Springer zhidkov p e korteweg de vries and nonlinear schroedinger equation LNM1756 springer 2001 (152 p)

Lecture Notes in Mathematics
Editors:
J.-M. Morel, Cachan
F. Takens, Groningen
B. Teissier, Paris

1756


3
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
Paris
Singapore
Tokyo


Peter E. Zhidkov

Korteweg-de Vries and
Nonlinear Schrödinger
Equations:
Qualitative Theory

123

Author
Peter E. Zhidkov
Bogoliubov Laboratory of Theoretical Physics
Joint Institute for Nuclear Research
141980 Dubna, Russia
E-mail:

Cataloging-in-Publication Data applied for

Mathematics Subject Classification (2000): 34B16, 34B40, 35D05, 35J65,
35Q53, 35Q55, 35P30, 37A05, 37K45
ISSN 0075-8434
ISBN 3-540-41833-4 Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part
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Law.
Springer-Verlag Berlin Heidelberg New York
a member of BertelsmannSpringer Science+Business Media GmbH

© Springer-Verlag Berlin Heidelberg 2001
Printed in Germany
Typesetting: Camera-ready TEX output by the authors
SPIN: 10759936
41/3142-543210 - Printed on acid-free paper

Contents

Page

Introduction

I

Notation

5

Chapter

1.

Evolutionary equations. Results

1.1 The

on

existence

9

(generalized) Korteweg-de

Vries equation (KdVE)
Schr6dinger equation (NLSE)
blowing up of solutions

10

1.2 The nonlinear

26

1.3 On the

36

1.4 Additional remarks.

Chapter

2.

37

39

Stationary problems

2.1 Existence of solutions. An ODE

42

approach

2.2 Existence of solutions. A variational method

49

2.3 The concentration- compactness method of P.L. Lions
2.4 On basis properties of systems of solutions

56

2.5 Additional remarks

76

Chapter
3.1
3.2
3.3

3.

Stability

Stability of
Stability of
Stability of

of solutions

79

soliton-like solutions

80

kinks for the KdVE
solutions of the NLSE

90

nonvanishing

as

jxj

3.4 Additional remarks

Chapter

4. Invariant

62

94
103

105

measures

4.1 On Gaussian

measures

4.2 An invariant

measure

in Hilbert spaces

for the NLSE

4.3 An infinite series of invariant
4.4 Additional remarks

oo

measures

107
118

for the KdVE

124

135

Bibliography

137

Index

147


Introduction
During
differential

large

the last 30 years the

theory

equations (PDEs) possessing

of solitons

the

-

solutions of

special

a

partial

of nonlinear

theory

kind

field that attracts the attention of both mathematicians and

-

has grown into

physicists

a

in view

important applications and of the novelty of the problems. Physical problems
leading to the equations under consideration are observed, for example, in the mono-

of its

graph by V.G. Makhankov [60]. One of the related mathematical discoveries is the

possibility of studying certain nonlinear equations from this field by methods that
these equations
were developed to analyze the quantum inverse scattering problem;
this subject,
are called solvable by the method of the inverse scattering problem (on
see, for

example [89,94]).

PDEs solvable

At the

by this method

is

time, the class of currently

same

sufficiently

narrow

and,

on

known nonlinear

the other

hand,

there is

The latter

of differential

called the qualitative theory
equations.
of various probthe
includes
on
well-posedness
investigations
particular
approach
such
solutions
of
as
the
behavior
stability or blowing-up,
lems for these equations,

approach,

another

in

dynamical systems generated by these equations, etc., and this approach
possible to investigate an essentially wider class of problems (maybe in a
of

properties
makes it
more

general study).
In the

present book, the author

qualitative theory

are

on

about twenty years.

So, the selection of the material

the existence of solutions for initial-value

travelling
problems

or

standing waves)

of the

stability

of

substituted in the

are

solitary

are

four main

problems

topics.

for these

equations,

special
(for example,
equations under consideration,

waves, and the construction of invariant

dynamical systems generated by

the

Korteweg-de

is

These

kinds

when solutions of

problems arising

studies of stationary

for

during

related to the author's scientific interests. There

results

and methods of the

equations under consideration, both stationary and evolutionary,

of

that he has dealt with

mainly

problems

some

surveys

Vries and nonlinear

measures

Schr6dinger

equations.
We consider the

following (generalized) Korteweg-de
+

Ut

and the nonlinear

f (U)U.,

+ UXXX

equation (KdVE)

0

Schr6dinger equation (NLSE)
iut + Au + f (Jul')u

where i is the

imaginary unit,

and

in the

complex

=

Vries

second),

u

u(x, t)

=

t E

R,

x

is

an

=

0,

unknown function

E R in the

case

(real in the first

of the KdVE and

x

E

case

R' for
N

the NLSE with

a

positive integer N, f (-)

is

a

smooth real function and A

=

E
k=1

P.E. Zhidkov: LNM 1756, pp. 1 - 4, 2001
© Springer-Verlag Berlin Heidelberg 2001

82
aX2

k


2

Laplacian. Typical examples, important for physics, of the functions f (s)

is the

As 2) respectively,

the

are

and

following:

2
as

Isl"

value

initial-boundary
for
u

travelling

e `O(w, x)

=

the

waves u

in the

equation (it

what

being

be called the

solitary

(as JxJ

for the

NLSE,

-+

oo

dealt

we

wt)

in the

NLSE,

0

is

supplement

with

_

Loo

some

+

A similar

of existence and

0(k)(00)

=

(k

0

=

nontrivial solutions

integer

any

argument

r

occurs

Let

kinds).

In this case, the

us

typical

Ix I,

has

can

the

0,

possessing limits

Ej

X

=

00

as x --+

into the

waves

of the second order:

le,

0.

be

solitary

waves

solution of

I roots

on

satisfying
solved

(see Chapter 2).

(for example,
on

f interesting for
our

r

>

for functions

the argument

us

problem which,

the half-line

proving

of

generally speaking, non-uniqueness

depending only

result for functions

1,

==

conditions of the

sufficiently easily

when such solutions exist

exactly

the method of the

of

consider solutions

a

into

function,

if necessary,

expression for standing

=

In this case,

We consider two methods of

are

real

a

waves

notation, specifying,
follows, the solutions of these kinds will

f(1012)0

uniqueness

I > 0 there exists
=

standing

boundary conditions, for example,

0, 1, 2)

Difficulties arise when N > 2.

the above

is

0

Chapter

expression

arises for the KdVE. For the KdVE and the NLSE with N

problem

problem

type

substitute the

c R and

w

elliptic equation

061--

the

we

bounded function

a

nonlinear

following

Cauchy problem and

this

just

In what

with).

if

of the

of the KdVE and

case

where

NLSE). Substituting

A0
which

-

of the

waves

obtain the

we

It arises when

problem.

O(w, x

case

positive constants).

v are

well-posedness

is convenient to introduce

is

equation

=

the

on

and

a

for the KdVE and the NLSE used further. In

problems

the stationary

qonsider

we

(where

1 contains results

Chapter
2,

e-a.,2

+S21

1

is the
as a

f

of

=

W.

following:

for

r

function of the

0.

the existence of

solitary

These

waves.

qualitative theory of ordinary differential equations (ODEs)

and the variational method.

As

an

example of

the

latter,

concentration- compactness method of P.L. Lions.

In

touch upon recent results

a

on

the property of

being

briefly

we

addition,
basis

in this

consider the

chapter

(for example,

in

L2)

we

for

systems of eigenfunctions of nonlinear one-dimensional Sturm-Liouville-type problems
in finite intervals similar to those indicated above.

Chapter
Lyapunov
set

3 is devoted to the

sense.

X, equipped

Omitting
with

a

some

distance

stability of solitary waves, which is understood

details, this

R(., .),

means

there exists

that,
a

if for

unique

an

arbitrary

solution

u(t),

uo

in the

from

t >

0,

a

of


3

to X for any fixed t >

equation under consideration, belonging

the

to X for any fixed

T(t), belonging
R, if for

any

>

satisfying R(T (0), u(O)) < b, one
Probably the historically first result on the stability

0.

that obtained

A.N.

by

for all

u(t), belonging to
R(T (t), u(t)) < C for

has

Kolmogorov,

the one-dimensional

of

solitary

stability

a

solitary

case a

of

kink for

a

is called

wave

in

our

nonlinear diffusion

a

kink if

a

waves was

[48]:

I.G. Petrovskii and N.S. Piskunov

terminology, they proved (in particular)
equation (in

solution

a

called stable with respect to the distance

X for any fixed t > 0 and

all t

0, then

0 there exists b > 0 such that for any solution

>

e

t > 0 is

0' (w, x) 0

0

X

x).

Let

introduce

us

functions of the

in the real Sobolev space H1

special distance

a

argument

of

consisting

by the following rule:

x,

p(u,v)=

JJu(-)-v(-+,r)JJHi.

inf
,ERN

If

we

for

call two functions

some 7-

E

and

u

from

v

H1, satisfying

set of classes of

R, equivalent, then the
a

stability of solitary

waves

smooth

family

of

any two solutions
t

0 in the

=

solitary

O(wl, x

sense

W2.

At the

same

distance p, then
T.B.

Benjamin

stability of

they

in his

the

many authors and

For
be

solitary

taken in the

and

wit)

same sense

time,
can

if two

O(W2, X

-

the parameter

f (s)

form
we

=

of

solitary

paper

first, because

usually

possesses

(a, b).

Therefore

E

have

close at t

=

velocities wi

0 in the

to be close for all t > 0 in the

has

proved

the

stability

of

solitary

wave.

Later, his

approach

point

Sobolev spaces,

or

non-equal

sense

solitary

was

of the

same sense.

with respect to the distance p. He called this

s

a

[7]

w

Lebesgue

as

they

waves are

with the

close to each other at the

L02t),

for all t > 0 if

easily verified

be

pioneering

the usual KdVE with

the

-

second,

on

of standard functional spaces such

cannot be close in the

and

depending

waves

the KdVE

r)

-

investigate the

of the KdVE with respect to this distance p;

the KdVE is invariant up to translations in x;

v(x

=_

equivalent functions

metric space. For several reasons, it is natural to

distance p becomes

a

u(x)

the condition

of

waves

stability

developed by

shall consider their results.

waves

of the

following

NLSE,

the distance p should be modified. It should

form:

d(u,v)=infllu(.)-e"yv(.--r)IIHI

(u,vEH')

T"Y

where H' is

only

now

the

complex

space,

-r

R' and

E

7 E R.

To

clarify

remark here that the usual one-dimensional cubic NLSE with

two-parameter family
ob (x,

where

w

> 0

t)

and b

family, arbitrary

=

this

f (s)

=

fact,
s

we

has

a

of solutions

V-2-w exp I i [bx

are

-

(b

real parameters.

close at t

=

0 in the

2
_

W)t]

Therefore,

sense

cosh[v/w-(x
two

-

2bt)]

arbitrary

solutions from this

of the distance p, cannot be close for all

4

t > 0 in the
any two

standing

close in the

of the

waves

close at t

NLSE,

parameter

of the distance p for all t

sense

above

family 40(x, t)

>

0 in the

=

0. At the

to each

cannot be

other,

time, the functions of

same

in the

stability

By analogy,

W.

-

of the distance p and

sense

nonequal

w,

the definition of

satisfy

V

to different values of

they correspond

to two values of the

corresponding
the

if

same sense

of the distance

sense

d.
In the two

cases

of the KdVE and the

condition for the

necessary")

stability

of

NLSE,

present

we

solitary

a

sufficient
lim

satisfying

waves

(and
O(x)

"almost
0 and

=

1XI-00

O(x)
for

a

is called the

0, that

>

nonlinearity

of

general type

Next,

consider the

Confirming

this

the function

f

of

point

of

prove the

for

non-vanishing

of

our

Chapter 4,

JxJ

as

-+

oo.

We present

theory

physics.

waves.

of

a

equations.

If

For the

NLSE,

energy

and, for

we

construct

higher

that kinks

many
our

always

are

stable.

assumptions

on

we

of

have

a

recurrence

an

present

case

an

a

new

constructing

attention

theorem

on

such

one

means

the

the

theory

application

con-

is well-known in

corresponding

measure

this

measures

in the

stability according

phenomenon

explains

measure

interesting

invariant

phenomenon which
it

of the

waves

and

important applications

bounded invariant

invariant

solitary

remain open in this direction.

the Fermi-Pasta-Ularn

we

of

stability of

dynamical system generated by

the KdVE in the

scattering problem,

a

Roughly speaking,

By computer simulations,

system, then the Poincar6

with

problem

We concentrate

trajectories

many "soliton"

is satisfied.

> 0

of kinks under

stability

questions

It is the Fermi-Pasta-Ulam

of nonlinear

Poisson of all
tion.

deal with the

we

dynamical systems.

the

many

equations. These objects have

nected with

stability

3 is devoted to the

Chapter

type. It should be said however that
In

dw

of kinks for the KdVE with respect to the distance

widespread opinion

a
we

d Q (0)

to the distance

general type.

The last part of

NLSE

view,

if the condition

NLSE)

stability

there is

Among physicists

p.

physical literature. Roughly speaking,

solitary wave is stable (with respect

a

p for the KdVE and to d for the
we

in the

Q-criterion

was

for

to

equa-

observed for

our

dynamical

phenomenon partially.

associated with the conservation of

when it is solvable

by

the method of the inverse

infinite sequence of invariant

measures

associated

conservation laws.

The author wishes to thank all his

colleagues

and friends for the useful scien-

tific contacts and discussions with them that have contributed
appearance of the

present book.

importantly

to the


Notation
Unless stated
the

otherwise,

of the KdVE and

case

the spaces of functions introduced

I for the KdVE and N is

(X1)

X

...

XN)

E

always real

in

for the NLSE.

complex

Everywhere C, C1, C2, C', C",...
N

are

denote

a

positive

constants.

positive integer

for the NLSE.

RN.

N

E

A

i=1

R+

=

For

a

8'Xi

is the

[0, + oo).
measurable domain Q C RN

of functions defined

Lp

=

Laplacian.

Lp(RN)

on

D with the

and

Mp

=

norm

for any g, h E

L2(9)-

00

00

12

ja

=

=

(ao,aj,...,a,,...)

:

an E

1: a2

R, 11al 1212

n

<

F, a nb

00b (a, b)12

n=O

where

a

=

For

infinitely

COOO

(ao, a,,..., an

i

...

)

b

=

(bo, bl,

b,,,,...)

...,

N
with
open domain Q C R

an

a

E

smooth

differentiable functions of the argument

X

boundary COOO(O)
E Q with

that,

if Q is

=

set

compact supports in Q.

a

bounded domain

or

Q

=

R'V,

Coc (Q)

then D is

in

L2(n)-

It is well-

positive self-adjoint

a

L2(Q)-

Let Q C

Q

is the space of

COOO (RN).

==

operator in

n

n=O

12-

D is the closure of operator -A with the domain

known

space

=

lUlLp(RN).

f g(x)h(x)dx

(g, h)L2(Q)

L,(Q) (p : 1) is the usual Lebesgue
i
JUILP(o)
ff lu(x)lPdx}p.

RN be

RN. Then, the

COOO(D) equipped

an

space

open bounded domain with

H-'(Q)

with the

is the

norm

11,U112H

a

smooth

completion, according
"

(0)

=

ID2!U122(o)
L

+

to

boundary

Hausdorff,

JU122(11) (3
L

E

R)

or

let

of the

and the


corresponding scalar product ('5 *)HI(Q). Then, H-(Q)
are

well-known Sobolev

For

RN,

open set Q C

an

(in fact, they

spaces).

H'=Hs(RN), 11* 11s= V IIHI(RN)

defined in Q with the

Hilbert spaces

are

C(Q)

julc(o)

norm

and

is the space of continuous bounded functions

Ju(x)j.

sup

=

E0

C

=

For

functions

order 1,

C(RN)

IUIC

IUIC(RN)-

=

Xk is the Banach

arbitrary positive integer k,

an
v

and

of the

(x)

2,..., k

argument

1 in any finite

-

x

R, absolutely

E

interval,

for each of which the
k

11JUJIlk
Let

Ck (I; X), where k

and I C R is

functions

tEPI IILdtk

SU

k

S is the Schwartz space
the argument

Pk,I(U)

SUP

=

I

E

x

k
u

max

R

t

dxi

12'

Banach space with

a

< oo,

a norm

continuously

k bounded in

0,

II

differentiable

I, with the

11'

consisting

rapidly decreasing

Xl dk-

is finite:

norm

-

X is

0, 1, 2,...,

following

du

+

X with derivatives in t of order

I'UI'C' (I;X)

norm

JUIC

connected set, be the space of k times

a

I

:

u

=

=

consisting of

space

continuous with their derivatives of

with the

of
as

infinitely differentiable functions u(x)

Ix I

---).

that for any

oo so

topology generated by

k,

I

=

0, 1, 2,

of
...

the system of seminorms

PA;,I(')For

an

open domain Q C

where p > I and I
of the set

=

1, 2, 3,

Coc (n) equipped

...,

R' with

a

sufficiently smooth boundary,

be the standard Sobolev space

with the

being

the

let

W'(Q),
P

completion

norm

P

IQ

Ilullw"(fl)
Let

W1

=

P

lu(x)lp+

u(x, t)

.

P

COO(I; S),

where I is

defined for

t E I and such that for any

(x, t)

an

sup

=

H',

interval,

E R x

integer k, 1, rn

XER, tEI

11kj
aX

kl,---,A;N: A;I+... +kN=-l

-

W'(RN) Then, clearly W21

The space

functions

IU(X)

=

1p]

dx

1, 2, 3,

infinitely difFerentiable

to the space S for any fixed

> 0

Oxiat-

aXklv
N

is the set of

1, belonging

IXk8I+mU(XI0
I

I

...

< 00.


V

=

8

8

8XI

8- N

)

is the

gradient.

A > 0, Hpn,,(A)
infinitely differentiable functions defined in
For

of

(

equipped

an

integer

with the

n

> 0 and

is the

completion

R and

norm

I
2

A

I I U I I Hpn,,, (A)

-U(
=10I -2(X)
+

dnu(x)

of the linear space

periodic with the period A,

2

_) 1

dx

1


Chapter

1

Evolutionary equations.
existence

on

In this

chapter

we

consider several results

lems for the KdVE and NLSE that
remarks to this
prove the

chapter,

we

on

the

[to, T], satisfy

segment

the

well-posedness of initial-value prob-

used in the next sections.

are

mention additional literature

result, generally well-known, which

Gronwell's lemma. Let
a

Results

is

on

this

intensively exploited

inequality

y (t) :!

aIy (s)

ds +

b,

[to, T],

t E

to
a

and b

are

positive

Then,

constants.

t

b

y(s)ds

b

a(t-to)

<

and

_

a

y(t)

<

be a(t-to)

a

to

for

all t E

[to, T].

Proof. Let

Y(t)

f y(s)ds. Then,
to

'(t

< I
'

Integrating

this

inequality from to

to

t E

-

aY + b

t,

we

[to, T].

get

I

ln

< t

b

a

-

to,

t E

[to, T].

Hence

y(t) :5 aY(t)
and the Gronwell's lemma is

+ b <

proved.0

P.E. Zhidkov: LNM 1756, pp. 9 - 38, 2001
© Springer-Verlag Berlin Heidelberg 2001

bea(t-to),

t E

[to, T]

Now

we

further.

nonnegative function y(t), defined and

a

t

where

In Additional

subject.

continuous

on


EVOLUTIONARY

CHAPTERL

10

(generalized) Korteweg-de
(KdVE)

The

1.1

tion
In this section

RESULTS ON EXISTENCE

EQUATIONS.

establish two results

we

the

on

Vries pqua-

of the

Cauchy problem

vanish

as

well-posedness

for the KdVE

f (U)ux

+

ut

U

At

first,

we

consider the

introduce the

following

+ Uxxx

(x, 0)

uo

=

(x).

when solutions of this

case

definition of

R,

X, t E

0,

=

generalized

problem

solutions in this

IxI

--

00.

We

0 be

ar-

case.

00

Definition 1. 1.1 Let

E H

uo(.)

2, f(-)

U C((-m, m); R)

E

and

T1, T2

>

M_`1

C((-Ti, T2); H2) n C'((-Ti, T2); H-1) a generalized solution (or a H -solution)
Cauchy problem (1. 1. 1), (1.1.2) if u(., 0)
uo(-)
2

in the space H and after the substitution of this function in (1. 1.1) the equality holds
in the sense of the space H-1 for any t E (-TI) T2)-

bitrary. We call

a

function u(-, t)
2

E

of the

=

u(-, t)
T2); H-'). We

Remark 1.1.2 Since in Definition 1.1.1

f (U (-, t)) U."; (-, t)

+

U '.'.' E C ((- T1,

E

C((-Ti, T2); H 2) clearly we have

,

also note that in view of Definition

(-, t) is a generalized solution of the problem (1. 1. 1), (1. 1. 2) in an interval of
time (- T1, T2), then it is also a generalized solution of the problem (1. 1. 1), (1. 1. 2) in
1, 2, so that it is correct to speak about
any interval (-T,, T2) where 0 < Tj' < Ti, i
1. 1. 1 if

u

=:

a

continuation of

u(., t)

can

a

generalized

be continued

any finite interval I
we

call this solution
The first result

on

solution onto

wider interval of time. If

the entire real line t E R

containing

zero

global (defined
on

a

the

it is

a

H 2 -solution

function of time

solution in this

generalized

for all t E

well-posedness

as a

a

so

that for

interval,

then

R).

of the

is the follow-

problem

ing.

Theorem 1. 1.3 Let

f (.)

be

a

twice

continuously differentiable function satisfying

the estimate

If M I
with constants C
there exists
solution
mapuol

a

C(1

+

iul')

(1-1-3)

(0, 4) independent of u E R. Then, for
H'-solution
unique global
u(-,t) of the problem
> 0

and p E

continuously depends
)

<

u(-,t)

is

on

the initial data in the

continuousfromH'

into

sense

that

for

2
any uo E H

This

any T > 0 the

C((-T,T);H 2) n C'((-T, T); H-').

In


THE

1.1.

(GENERALIZED)

addition, if u(., t)

is

KORTEWEG-DE VRIES

H'-solution of the problem

a

(1. 1. 1), (1. 1. 2),

00

I 2 u'(x, t)

El(u(.,t))=

and

f 7(s)ds,

F(u)

0

t G

R,

i.

the

e.

junctionals E0

and

El

place
we

we

initial data. Instead of it

with respect to the

shall

exploit this

solutions in the

case

2 and

Chapter

TI, T2

=

independent of

Cauchy problem
certain result for the

for the standard KdVE with

x

problem

f (u)

=

u;

definition of

following

periodic.

are

Hpn,,(A) for

G

uo

u,

We call

0.

a

We introduce the

4.

when the initial data

f (u)

for the

consider

spatial variable

result in

Definition 1. 1.4 Let
>

determined and

conservation laws.

are

with

periodic

are

0

A result similar to Theorem 1.1.3 takes

periodic

F(u(x, t)) dx,

U

f f (s)ds

=

-

x

-00

U

7(u)

quantities

I

and

00

where

then the

11

00

u'(x,t)dx

Eo (u (-, t))

EQUATION (KDVE)

function u(-, t)

A

some

0 and inte-

>

C((-Ti, T2); Hln,,,(A)) n
C1 ((_ T1 T2 ). Hne-3 (A)) a solution of the problem (1.1.1),(1.1.2) periodic in x with
P
the period A > 0 (Or simply a periodic Hn-solution) if u(., 0)
uo(.) in the space
t
1.
holds
the
G
in
and,
for
sense
(-Ti, T2), equality (1. 1)
any
Hpn,,,(A)
of the space
-3
Hpn,,r
(A) after the substitution of the function u in it.
ger

n

,

7

>

a

E

r,

=

As
onto

earlier,

it is correct to

The result

the

on

well-posedness

considered in this book is the

Theorem 1.1.5 Let
any

integer

into

about

> 2

n

and

f (u)

uo E

sense

of the

global

for

a

periodic Hn-solution

solution

(defined for all t E R).
problem (1.1.1),(1.1.2) in the periodic case

=

u

so

Hpn,,(A)

that

deal with the standard KdVE. Then

we

there exists

This solution

that

continuation of

a

a

following.

of the problem
in the

speak

wider interval of time and about

a

a

for
unique global periodic H'-solution

continuously depends

any T > 0 the map uo

i

)

C((-T, T); Hpn,,,,(A)) n C'(( T, T); Hpn,,,-3 (A)).
-

u(-, t)

on

the initial data

is continuous

from

In addition there exists

a

Hpn,,(A)
sequence

of quantities
A

Eo (u)

A

I u'(x)dx,

Ej(u)

0

I

2

U2(X)

U

X

6

3(X) dx,

0

A

E.,, (u)

1 2 [U(n)12
X

+ CnU

[U(n-1)]2
X

qn(U7

...

(n-2))

)U X

dx,

n

=:

2,3,4,...,

0

where Cn

periodic

are

constants and qn

Hn-solution

u(.,t) of

are

polynomials,

the

problem (1.1.1),(1.1.2) (with f(u)

such that

for

any

integer
=

n

u)

> 2

and

a

the quanti-


EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE

CHAPTERL

12

ties

Our

do not

E,,(u(., t))

Eo(u(.,

conservation laws

proof

depend

for periodic

Wt

of the

f(W)W.'

+

+ WXXX +

1.1.6 Let

be

f(-)

get

(x, 0)

=

Wo

we

consider the

(1.1.4)

e>O,

(1.1.5)

(x)

the

infinitely differentiable function satisfying

E S and

following

statement which

f(-)

Then, for

(1.1.3).

be

E

E

to Coo ([0,

take the limit

we

1.1.7 Let

Proposition
estimate

first,

xER, t>O,

0,

=

X

an

(1. 1. 3).

At the second step,

fact,

(4)

6W

Then, for any uo
unique global solution which belongs

we

are

following:

Proposition

a

junctionals Eo,...' E,,

problem (1.1.1),(1.1.2):

W

estimate

the

e.

of Theorem 1.1.3 consists of several steps. At

following regularization

and prove the

t, i.

on

Hn-solutions.

an

any uo

c --4

is,

problem (1. 1-4),
(0, 1]
n); S) for an arbitrary n 1, 2,3,....
=

+0 in the problem

of course, of

an

(1. 1.4),(1.1.5).

independent

S there exists

a

In

interest.

infinitely differentiable function satisfying
E

the

(1. 1. 5) has

the

unique solution u(.,t)

the
E

00

U C-((-n, n); S) of the problem (1. 1. 1), (1. 1.2).
n=1

At the third step,

Now

we

using Proposition 1.1.7,

turn to

proving Proposition

Lemma 1.1.8 The system

we

prove Theorem 1.1.3.

1.1.6. We

begin with

the

following:

of seminorms
I

00

P1,0(u)

II (

=

2

2

dx1

)

dx

00

generates the topology in the

00

I

and

1

po,,(u)

x21u2(x)dx
-00

1

I

=

0, 1, 2,...

space S.

Proof follows from the relations
00

2

PM

21

1(u)

x

U(X)

(dM )

00

2

dm
dx

u(x)

dxm

,

00

Cl""

[X2,dmu(x) ]
dxm

-00

min m;211

:5

dx-

E
k=O

jI
-.

2

x

2(21-k) u 2

(x)dx

+

d2m-kU)
(dX2m-k

f

dx. 0

dx

<


THE

1.1.

Let

(GENERALIZED)
take

us

(1.1.4),(1.1.5) by

arbitrary

an

the iteration

Wnt + Wnxxx +

KORTEWEG-DE VRIES EQUATION

IEW(4)

(0, 1].

E

c

(x, t; c)

=- uo

C- ([0,

m); S),

w,

(x)

13

problem

procedure

Wn(Xj 0)
where

We construct solutions of the

-f(Wn-1)W(n-1)xi

==

nx

(KDVE)

C S.

=

t >

0,

n

(1.1-6)

2,3,4,...,

=

U0(X)j

the Fourier

Using

R,

E

X

(1.1.7)

transform,

easily show

one can

that

00

U

Wn E

n

=

2,3,4,....

M=1

into account

Taking
from

get

(1.1.3)

and

00

00

+

(194 )
Wn

dX

2

-00

-

-C0

I(

c

92 Wn
0XI

)2

dx-

-00

00

I (Wn

a4
+

Wn)

OX4

19Wn-1

f(Wn-1) -51- dx

<

-,E

-00

a2Wn

I

YX2

2

) (,94 )2
Wn

+

9X4

dx+

-00

+C1(jjUn-1jjP2+1

+

1)(IW(4) 12 + JWn 12)

to(,E)

>

<

nx

In view of the Gronwell's

C2(f)(1

<
-

21luol 122

ax,
for all t E

2,3,4,...,

n

,

axl

(1-1.8)

the existence

t E

[01to]-

(1.1.9)

2

2

c(E,

<
-

I

=

3, 4, 5,

[0, to] and n
2, 3, 4,
By using the
(1. 1.9) and embedding theorems, we get:

2 dt

11 W j 122).

lemma, inequality (1.1.8) immediately implies

19'Wn

a IWn

+

obtain the estimates

us now

I d

I JWn-1 112(p+l)
2

+

0 such that

JjWnj 122

estimate

dx

OX4

(X)

Let

we

0"

2

192Wn

I [Wn ( -WX ) 21
2

2 dt

=

embedding inequalities,

(1.1.6):

1 d

of to

Sobolev

applying

....

induction in

1, equation (1. 1. 6),

00

2

'9X 21

2

(f(Wn-1)W(n-1)x+Wnxxx+ EW(4) )dx
nx

<
-

-c

I

9X1+2

122_

00

W(1+2)
nx

and the estimates

Now,
m,

n

=

let

us

(1.1.10)

al-2

f( Wn-1)W(n-j)x]

ax 1-2

are

dx <

C(c, 1)

proved.

show the existence of tj

=

ti(c)

E

1, 2,3,...
00

0:5tsup

00

X

2mW 2dx

n

< c,

(m, c)

(O,to(c)]

such that for any


EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE

CHAPTER1.

14

where cl (m,
because the

c)

is

1

case m

can

independent of

constant

positive

a

be treated

(

X

2mW2dx

-c

=

n

of the

integrals

are

I

X2m W 2

=

> 2

derive

we

Pi

+

kinds:

following

00

I

Ci

xdx

n

00

Pi

m

-00

-00

where Pi

that

00

j

2 dt

can assume

by analogy. By integrating by parts,

00

d

We

n.

X2m-kW(,)
nx

101Wn
dx,

aXI

Pi

I

ci

=

X

2m-17(Wn_l )Wndx

-00

00

and
00

Pi

=

ciI X2m7(Wn-1)Wn.,dx
00

with

k,

I

0, 1,

=

where 0 < m'

2 and

r

0,

=

1. In view of the

(1.1.9)

and the estimates

< m,

the terms P- of the first type with I

2 and

=

inequality x2?n'

r

=

In

:5

I

-

K

are

is

> 0

a

+ C2 (1611

7n'7 M)

7

get the following for

we

00

X2rn (Wnxx) 2dx + c(c, m)

I

X2mW2dx + C, (c, m),
n

-00

-00

where K

E,X2m

0:

00

,

(1.1.10),

and

<

sufficiently large

trivial. Consider also the

constant. The estimates for I

case r

I

=

=

I and k

=

0

or

k

=

=

1.

0

or

Then,

I

=

1,

r

=

0

have

we

d
00

00

Pi :

X2mW2 xdx

E

C2 + ICil

d=-oo

<

X2m

C2'
d

d-i

[C3(c, K)W2

6

+
K

W2

xx] dx

+ -C2"

=

=-('Od-I

00

X2m

C21
where the constant K

estimated

by analogy.

>

0 is

For

[C3(c,

6

K )W2 +
n

K

n

+

C22

The terms Pi of other kinds

arbitrarily large.

example,

W2xx] dx

for the terms Pi of the second kind

we

can

be

have

00

Pj :5 C + C

f X2m(W2_,
n

+

W2n)dx.

-00

00

So,

we can

choose the constant K > 0

so

large that

the term

e

f

X

2mW 2x.,dx becomes
n

-00
00

larger

than the

sum

of all terms of the kind 2z

f

K

X

2mW2x.,dx. Therefore,
n

we

get

-00

00

(X,

I d

2 dt

I

X2m W 2dx
n

<
-

C(c, m)

1 +

00

I
-00

-

X

2m(W2-,
n

+

W

2)dx

n

(1.1.12)


THE

1.1.

(GENERALIZED)

The estimate

(1.1.11)

KORTEWEG-DE VRIES EQUATION

follows from

fWn}n=1,2,3,...

C([O, ti(r-)]; S). Also,

00

I d
2

the compactness of the sequence

the estimate

00

f gndx

Tt

15

(1.1.12).

Inequalities (1.1.9)-(1.1.11) immediately yield
in the space

(KDVE)

f [g2

C3 (E)

<

+

n-1

gn2]dx,

gn

=

Wn

Wn-1,

-

-00

is

implied by equation (1.1.6)

and the estimates

(1.1.9),(1.1.10). Therefore, the sequence fWn}n=1,2,3....
w(x, t; c) in the space Q0, t'(c)]; L2)
where t' E (0, ti] is sufficiently small. Hence, due to its compactness in C([O,
ti(c)]; S),
this sequence converges to w(x, t; c) in the space Q0, t'(c)]; S). Thus, taking the limit
in (1.1.6),(1.1.7) as n
oo, we get the local solvability of the problem (1-1-4),(1.1.5)
in the space C([O, t'(c)]; S).
converges to

a

function

-+

To show the

solutions

w'(x, t;,E)

I

W

=

W

uniqueness

_

w

2,

W2(X, t; 6)

and

easily

we

of this

solution,

derive from

Q0, T]; S)
equation (1.1.4):

t;

6)]2 dx

<

C(c)

-00

where
1
W

a
2

=

W

constant

C(c)

according

above class of the

Now

we

>

to the Gronwell's

depend

be

T > 0.

some

on

t; 6)] 2dx,

E R and t E

x

uniqueness

[0, T]. Therefore,

of

is proved.
estimates, uniform with respect

(1. 1.4), (1. 1. 5) of the class

solution

of

the

a

solution of the

and T > 0 there exists R2

>

of Proposition

t > 0.

Also, for

0 such that

for

to

c

E

an

Then

any C >

w(x, t; 6) E
lw(., t; 6)12 is a nonin0, p E (0, 4), R, > 0

arbitrary infinitely differentiable

condition

these constants C and p, any

and

C

T' E

solution

w(x, t; c)

(1-14),(1.1.5), satisfying
for

all t E

the condition

for

L 1. 6 be valid and let

function f (-), satisfying

(1. 1. 3) with
C([O, T']; S) (where

(0, 1],

C([O, T]; S).

problem (1-1.4), (1.1.5).

creasing function of the argument

a

Setting

problem (1.1.4),(1.1.5)

problem

a

I [W(X,

and the

lemma,

Lemma 1. 1.9 Let the assumptions

C([O, T]; S)

some

00

0 does not

want to make

solutions of the

with

00

I [W(X,

dt

suppose the existence of two

us

of the class

00

d

let

Jjw(-,0;c)jjj

E (0, 1]
arbitrary) of the problem
R1, one has Jjw(-,t;'E)jjj < R2

(0, T]
<

E

is

[0, T].

Proof. To prove the first statement of

our

00

I d
2 dt

I

Lemma,

it suffices to observe that

2. (X,
WX.

E) dx

CIO

2

W

(X, t; c)dx

=:

I

-e

t;

< 0.

-00

Let

problem

us

prove the second statement. For

a

solution

w(x, t; 6)

E

Q[0, T]; S)

by applying embedding theorems and the proved

of the

statement of


CHAPTERI.

16

the

lemma,

get:

we

00

00

I d
-

2

Tt

f

EQUATIONS. RESULTS ON EXISTENCE

EVOLUTIONARY

W2 dx

=

f

-e

X

-00

00

00

d

W2

dx +

X

F(w(x, t; r-))dx

Tt

-00

-6

I

<

-00

-00

00

00

<

5X rf-(w)]wxxxdx

r-

-

W

2,.,dx

X

+

CCc(l

+

d

1+1.+P3

6)

IWXXX12

+

I F(w(x,

-

dt

c))dx

t;

<

-00

00

00

d

f F(w(x,

dt

c))dx

+

C2F-

C1, C2

>

0

t;

-00

because I + 1 +
3
on

P- <
6

C', C"

constants

2, and where constants
from the

> 0

IUIC

<

Since due to condition

depend only

on

C,

R, and

p,

multiplicative inequalities

i

I

6

6

and

:5

C3(U

+

C4IUIPF2+2 1UXI

C'IU12 IUXXX12

(1.1.3) F(w)

JU, 12
2

+

a

i

3

3

CIIIU12 IUXXX12

!5

JUIp+2)

where p E

(0, 4),

have

we

by

theorems

embedding

00

1

F (w (x, t;

E)) dx
2

P2

IIU.12 + C5,

<

2

(1.1.14)

2

4

00

where the

following inequality

has been used:

JUIp+2
and where

C3, C4, C5 and C6

are

<

1
1+
2
p+2

positive

1

i-

1 U 122

C61U12

constants

p+2

depending only

on

Rj,C

Now the second statement of Lemma 1.1.9 follows from the first statement,
and

(1.1-13)

(1. 1. 14). n
Lemma I.1.10 Let C >

R2

and p.

=

R2 (C7 p, R1, T)

arbitrary

twice

>

0,

0 be the

p E

(0,4), R,

>

corresponding

0 and T > 0 be

constant

continuously differentiable function f (-)
sup

from

we

I u 1,, F2 (C, p, f, R1, T)

=

u(=-HI: jjujjj:5R2

arbitrary

Lemma L 1. 9.

and let
For

an

set:

sup

I f'(u) I

1U1
and

F3 (C, p, f, R1, T)

=

sup

If" (u) I

lul
(here W
large R3

<

>

oo

0.

in view

Then,

of the embedding of H1

into

C).

there exists R4 > 0 such that

Take

for

any

an

6

arbitrary sufficiently
E

(0, 1],

an

arbitrary

1.1.

KORTEWEG-DE VRIES

(GENERALIZED)

THE

EQUATION (KDVE)

infinitely differentiable function f(.), satisfying (1.1.3) with
f, R1, T) :! , R3

and p and such that F2 (C, p,
an

solution

arbitrary

w(x,t;c)

and F3 (C, p,

C([O,T];S) of

E

the above constants C

f, RI, T)

<

R3,

and

for

problem (1.1.4),(1.1.5) (T'

the

(0,T]), obeying the conditions Ijw(-,0;c)IIj :5 R,
I I W (') t 6)112 :5 R4 for all t E [0, T'].

17

IIW(',O;'E)112

and

:5 R3,

one

E

has

1

Proof. Take
c

E

and let

(0, 1]

constants R1, sufficiently large R3 C > 0, p E (0, 4), some
infinitely differentiable function f (.) satisfy condition (1. 1. 3) with

arbitrary

an

7

these constants C and p, F2 (C, p,
Lemma 1.1.9 and

inequality (1.1.15),

2

Tt

dx

=

dx

-

x

-00

CIO

00

a2

I

WXX

1 (W(4))2

(9X2

If (w)w.,]dx

-00

00

dx

00

I f1l (w)wxwxxdx , I f'(w)wxwx.,dx
2

3

-

-

x

00

R3. Using

00

I (U(4))2

-c

9X2

<

get

we

2

f (a2W )

-6

R3 and F3 (C, p, f, R1, T)

<

00

00

d
-

f, R1, T)

-

-00

-00

00

-6

I (W(4))2

(1.1.16)

dx + I, (w) + 12 (w).

x

00

Let

estimate the terms

us

inequality (1.1.15)

and Lemma

I, (w) :5 F31W

1.1.9,

13 IWxxI2

<

6

where the constant C,

and

we

I2(w) separately.

11(w), applying

For

get

CjF31WX122

depends only

0

>

Il(w)

IWXX12
2

<
-

CIR 2F
31W
2

=

12,

the constant from the

on

(1.1.17)

2

embedding

in-

equality (1. 1. 15).
Let

estimate

us

I2(w).

We have

00

5 d

12 (W)

6dt

CX)

j f (w)wx.,dx

5
+
6

00

1 (f (w) f, (w)

w3 +
x

f"(W) W3 w.,.,)dx+
x

-00

00

5

+

c

6

I [2f(w)w

2
x xx

+

fll(w)w'wxxxdx
x

+

4f(w)wxw.,xwxxx]dx.

(1.1.18)

-0.

The second term in the
as

from

II(w)

(1.1.16),

right-hand

so

that

we

side of this

equality

can

be estimated

completely

have

00

5
6

1 jf(W)f,(W)

3

W

-00

where F,

=

sup

IUI
If (u) 1.

x

+

fll(W)W3W. } dx
x

<

C2 (F, F2 + F3) (I wxx 122 + 1)

(1.1.19)


CHAPTERL

18

Due to

EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
the term from the

embedding theorems,
c

the coefficient

can

6

be estimated

right-hand

side of

(1. 1.18)

with

as

00

I (W(4))2

dx +

x

C3 (Fl, F2, F3, R2)

(1.1.20)

-

co

00

for

Finally,

43(w)

f 7(w)w.,xdx

6

have

we

-00

I3(W)

<

C(I

+

where the constant

:5 4 C2 R22(1 + C4R 2)2 +

C4jjWjjP1)jWj2 lWxxl2
C4

0

>

depends only

Lemma 1. 1. 11 Under the

constants from

on

In view of Lemma 1.1.9 and the estimates

(1.1.16)-(1.1.21),

1

4

1 WX. 12,
2

(1.1.21)

embedding inequalities.

Lemma 1.1.10 is

assumptions of Theorem L 1. 3 for

integer

any

proved.E]
I > 2 and

T > 0 there exists

c(l, T) > 0 such that for any c E (0, 1] and an arbitrary solution
E
w(x, t; 6)
C([O, T']; S) (here T' E (0, T] is arbitrary) of the problem (1. 1.4), (1. 1.5)
one has
:5 c(l, T) for all t E [0, T'].
9xj

121W 12

Proof. We

proved
case

I

the induction in 1. For I

use

2 the statement of Lemma is

with Lemma 1. 1. 10. Let this statement be valid for I

=

r

+ 1.

Using

00

d

I

=

(Or+lw

2 dt

axr+1

the

integration by parts

dx

-c

=

-0.0

I(

W

oXr+3

)

dx

I

-

-00

gXr+l

Cl(JJW112)

Lemma 1.1.12 Let the

integer
and

a

m

> 0 be

solution

(1. 1.4), (1. 1. 5)

the

C2(11W112)

+

E

+_1
_5XIWI

dx <

axr+l

2

2

assumptions of Theorem L1.3 be valid and let T

arbitrary. Then,there

w(x, t; 6)

If (W)WX]

-00

ar+IW
<

get

we

00

2

ar+

C([O, T'j; S),

following

already

2,..., r. Consider the

embedding theorems,

00

2

)

and

=

exists

c(m)

where T' E

estimate takes

> 0

(0, T]

such that
is

for

any

arbitrary, of

f

the

>

0 and

(0, 1]

E

problem

place:

00

f

x

2mW2 dx

c(m),

<

[0, T'j.

t E

00

Proof. First of
and

integer

r

> 0

all,

we

shall show that for any

such that for

u

E

S

we

00

IX12m-1
00

dnU
dXn

=:

1, 2, 3,

...

there exist C

>

0

(X)

2

)

m

have

dx < C

1

JJU112
r

+

IIUI12 +
2

f
-00

X2m u2(x) dx

1

(1.1.22)


where I

0

=

1

or

I

=

or

EQUATION (KDVE)

KORTEWEG-DE VRIES

(GENERALIZED)

THE

1.1.

1

2 and

x

(0, 1),

n

1

=

or n

2. For this

=

aim,

we use

19

the obvious

estimate

Jkl
Jk+xj

I
<
-

2

<

-

2,

k

-2,-3,-4,...

k

or

(1.1.23)

1, 2,3,...

multiplicative inequality

and the

a+1

2

dnu

WX_;)

dx

C(r)ju IL2
1- a,a+l)

<

(I

U

X(r) IL2(a,a+l)

(1.1.24)

1U1L2(a,.+1)

+

a

where

a

Due to

=

0, 1,::L2,... is arbitrary,

(1.1.24),

and

(1.1.23)

n

I

=

n

or

get for integer

we

I 1XI 2m-1(U(n))2

dx

=

I 1XI 2m-1(U(n))2

E +E ) I jXj2m-1(U(n))2
(k=-00

dx +

2

+

(JUjL2(k,k+1)

X

-1

1-n

U

2dX

X

k
r

.5 C111U112 + C"(r)2

2m-1

X

2

1-n
r

1

+

k=-00

f

2m-1

jkj

k=1

k+1

( 1:

E

I U (r) IL2(k,k+l)) 1

+

-2

X

+

-

k

k+1

1:
(k=-oo

C"(r)

dx <

X

k=1

C11 JU 112

arbitrary integer.

C o

2

<

is

k+1

2

X

X

> 2

r

2mnl-':

>

r

1

00

2 and

=

2

dx

jjUjj2

+

2M
x

u

(JUjL2(k,k+1)

+

I U X(r) 1L2(k,k+1))!!

<

k
00

C (r)

r

IJUI12
2

1

+

2m
X

U2dx

00

where
k

=

we

have used the trivial

1, 2,3,... and

x

Consider the

G

(k, k

+

inequality Ik 12m-1

1),

2 dt

I

(1.1.22)

I

2

wdx=-

00

2m

x

wf (w)wxdx

+ 2m

I

2m-1
X

W2dx
X

-c

third terms in the

the H61der's

right-hand

wwxxdx-

I

X2m WW (4)dx.
X

-00

00

1.1.94.1.11,

2m-1
X

00

00

_M

I
-00

-00

00

Due to Lemmas

and

-2, -3,

follows.

00

2m

x

=

expression

00

1 d

and

22m-1X2M-1 for k

<

side of this

inequality

equality

and

can

00

C1 + C2

I
00

2m
X

W2dX

(1.1.22),

be

the

obviously

first,

second and

estimated

as


20

CHAPTER1.

with

some

EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE

C1, C2

constants

integration by parts. So,

0. The last term

>

can

'X,

1 d

Tt

2

be estimated

by analogy after

an

to the estimate

we come

00

I

2mW2dX

X

<
-

f

C3 + C4

X2m W2 dx.

_C0

the statement of Lemma 1. 1.12 is

Thus,

Lemmas I.1.1 and 1.1.9-1.1.12
of the

ability
T*

whose

S-solution

and cannot be continued

[0,T*)

point

t

T*.

=

lim
t

corresponding

uniqueness has already been proved,

of time
the

immediately imply

problem (1.1.4),(1.1.5). Indeed,

0 such that the

>

proved. El

w(x, t; E)
T*-O

w(x, t; c)

the

Cauchy problem

for

6

0, i.

contradiction.

for all t > 0, solv-

Suppose

of the

the existence of

problem (1. 1.4),(1.1.5),

be continued onto the half-interval

can

on an

S understood in the

global,

let uo E S.

arbitrary right half-neighborhood

due to the above-indicated

Then,
ul E

=

the

sense

results, there

of the space S.

exists

of

limit

a

Thus, considering

equation (1. 1.4) with the initial data w(x, T*; E)
ul (x), we
the
local
of
this
interval
of
get
time [T*, T* + 6) with some
solvability
problem on an
>

e.

Let

get

we

us now

the existence of

obtained

a

prove

u(x, t)

the limit

by taking

So, Proposition

Proposition

solution

a

=

C([O, T); S)

as

for the

belonging

of the

C([O, T); S) for
problem (1.1.4),(1.1.5). The

Now,

Thus, Proposition

turn to

existence and

1.1.7 is

proving Theorem

can

can

be

proved,

be

> 0

can

(1. 1.4),(1.1.5).

any T > 0

to S for any fixed t in the domain t < 0

above construction.

problem

+0 in the problem

of this solution of the class
way

proved.0

1.1.7. Due to Lemmas 1.1.9-1.1.12 for any T

E

as c --+

1. 1.6 is

The

proved

uniqueness

uniqueness

in the

of

be

a

same

solution

proved by analogy with the

too.E1

1.1.3. Let

us take an arbitrary twice continuously differentiable function f () satisfying the estimate (1-1.3) and let If,,(')jn=1,2,3....

be a sequence of infinitely differentiable functions
satisfying the estimate (1.1.3) with

the

same

for any

we

constants C and p and

I, m

f Un0 1 n=1,2,3....

1,2,3,....

C S be

For each

n -+ oo.

of the

=

n

a
=

Let

us

sequence

1, 2, 3,

problem

...

converging

also take

C2((_M, M)

x

(- 1, 1); R)

2
and T
uo E H

>

0 and let

f (.)

arbitrary

in

converging to uo weakly in H 2 and strongly in H' as
by Un (X, t) E C- ((- T, T); S) we denote the solution

taken with

f

=

fn and

uo

=

Un.
0

It is clear that the

JR2(CIP) JjUnjjj,T)jn=1,2,3 where the function R2 > 0 is given by Lemma
bounded and let R2
sup R2 (C7 Pi Un 1, T) > 0 Let also R3
Sup I I Un
0 1 12-

sequence

....

1. 1. 9, is

to

,

=

-

n

Then, clearly W3

E

(0, oo).

We set R4

n
=

R4(R3)

where the function R4

R4(R3)

> 0