkj2x þ 2gDx2Xn

k¼1

DtjDþD�D�vkj2 � g½D�D�vk�20

¼ jv0j2x þXn

k¼1

Dtjvkj2: ð6:13Þ


On the other hand, noting that

jDþD�vkj2x � DxjDþD�vkj2 ¼XN�1

i¼1

ði� 1ÞDx

ðvkiþ1 � 2vk

i þ vki�1Þ

2 P 0;

jDþD�D�vkj2x � 2DxjDþD�vkj2 þ 1

Dx½D�D�vk�20 ¼

XN�1

i¼2

ði� 2ÞDx

ðvkiþ1 � 3vk

i þ 3vki�1 � vk

i�2Þ2 P 0

and replacing these inequalities in (6.13), we deduce

jvnj2x þ DtXn�1

k¼0

Dtvkþ1 � vk

Dt

��2

x

þ 3Xn

k¼1

DtjD�vkj2 � 5gXn

k¼1

DtjDþD�vkj2 6 jv0j2x þXn

k¼1

Dtjvkj2:

Finally, using the inequalities of Proposition 6.1 and the fact that we have in a boundary domain ð0; LÞ, wemay conclude the proof. h

Now, let us introduce the non-homogeneous linear scheme approximating the solution of the (KdVK)NH

problem:

vnþ1 � vn

Dtþ Avnþ1 ¼ f n:

The existence proof of the continuous case studied in the previous sections applies in the discrete non-homo-geneous linear case and the discrete nonlinear case for any discretization of the nonlinear part, in particular forfn ¼ 1

2D�½un�2. Thus, we obtain the following result of convergence:

Theorem 6.1. For any un 2 X N satisfying the nonlinear scheme (6.1), with a ¼ 1, and g 6 0, there exists e0 > 0such that, if Dt 6 e0, then there exists T > 0 and a constant C ¼ CðT Þ > 0 (independent of Dt and Dx) such that:

pk¼0;...;p

jvkj2 þ DtXp

k¼0

jD�vkj2 � gDtXp

k¼0

jD�D�vkj2 6 Cjv0j2:

This result means that the scheme is unconditionally stable. Let us observe that in agreement with the gainof regularity of the (KdVK) equation, we obtain an additional estimate respect to the analogous numericalscheme of the KdV equation, studied in detail in [19]. On the other hand, in [21] it is described a similar schemewith application to KdV and Benney–Lin equations.

7. Some numerical results

First we compare our numerical solution with an explicit solution obtained by the Adomian decompositionmethod for a KdVK equation with initial condition in the unbounded domain x 2 R [22,23]. The tanh methodto obtain explicit solutions of the KdvK equation is proposed by Wazwaz [24] in a slightly different way. Tocompare our numerical solution in a bounded domain ð0; LÞ with an explicit solution in all x 2 R we considersolitons moving between x ¼ 0 and x ¼ L no touching the boundaries. Let the KdvK equation

ut � uxxxxx þ uxxx þ uux þ ux ¼ 0; x 2 R; t > 0 ð7:1Þ

with the initial condition

uðx; 0Þ ¼ 105

169sech4 1

2ffiffiffi1p

3ðx� x0Þ

�;

where it is known that the explicit solution is given by the following travelling wave (see [22,23]):

uðx; tÞ ¼ 105

169sech4 1

2ffiffiffi1p

3ðx� 205t

169� x0Þ

�:

This result can be verified through substitution.

0 20 40 60 80 100 120 140 160 180 200-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7exactsimulated

Fig. 1. Comparison between the exact solution and the simulated for Dx ¼ 10�3, Dt ¼ 1:2� 10�4 (T ¼ 0:0 s, T ¼ 60:0 s, and T ¼ 120:0 s).


We make the simulations in Fortran90, using a factorization A ¼ LU with a generalization of the Thomasalgorithm for a 6-diagonal matrix like (6.2), and a posteriori error correction using the residual. We choosex0 ¼ 20:0, L ¼ 200:0, T ¼ 120:0 and we fix Dt=Dx ¼ 0:12. We compute different simulations on the time inter-val ½0; T � for n ¼ 2� 101; 2� 102; . . . ; 2� 105. The comparison between exact solution and the best simulation(with n ¼ 200; 000) is represented in Fig. 1 for three different times. The error, that is the normL1ð0; T ; L2ð0; LÞÞ of the difference between the exact solution and the simulation for different n is representedin Fig. 2.

The second numerical test is an intersection of two solitons. We consider Eq. (7.1) with the following initialcondition:

uðx; 0Þ ¼ 105

169sech4 1

2ffiffiffi1p

3ðx� 20:0Þ

�þ 1

4sech4 1ffiffiffi

1p

3ðx� 60:0Þ

�� :

This corresponds to the superposition of two solitons with different speeds given by the nonlinear termuux þ ux of Eq. (7.1). For this example, we choose L ¼ 1000:0, T ¼ 800:0, n ¼ 100; 000, Dt ¼ 8� 10�4,Dx ¼ 1� 10�2. Fig. 3 shows three-dimensional plots of the solution.

101 102 103 104 105 10610-3

10-2

10-1

100

101

Fig. 2. Decreasing of the error between exact solution and simulated solution as function of n ¼ L=Dx.

Fig. 3. Interaction of two solitons for the KdV–Kawahara equation.


Acknowledgements

JCC and OVV acknowledge support by Universidad del Bıo-Bıo. DIPRODE projects, # 052808 1/R and #061008 1/R. MS has been supported by Fondecyt project # 1030718 and Fondap in Applied Mathematics(Project # 15000001). Finally, OVV and MS acknowledge support by CNPq/CONICYT Project, #490987/2005-2 (Brazil) and # 2005-075 (Chile).

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Documents

The Korteweg–de Vries–Kawahara equation in a bounded domain and some numerical results