Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory

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: [email protected]

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 of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © 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

(generalized)

1.1

The

1.2

The nonlinear

1.3

On the

1.4

Additional

Chapter

2.

equations.

Results

on

existence

Vries equation Korteweg-de equation Schr6dinger (NLSE)

blowing

36 37

problems

Stationary

39

Existence

of solutions.

An ODE approach

Existence

of solutions.

A variational

2.3

The concentration-

2.4

On basis

2.5

Additional

3.3 3.4

Additional

3.2

Chapter

4.

of P.L.

49

Lions

56

of solutions

62

remarks

76

of solutions

79

of soliton-like

of kinks

for

of solutions

Invariant

42

method

method

compactness of systems

properties

Stability

Stability Stability Stability

3.1

10 26

up of solutions

2.2

3.

(KdVE)

remarks.

2.1

Chapter

9

solutions

the

80

KdVE

of the

90

NLSE

nonvanishing

as

jxj

remarks

oo

94 103

105

measures

4.1

On Gaussian

measures

4.2

An invariant

measure

4.3

An infinite

series

4.4

Additional

remarks

in Hilbert

for

the

of invariant

107

spaces

NLSE measures

118

for

the

KdVE

124

135

Bibliography

137

Index

147

Introduction

that

field

leading

are

approach properties makes more

possible

it

general

study).

the

present

In

qualitative

results

studies

dealt

with

on

of

travelling

investigate book,

author

to

the

problems

waves)

dynamical

stability

systems

twenty

substituted

solitary

of

in

by

following

(generalized)

main

material

is

These

topics.

these

equations,

(for example, consideration,

under

of invariant

construction

Vries

for

kinds

special

equations

Korteweg-de

generated

the

four

are

of

the

and the

waves,

of the

problems

initial-value

for

when solutions

arising are

There

a

evolutionary,

and

stationary

So, the selection

years.

interests.

of solutions

existence

both

in

of the

methods

and

problems

approach

(maybe

problems

of

class

some

surveys

scientific

author's

the

standing

of the

about

during

wider

blowing-up,

or

and this

etc.,

is

prob-

of various

stability

as

there

The latter

equations.

equations,

these

consideration,

under

equations

such

of solutions

hand,

other

well-posedness

the

subject,

this

known nonlinear

the

on

of differential on

by generated an essentially

the

and,

narrow

(on

problem

that

equations

these

currently

of

class

theory

behavior

systems

stationary or

problems for

of

related

mainly are

theory

he has

that

to

sufficiently

investigations

the

dynamical

of

the

time,

the

is

methods

by

problem;

scattering

inverse

qualitative

the

equations,

these

same

is

includes

particular

in

for

lems

called

approach,

another

At the

method

by this

PDEs solvable

of the

method

the

[89,94]).

example

for

see,

by

solvable

called

from

mono-

discoveries

field

this

problems

in the

example,

for

scattering

inverse

physicists

a

in view

Physical

mathematical

related

equations

nonlinear certain of studying possibility the to were quantum analyze developed

has grown into

-

and

observed,

are

One of the

partial

of nonlinear

kind

problems.

of the

novelty

consideration

[60].

special

a

mathematicians

of both

and of the

Makhankov

of

theory

the

-

solutions

attention

under

equations

V.G.

by

the

applications

the

to

graph

(PDEs) possessing

attracts

important

of its

of solitons

theory

the

30 years

equations

differential

large

last

the

During

measures

Schr6dinger

and nonlinear

equations. the

We consider

Ut

and the

Schr6dinger

nonlinear

+

Korteweg-de

f (U)U.,

+ UXXX

i is the

imaginary

and

complex

the

NLSE with

in the

unit,

second),

u

u(x, t)

=

t E

R,

x

0

(NLSE)

equation

iut + Au + f where

=

(KdVE)

equation

Vries

is

(Jul')u an

0,

unknown function the

E R in

=

case

of the

(real

in the

KdVE and

first

x

E

A

=

case

R' for N

a

positive

integer

N, f (-)

is

a

smooth

real

function

and

E k=1

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

82 aX2

k

2

Laplacian.

is the

As 2) respectively,

Chapter

u

qonsider

the

x) (it

equation

what

(as JxJ

oo

NLSE,

for

following A0

which

we

0

if

the

in

NLSE,

In what

is

supplement

with

nonlinear

Loo

_

elliptic

boundary

some

problem

A similar

problem

0(k)(00)

type

Difficulties

=

nontrivial

solutions

the

kinds).

In this

integer

any

argument

r

I > 0 there

exists

Ix I,

method

are

the

and

of

in finite

intervals

Chapter Lyapunov set

of the

qualitative

method.

results

sense.

X, equipped

to

3 is devoted

to the

Omitting a

those

some

distance

of

of P.L.

Lions.

being

differential

indicated

stability

2).

Chapter

of

functions

argument

is

the

as

a

f r

=

following:

of

W. for

function

of the

waves.

These

addition,

briefly

(ODEs)

in

consider

this

(for example,

the

chapter

we

L2)

in

Sturm-Liouville-type

one-dimensional

1,

==

of the

equations

we

basis

a

for

solitary

of

latter, In

N

0.

>

r

us

which,

existence

of the

of

property

the

non-uniqueness the

on

for

ordinary

example

of nonlinear

similar

with

the

half-line the

(see

solved

only

problem

our

the

into

order:

conditions

(for example,

f interesting of

00

NLSE with

speaking,

exist

proving

an

easily

depending

theory As

on

on

of

method

eigenfunctions

systems

solution

the

satisfying

waves

generally

functions

waves

second

will

--+

x

example,

for

KdVE and

solitary

case,

standing

kinds as

le,

Ej

X

sufficiently

solutions

I roots

methods

compactness recent

upon

a

exactly

the

solutions for

two

variational

concentrationtouch

has

this

In

when such

result

=

be

can

consider

us

typical

We consider the

0, 1, 2)

=

occurs

Let

the

case,

(k

0

of

uniqueness

of these

of the

into

if necessary,

0.

=

KdVE. For

the

and

when N > 2.

arise

above

for

arises

of existence

for

waves

function,

limits

possessing

conditions,

061--

the

0,

real

a

expression

standing

specifying,

equation

=

is

solutions

expression

f(1012)0

+

the

function

the

0

notation,

and

Chapter

In

the

KdVE and

c R and

follows,

bounded

a

w

this

just

substitute

we

of the

case

where

NLSE). Substituting

the

and

problem

Cauchy

of the

when

arises

introduce

with).

waves

the

f (s)

functions

constants).

positive

are

v

well-posedness It

wt)

-

to

dealt

solitary

obtain

we

x

of the

case

being

of the

for the KdVE and the NLSE used further.

O(w,

=

the

on

and

a

problem.

is convenient is

the -+

u

in the

equation

be called

results

stationary

waves

e `O(w,

=

the

(where

problems

value

travelling

for

e-a.,2

+S21

1 contains

initial-boundary we

2

as

1

physics,

for

important

following:

the

are

Isl"

2,

examples,

Typical

for

problems

above. of

details,

this

R(., .),

there

solitary means

exists

waves,

that, a

which is understood if for

unique

an

arbitrary

solution

u(t),

in the

uo

from

t >

0,

a

of

3

the

under

equation

T(t),

belonging

R, if for

that

by

obtained

equation

(in

the

us

introduce

Let

of the

functions

(in particular)

call

we

distance

special

a

argument

by

x,

functions

two

some

p becomes

stability

of

a

solitary

family

two

t

0 in the

=

metric

W2.

time,

they

can

usual

the

stability

of

many authors

For

taken

and

solitary in the

wit)

-

we

shall

with

s

=

of

following

if

they

H'

is

remark

the

here

that

family

two-parameter ob w

now

> 0

arbitrary

(x, t) and

=

b

are

close

at

t

all

for

the

=

approach

the

at

Sobolev

or

point spaces,

velocities

0 in the

wi

sense

of the

same

sense.

solitary

waves

developed

was

of

stability

this

He called

p.

his

(a, b).

other

t > 0 in the

stability

distance

possesses

Therefore

E

of

the

because

usually

non-equal

have

close

proved Later,

Lebesgue

as

are

to the

their

NLSE,

such

w

each

to

the

with

first,

p;

KdVE

r)

-

investigate

to

distance

the

close

to be close

has

wave.

consider

of the

waves

of

v(x

=_

functions

parameter

L02t),

> 0

respect

solitary

a

-

waves

[7]

paper

the

on X

t

verified

easily

to this

by

results.

distance

the

p should

be modified.

should

It

form:

(u,vEH')

T"Y

family,

0

X

u(x)

condition

it is natural

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

where

0' (w, x) 0

if

H1 consisting

space

equivalent

of

second,

x;

spaces

all

solitary

if two

f (s)

O(W2,

for

sense

the

respect

depending and

Sobolev

real

reasons,

in

functional

be

form

the

For several

KdVE with

pioneering

KdVE with

the

only

our

rule:

of classes

set

translations

same

same

in his

the

waves

x

the

in

then

p,

Benjamin

where

kink

a

was

in

diffusion

nonlinear

a

is called

H1, satisfying

from

v

of standard

sense

At the

distance T.B.

waves

[48]:

Piskunov

for

to

for

C

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

inf

space.

solitary

O(wl,

in the

,ERN

of the

up to

of

be close

cannot

and

solitary

of

N.S.

kink

a

wave

following

the

then

waves

solutions

any

be

a

KdVE is invariant

smooth

and

u

R, equivalent,

E

7-

distance

the

stability

of

solitary

a

has

one

distance

u(t), belonging R(T (t), u(t)) <

solution

and

stability

a

case

p(u,v)=

for

the

on

Petrovskii

respect

solution

a

the

to

x).

all

If

result

I.G.

b,

0, then

t >

with

any

R(T (0), u(O)) first

one-dimensional

for <

Kolmogorov,

A.N.

they proved

terminology, for

satisfying

any fixed

stable

that

0 such

>

historically

the

called

t > 0 is

b

exists

t > 0 and

Probably

t > 0.

any fixed

0 there

>

e

any fixed

X for

all

any

to X for

belonging

consideration, X for

to

complex

space,

usual

the

-r

E

R'

and

one-dimensional

7

E R.

To

NLSE with

cubic

fact,

this

clarify

f (s)

=

we

has

s

a

of solutions

V-2-w real at

t

exp

I i [bx

-

parameters. =

0 in the

(b

2 _

W)t]

Therefore, sense

of the

cosh[v/w-(x two

-

arbitrary

distance

p,

2bt)] solutions

cannot

from

be close

for

this all

4

t > 0 in

the

any two

standing

family

NLSE, of the

sense

of the

distance

40(x, t)

above

satisfy

in the

the

two values

to

correspond

of the

waves

corresponding close

they

if

sense

same

different

to

close

at

t

parameter all

p for

At the

to same

stability

of

By analogy,

W.

-

distance

of the

sense

nonequal

w,

t > 0.

definition

the

of V

values

0 in the

=

each

other,

time,

the

in the

p and

of the

sense

be

cannot

functions

of

distance

d. In the two

necessary")

O(x)

>

for

nonlinearity

a

0, that

Next,

for

In

Chapter

theory

For the energy

we

and, for

the

higher

problem,

tific

contacts

appearance

and of the

Roughly

literature.

(with respect d Q(0) > 0 dw

condition

opinion

that

stability

3 is

devoted

oo.

We present

that

"almost 0 and

=

speaking,

to the

distance

is satisfied. to the

respect

kinks

of kinks

distance

always

are

under

stable.

assumptions

objects

many

on

dynamical the

If

recurrence

construct

KdVE in the we

wishes

present

theorem

explains

measure

associated

when it is solvable

infinite

corresponding for

by

our

phenomenon with

the

the

equa-

for

dynamical partially.

conservation

method

of the

of invariant

measures

colleagues

and friends

for

that

contributed

sequence

to

observed

was

measure

in

according

stability

the

this

con-

is well-known

phenomenon

invariant

theory

application

which

by

the

of

inverse

associated

laws.

to

thank

discussions

present

case an

generated

measures

in

such

the

means

bounded

invariant

an

it

system

a

one

phenomenon

Fermi-Pasta-Ularn

have

we

on

direction.

invariant

applications

important

interesting

in this

open

of the

waves

and

new

a

constructing

attention

speaking,

Roughly

a

simulations, Poincar6

our

of

remain

of

Fermi-Pasta-Ulam

waves.

of

questions

solitary

of

stability

a

problem

the

have

stability

the

to

many

with

It is the

conservation

The author

lim 1XI-00

for the KdVE with

the

prove

We concentrate

equations. the

NLSE,

scattering

deal

trajectories

then

-+

however

we

of nonlinear

many "soliton"

system,

JxJ

These

By computer

tion.

with

4,

if the

widespread

we

satisfying

is stable

(and O(x)

sufficient

a

type.

as

physics.

of all

view,

present

physical

wave

of kinks

a

Chapter

systems.

with

Poisson

of

equations.

dynamical

the

is

be said

should

nected

general

part

non-vanishing

It

our

there

we

waves

in the

NLSE)

the

NLSE,

solitary

of

solitary

a

stability

of

point of

f

The last

type.

type

the

consider

this

function

NLSE

stability Q-criterion

general

of

we

Confirming the

KdVE and the

the

the

is called

Among physicists

p.

of

for

KdVE and to d for

the

p for

of the

cases

condition

with book.

all

his

them

have

the

useful

importantly

sciento

the

Notation

the

of the

case

otherwise,

stated

Unless

KdVE and

I for

(X1)

X

N

8'Xi

i=1

R+

[0,

=

For

E

=

positive

denote

positive

a

always

are

constants.

for

integer

NLSE.

the

Laplacian.

Q C RN

domain

defined

on

Lp(RN)

and

D with

Mp

=

the

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

norm

Lebesgue

lUlLp(RN).

f g(x)h(x)dx

for

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

=

g, h E

any

L2(9)-

11al 1212

R,

E

an

:

00

1: a2

<

n

F,

(a, b)12

00b

n=O

where

a

(ao,

=

For

==

that,

operator

in

Let

Q

=

RN.

...

)

b

=

(bo, bl,

Q C RN with

functions

of the

b,,,,...)

...,

a

E

argument

X

nb

n

12-

COOO(O)

boundary

smooth

a

n=O

E Q with

the

is

compact

space

supports

of

in Q.

COOO(R N).

D is the

known

i

domain

differentiable

COOO

set

an

open

an

infinitely

a,,...,

space

=

00

ja

=

in

+ oo).

(g, h)L2(Q)

12

real

NLSE.

the

for

introduced

RN.

is the

measurable

a

of functions

Lp

XN)

...

E

A

complex

KdVE and N is

the

of functions

spaces

C, C1, C2, C', C",...

Everywhere N

the

closure if Q is

-A with

of operator a

bounded

domain

the Q

or

=

domain

R'V,

then

Coc (Q) D is

in

L2(n)-

is well-

self-adjoint

positive

a

It

L2(Q)-

Q C RN be

Then,

the

COOO(D) equipped

an

space

with

open

bounded

H-'(Q) the

norm

is

the

domain

completion,

11,U112H (0) "

=

with

a

smooth

according

ID2!U122(o) L

+

to

boundary

Hausdorff,

JU122(11) (3 L

E

R)

or

let

of the

and the

are

H'=Hs(RN), For

defined

C

C(R N)

functions

an v

Q C norm

IUIC

and

of the

argument

is the

and

C R is

I

functions

a

I'UI'C' (I;X)

norm

R, absolutely

E

x

Xk is the

k,

interval,

for

of which

argument

Pk,I(U)

=

I

SUP

R

E

x

of

space

k

t

u

of

their

derivatives

of

following

is finite:

norm

12' with

space

a

k bounded

0,

norm

II

differentiable

continuously

k times

I, with

in

the

11' of

consisting with

consisting

space

with the

Banach

a

in t of order

decreasing

< oo,

dxi

X is

tEPI IIL dtk

rapidly

du

+

be the

SU

space

Xl dk -

functions

-

0, 1, 2,...,

set,

k

Schwartz

=

Banach

continuous

each

JUIC

=

derivatives

max

S is the the

k

X with

I

:

u

where

connected

bounded

E0

11JUJIlk Ck (I; X),

of continuous

space

k

Let

they

Ju(x)j.

sup

=

integer

1 in any finite

-

(in fact,

spaces

IUIC(RN)-

=

positive

k

Hilbert

are

and

C(Q)

julc(o)

arbitrary

(x)

1, 2,...,

RN,

set

the

open

an

=

For

order

11* 11s= V IIHI(RN)

in Q with

H-(Q)

Then,

spaces).

Sobolev

well-known

('5 *)HI(Q).

product

scalar

corresponding

infinitely

Ix I

as

topology

the

differentiable oo

---).

so

for

that

generated

by

the

u(x)

functions I

k,

any

0, 1, 2,

=

of ...

of seminorms

system

PA;,I(')For where

p > I

of the

set

and

I

=

1, 2, 3,

Coc (n) equipped

IQ

Ilullw"(fl) Let

Q C R'

domain

open

an

W1

with

...,

with

the

lu(x)lp+ kl,---,A;N:

-

COO(I; S),

The space

functions

u(x, t)

t E I and

such

defined

that

for

where

for any

I is

(x, t)

sup

an

=

+kN=-l

H',

interval,

E R

x

k, 1,

rn

integer

XER, tEI

A;I+...

W21

.

P

smooth

Sobolev

space

W'(Q),

boundary,

let

being

completion

the

P

norm

W'(R N) Then, clearly

=

P

sufficiently

a

standard

be the

=

aX

> 0

Oxiat-

...

aXklv N

1p]

P

dx

1, 2, 3,

is the

1, belonging

IXk8I+mU(XI0 I

I

11kj

IU(X)

< 00.

set

of

to

the

infinitely space

difFerentiable S for

any

fixed

V

=

For

of

infinitely

equipped

( an

8

8

8- N

8XI

integer

n

)

> 0 and

differentiable with

the

is the

gradient. A

>

0,

Hpn,,(A)

defined

functions

is the in

R and

completion

of the

periodic

with

norm

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

I I U I I Hpn,,,

(A)

+

_)

dnu(x)

2

1 1 dx

I 2

the

linear

period

space

A,

Chapter

1

Evolutionary

In this

chapter the

for

remarks

to

the

prove

we

consider

NLSE that

chapter,

this

result,

we

generally lemma.

a

the

inequality

the

used

well-posedness

which

the

in

additional

mention

Let

on

are

well-known,

[to, T], satisfy

segment

results

several

KdVE and

Gronwell's a

Results

existence

on

lems

equations.

on

this

exploited

intensively

y(t),

function

nonnegative

sections.

next

literature is

of initial-value

defined

t

y

(t)

aIy

:!

(s)

ds +

b,

[to, T],

t E

to

where

a

and b

are

positive

Then,

constants.

t

y(s)ds

<

b

b

a(t-to)

and

_

a

a

y(t)

<

be a(t-to)

to

for

all

t E

Proof.

[to, T]. Let

f y(s)ds.

Y(t)

Then,

to

'(t aY + b

Integrating

this

inequality I

< I -

t E

'

from to to t,

we

ln

< t

b

a

[to, T].

get -

to,

t E

[to, T].

Hence

y(t) and the

Gronwell's

:5

lemma is

aY(t) proved.0

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

+ b <

bea(t-to),

t E

[to, T]

prob-

In

Additional

subject.

Now

we

further.

and continuous

on

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

10

(generalized) (KdVE)

The

1.1

tion In this

section

the

for

two results

establish

we

Korteweg-de

pqua-

of the

Cauchy problem

vanish

as

KdVE ut

f (U)ux

+

U

first,

At

well-posedness

the

on

Vries

we

consider

following

the

introduce

the

+ Uxxx

(x, 0)

uo

=

when solutions

case

definition

0,

=

(x). problem

of this

solutions

in this

U C((-m,

m); R)

generalized

of

R,

X, t E

IxI

We

00.

--

case.

00

Definition

1. 1.1

2, f(-)

uo(.)

Let

E H

E

and

T1, T2

0 be

>

ar-

M_`1

We call

bitrary. in the

space

the

sense

in

(or

solution

ized

1. 1. 1 if

any a

E

of

space

H-1 for

1.1.2

Since

in Definition

of

n

1.

substitution

the

the

any

u(-, t)

1.1.1

E

U.";

can

finite call

generalized

a

be continued interval

I

result

The first

the

containing

global

solution

this

on

solution

on

the

onto

real

entire it

zero

(defined

is

all

for

a

line a

wider t E R

(1.

1.

of the that

in view

1), (1.

1.

2)

of time. function

an

(1.

problem

interval

1), (1.

of

2)

1.

speak

in

about

H2 -solution

a

of time in this

solution

1.

to

If

have

we

of Definition

in

it is correct

a

clearly

,

that

interval as

H2)

that

so

interval,

for then

R). problem

of the

well-posedness

T2);

note

generalized t E

=

C((-Ti,

u

of

C'((-Ti,

T2); H-1) a generalCauchy problem (1. uo(-) 1), (1.1.2) if u(., 0) holds in (1. 1.1) the equality of this function t E (-TI) T2)T2); H2)

C((-Ti,

the

=:

u(., t) any

after

H and

continuation

we

H2 -solution)

We also (-, t) + U '.'.' E C ((- T1, T2); H-'). of the solution is a problem generalized (-, t) solution (- T1, T2), then it is also a generalized interval 1, 2, so (-T,, T2) where 0 < Tj' < Ti, i

f (U (-, t)) time

function

a

2

Remark

u(-, t)

a

is the

follow-

ing. Theorem

the

1. 1.3

Let

f (.)

be

a

twice

If MI < C(1 with

differentiable

continuously

function

satisfying

estimate

C

constants

there

exists

solution mapuol

a

> 0

unique

)

u(-,t)

is

iul')

(1-1-3)

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

and p E

global depends

continuously

+

on

the

continuousfromH'

initial

data into

in

the

C((-T,T);H

sense

2)

that n

for

C'((-T,

any

uo

2

E H

This any T > 0 the

T); H-').

In

(GENERALIZED)

THE

1.1.

if u(., t)

addition,

is

KORTEWEG-DEVRIES

H'-solution

a

of

(1.

problem

the

00

u'(x,t)dx

I

R,

i.

A result with

periodic

f 7(s)ds,

F(u)

and

junctionals

shall

solutions

data.

respect

exploit

case

Definition

takes

the

F(u(x,

-

determined

are

t))

dx,

independent

and

of

when the

TI, T2

Cauchy problem

the

for

x

data

standard

the

the

for

following

the

problem f (u)

KdVE with

=

definition

u;

of

periodic.

are

Hpn,,(A)

G

uo

result

certain

a

We introduce

4.

u,

=

for

consider

variable

initial

f (u)

Let

place we

Chapter

in

laws.

conservation

are

of it

spatial

result

1. 1.4

and

El

1.1.3

Instead

to

this

in the

E0 and

Theorem

to

initial

with

quantities

0

similar

periodic we

the

e.

the

-00

0

t G

then

U

f f (s)ds

=

t)

x

00

7(u)

2),

1.

I 2 u'(x,

El(u(.,t))=

and

U

where

1), (1.

1.

11

00

(-, t))

Eo (u

EQUATION(KDVE)

for

A

some

and

0

>

inte-

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

2

>

n

C1 ((_ T1 , T2 ). 7

We call

0.

>

a

solution

r,

function

E

the

=

As onto

earlier,

The result

interval

on

considered

the

integer

of

the

in

the

into

n

speak

to

of time

1.1.5

f (u)

Let

and

uo E

=

C((-T,

u

so

Hpn,,(A)

for

n

a

periodic

(defined

(1.1.1),(1.1.2)

C'((

-

deal

we

exists

solution

T > 0 the

any

T); Hpn,, ,,(A))

that there

This that

global

of

solution

Hn-solution for

the

in

all

t

R).

E

periodic

case

following.

problem sense

continuation

a a

problem

of the

book is the

> 2

about

about

and

well-posedness

in this

Theorem any

is correct

it

wider

a

Hpn,,,

the

standard

global

unique

a

-3

i

u(-, t)

)

(A)).

KdVE.

periodic

depends

continuously

map uo

T, T);

with

is

on

there

for

H'-solution the

initial

from

continuous

In addition

Then

exists

a

data

Hpn,,(A) sequence

of quantities A

Eo (u)

A

I u'(x)dx,

Ej(u)

0

I

2

U2(X) X

6

3(X)

dx,

U

0

A

E.,,

(u)

12

[U(n)12 X

+ CnU

[U(n-1)]2 X

qn(U7

...

(n-2))

dx,

)U X

n

=:

2,3,4,...,

0

where

periodic

Cn

are

constants

Hn-solution

and qn

u(.,t)

of

are

polynomials,

the

problem

such

(1.1.1),(1.1.2)

that

for

(with

any

integer

f(u)

=

n

u)

> 2

the

and

quanti-

a

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

12

E,,(u(.,

Eo(u(.,

ties

regularization

following

Wt

1.1.3

f(W)W.'

consists

Proposition

(1.

3).

1.

At the

fact,

1.1.6

global

unique

(4)

+ WXXX+

step,

6W

first,

At

steps.

xER,

0,

=

X

E,,

consider

we

t>O,

are

the

(1.1.4)

e>O,

(1.1.5)

(x)

Wo

=

take

f(-)

to

the

limit

E

uo

function

satisfying

the

(0, 1] the problem (1. 1-4), (1. 1. 5) has n 1, 2,3,.... ([0, n); S) for an arbitrary

c

E

=

is,

of course,

of

exists

independent

an

differentiable

S there

(1. 1.4),(1.1.5).

problem

+0 in the

--4

infinitely

an

any

differentiable E

Coo

which

be

for

Then,

S and

belongs

statement

Let

infinitely

an

uo E

any

we

1.1.7

(1.1.3).

be

which

following

Proposition estimate

for

Then,

second

f(-)

Let

solution

the

get

we

Eo,...'

following:

the

estimate

junctionals

the

(1.1.1),(1.1.2):

problem

(x, 0)

a

e.

of several

W

and prove

i.

t,

on

Hn-solutions.

of the

+

depend

do not

of Theorem

proof

Our

t))

for periodic

laws

conservation

function a

satisfying

the

u(.,t)

solution

unique

In

interest.

00

U C-((-n,

n); S) of

the

(1.

problem

1.

1), (1. 1.2).

n=1

At the

third

Now

using

step,

II ( 00

P1,0(u)

=

)

dx1

2

dx

00

the

generates Proof

of

The system

Lemma 1.1.8

topology

follows

in

from 00

2

PM 1(u)

x

,

21

the the

we

Proposition

proving

to

turn

we

1.1.7,

Proposition

I

I

1

2

and

following:

the

po,,(u)

1

00

x21u2(x)dx -00

I

00

2

u(x)

dx

dxm

dm

dx-

[X2,dmu(x) ] dxm

-00

k=O

0, 1, 2,...

S.

space

(dM ) E

=

relations

U(X)

min m;211

Cl""

with

seminorms

00

:5

We begin

1.1.6.

1.1.3.

Theorem

prove

j -.

I

x

2(21-k)

u

2

(x)dx

+

d2m-kU) (dX2m-k

2

f

dx.

0

dx

<

E

(GENERALIZED)

THE

1.1.

Let

take

us

(1.1.4),(1.1.5)

by

arbitrary

an

the

Wnt + Wnxxx +

KORTEWEG-DE VRIES EQUATION(KDVE) We construct

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

==

nx

(x, t; c)

w,

=-

(x)

uo

Using

C S.

the

R,

E

X

Wn(Xj 0) where

solutions

the

of

problem

procedure

iteration

IEW(4)

(0, 1].

E

c

13

0,

t >

n

(1.1-6)

2,3,4,...,

=

U0(X)j

=

Fourier

(1.1.7)

transform,

one

easily

can

show that

00

U

Wn E

([0, m); S),

C-

n

2,3,4,....

=

M=1

Taking

into

(1.1.3)

account

applying

and

Sobolev

embedding

inequalities,

we

get from (1.1.6):

[Wn ( ) 21

00

I

1 d 2 dt

2

-00

00

192Wn

+

-WX2

(194 ) Wn

dX

I (Wn

OX4

+

a4

00

Wn) f(Wn-1)

19Wn-1

dx <

-51-

OX4

In view of the

to(,E)

=

>

+

1)(IW(4) 12

Gronwell's 0 such

Let

us

now

JWn 12)

+

nx

obtain

21luol 122

< -

the

I

-,E

YX2

C2(f)(1

+

) (,94 )2 2

Wn

+

a IWn axl

9X4

'9X

2

(1.1.10)

estimates

let

112(p+l)

dx+

immediately

2,3,4,...,

t E

11 W j 122).

+

2

implies

the

n

,

[01to]-

us

(1-1.8) existence

(1.1.9)

2

< -

c(E,

I

=

3, 4, 5, induction

in

1, equation

00

2

nx

Now,

I JWn-1

(1.1.8)

....

W(1+2)

n

dx-

2

E

00

m,

a2Wn

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

t

estimate

=

0XI

estimates

ax,

and the

<

lemma, inequality

19'Wn

2 dt

)2

92 Wn

that

JjWnj 122

I d

I (

-00

+C1(jjUn-1jjP2+1

all

c

-

-00

-00

for

dx

-C0

(X)

of to

0"

2

show

are

the

21

EW(4) )dx

(f(Wn-1)W(n-1)x+Wnxxx+

al-2 ax

1-2

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

nx

dx <

< -

-c

I

(1.

9X1+2

1.

6),

122_

C(c, 1)

proved. existence

of

tj

=

ti(c)

E

1, 2,3,...

0:5t
00

2mW2dx

sup

X

00

n

< c,

(m, c)

(O,to(c)]

such

that

for

any

EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

CHAPTER1.

14

where

(m, c)

cl

the

because

is

positive

a

case

1

m

independent

constant

can

d

j

(

2mW2dx

X

I

-c

=

n

-00

Pi

where

integrals

are

I

that

assume

by parts,

By integrating

m

> 2

derive

we

X2mW2 xdx

Pi

+

n

kinds:

following

of the

Ci

=

can

-00

00

Pi

We

n.

00

00

2 dt

of

by analogy.

be treated

101Wn

X2m-kW(,)

nx

aXI

00

dx,

Pi

I

ci

=

2m-17(Wn_l )Wndx

X

-00

00

and Pi

=

00

ciI

X2m7(Wn-1)Wn.,dx 00

k,

with where the

I

=

0, 1,

0 < m'

terms

2 and

r

and

< m,

P- of the

0,

=

first

the

with

type

(1.1.9)

estimates

I

r

In

:5

K

I

are

K > 0 is

trivial.

large the

also

d=-oo

I

X2mW2dx + C, (c, m), n

r

I

=

I and

=

<

k

=

X2m

C2' d

d-i

0

k

or

=

I

0

=

I

or

Then,

1.

=

1,

r

0

=

have

we

d

00

X2mW2xdx

for

The estimates

constant.

case

00

E

C2 + ICil

Pi :

7

for

following

-00

sufficiently

a

Consider

7n'7 M)

+ C2 (1611

the

get

we

00

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

-00

where

E,X2m

<

0:

=

00

, -

(1.1.10),

and

2 and

=

x2?n'

inequality

In view of the

1.

[C3(c,

K)W2 +

6

K

xx]

W2

dx + -C2"

=

=-('Od-I

00

where

the

estimated

K

constant

by analogy.

[C3(c,

X2m

C21 0 is

>

For

the

for

W2xx]

K

large.

arbitrarily

example,

6

K)W2 + n

n

dx +

The terms

Pi of the

terms

C22

Pi of other second

kinds

kind

we

can

be

have

00

Pj :5 C + C

f

X2m(W2_,

+

large

the term

n

W2 n)dx.

-00

00

So,

we can

choose

the

K> 0

constant

so

that

e

f

2mW2x.,dx

X

n

becomes

-00 00

larger

than

the

sum

of all

terms

of the

kind

f

2z K

2mW2x.,dx.

X

Therefore,

n

we

get

-00

(X,

I d

2 dt

I 00

00

X2mW2dx n

< -

C(c, m)

1 + -

I -00

2m(W2-,

X

n

+

2)dx

W n

(1.1.12)

(GENERALIZED)

THE

1.1.

KORTEWEG-DE VRIES EQUATION(KDVE)

(1.1.11) follows from (1.1.12). Inequalities (1.1.9)-(1.1.11) immediately yield in the space C([O, ti(r-)]; fWn}n=1,2,3,... S). Also, the

15

The estimate

00

Tt

2

of the

compactness

sequence

00

f

I d

the

estimate

gndx

f

C3 (E)

<

[g2

gn2]dx,

+

n-1

gn

Wn

=

Wn-1,

-

-00

is

implied

quence

where this in

t'

sequence

as

n

C([O, t'(c)];

space

To show

the

I W= W

2,

we

w

_

we

estimates

function

a

small.

Hence, in the

the

get

(1.1.9),(1.1.10). w(x, t; c) in the

due to its

solvability

easily

solution,

of this

W2(X,

t; 6)

derive

from

and

of the

I

dt

above

C(c)

constant

a 2 W

=

according

class Now

solutions

of the

>

dx <

I

C(c)

f (-), solution

a

(1-14),(1.1.5), for

all

of

the

R2

satisfying

w(x, t; c)

Setting

T > 0.

some

5)

t

(1.

uniform

of the

1.

with

of

Therefore,

solution

a

for

for

an

C([O, T']; S) (where T' the condition Jjw(-,0;c)jjj

C

E

Then

C

any

these

with

respect

L 1. 6 be valid

(1.1.5).

Also,

that

3)

[0, T].

E

of the

to

E

c

(0, 1],

for

C([O, T]; S).

class

(1-1.4),

> 0.

t

uniqueness

of Proposition

0 such

>

and the

estimates, 1.

E R and

x

proved.

is

problem

condition

satisfying

t E

the

argument

exists

of two

[W(X, t; 6)] 2dx,

on

lemma,

assumptions

solution

of

T > 0 there

function and

a

function

creasing and

be

some

(1. 1.4), (1.

problem

depend

not

(1.1.4),(1.1.5)

Lemma 1. 1.9 Let the

C([O, T]; S)

0 does

Gronwell's

make

to

with

existence

00

problem

want

we

the

suppose

us

00

[W(X, t; 6)]2

the

to

of the

S). Thus, taking the limit problem (1-1-4),(1.1.5)

Q0, T]; S) (1.1.4):

equation

-00

1 W

se-

L2) S),

of the

let

class

00

d

where

the

Q0, t'(c)]; C([O, ti(c)];

in

compactness

Q0, t'(c)];

space

local

Therefore, space

S).

uniqueness

w'(x, t;,E)

solutions

oo,

-+

to

w(x, t; c)

to

converges

the

and

fWn}n=1,2,3.... converges E (0, ti] is sufficiently

(1.1.6),(1.1.7)

in the

(1.1.6)

by equation

>

w(x, t; 6)

and let

lw(., t; 6)12 p E (0, 4), is

nonin-

a

0,

R,

> 0

arbitrary

infinitely

differentiable

constants

C and p,

any

(0, T] <

is

R1,

arbitrary) has

one

of

the

E

E

E

(0, 1]

problem

Jjw(-,t;'E)jjj

<

R2

[0, T].

Proof.

To prove

first

the

of

statement

our

00

I d 2 dt

I

Lemma,

it

2. (X, WX.

E) dx

suffices

to

observe

that

CIO

2

W

(X, t; c)dx

=:

-e

I

t;

< 0.

-00

Let

problem

us

prove

the

second

statement.

by applying

For

embedding

a

solution

theorems

w(x, t; 6) and the

E

Q[0, T]; S)

proved

of the

statement

of

EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

CHAPTERI.

16

lemma,

the

get:

we

00

I d

Tt

-

2

00

f

W2dx

f

-e

=

X

-00

00

00

d

W2 dx +

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

Tt

X

-00

-00

00

I

-6

<

-00

00

<

5X rf-(w)]wxxxdx

r-

-

2,.,dx

CCc(l

+

W

X

1+1.+P3

I

d

6)

IWXXX12

+

+

-

dt

00

F(w(x,

t;

c))dx

<

-00

00

f

d dt

F(w(x, t; c))dx

+

C2F-

>

0

-00

I + 1 + 3

because on

P-

2, and where

<

6

C',

constants

C"

> 0 from

IUIC Since

due

i

I

6

6

and

:5

C3(U

+

C4IUIPF2+2 1UXI

F(w)

depend

only

a

i

3

3

on

C,

R, and

p,

inequalities

multiplicative

C'IU12 IUXXX12

(1.1.3)

condition

to

embedding

<

C1, C2

constants

the

JU, 12 2

CIIIU12 IUXXX12

!5

JUIp+2)

+

where

(0, 4),

p E

have

we

by

theorems 00

1

F (w

(x, t; E))

dx


2

IIU.12

P2

<

2

4

2

(1.1.14)

C5,

+

00

where

following

the

inequality

has

JUIp+2 C3, C4, C5 and C6

and where Now the and

(1.

1.

second

14).

R2

R2 (C7

arbitrary

<

1+ 2

C61U12

positive

are

1

1 U 122

p+2

depending

constants

follows

of Lemma 1.1.9

statement

i-

1

p+2

from

the

only first

on

Rj,C

and p.

(1.1-13)

statement,

n

Lemma I.1.10 =

used:

been

p,

Let

R1, T)

C

continuously

twice

0,

>

> 0 be the

u(=-HI:

p E

(0,4),

>

corresponding

I u 1,,

0 and

constant

f (-)

function

differentiable sup jjujjj:5R2

R,

F2 (C,

f, R1, T)

p,

T > 0 be

from we

=

arbitrary

Lemma L 1. 9.

and

let

For

an

set:

sup

I f'(u)

I

1U1<W

and

F3 (C,

p,

f, R1, T)

sup

=

If" (u) I

lul<W

(here W< large

R3

>

oo

0.

in

view

Then,

of

the

there

of H1

embedding exists

R4

>

into

0 such

that

C). for

Take any

an

6

arbitrary E

(0, 1],

sufficiently an

arbitrary

(GENERALIZED)

THE

1.1.

differentiable

infinitely and

such

p and

arbitrary

an

(0,T]), I I W(')

t

1

6)112

c

(0, 1]

E

the

conditions

:5 R4 for

all

and let

infinitely

an

00

f (a2W )

d

Tt

-

2

p,

we

dx

9X2

(U(4))2

dx

x

I

-

CIO

00

1

dx

x

00

(T'

<

one

(0, 4), (1. 1. 3) R3.

E

has

some

with

Using

a2 WXX

(9X2

If (w)w.,]dx

00

I

-

f, R1, T)

for

and

-00

00

(W(4))2

p E

condition

p,

C

constants

R3,

<

0,

>

17

get 00

I

-c

=

-00

-6

f (.) satisfy

00

2

f, RI, T)

R3 7 C

large

R3 and F3 (C,

<

above

(1.1.4),(1.1.5) :5 R3, IIW(',O;'E)112

and

R,

:5

p,

the

problem

the

function

f, R1, T)

(1.1.15),

inequality

and

Lemma 1.1.9

of

R1, sufficiently

differentiable

F2 (C,

C and p,

constants

C([O,T];S) Ijw(-,0;c)IIj [0, T'].

constants

with

R3 and F3 (C,

:! ,

E

t E

arbitrary

Take

Proof.

these

w(x,t;c)

(1.1.3)

satisfying

f, R1, T)

p,

solution

obeying

f(.),

function F2 (C,

that

EQUATION(KDVE)

KORTEWEG-DE VRIES

3 f1l (w)wxwxxdx

,I

2

f'(w)wxwx.,dx

-

-

-00

-00

00

I (W(4))2

-6

(w)

dx + I,

x

+ 12

(1.1.16)

(w).

00

Let

(1.1.15)

inequality

the

where

(1.

equality

and Lemma

C,

constant

Let

Il(w)

terms

:5 F31W 136

(w)

I,

the

estimate

us

1.1.9,

IWxxI2

we

separately.

11(w),

For

applying

get

CjF 31WX122

IWXX12

only

the

depends

0

>

<

I2(w)

and

on

2

< -

CIR 2F 31W 2

=

constant

from

12,

(1.1.17)

2

the

embedding

in-

15).

1.

estimate

us

I2(w).

We have

00

5 d

12 (W)

6dt

CX)

j

f (w)wx.,dx

+

00

5 6

1 (f (w) f, (w)

w3 + x

f"(W) W3w.,.,)dx+ x

-00

00

+

I

5

6

c

[2f (w)w

2 x xx

+

fll(w)w'wxxxdx

+

x

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

(1.1.18)

-0.

The second as

II(w)

term

from

right-hand

in the

(1.1.16),

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. x

} dx

<

C2 (F, F2 + F3) (I

wxx

122

+

1)

(1.1.19)

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

18

embedding

Due to

the

c

coefficient

6

theorems,

the

be estimated

can

from

term

right-hand

the

side

(1. 1.18)

of

with

as

00

I

(W(4))2

C3 (Fl,

dx +

x

F2, F3, R2)

(1.1.20)

-

co

00

Finally,

43(w)

for

f 7(w)w.,xdx

6

have

we

-00

I3(W) where

the

In view

C(I

<

C4jjWjjP1)jWj2

+

C4

constant

and the

Lemma 1. 1. 11 Under

E

9xj

Proof.

proved I

case

We use the

with =

00

axr+1

2 dt

Let

m

0 be

>

this

dx

ar+

oXr+3

)

1.

Let

5)

the

+

dx

I

-

proved.E] I > 2 and

arbitrary

solution

(1. 1.4), (1. 1.5)

problem

of Lemma is =

2,...,

already

Consider

r.

theorems,

the

get

we

dx If (W)WX] +_1 _5XI WI

<

axr+l

c(m)

> 0

where

T' E

(0, T]

takes

estimate

2

2

of Theorem L1.3

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

following

I

gXr+l

exists

Then,there E

for

ar+IW

C2(11W112)

assumptions

arbitrary.

an

is

-00

Cl(JJW112)

the

inequalities.

00

2

W

the

embedding

and

(1.1.21)

integer

any

statement

be valid

by parts

for

and

of

2 the

=

statement

I (

-c

=

(0, 1]

E

c

L 1. 3

arbitrary) I

For

00

w(x, t; 6)

solution

(1. 1.4), (1.

in 1.

any

is

1 WX. 12,2

Lemma 1.1.10

Theorem

-00

Lemma 1.1.12

a

for

(0, T] [0, T'].

integration

<

and

E

t E

-0.0

integer

of

that

such

T'

2

)

(Or+lw

d

I

the

Using

+ 1.

r

(1.1.16)-(1.1.21),

4

embedding

from

constants

on

assumptions

induction

Lemma 1. 1. 10.

only

estimates

121W 12

has

one

the

exists c(l, T) > 0 C([O, T']; S) (here :5 c(l, T) for all

T > 0 there

w(x, t; 6)

depends

0

>

of Lemma 1.1.9

1

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

lWxxl2

be valid such is

that

and let

for

arbitrary,

T

any

of

the

f

0 and

>

E

(0, 1]

problem

place:

00

f

x

2mW2dx

c(m),

<

t E

[0, T'j.

00

Proof. and

integer

First r

> 0

of

all,

such

we

that

00

IX12m-1 00

d nU dXn

)

shall for

show that u

E

2

dx < C

1

S

we

for

any

m

=:

1, 2, 3,

...

(X)

JJU112

r

there

exist

C

>

have

+

IIUI12

2

+

f

-00

X2mu2(x) dx

1

(1.1.22)

0

THE

1.1.

I

where

(GENERALIZED) 0

=

1

or

I

=

1

2 and

x

(0, 1),

or

EQUATION(KDVE)

VRIES KORTEWEG-DE

n

1

=

n

or

this

For

2.

=

aim,

we use

19

obvious

the

estimate I

Jkl Jk+xj

<

2

-

< -

2,

a+1

WX_;) dnu

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

k

or

(1.1.23)

1, 2,3,...

inequality

multiplicative

and the

k

2

dx

IL21-

C(r)ju

<

(I

a,a+l)

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

(1.1.24)

1U1L2(a,.+1)

+

U

a

where

a

Due to

=

(1.1.23)

(1.1.24),

and

00

I

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

dx

X

arbitrary,

is

0, 1,::L2,...

we

I

=

n

get

I

=

n

or

integer

for

2 and

=

2

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

dx +

X

Co

k=1

2

C11 JU 112

X

+

(k=-1:oo

C"(r)

(JUjL2(k,k+1)

( 1:

1

+

k=-00

-1

E

2M 2 x u

-

1-n

2dX

X

U

k

1.

C111U1122

5

r

1-n

dx <

X

k

f

2m-1

jkj

k=1

k+1

-2

X

+

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

+

jXj2m-1(U(n))2

k+1

2

<

integer.

k+1

(k=-E00 +E ) I

1

arbitrary

is

> 2

r

2mnl-':

>

r

2m-1

C"(r)2

+

X

r

(JUjL2(k,k+1)

dx

+

I UX(r)

1L2(k,k+1))!!

<

k 00

jjUjj2

C (r)

+

r

IJUI12

2

1

+

2m X

U2dx

00

where k

=

we

have

used and

1, 2,3,... Consider

the

G

inequality

trivial

the

x

(k,

k +

1),

2 dt

Ik 12m-1 < 22m-1X2M-1 (1.1.22) follows.

x

2m

I

2

wdx=-

x

2m

wf (w)wxdx

_M

1.1.94.1.11,

in the

and

-2, -3,

right-hand

I

2m-1 X

wwxxdx-

-00

I

2m-1 X

W2dx

I

-c

X

(4)dx. X2mWW X

-00

00

terms

=

00

00

third

+ 2m

-00

00

Due to Lemmas

k

00

00

I

for

expression

00

1 d

and

the

H61der's

side

of this

inequality

equality

and can

00

C1 + C2

I

00

2m

X

W2dX

(1.1.22), be

obviously

the

first,

second

estimated

as

and

20

CHAPTER1.

with

some

EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

C1, C2

constants

integration

So,

by parts.

The last

0.

>

term

to the

we come

be estimated

can

'X,

Tt

2

after

an

00

I

1 d

by analogy

estimate

2mW2dX

f

C3 + C4

<

X

-

X2mW2dx.

_C0

Thus,

the

of Lemma 1. 1.12

statement

Lemmas I.1.1

ability T*

of the 0 such

>

whose

[0,T*)

point T*-O

t

t

0, i.

>

us

way

for

as

belonging above

to

Now,

the

we

turn

I,

any

m

-+

oo.

of the

infinitely

1.1.7

For each

n

=

and

Let

us

sequence

1, 2, 3,

...

9, is bounded

the

and let

R2

=

with ....

sup

us

,

where

R2 (C7

Pi

=

Un

f (.)

estimate

2 E H and 2

clearly

W3 E (0, oo).

We set

> 0

be

same

solution with

the

in H and we

>

R2 0

-

>

Let

is

with

x

(- 1, 1); R)

>

0 and

let

in H'

strongly

denote It

=

function

continu-

If,,(')jn=1,2,3.... (1.1.3)

T

the

as

solution

clear

that

0 is

given

by

also

R3

Sup

n

Then,

a

twice

C2((_M, M)

uo

T)

the

of

and let the

in

weakly

1,

T

uniqueness

by analogy

((- T, T); S) Un. fn and uo 0

the

some

any

in

arbitrary

an

(1-1.3)

E C-

f

proved

uniqueness

satisfying

to uo

with

we

can

proved

take

estimate

to

(x),

ul

too.E1

arbitrary

take

converging by Un (X, t)

JjUnjjj,T)jn=1,2,3

Let

converging also

be

can

6)

The

be

can

and

proved,

functions

taken

JR2(CIP)

t < 0

considering =

for

(1. 1.4),(1.1.5).

T > 0

limit

a

problem

existence

1.1.3.

f () satisfying

problem

sequence

Theorem

p

of the

any

is

T* +

of

proved.0

Lemmas 1.1.9-1.1.12

+0 in the problem

differentiable

1,2,3,.... a

C([O, T); S)

E

c --+

is

exists

Thus,

[T*,

of time 1. 1.6

Due to

Proposition

C and

C S be

1.1.7.

domain

function

=

an

half-interval

w(x, T*; E)

data

interval

of

(1. 1.4),(1.1.5),

the

there S.

space

initial

So, Proposition

t in the

proving

to

on

the

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

any fixed

constants

same

as

Thus,

of

sequence

Un 0 1 n=1,2,3....

1. 1.

class

differentiable

a

for

of the

with

problem

u(x, t)

limit

construction.

(1. 1.4)

existence

half-neighborhood

results,

of the

sense

0, solv-

t >

the

onto

right

above-indicated in the

Proposition

solution the

S for

the

to

contradiction.

a

problem

the

due

all

Suppose

arbitrary

an

for

t; c) of the problem

be continued

can

on

global,

the E S.

uo

w(x,

proved,

be continued

equation

prove

a

been

of this

let

S-solution

E S understood

for

by taking solution

ously

get

of

of this

n

Then, ul

=

now

existence

obtained

f

T*.

solvability we

e.

Let

the

be

cannot

=

get the local 6

and

Cauchy problem

the

already

imply

Indeed,

corresponding

has

El

immediately

(1.1.4),(1.1.5).

the

w(x, t; E)

lim

1.1.9-1.1.12

problem that

uniqueness

of time the

and

proved.

is

the

Lemma

I I Un0 1 12-

n

R4

=

R4(R3)

where

the

function

R4

R4(R3)

> 0

is

(GENERALIZED)

THE

1.1.

by

given

Then,

Lemma I.1.10.

jju,,(-,t)jjj For x

t

[-T,O)

E

these

and t

-x

--

(I.1.1).

equation

t)112

5

W4)

(-T,T).

t E

Therefore,

have

we

21

1.1.10,

and

(1.1.25)

change

by the simple

be obtained

can

for

of variables

t > 0

00

0"

I

JjUn(')

and

estimates

in

-t

--+

due to Lemmas 1.1.9

R2

!5

EQUATION(KDVE)

KORTEWEG-DE VRIES

d

(Un

2 dt

Um) 2dx

-

=

(Un

-

Um)(fn(Un)Unx

-

fm(um)um.,)dx

-

-00

00

I

J(Un

Um)[f(Un)(Unx

-

Umx)

-

+

Umx(f(Un)

f(Um))+

-

0.0

+(fn(Un)

f(Un))Unx

+

Umx(f(Um)

fm(um)jjdx

-

00

f

C(T)

(Un

um)2dx

-

+ an,m)

00

where

an,m

--->

+0

n,

as

convergence

of the

Due to the

estimates

m

sequence

let

us

take

fUn(*) t)}n=1,2,3....

+oo and

by analogy

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

in the

for

t < 0.

These

TI; L2)

C([-T,

space

yield

estimates to

the

u(x, t).

some

(1.1.25),

u(-, t) Indeed,

---

weakly

t E

[-T, T].

2

E H

loss

of the

(1. 1.25),

Due to

generality

I JU('7 t) 112:5

and

tE[-T,T].

H2 , hence

in

compact

the (without Therefore, JUn(') t)}n=1,2,3_.).

subsequence

u(-, t)

JjU(',t)jj2<W4,

and

arbitrary

an

is

2

E H

it

the

contains

sequence a

weakly

that

accept

we

liM illf

(1.1.26)

I JUn('7 t) 112

it

<

converging the

is

sequence

W4

n-oo

and the

The

for

following

are

statement

For

Lemma 1. 1. 14

If

Un 0

uo

as

n

oo.

any t E

weakly 1 2

[-T, T]

proved. can

Lemma 1.1.13

Proof. oo

(1.1.26)

properties

any T > 0

be

Un(',t)

in H for

JUnz,&i t) 122

any

2

t E R.

u(.,t)

--+

in H2

strongly

Due to Lemma 1. 1. 13 and the 2

by analogy.

proved

Further,

n

above

we

1 Unxx ('1 0) 122

as

-00

--+

n

--+

have

from

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

I JU(*) t)

(-, t) (1. 1.16),(1.1.18) un

fIf( Un( 3))Un n

in

oo

then

oo,

arguments

6 0

as

I

3

-

Un(* t) 112 i

---+

u(-, t) with

-9)Unxx(

i

as c

-9)

=

+

--+

n

0

0

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

22

0"

(.,s)ldxds+'

5

+

6

nx

f f f- (U- (" 0) Un'x (',

-

6

t)

-00

-fn(Un (.' 0))U2 nx(*, 0)JdXWe want n

show that

to

Consider

the

right-hand

the

Obviously,

oo.

--->

the

for

last

R(f,,,u,,) (1.1.27)

side in

term

3

0

f"(u)uxuxxj

dxds

the

R(f,u) is

as

valid.

3

-

-00

00

I [fn (Un) 0

3

f"M]

-

3) + Ux

3

ff(U) [Unxx (Unx

UnxUnxx +

_

U3X(Unxx

uxx)]Idxds.

oo

-

Due to

embedding

C((-T,

T)- JJ2)'

and

its

weak

this

equality

(1.1.27)

theorems, its

to

--+

Let

Un

I

by

R(f, u) 0)

U

t) 12

u(., t) as

n

-4

in H2

1 2

(1.

problem

1.

1), (1.

1.

all

2) by +

1vt

an

arbitrary

Then,

we

above

problem,

C ((-

T, T); H')

get

Wn(.) 0

addition

2

the

that

taken

and, ___

the

any

f

< -

1liminf

2

n-c*

1 2

.

-+

the

right-hand

side

of

the

right-hand

side

of

that

of the

integral

of

and

n

Then,

oo.

lim inf

S.

we

in

view over

Therefore,

of

em-

(0, t)

in

indeed

0)12

I Unxx

have

+

n-oo

+

R(f, u)

R(f, u).

considerations

+ WXXX=

(1.1.28) also

are

fixed

W(-, t)

0, C S

infinitely

of =

U(., t) strongly

1 U x (.' 0) 122

in in

observe

integrand

n

-

I Wn o}n=1,2,3....

sequence

for

we

in

Lemma 1. 1. 13

valid

if

we

change

the

following:

f (7v)u)x

with

as

1 Uxx (.' 0) 122

above

the

sequence

term

independent

I Unxx (', t) 12

lim inf

that

addition,

In

of the

constant

(*1 0) strongly

n-oo

observe

u(-, t) following expression

from

oo.

_

we

the

fUnjn=1,2,3....

sequence

The other

oo.

value

positive

n

<

H2,

by analogy.

a

of the

in H1 to

in

--+

absolute

as

-+

1 U '.

2

zero

the

is bounded

boundedness

convergence to

be considered

inequalities

(1.1.27) R(fn, Un)

the

strong

convergence

tends

can

bedding

Take

passage

to

limit

tends to

00

jf.(Un)Un unxx

Now,

(1.1.27)

in

the

expression t

t

(1.1.27)

f, s

in H2

lWnxx(')

R,

E .

=

converging Wn' 0

to

we

2

get

lim inf n-oo

u(., t) weakly in H2. Wn(')S) Of the to u(.,s) in strongly H' to u (., S). Let in

solutions

converges

weakly

converges

Then,

0) 122

Wo E S.

differentiable

wo

and

=

in

as

earlier

JUn.

Wx

(.' t) 122

-

R(f, u)

(GENERALIZED)

THE

I J.

1

(.' t) 122

21 U. This

inequality

(1.

with

together

EQUATION(KDVE)

KORTEWEG-DE VRIES

R(f, u).

-

28) yields

1.

23

equality

the

t) 12

I Unxx

liM

I UXX(') t) 12

=

n-oo

and Lemma 1. 1.14

(. t) u,,(., 0)

Lemma 1. 1. 15

Proof. n

--+

Let

u

(1.1.27),

in

oo

proved.

is

C ((-

E

7

we

El

plies

the

1

(. t) 122

as

1.1.9,

0) 122

21 UxX(.'

=

I

lUxx(*7t)12

of

H2

in

n

the

Taking

oo.

--+

limit

as

1.1.10

immediately

theorems

embedding

and

JIU(*,t)112

Therefore,

in t.

R(u).

+

is

function

continuous

a

im-

[-T, T1.

of t E

Let

that

UxX

due to Lemmas

continuity

any T > 0.

get 1

equality

for

u(., 0) strongly

--+

21 This

H2)

T, T);

take

us

liM tn

arbitrary

an

to.

=

Then,

to

[-T, T]

E

according

to

the

and

Itnln=1,2,3....

sequence

a

proved

facts,

the

[-T, T] such fU(*, tn)}n=1,2,3.... C

sequence

n-oo

converges

fore i.

any

of

this

e.

=

and this

in the

weakly

converges

I JU(*) to) 112,

hence

sequence

I JU('i tn) 112

lim

in H'

u(., to) strongly its limiting points

to

sequence 2

equal

H is

space

weakly

is

u(-, to)

to

u(., to) in H2. I JU(* tn) U(*) to) 112

But

to

7

--+

0

in H2, there-

compact in the

as

it

is

as

n

--+

weak sense,

proved,

already 00.

n-oo

Lemma 1.1.16

For

Suppose

Proof.

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

C

any T > 0

this

[-T, T]

is

statement

such

Un(-,t)

to

ftnk}k=1,2,3.... E [-T, T].

be But

Lemma 1. 1. 14 that and Lemma 1.1.16

in

the

sense

existence

uniqueness problem have

for

of

U(', tnj un,, (. t,,,)

--+

of the a

of this

u(-, to) u (., to)

2 dt

>

exist

oo

in

E

>

H2).

Q-T,T]; 0 and

sequence

a

as

k

as

k

--+

and

oo

-*

oo

6-

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

sequence

in H2

from space

Lemma 1. 1. 16 that

H-1 for

solution.

Let

defined and

U'

=

ul

in U1

I

'0

-

t

any

E

of the

solution

00

1 d

-4

in H2

one

So,

converging can we

easily

get

a

to

some

prove

as

in

contradiction,

proved.0

generalized

(1.1.1),(1.1.2) t E [0, T2)

n

there

U(',tn)112

-

-+

7

is

follows

easily

It

and

invalid

of the

subsequence

a

as

that

JjUn(* tn) Let

u(.,t)

(., t) an

-

7

problem

and

interval

f (u (-, t)) u', (., t) uxxx (., t) (. t) [-T, T] and, thus, we have proved the ut

U2

('7 t)

Let

be two

of time

(-Tl,

generalized

T2)

where

U2: 00

2 W

(X, t)dx

W(f(U1)U1x -00

-

f(U2)U2x)dx

us

prove

solutions

T1, T2

the

of the > 0.

We

(GENERALIZED)

THE

1.1.

jA j (

)2

d Xn

2

+ CnU

KORTEWEG-DEVRIES

(

)2

q,,

u(x),...'

polynomials,

such

d Xn-1

-

EQUATION(KDVE)

dx

dx,

n-2

25

n>2,

0

where

Cn

real

are

differentiable

period oft, of

i.

e.

the

the

quantities

the

junctionals

statement

with

f(u)

readers

consisting

weakly

to uo

infinitely period

and t E

Then,

all

strong

I

integer

Hpnr (A)

in

I

as

2,

result

(1.1.1),(1.1.2)

questions,

(see

is obtained

of the

Hn,,(A)

uo E

and

periodic

(x, t),

Let ul

oo.

--+

proof

the

in

as

1, 2, 3,

=

in

I

a

with

x

1, 2, 3,

=

we

refer

Additional

1.1.3,

one

can

fU0(1)}1=1,2,3. .

sequence

period

the

A converg-

corresponding

be the

...,

periodic

in

show that

for

problem

of Theorem

and

...

JU(*) 0)12

=

that

and

ul 0

now

--+

is

and,

A

I

A

3(x)

dx

U

f

and

0

weakly

on

En (u (.,

addition,

here in

Hpnr(A)

Suppose

the as

this

0))

I

---+

any

R,

strongly

and

continuous

CnU (dn-lU)

space,

=

we

of the

with

x

the

any T > 0

to

2 _

gn

I-00

Hpner(A);

in addition

:

0 be

(U'. .'

arbitrary.

and the

dn-2

u

dXn-2

Since

the

functionals

dx

have

En (ul (., 0))

lim inf

f U1 (X, t) 1 1=1,2,3....

5 C1-

Hpnr(A)

on

:5 C1

sequence

weak in

JjU&)t)jjHPn r(A)

I-00

equality

strong

(x, t)

dXn-1

0

continuous

u

t E

:5 liminf

& are obviously

Eo,...,

limit

a

for

HPn, ,r(A)

Uo in

jjuj(-,t)j1Hpn,,(A)

max

tE[-T,71

there

T); Hpn,-r'(A))

C((-T,

in

functionals

strongly

this

other

to

(-T, T)

Let

in

the

H' -solutions

problem

of the

devoted

is

functions

1JU(*)t)11HPn_(A)

are

book

>

n

solutions

U(*) t)12 for

with

independent

and

for periodic

laws

integrability

where

differentiable

differentiable A.

complete present

determined

are

infinitely x

u.

=

literature

arbitraiy

infinitely

of

the

t)),...

in

chapter).

take

us

since

arbitrary

an

periodic

conservation

are

f (u)

with

for

that

problem

E,(u(.,

to the

corresponding

this

to

Let

and,

u

the

to

1), (1, 1.2)

1,

the

&,...

Eo,...,

is related

=

remarks

ing

(1.

problem

This

are

of u(x,t) Eo(u(.,

solution

A,

and qn

constants

takes

place

if

En (u

(., to));

and

only

if

u&, to)

--+

u(., to)

oo.

that

En (u (.,

0))

>

En (u (, to)). 7

(1-1-31)

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

26

IUI,}I=1.2,3....

Let

argument

strongly

Hp' JA).

Hpn,,(A)

in

(-, t), (1.1.31),

of the

I

ul

of

...

get

as

differentiable

converging

functions to

of the 1

as

-4

(1.

1.

infinitely

differentiable

1) satisfying

ul

(., to)

lim inf &(u,

(-, 0))

periodic

ul, (-).

=

solutions

Therefore,

due to

have:

we

we

e.

infinitely A and

of

sequence

equation

E,, (u (-, 0)) i.

period

00 u(-, to) autonomy of equation (1.1.1), the function CQ-T, T]; Hpn ;'(A)) and, for any fixed t E [-T, T], weak

corresponding

1, 2, 3,

=

the

in view of the

in

strong

of

sequence

with

Then,

limit,

is the

arbitrary

an

periodic

E R

x

in

u(-, t)

be

E,, (u (-, to))

>

contradiction.

a

Thus,

=

1-00

for

ul(.,

any t E R

t)

(u (., 0)),

! E,,

u(., t) strongly

--+

HPn,

in

oo.

One

that,

if

again

then

ul(.,

t)

t)

ul(.,

complete analogy that u(-, t) E C([-T, T]; Hpn,,(A)) and u. C Hpnr(A) in Hpn,,,(A) I fUoj1=1,2,3.... as uo strongly in T > 0 u(., t) strongly C([-T, T]; Hpn,(A)) for an arbitrary the

-+

corresponding

are

proof

periodic

En(u(-,

t))

Hpn r(A).

I 1 .5 about

follows

The

In this

the

-

from

Theorem

1.2

the

1.1.5

time-independence

continuity

completely

is

nonlinear

section,

we

prescribed

consider

We shall

results

iUt + AU +

f(IU12)U

initial

complex

plane

f(JU12)U

tion

for

the

is

k times

the

complex

n)

x

En

t)),

the

on

...,

space

=:

0,

=

Uo

(NLSE)

equation of solutions

existence

N,

of the

NLSE

(1.2.1)

t E R

E R

x

(X, 0)

(X).

(1.2.2)

operator

A in

equation

operator

-D.

Here

function

continuously

f(JU12)U

:

linear

differentiable

00

U C'((-n,

Eo,...'

The state-

Eo (u (.,

quantities

functionals

the

on

two-dimensional

the

as

Laplace with

it

the

C

H'-solution uo.

data

identifying

smoothness

of

the

of the

Schr6dinger several

understand

sense

of

where

proved.0

U

ized

a unique u(-,t) global periodic depending on the initial data

1.1.3,

oo,

--+

problem

of the is

continuously

of Theorem

ment

Hn-solutions

of Theorem

problem

of the

and

-->

As in the

with

by

prove

can

R2)))

(-m, m);

if

f(JU12 )u

as

a

(1.2.1)

we

C

1

space

R2,

(we

write

map from

shall

conditions

accept

Considering

C.

)

we

say

this

in

2

R

general-

the

in

into

that

the func-

the

f(I

case 2

R

is

U

12 )U

E

k times

m,n=l

continuously

differentiable.

To formulate

jxj

--+

oo,

we

need

a

the

result

on

following

the two

existence

of solutions

assumptions.

of the

NLSE

vanishing

as

THE NONLINEARSCHR6DINGER EQUATION(NLSE)

1.2.

(fl) f (s),

and

Co N

where

such

2,

>

s

be

0,

-

differentiable

continuously

a

a

real-valued

1),

where

be

(0, p*

and p E

> 0 =

f(JU12 )u

Let

function p*

if

N-2

of

in the

3 and

N

the

p*

argument

(9

Co(I

i)u

>

Let there

Remark u

C

exist

0 and pi

>

a

Under

1.2.1

iU2

U1 +

-=

exist

arbitrary

1 is

(0,

E

+ '

N

)

JUIP), such

U

for

(1.2-3)

E C.

f(S2)

that

<

C(j+spj)'

au

[f(JU12)U]

aU2

af(U2j+U22)U2

i9f(U2 I +U22 )U2

aU2

.9ul

UI) U2 E R.

R+.

qf( 2 I+U2)Ul 2

aul

the matrix

we mean

E

s

2 af(UI+U2 2)U1

where

E C

u

N > 2 there

case

that

I [f(JU12)U]1:5

(f2)

N 22

=

function and let

27

(fl) and let T1, T2 > 0 be arbitrary. We call a function T2); H-1) a (generalized) u(-, t) E C((-Ti, T2); H1) n C'((-Ti, the 2. solution uo in the of problem (1. 1), (1. 2.2) if u(-, 0) (or simply a H'-solution) sense of this function of the space H1 and equation u(-, t) (1. 2. 1) after the substitution H-1. the the in becomes the equality t sense of for any E (-Ti, T2) space Definition

1.2.2

f (-)

Let

satisfy

condition

=

this

In

section

existence

for

following

integral

for

NLSE for

the

simplicity

the

N

=

1.

all

present

we

We convert

proofs

the

on

(1.2.1),(1.2.2)

problem

the

results

of the

the

into

equation:

U(., t)

=

6-itD

UO +

I

t

e-i(t-s)D[f(jU.'s

12) U(.' s)]ds.

(1.2.4)

0

In

the

fact,

following

H'-solution

arbitrary

equation

C((-Tl,

T2); H')

interval

of

Theorem

depending unique

+oo) if

1.2.4

Hl-solution

(-TI,

this

is

the

in

the

of

the

T1, T2

any

> 0

of

that

(fl)

the

oo

the

in

time

solution

a

problem

if

any uo E

of *time

(1.2.1),(1.2.2) I Ju(-, t) 111

problem

(resp.

for

interval

lim sup t-T2

H'

there

(-TI,

and T, <

T2) =

TI, T2

exist

+00

there

1 U(X, t) 12 dx, RN

E

in

(resp.

E (u

-0

(-, t))

1 RN

11 VU(.,

2

t) 12

T2

For any H'-solution

oo).

-

F(ju(-,

> 0

exists

quantities

P(u(-1 0)

an

(-T,,T2) u(-, t) (1.2.1),(1.2.2)

of

interval

Conversely,

time.

H'-solution

a

assumption

,--Tl+o

the

of

(fl) for

assumption

-

such

<

the

(1-2.1),(1.2.2)

interval

(1.2-4)

T2)

Under

I luol 11 u(x,t) I Ju(-, t) 111 on

Under

problem

in

of equation

time

only lim sup

of

the

(1.2-4)

satisfies the

Let uo E H'.

1-2-3

Proposition

place.

takes

statement

t) 12) } dX

a =

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

28 and

MM-It))

u(-, t)V-u(-,

t)dx,

RN S

F(s)

where

f f (r)dr,

2

determined

are

and

independent

of

(i.

t

the

e.

functionals

0

P, E, M are

and

zero

global

is

Proof

of

g(-)

Proposition E

the

1.2.3.

C(I; H).

We understand

Let I

Consider Dw+

-

solutions

its

condition

addition,

In

under

the

(f2)

assumption

above

any

time.

in

iwt

initial

laws).

conservation

H'-solution

as

in Hs and

the

=

the

g(t)

=

T2)

(-Ti,

linear

0,

be

arbitrary

an

interval

containing

problem

t E

functions

from

equation

in the

w(O)

I,

=

C(I; Hs)

n

C'

of Hs-2

sense

(1.2.5)

H.

wo E

(I; H'-2) for

the satisfying Clearly,

each

t E I.

function

W(t)

=

e-itD

I

i

WO+

t

e-i(t-s)D

g(s)ds

0

the

problem

suffices

to

satisfies

aim,

it

(1.2.5).

Let

show that

the

us

homogeneous

iwt-Dw=O, has

w(t) =- 0. w(-) E C(I; Hs) n C' (I; H'-') function e-i(t-r)Dw(r) p(t, r)

the

=

For

this

w(O)=OEH'

(1.2.6)

be E

a

solution

C(I; Hs)

of the

U

C'(I;

(1.2.6).

problem

H3-2)

.

Then,

Consider have

we

for

0_ r
=

,9r

the

Hs-2

space

uniqueness

of the

I

hence

solution

a

5r-

w(t)

[,-i(t-r)D

=

of the

u(-, t) E C(I; HI) n C'(I; due to embedding theorems

Let

Then,

i

-De-i(t-r)DW(r)

-

in

solution.

solution

unique

r,tEI,

of this

uniqueness problem

tEI,

Let

the

the

prove

p(t, t) problem

H-1) f(JU(.,

] w(r) + =

+

i.-i(t-r)Ddw(r)

e-i(t-r)DDw(r)

=

0 and p(t, 0) (1.2.5) is proved. =

be

a

solution

t)12)U(.,

t)

0

thus

of the E

dr

w(r)

problem

C(I; H')

(we

=-

0,

i.

e.

the

(1.2.1),(1.2.2). remind

that

we

of the linear as in the case 1). Therefore, problem (1.2.5) the function satisfies let E u(-, t) equation Conversely, (1.2.4). u(-, t) C(I; H1) be a Hl-solution of equation Then, since as earlier (1.2.4). f( I U(.' t) 12) U(.' t) E C (1; H'), we have -at!-u(-, t) E C(I; H-1) and the function u satisfies equation (1.2. 1) and the initial condition 1.2.3 is proved.0 Proposition (1.2.2). consider

only

the

case

N

=

THE NONLINEARSCHR61)INGER

1.2.

Remark

[451

where

of

essentially

an

of Theorem

Proof

proof

In the

1.1.5

in fact

Proposition

1.2.3

As noted

we

result

strong

more

1.2.4.

EQUATION(NLSE)

above,

the

by

paper

T. Kato

presented.

is

we

followed

29

this

prove

only

theorem

for

N

=

1.

00

Also,

f(JU12 )u

that

we assume

U C2((_rn,,rn)

E

x

(-n,

to

the

n);

R2) for

simplicity.

Due

m,n=l

to we

Proposition

1.2.3, this

consider

(., t)

and ul

(*) t)

U2

equivalent

is

We first

equation.

and

by embedding

(1. 2.4)

equation

show the

be two solutions

theorems

for

uniqueness

of this

from

equation

(1.2.1),(1.2.2).

problem

So,

solution.

Let

C(I; H').

Then,

of its

uo E

H'

we

get

t > 0 t

U1

('7 t)

fIUI(*7

t) 12 :5 C1

U2

-

S)

U2(*7 8)12ds7

-

0

hence

ul

and the

(-, t)

Let

is

right-hand

side

of

existence

of

infinitely

T(R) (1.2.4),

=

of

For t < 0 this

t > 0.

solution

a

the

an

of T

existence

for

of

prove

us

f(JU12)U

that

(*, t)

_= U2

uniqueness

(Gu)(t)

=

solution.

a

differentiable

>

e

-itDUO

IluoIll

if

I

+

I

Let

any

R, then

Suppose

integer.

be

I

>

For <

by analogy,

proved

be

can

proved.

is

function.

that,

0 such

statement

(1.2.4)

equation

R

0 let

>

the

us

now

show the

G in

operator

the

t

e-i(t-s)D[f

(jU."9)j2)U(.'

s)]ds,

0

the

maps

set

MT

mutes

with

C([-T,TI;Hl)

E

u(O)

:

G(MT) C C([-T, by applying embedding

Indeed,

itself.

into

Ju()

=

uo E

jju(-)jjj

H,

T]; H') and,

clearly

e",

=

theorems

we

21luolll}

:! , the

since

-2-

operator

get from (1.2.4)

ax

for

com-

u(-, t)

E

MT: t

jj((;U)(t)jj1

JIU0111

:

+I7j(j u(-,s)jj )ds, 0

where

-yi(s)

implies

is

continuous

a

the existence

function

positive

of theaboveT

>

of the

argument

s

Iluolli,

0, dependingonlyon

This

> 0.

inequality

G(MT)

for which

MT. Now let

the

map G is

theorems

for

us

show the

a

contraction

of the

infinite

differentiability

and the

existence,

set

any R >

MT, of

0, of

Iluolli f(JU12)U,

if

a

constant

:5 R. we

Ti

Indeed, easily

E

(0, T]

using

derive

t

II(Gul)(t)

-

(GU2)( )III

:5

C21 jjU1(,9)-U2(8)jjjds, 0

U1(*,t),U2(',t)EMTj,

such

that

embedding

c

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

30

where

the

C2

constant

depends

> 0

only

Iluolli,

on

and therefore

such

T,

constant

a

exists.

Since we

have

that

for

the

interval

of

fixed

a

proved, any

E

H' there

uo

the

on

(-T, T)

(1.2.4).

Since

the

not

length

by

and

of the

I Ju(-, t) 111

limsup

T >

there

t

is

of

of the

TF

existence

of

again uo H'-solution

unique

Tk+,

be the

and

0 <

:5

I

1,

-

:5

T2+ (E

I IU(') t) I Jk+1

of its

Tj+

:5

k

54

is

bounded

=

from

initial since

from

Tj-

of

below and

< 0

(Tj-, T,+) < 00).

and

Tj+

+oo if

1, 2, ...,1

there

where

Tj- :5 T - :5

<

one

Therefore,

interval =

the

pose

0.

(T;, T,+,),

-oo

(tE[O,Tkl

)I

Gmax JjU(',t)jjk

+

we

to

the

each

follows

it

if

above

exists

T-

(1.2.4)

equation

Tj-

<

...

a

0 <

<

0

<

that

for

[0, Tk+11

and t (E

1 1U01 jk+1

for

the

to

existence

I Ju(-, t) 111

Then,

Since

=

onto

and let

existence.

:5 +oo.

(0, Tk+)

:5

obviously

(1.2.1),(1.2.2)

problem

interval

Tk+j

...

Then

I > 2.

of the

maximal

Tj+

=

H1,

E

t

the

get

we

t-T,+-o Let

k

a

lim sup

t-TF+0

place

point Hl-solution

be continued

(resp.

-oo

similar

takes

IluoIll,

on

on

result

a

arbitrary

an

can

>

constant),

(1.2.4), f(IUI 2)U,

equation function

such that in Iluolli, E u(-, t) C((-T, T); H1) with respect to (invariant

only

autonomous

H'-solution

a

only

H-solution

+oo if

is

of

of the

solution

unique

a

a

at

solution

a

differentiability

(1.2.1)

0 but

=

obviously

0, depending

exists

c

depending

our =

exists

uniqueness

interval

that

infinite

where

c

point

T > 0

such

> 0

of the

equation

t +

---

the

at

constant

a

Tj+

t

existence

condition

case

of time

substitution

map G is

of the

the

equation

the

point in

t

JjU(*i3)jjk+jds

0

by analogy

and

for

I > 3,

Let

0,

t <

H',

E

uo

Ifn(JU12 )Ujn=1,2,3.... (1.2.3) with 1,2,3,... fn(JU12)U

let

be

condition

the

__+

be

a

H'-solution

T2n) Let

Tjn n

t2

> --+

E

t'

us

and

T2n

in

C([-t',

(0, t"]

such

oo

continued

onto

>

t" for

all

the

for

n

=

with

E

oo

1, 2,3,...) =

segment

sufficiently

[-tl,

t2l

and Un 0

and

large large are

k)

t"

as

n

E

--+

that

numbers

Let

solutions,

We want

arguments

numbers

n

bounded

in

there

k,

any

R2)

m

u(-, t)

Let

.

=

of the

(-Ti,

T2)

and

respectively.

and that

n

and

satisfying

for

Hl-solutions

of

fn (-).

(0, T2).

H',

in

oo

(-m, m);

x

T1+-

functions

sequence

f

T2+

=

such

of these

above

and

p and

be the

existence

(0, Tj)

uo

-4

C2 ((- k,

in

Tj+

and

differentiable

C and

In view of the

all

u0n

infinitely

--*

uo

sufficiently

H',

C

of the

t'

t"]; H1). that

as

intervals

TI-

constants

same

f(JU12)u

arbitrary

take

T-

=

of

sequence

a

taken

be maximal

T;-

f u0'j,,,=j,2,3....

Un(') t) (n

(1.2.1),(1.2.2)

problem

(-T,n,

and

have

we

un

to

(-, t)

exist

H'-solutions H' uniformly

prove

that

u(-, t) as tj (0, t'] and Un (' t) can be ---

E

i

with

respect

THE NONLINEARSCHR6DINGEREQUATION(NLSE)

1.2.

to

t2l.

[-tl7

t E

Then,

for

t2l

[-t1j

t E

(t

have

we

31

0):

>

t

Un(*it)12

JU(* t) I

Cn

'5

+

11

C4

8)

U

(*) 5) 12ds,

Un

-

0

Cn1

where u,,

(-, t)

+0

-+

(-, t)

u

-+

Further,

as as

n

constant

=

>

for

by analogy

0 and

0, hence

t <

t2l; L2)

in C ([-tl)

oo

theorems

by embedding

t

12

(U(',t)-U-(',t))

0X

C4

and

oo

n

2


G

+

ax

(U(

1)

12

Un(*) s))

-

ds,

0

C,,2,

where

jju(-,tj)jjj, (resp. t2

n

the

--*

(t2) t"j)

as

n

only n

number

of steps

initial

data

Since

H1 and

P, E,

on

of

a

t'

t2

or

<

t",

then

I Ju(-, ti) 111 (resp. Un (* t) can be to u(-, t) converge since

process,

latter

u,(.,

t)

as

n

as

analogy

n

of this

shows

the

0,

<

implies, 1, 2,

=

tj

E

the

to

(ti7t']

that

for

segment

CQ-tj, 0]; H') I Ju(- t) 111 < oo, after a in

00

--+

,

I

oo

--+

C([-t, (1.2.1),(1.2.2) in

t"]; H'). on

its

construction.

differentiable

continuously

onto

problem

of the

t

such

7

continued

max,,

u(-, t)

--->

7

i

exists

I JU(* t2) 111)

on

for fact

....

there

tE[-tl,t

Hl-solution

complete

the

verification

direct

C([-t1,t21;H'). fjjUn(-,ti)jj1}n=1,2,3

i

get that

obviously

Mare

the

since

shall

proved

be

can

we

by analogy

0 and

The

in

<

on

solutions

dependence

The continuous

>

sequences

[-t'j, 0] (resp., on [0, t2]) and, as above, (resp., in Q0, t']2; H1)). Continuing this finite

constant

=

oo

if tj

Hence, numbers

C5

--+

of the

depending

large

sufficiently

and

convergence

respectively. E

oo

u(.,t)

--+

particular,

all

as

u,,(-,t)

therefore in

+0

--

functionals

on

of t of the

independence

the

space

quantities

H'-solution u(-, t) of the problem P(u(., t)), E(u(-, t)) and M(u(., t)) for an arbitrary > also the these above arguments to are quantities 3), according (1.2.1),(1.2.2) (1 of t if u(-, t) is a H'-solution of this problem. independent the under we get Finally, assumption (f2) using inequality (1.1.15), 00

1 F(ju(x)

12 )dx

<

C61ul'

2

+

C71ul

p+2 2

+1

2

lu"I

p+2 2

_1

2

00

where

0 < P+2 2

JU(*7 t) 12 from

the

=

interval

2, hence,

1 <

JU(* 0) 12

(1.2.1),(1.2.2).

for

any

of the

Therefore,

Remark

ficiently

-

1.2-6

smooth

With

function

if uo E H1 ,

t, there

existence

this

the

f

proof

>

then, 0 such

E(u(., 0)) and E(u(., t)) C'for all t E R lu,,(*, t) 12 solution u(-, t) of the problem

since

that

corresponding

of the

solution

the

C'

exists

is

global,

of Theorem

existence

and Theorem

1.2.4

we

of H-solutions

in fact

1.2.4

is

have

(where

proved.0 shown

I

=

for

2,3,4,...)

a

suf-

of

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

32

if uo E Hi.

(1.2.4)

equation

Now we consider

functions

problem the

for

C(R).

space

with

argument in

show the

when for

initial

satisfies

data

following

the

our

all,

the

for the

should

it

f

0 and

that

fact

the

problem

conditions

e-itD

of the in the

acts

this

the

as

the

[80]).

from

Xk (i.

spaces

e.

assumed).

are

L2 and Hk, k

spaces

of the

unbounded

is

taken

is

in 47 -tT:: t7

e

function

regularity

Cauchy

function

a

of

ill-posed

1 is

=

function

(this example in (1.2.1),(1.2.2)

to

=

that

N

C(R)

to

NLSE because t

NLSE in classes

be noted

=-=

[0, to) belongs

point

of the

operator

integral

E

of the

additional

uo

from t

linear

well-posedness

It is known that

the

each

half-neighborhood

We shall

as

0 for

>

E R and

x

left

any

to

of

NLSE with

follows

this

of solutions

existence First

oo.

--+

one-dimensional

Indeed,

arbitrary

an

the

on

jxj

as

linear

the

result

a

nonvanishing

=

1, 2, 3,

operator: 00

Gto

I

=

K(x

yt)o(y)dy

-

(t 54 0),

Goo

0,

=

-00

where

for

t

quadrant

0).

if t < L

=

:

12. 82

the

exp( 0-2)

(47rit)72

(here

4t

K(x, t) is f(JU12)U.

function

We also

g(U)

set

be written

can

=

Imz > 0 if t > 0 and in the

0,

>

Of course,

i -2- + at

formally

K(x, t)

0

Rez

as

the

the

fourth

(4rzt)12

quadrant

fundamental

In this

=

root

way the

lies

Rez

in the

>

first

Imz

0,

solution

of the

integral

equation

<

0

operator

(1.2.4)

follows:

u(x, t)

=

Gtuo

+ i

f

t

Gt-,g(u(.,

s))ds.

(1.2.7)

0

We begin

the

with

Proposition the

following

1.2.7

For any k

bounded

any

bounded

(G2) for in the

statement.

=

1, 2, 3,

the

...

operators

Gt considered

on

Xk satisfy

properties:

(Gl) for uniformly

following

sense

Proof.

with

0

any

of

the

Let

t

=

(-=

I C R the

interval

t E I; respect X1' the function Gt 0

family

of operators

Gt

:

Xk

__+

Xk is

to

space

X'.

0.

Consider

:

I

-*

Xk is continuous

and lim t-0

0

E

Gto

=

X' and 00

Gto =(Ki)-2

e

iZ2

O(X

+

2-v/-tz)dz

e'z

+

2

O(x+2Vt-z)dz+

e

iZ2

O(x+2Vt-z)dzl

00

(iri)_2

i

f1l

+

12 + mi

(1.2-8)

0

THE NONLINEARSCHR6DINGEREQUATION(NLSE)

1.2.

P

where are

(in

arbitrary

0 is

>

converging).

follows,

what

the

that

prove

we

33

integrals

improper

here

We have

1111

I

:5

+00

,2

I

+

I

lim

2

-+oo

[,

lim a

O(x

i,,

S=Ce2 j

2v't-s)+O(x

It-01(x

isf

2V-t-s)

-

2 Vs

+

2v't-s)

-

2s

2

+

z)l

Ids

<

#2 2,6

<

CIP-1101C

Vt-

+

1

lim +00

ce

v'-t

10'(x

dz <

IZI

-2aN,rt

-1101C

C,

+

C2P

-i

2t4

1

10'12-

(1.2.9)

By analogy, :51 C30 -1101C +

1131

JGt0jc all

:5

for

since

0

Coc (R),

E

we

C51110111k

get

d

-X [Gto] the

since

I E R

Gt

operator

statement

Concerning to

of this

inequalities

(the

(Gl) (G2),

statement

(1.2.9),(1.2.10),

7-2

last

=

GtO'

L2 is unitary,

--+

for

0 EXk

any

and

a

finite

interval

have

we

and the

L2

:

Id'dxl part

(1.2.10)

4

t E I

Further,

and,

2

by (1.2-8)-(1.2.10)

Thus,

for

C40-4110112-

take

1110111k

:5

(1

1, 2,

=

k),

...,

follows. we

only

can

there

GtO12

be

exists

that

prove

proved

Gto

--+

0

by analogy.

> 0

such

that

as

Fix

for

all

t

--+

arbitrary

an

t

Xk because

0 in

:

Itl

6

< I

> 0.

the

+

JI31)

one

is valid

<

(k

because

1(ir)

and +

(i7r)-126

2

eiX2

00

f ei. ,;2 -00

dx

1).

dx

-

11

<

3(k

+

1)

other

According

following

place:

(1111

the

two

CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

34

Then, respect

z

O(x

function

the

to

I (i7r)2 Further,

by

the

2Vt-z)

+

Therefore,

E

iZ2

e

O(X

continuity

strong

sufficiently

for

t and

small

Proposition

of the

1.2.8

function

of

function

u(-, t)

the

proved.

Let

N

complex E

C(J; X3) .a

Zat

U

<

be

and

u

and

only if u(x, 0)

02

+

ix2U

uo(x) for

=

e"'

011 JA;

-

uniformly

0

--+

Itl

if

3(k

<

(1.2.12) L2

space

k)

<

(1.2.13)

for

6

with

to, then

1)-

+

in the

1, 2,...,

complex-valued

a

be

I

g(u)

any

x

the

0,

=

=

interval

an

(I; XI) satisfies

U(X, 0) if

<

group

11 jGtO

t

6

O(x) I

-

T_+1

1, g(u)

=

C'

+

as

that

sufficiently

all

small

D

argument n

O(x) such

6

12

is

to

unitary

By (1.2.11)-(1.2.13)

t.

1.2.7

Proposition

exists

2Vt-z)dz

+

Id'dx1 Gt 0

to > 0

converges

there

infinitely containing

differentiable

Then,

zero.

problem E

x

R,

1,

t E

(1.2.14)

X3

uo E

E R and this

a

(1.2.15) function

is

a

solution

of equation

(1.2.7). Proof.

prove

first

Let

show that

it

that

is

this

Clearly,

a

u(., t)

function

a

solution

of the

E

problem

satisfies'equation t)) E C(I; X3). Let

function

g(u(., ,92

=

aX2

=

00

02

_tGtUoj

us

X3) satisfy equation (1.2.7). For this (1.2.14),(1.2.15). aim, (1.2.14). show that L[Gtuo] 0 for t 54

C (I;

I

_X2

K(y, t)uo(x

+

y)dy

=

(

Gt

We shall we

0.

have

to

We have

d2 UO dX2

00

where

the Let

the

right-hand us

side

show the

definiteness

that

is continuous.

(the

-2-[Gtuo]. at

of

existence t > 0

case

t < 0

00

Gtuo

=

(47ri)-l'

I

uo e

(x

By setting can

+ 2 v"t_z)

relation

yields

at

+ uo

(x

and

by analogy), -

2

vft-z)

dz.

formally 00

-[Gtuo]

4t

N/"Z-

0

This

z

be considered

=

(4ri)-12

f 0

e' 'u'(x

0

+ 2 vft-z)

-

VIt-

u' 0 (x

-

2

vltz).dz.

accepting we

get

for

THE NONLINEARSCHR6DINGEREQUATION(NLSE)

1.2.

For

> 0

c

I

Ci-. e

we

have

U0(x

+

2v t_z)

u' 0 (x

-

2v'-t-z)

-

dz

-\/t-

icu'(x0

-ie

=

2 v'tc)

+

,

35

u'(x -0

-

2v1t_`c)

-

+

-

Vt_

0

I

C

j

'(x+2v' t_z)+u0'(x-2-\/t_z)

uo

_.

dz,

V"Z-

0

where

uo(x)

of this

equality

interval. is

an

0

--->

tends

to

zero

integral

improper zero,

for

consequently,

and,

oo

--

which

(1.2.8)-(1.2.10)

t > 0 and

any

,t[Gtuo]

i(4ri)

=

to

right-hand

any bounded

58-t [Gtu0j

uo'(x VZ_

2vft-z

-

side

interval

not

and

is determined

at

+

side

any bounded

t from

in the

term

in t from

2vlt-z)

u"0 (x +

2

respect

derivative

00

right-hand

in the

term

second

the

R the

E

x

with

uniformly

converges

first

the

uniformly

oo

c --+

as

due to estimates

Since

containing

jxj

as

)dz

0 00

i(47rit)-

e

JEWL, U/1 (y)dy

=

0

a2

-[Gtuo] X2

i

Co

for

t < 0

at

t=o

=

a

[Gtuo]

=

there

lim

exists

These

0.

L

first

statement

(1.2.14),(1.2.15) Now

equation

is we

homogeneous

any

X3_SolUtion

that

In view

any

of the

X'-solution

only

X'-solution

the

=

of

iu".0

Hence,

imply

also

arguments

of

arguments there

exists

that

=-g(u(xt)). (1.2.7)

equation problem

the

above

arguments,

the

satisfies

problem

Lu=O,

xER,

it

suffices

(1.2.14),(1.2.15) to

that

prove

satisfies the

linear

problem

U(X, 0) has

at

[Gtuo]

above

proved.

prove

(1.2.7).

that

a

I

I ijGt_,(g(u(-,s)))ds 0

our

due to the

Further,

u".0 t-0

t=o

L[Gtuol

iu" 0 and

2

a2

t

Thus,

relation

of this

proof

the

earlier,

noted

we

by analogy.

iGtu" 0 and therefore

[Gtuo]

As

t > 0.

obviously

0, then

-12jGtuo] at

0 for

=

be made

can

If t

-2-

L[Gtuo]

indeed

that

so

trivial

solution

of this

problem

u

=-=

in the

0 from interval

=-:

C (I;

tEI, 0

X3)

of time

Let

.

Simple

I.

0')

d

Tt

j

I ux (x, t) I'dx

=

us

0

suppose

that

calculations

u(x, t)

is

show that

a

Concerning in

the

For

simplest

results

the

on

[102]. f (s)

case).

with

sP

=

NLSE

essential

an

periodic periodic =

and

u)

we

no

uniqueness

KdVE and for

of

As for

the

phenomenon

is unknown

for

the

this

[69,70]). blowing

For up

[38], f (s)

this

in

as

solutions,

NLSE) to

paper

NLSE with

=

the

sP with too.

paper

result

is p

recall

proved

that

we

1.2.10

well-

(see,

one

of the

considered

only

first

are

proved

[91] stating

Cauchy problem

of the

non-smooth

in

for

for

by

J.

problem

Bourgain in

the

solutions,

proved

for

have

(see, in

of the

one we

rigorously proved

[16] (see

[17])

the

for

to

the

where

the

(with Is IP.

usual

the

f (8)

like

it is known for

considered

vanishing

devoted

literature

is

problem

data

also

are

x

there

of the

initial

nonlinearities

Although is

with

whole

superlinear

[38].

data,

initial

well-posedness

the

review

have

it

of the

smooth

more

of Y. Tsutsumi

of the

and the

justified

2-, N

or

have

we

Theorem

with

of the blow up of we

H'

L2.

periodic

L2-solutions

KdVE. Above

and followed

subject

technique

from

possibility

the

uo

result

investigations

the

of

from

1.2.4

form is contained

investigations

lot

well-posedness

the

consideration

the

mention

and

general

in its

(we

important

data

between

Having

problem,

existence

f (u)

(with

x oo.

--+

-) (0, -N

N = I for

Theorem

1.2.7

the

initial under

difference

in

IxI

P E

equations

the

Proposition

with

a

data

of Hl-solutions

mention

some

(1.2.1),(1.2.2)

For

With

1.2.4

are

initial

with

existence

especially

We also

there

equation,

[33,37,69,70,79,88]).

one-dimensional

the

this

of Theorem

proof

the

Cauchy problem

of the

example,

for

NLSE,

the

[45].

paper

posedness

as

EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE

CHAPTER1.

38

=

NLSE but

the

simplest

results

on

formal

presented

only

example,

[90] and, also,

paper

[90]

that

a

there

exist

Chapter

2

problems

Stationary As

noted

already

have

we

standing

for

sentation

AO 0

is

a

real-valued

supply

their

_

(IJ.0-1)

Usually

the

problem

N

1,

=

with

following

jxj

as

11. 1_00

QC

RN is that

Suppose function

and

problem

pairwise

By analogy,

a

oo

N

02)0

f(X, 02)

as

it

will

it

Q.

It

has

solutions.

then, as

domain

3 and in

is

=

(11.0.1)

O(x).

O(x)

equation,

of this

Rlv,

E

(11.0.2)

conditions

jxj

as

infinity

the

on

vanish

--+

oo,

i.

(11.0.3)

be reduced

assumptions

natural

KdVE under

can

for

e.

0.

(11.0.1),(11.0.3)

bounded

>

k(x) > 0 (11.0.4),(11.0.5)

different

starshaped,

now

x

some

=

the

for --+

If

is

f

proved

be shown

=

IOIP-',

in

[741,

further

P.E. Zhidkov: LNM 1756, pp. 39 - 78, 2001 © Springer-Verlag Berlin Heidelberg 2001

X

equation

to

(11.0.1)

(11.0.2),(11.0.3)

and

on

with we

E

Q,

0

with

also

(11.0.4)

O(X),

(11.0.5) a

smooth

sufficiently

k(x)jOjP-1,

=

=

0,

=

p >

that

if p E

positive

in

p >

7Nq j2,

known

solution

a

0,

=

Olau

the

the

to

problem:

WO+ f(X,

_

0,

=

solutions

problems

the

similar

AO

where

repre-

NLSE leads

generalization

a

with

that

waves

solutions

Along

too.

the

consider

solitary

finding

of

of these

behavior

RN, 0

E

consider

(11.0.2)

and

O(X) The

x

02)0

WO+ f(X,

suppose

we

0,

=

We also

function.

equations

solutions.

f(02)0

wO +

-

AO and

e"'O(x),

general

of the the

into

R,

E

w

equation

stationary

Here

u(x, t)

=

substitution

the

Introduction,

in the

waves

N-2

this

problem

(see

Example

1,

(1, N 2)

and

2

Q and w

has 11.0.1

boundary.

where

an

w

infinite

0 and no

k(x)

the

a

C'-

0, then of

sequence

domain

nontrivial

and

is

Section

fl

is

solutions. 2 of

this

chapter), critical

the

fying

condition

f(02)

lim

if

and

infinite

an

JxJ)

=

been

have

problem

(11.0.2),(11.0.3),

voted

it.

to

simple

k(.)

11.0.1

tends

rapidly

Let

to

zero

as

f (x, 0')

above,

under

similar

:N 22

p <

P(xi) Ix, I

2

nontrivial

no

the

maximum

of xi

solutions.

some

positive

(11.0.2)

by

+

by parts,

integration

we

be

a

easily

can

solution

positive

only

and

too,

this

in

on

11.2.1).

handled,

direction.

nonlinearities. existence

As for

[57,58]

0 and

--

Theorem

and

satis-

(11.0.4),(11.0.5)

with.

of this

of radial

so-

non-radial

the

publications

among

problem

with

>

0 and

1: I

N > 3,

x,

C, :5 k(xi)

<

R.

We recall

that,

we

C2

de-

following

the

V(xi)

it is

as

show that

in the

1, where

sufficiently positive

some

already

mentioned

bounded

a

problem

problem.

of this

p >

+oo for

<

(11.0.4),(11.0.5)

solution

domain

(11.0.2),(11.0.3)

Using,

example,

for

get the estimate +

aXk

integrating

w

derivative

the

R,

E

11 'O(X)J

:5

C3 and C4 independent

constants

80('aXJ)

called

JOIN-2

of the

deal

(02)

depending

e.

results

<

Here

O(x)

Let one

and

a

supercritical

papers

problem

the

k=1

tion

E

solutions.

10WI with

all

and 0

assumptions

principle,

the features

k(x1)J0JP-1,

=

oo

--

has nontrivial

has

here

0 for

>

C, and C2 independent

constants

i.

important

shall

we

N-2

0

W>

is

[99].

from

C'-function,

a

has

[1,77,78]

investigations

of

specific

some

radial

with

deal

that

mention

we

taken

Example

lot

a

are

f(.)

function

and

not

(11.0.1),(11.0.3)

We illustrate

example

is

there

problem

of the

lim

f (.) have been intensively

shall

we

f

problem

the

example,

interesting

many

book

present

literature,

In the

see,

nonlinearities

obtained

in the

However, lutions

(on

if

speaking,

(resp.

for

N+'

=

1951-oo

different

subject

p*

f(o 2) 0 (or f(x, 0 2) 0)

subcritical

subcritical

a

pairwise

of

this

superlinear

with

0 and

>

w

sequence

solutions

Problems there

with

N-2

1, and

p >

The exponent

called

is

OP',

=

nonlinearity

Roughly

+oo.

=

(0')

3, f

N+'.

the +oo

=

101-

(or (11.0.1),(11.0.3))

with

f(02)

Jim k61-00

N>

<

p

Respectively,

literature.

supercritical

with

only if

if and

solutions the

in

(11.0.1),(11.0.3)

problem

the

nontrivial

has

r

STATIONARYPROBLEMS

CHAPTER2.

40

obtained

the

C31_ _C'('X1 of

x

equality

E RN. over

(11.0.6) Multiplying RN with

the

equause

of

get

f

k(xl)10(x)IP+'dx

=

0.

RN

Hence,

since

So, this

we

connection,

k'(xl) shall

>

0,

usually the

we

O(x)

have

look

important

for

radial

results

-=

0.

solutions

by B. Gidas,

of the Ni

problem

Wei-Ming

(11.0.1),(11.0.3). and

L.

Nirenberg

In

41

[34,35]

be mentioned:

should

solution

problem

and of the is shown we

also

[9]

on

above

in the

assumptions

authors

under

the

with

assumptions

of

by

problem

a

function

of

general

similar

of

this

is not

and

P.L.

Now

so.

from

Lions

(H.0.1),(H.0.3)

to

x

As it

is radial.

type

sign

positive

independent

f

Berestycki

H.

arbitrary

an

a

alternating

with

result

that

proved

ball

a

solutions

solutions

general

very

a

have

in

proof)

a

of radial

existence of

for

papers,

(without

present the

these

(H.0.4),(H.0.5) (11.0.1),(11.0.3)

problem

of the

under

kind. 00

Theorem

11.0.2

9(-)

Let

U C((-n,

E

n); R)

be

odd,

3 and let

N :

n=1

(a)

'()

0 < lim inf

(b)

lim inf

(c)

there

'()

Jim sup 9W U

where

(0, +oo)

71 E

p*

again

-

exist

+00;

<

U-0

> 0

up

1-1-+00

<

U

U-0

such

N+2; N-2

=

G(,q)

that

0, where G(0)

>

=

f g(s)ds.

-2

0

Then,

problem

the

AO has

a

countable

In this

chapter,

qualitative

u(jxj),

=

of pairwise

different

we

shall

this

prove

establish

same

form

(N

2)

-

N,

solutions

radial

Theorem

011xj_.

E R

x

and

=

0

solution.

radial

positive

a

particular

in several

and

cases

study

the

of solutions.

also

we

in the

u

set

behavior

Now, taken

g(u),

=

as

Pohozaev

the

for

identity

(H.0.1),(H.0.3)

problem

the

[87]:

in

1 IV012

dx

=

-(N

1 Og(O)dx

2)

-

=

I

N

RN

RN

G(O)dx

RN

0

g(o)

where

Wo

=

f(02) 0

_

G(0)

and

-2

=

f g(s)ds.

equality

A similar

also

was

0

[74]

obtained

in

bounded

and

for

the

the

equation.

the

above

For

equality

equation

(11.0.1)

multiply

the

RN with

only

the

the

the

N 2 (N 2

independent

this

of

w

the

>

in

O(x) is

valid,

.5)

taking

into

integration

80('ax,),

sum over

i

1, 2,...,

case

the

function

1,

are

sufficient.

(11.0.6)

=

our

on

p >

of the

in

domain

one

result

may,

over

N and

first,

RN

integrate

and,

Q is of

the

f(o 2) To

in

get

multiply second, the result

by parts.

particular =-

jolp-1,

=

when the

case

Of course,

x.

account

further xi

the

assumptions

f(02)

0 and

in

of

additional

integration

identity This

3).

needs

by

solution

sign.

is

case,

with

equation use

The Pohozaev

change

f

example,

by 0

trivial

(11.0.4),(11.0

one

in

same

over

problem

function

(H.0.1),(H.0.3)

problem

p >

the

0 if

for

yields the

that

function

example,

for

(H.0-1),(H.0-3) NG(0) + (N 2)0g(O) does w > 0 and 101P-1 f(02) the

problem

-

=

has not

with

CHAPTER2.

42

Existence

2.1 first,

At

we

solitary

consider

u(x, t)

substitution

of solutions.

O(x

=

for

waves

ct),

-

-Wol Assuming

0(oo)

that

following

the

to

we come

a

NLSE with

a-

to

The

1.

=

equation

0.

=

0"(00)

and that

constants,

are

the

N

0,

=

equation: +

7(0)

0"

+

(II.1.1)

a

=

0

7(0)

and

-wa-

=

the

KdVE leads

the

into

a+ and

-wO with

KdVE and for

f M01 + 01"

+

where

a,

=

An ODE approach

the

R,

E

c

STATIONARYPROBLEMS

f f (s)ds.

is clear

It

that

substitution

the

general

of the

a-

for

representation

of

First

of this

solution

onto

(11.1.1) is bounded, 0"(-) is bounded, too,

equation

derivative

similar

a

However,

is

a

on

a

continuously

equation.

it

to

make

a

function

and

a

then

[a, b),

of the

follows

it

therefore

point

from first

the

where

this

if

equation

also

Setting

bounded.

Oo

0(a)

=

+

Oo, 0'(b) similar

Cauchy

the

00,

=

we

reasoning Let

fj(0)

problem

immediately several

=

for

get

times

f 0'(x)dx

the

0

second

solution

and

00

0'(a)

=

+

f 0"(x)dx

and

a

(II.1.1)

with

In this

statement.

this

specifying

not

can

b

equation

our

it

solution

of this

a

considering

then a

that

0'(.)

derivative

b,

<

a

Indeed,

b.

b

is

Of course,

simpler

is

differentiable

half-interval

half-neighborhood

right

a

f

if

is bounded

equation

be continued

to

here.

that

note

we

NLSE leads

the

by quadratures.

consider

we

all,

into

waves

be solved

can

analysis

qualitative

of

solitary

(II.1.1)

equation

the

section,

in each

apply

we

0(b)

data

initial

implicitly

case.

.0

7(0)

wO

-

f fi(p)dp.

FI(o)

and

a

-

Then,

the

equality

a-

(01)2

2

follows

0 if

<

(IT.1.1)

equation an

(11.1.1).

from

FI(o)

and

arbitrary

point

Suppose 0

E

has

a

(a, b).

that

solution

Xo E

a-

Let

by (1.1-1)

x0.

There

function

0"(xo)

cannot

E

=

1

exist

2 [01(X)]2

a

+

a+

satisfying

R and the

<

=

prove

us

0, thus point

FI(O(X))

=

0'(x) x

b, <

=

if these

initial

data

0'(xo) 0 in

we

=

a

right

that call

0, fl(b)

>

0 for

assumptions

0(oo)

conditions

> xo such

(which

Fi(b)

a,

=

0

=

that the

following

O(xo) Then,

+ Fl(O)II

for

=

some

a.

equation

b

valid,

are

Indeed,

>

a

then we

take

(II.1.1):

0.

half-neighborhood

O(x) the

=

a

energy)

because must

of the

otherwise be

positive

point the at

EXISTENCE OF SOLUTIONS. AN ODEAPPROACH

2.1.

the

point any

the

energy

particular, line

that

Also,

> x0.

x

horizontal

left

the

of the

as

x

point

It

x0.

can

10(x) for

positive

some

By

right

the

(or (resp., for

of a, x

>

a,

0(-oo)

a,

>

such

a

is

X__00

second

is valid

<

monotone

0

E

the

on

all

in

O(x)).

lim X_+00

if,

since

Indeed,

example,

for

this

where

points

solution the

a

is

0, then

and

0

and

=

point

no

0'(x)

limit

and

(11. 1.2)

0

<

finite

no

(11.1.2)

from due to

that

0(+00) is

then is

0

=

above

as

> a,

+00

--+

decreasing

that

there

a, then

x

F, (a,)

=

If there

follows

>

a,

d)

as

a

0 from

<

and

such

0 for

<

to

proved

is

a,).

=

claim =

F1(0)

f, (a,)

be

can

exists

first

O(x)

lim

it

0(+oo) FI(o)

a,

=

halfhas

H1.

<

x

in addition

line

real

entire

in

0 from

>

<

Lo

0 and

0 tending

(resp.

d

(- oo, d)). If (a, a,), then

on

0 for

fl(a)

>

x

0(-oo) (or conversely 0 and consequently Fl(al)

a,

that >

lim

(resp.,

0'(x)

-

E

a

-

=

solution

a

half-line

a

0

considerations

and

f (a)

0

of

whole

the

above x

0

imply,

easily function

=

0

case

(II.1.1)

=

0) 0'(x) O(x) (resp.

(resp.

+ oo)

F, (0)

and

a

=

(d,

on

in this

possesses

on

0'(x)

of the

all

the

Cle-C21XI

<

equation

(II.1.1)

equation

O(x)

solution

a,

analogy,

of the

(a, b) for a, O(x) that if fjf(a) proved

by

x

conservation

no

onto

graph

the

corollary

10'(x)l

+

if in

continuable

--*

some

this

then

-oo), increasing)

as

al

-

is

facts

be continued

can

point

Further,

there

E

=

C, and C2. Thus,

constants

complete

the

simple

easily

be

equation).

The above

arguments,

0(-oo)

this

0 for

otherwise

it

e.

:

of the

> xo.

x

i.

A

+oo.

--

all

(0'(x)

xo

because

above

of the

at

solution

a

global,

is

By analogy,

a.

=

0

in view

asymptote

0(+oo)

that

O(x)

is

a

E

E

solution

the

itself

to

-=

(a, b) (a, b) for

O(x)

that

Hence

E.

0

because

such

> x0

x

non-equal

hence,

theorem

uniqueness for

and,

x

43

the

it should

X__00

Fi(al)

be and

=

(0)

F,

is

a

for

0

E

< 0

0'(x)

above,

as

0 that

(a, a,) exists, 0'(x) (resp.

0

<

If the

contradiction.

then

0)

>

above

two

cases

all

for

point

a,

a

X

+00

as

jxj

O(x) oo

and

f, (a,)

>

--+

always

a,);

=

fi(al)

if

two

Let

0 is x

E

us

and

(resp.

a,

00

O(x) tending

solution

(since

of maximum

point

We shall

call

following

solutions

from

the

and

the

corresponding

ove-

constructed

above

limits

possessing

solutions

monotone

are

us

bounded

have

can

prove

the

continuously R

above

fi(al)

=

to

a

Fl'(al),

solutions

solitary

with waves

as

x

-+

kinks

and

only

of

00

precisely

that

is

arguments

one

point

soliton-like

respectively.

solutions,

(11.1.1)

for

important

These

of extremum.

one

the

get

we

0, then,

=

=

0

=

0).

(11.1.1)

types.

0,

then

precisely

possessing

An observation

equation

>

O(x)

lim X

lim

f, (a,)

if

R and

Fl(al)

satisfying

possible:

are

E

x

>

(in

view

that

the

ab

O(w,oo)

conditions

differentiable of

the

invariance

as

=

a

function

of the

O(w, x) satisfying

solution

0(w,x)

a,

of

equation

>

the

a

for

x

argument with

respect

E R and W

to

for

equation

0'(W,xo) X

an

arbitrary

translations

in

x,

depending

b introduced wo

b is

for

:

a

-

respect point

>

0.

But this

thus,

w).

follows

from the

F, (w, b)

by 0'

W

we

of

W

to

implicit

a

denote

the

function

of

a

0, f (0)

=

wo b >

-

(wo, x)

that

0

0 for

<

there

and

wo < 0

-

F, (0)

0 and

prove

P

-

because =

r=b

7(b)

-

solution

O(W, x)

with

above

function

with

fixed

the

a

(0, b)

E

b

exist -

such

> 0

Then,

that

above

the

using

f (0)

If

0"

=

0(wo,x)0'(wo,x)dx W

and

>

v

They

sech(z)

verify

this

2

N

I is

A > 0 is

considerations

problem

simple

NLSE with

a

I

1'(0)

in

a

existence

sense.

Now,

In this

N > 2.

takes

and

real

parameter

_

following

the

g(y)

=

with

f (y2)y

0y

initial

obtain

First

we

of the

functions

0'(w,x)dx KdVE

2))]

2A 2t _

1)(v

and

+

2A2

w

as

family

one-parameter

+

vanishing

One

(v+l)(v+2)

easily

can

a

point 0 <

result r

a

to

study

sections,

next

for

waves

the

multiconsider

we

O(x)

substitution

the

=

y(r),

(II.1.1)

equation

rele-

KdVE and NLSE with

dimensional radial

of the

case

of the

solutions

where

r

=

jxj, equation

N- I

and the

y'= g(y),

prime

>

r

the

means

(11.1.4)

0,

derivative

in

r.

We supply

this

data

continuously

where

one-dimensional

form:

y(0)

hood

the

solitary

begin

we

the

After

of

r

equation

following

( ! )_ )(v

the

yll+ where

of the

solutions

_v-

show that

of the

(11-0.1),(11-0-3).

problem

(11.0.1)

,

can

one

formula.

the

to

the

Av

C.+e-.

The above vant

form

jv

-

where

soliton-like

0, then

[ V\/ -)-(( (_2( _+1

A sech

0, f, (b)

methods,

-00

known.

well

are

=

00

here

-00

oo

(b)

F,

H1 and

E

f

2

TW

(0, a)

I

(Wo

E

of the

0"

d

=

parameter W

theorem

-2-F, (wo, r) i9r

of

x0.

easily

x

family

that

function

0, where

=:

arbitrary

an

prove

as

differentiability

continuous

Everywhere

to

differentiable

equation

the

respect

It is sufficient

continuously

algebraic

proved.

is

w

Let

f (b)

locally

of the

with

parameter

is 8

some

0, and,

to

the

on

above

solution

a

wob

8)

+

differentiability

the

state

cannot

one

solutions

6,

STATIONARYPROBLEMS

CHAPTER2.

44

=

on

0.

the

yo E

local

The class

differentiable

< +oo.

=

R,

y'(0)

=

well-posedness

of solutions in

[0, a)

y(r)

(11.1.5)

0. of this

we

and twice

admit

problem consists

continuously

in

a

neighbor-

of real-valued

differentiable

in

EXISTENCE OF SOLUTIONS. ANODEAPPROACH

2.1.

45

"0

Theorem

I1.1.1

Let

9(-)

U C'((-n,

E

n); R).

Then,

for

any yo E R there

exists

n=1

0 such

>

a

sponding the

solution

problem C'

into

2

=

(11.1.2) for [0, b] and the

all

problem

by analogy.

One

h E C ([0,

with

neighborhood

Further,

of

the direct

of the

[0, b],

y(r)

)

('go

is

0,

>

This

Vo

E R the

then

solutions

some

b

[O,a).

in

corre-

y(r)

6), where 6 from (y, 6, -go

8, F0

-

continuous

+

-

>

0,

+

6)

has

zero

verification

integral

nonlinear

only

the

by solving

deduce

easily

can

N > 3 because

case

the

the

case

that

equation

a); R)

N-IWI)l

w(O) a

i

if for

interval

an

consider

r1-N (r

in

from

yo

that,

segment

a

solution

unique

a

sense

on

map yo

We shall

proof.

of the be studied

can

linear

the

on

the

yo in

on

be continued

can

has

Q0, b]; R).

Sketch N

depends

V(r) (11.1.1),

be continued

can

(11.1.4),(11.1.5)

problem

the

continuously

solution

of

that

at

most

shows

an

0

=

solution

one

that

0,

>

r

w'(0)

R,

wo E

=

h(r),

=

above-described

of the

arbitrary

solution

sg(y(s))ds

(r

class.

C([O, a); R)

class

of the

equation r

y(r)

(N

yo +

=

1 [1 (S)N-21

2)-l

-

-

r

0),

>

y(O)

=

(11.1-6)

yo

0

satisfies is

Let a

of the

integral

to

Ma

integral

(0, a]

x

[0, a],

for

0, the operator

is

a

contraction.

Thus,

local

solvability

of the

follows

lemma. of

After

approach

we

this

Theorem

g(al)

=

that

g(a2)

has

shall

=

0

on

for

E

by

of any

with

is devoted

to.

a

a-,

a

(in

a

model

We shall

0 < a,

we

is

prove

a

of

9(y)

corollary

(r, 5)

proved

5

E

small itself

the

and

unique

dependence on

I

of

neighborhood the

of the

Gronwell's

regularity

0

ODE approach,

i.

e.

the

following. function,

differentiable < a2,

have

small

zero.

the

the

r

of

Ma into

continuous

based

where

(I)N-2

-

sufficiently

set

sufficiently

example

I

1

a

for

methods

dependence

[0, a] }

E

that, So,

neighborhood

continuously <

yo

standard

-1

function

maps the

X.

r

2)

-

bounded

The local

parameter

continuous

outside

some

y(r)

point

(11.1.6)

Let g be

(11.1.6)

of

(N

=

methods

1,

yo

-

and

continuous

(H.1.4),(H.1.5).

the

illustrate

H.1.2

a

kernel

side

fixed

a

global

from

section

(H.1.4),(H.1.5)

Cauchy problem

our

I u (r) k (r, s)

yo,

=

by standard

right-hand

equation

the

(11.1.4)

equation Here

from

is

get

problem

problem

of this

zero)

(H. 1.6)

from the it

0 the

>

a

easily

can

>

of

any

equation

one

a

solutions

E

since in

Thus

equation

I u (r)

=

Then,

0.

>

(H.1.4),(H.1.5). (11.1.6). C Q0, a]; R) : u (0)

Cauchy problem

the

equivalent

>

0

if

y E

g(a-,) (a-2, a-,)

g(O) U(O, a,)

=

STATIONARYPROBLEMS

CHAPTER2.

46

for

g(y)

< a-,,

a-,

some

if

0

<

y E

(a-,,

0) U(aj,

1

0, 1, 2,3,...

a2)

G(a-2)

and

G(a2)

>

> 0

S

G(s

where

f g(r)dr.

-2

for

Then

any

=

(11. 1-4), (11.

problem

the

5)

1.

0

has

solution

a

continuable

precisely

possessing

Our Proof

large

arbitrary

bound

yo the

greatest

of the

Cauchy problem

integer.

We show that

I roots

and tends Let

Take

lower

to

(11.1.4),(11.1.5)

r

first

Hence,

since

of the

G(y)

argument

< r

G(a2)

(11.1.4),(11.1.5)

problem

G(y)J1

+

G(a-2)

satisfies

correspondingly

clearly

due to

it

(11.1.7)

there

of

any

of these

solution

:5

a

Since

y(r). y(r)

(a,, a2) sufficiently there

exist

yo E

=-

is

a2

(a,, a2)

y(r)

a2,

for

r

E

(a,, a2) for (ri, r2). Then,

all it

r

we

>

1)

the

an

for

solution

where

Cauchy problem this

we

follow

I > 0 is

exactly

has

the

y(r)

of

[98].

paper

problem

the

a2,

r

[y'(r)]2

function

and the

yo E

the

such

:5 Co,

(a,, a2)

the

+

G(y (r))

y (r)

solution

of the

get by

the

-

(11-1.8)

0, half-line

entire

>

r

r

addition,

In

> 0.

(11.1.9)

0,

E(r)

=

(y'(r)

(11.1.4),

theorem

large

2Co(a2

>

that

equation

[0, ro].

E

a-2)(l

+

)2

taking on

> 0

ro

from

Let

ri

and

r2

(11.

1.

8), (11.

1.

1*2

2(N

we

+

yo from

continuous

such

2)(N

G(y(r))

the

the

energy

interval

dependence

that

that -

1)

ro

follows

-

step,

take

roots,

solution

any

on

sufficiently

a

with

of which

1)

+

(a,, a2),

Y/2.

function

of

and

E

1

solution

G(y(ro)) and

:

> 0

the

to

With

(0, a2)

y E

y(r)

We call

a

close

E

yo

show that

we

estimate

Co

exists

solutions.

of the

step

second

each

(I

than

-

for

I y'(r) for

and

r

be continued

can

+00

--4

place:

=

for

the

a-2

and

for

arbitrary

an

takes

nonincreasing,

is

> 0

<

less

proof.

For

0.

identity

IY12

At the

(a,, a2)

no

first

which

for

solution

of the

>

following

x

as

+oo.

--*

I

zero

At the

0.

>

r

from has

part

integer the

values

corresponding as

the

arbitrary

an

zero

sketch

us

of its

half-line

(11.1.4),(11.1.5) the

solutions,

has in the

of roots

to

(0, oo). of two steps.

consists

(11.1.4),(11.1.5)

number

0, tending

>

r

half-line

the

in

theorem

problem

Cauchy

the

I roots

of this

half-line

the

on

dr < r

2(N

-

be two

9)

points

that

I)Co(a2 ri

-

a-2)

such

that

y'(r)

0

0

EXISTENCE OF SOLUTIONS. AN ODEAPPROACII

2.1.

Now, and

y(r)

let let

r,

solution

a

y'(r) (11.1.10),

inequality

y'(r)

ri

asymptote

an

y

then

+oo,

=

contradicts 0.

y

0,

a-,,

=

a,,

the

y'(r)

nearest

to

Since

the

ro

by

and

r,

right

the

Now, follow

the

ro

>

y(r)

Second,

y(O)

with

f(y)

equal

to

and

the

(a-2,a2)

G(a-,),

is the

(any

principle

maximum

if

this

But

G(O), G(aj)

> ro

r,

has

only

numbers.

that

We assume

graph

its

interval three

because

> ro

r

Therefore,

of these

and due to the

> 0

:5 of

root

solution

(a-,,

domain

of Theorem

all,

of

we

y(r)

a

unique

In this

proving. problem

of

arbitrary

an

any two nearest

a

y(r).

solution

of the

part

First,

since

as

it

(11.1.4),(11.1.5) of

was

also

for

the

we

noted

a

point

problem

our

are

solution

nontrivial

a

Indeed,

solution.

of this Let

proof,

and

of

solution roots

of extremurn

point of

roots

proved.

is

of the

roots

between

positive

11.1.2

our

make two remarks.

we

have that

lies

of

part

solution

show that

(a,, a2)

1)

-

ro.

0 in

=

y'(r).

>

r

one

all

for

of

> ro

r,

part

a

: - 0,

be two nearest

< r2

y(r) y'(T) E(T)

E

for

Since

energy

second

First

we

inequality

latter

have

first

0 for

=

ri

+00 if there

=

0) U(a,, a2) and only a point Hence, (a-2, a-,) U(O, a,)), we y(ri) C- (a-2, a-,). than ro. Repeating these arguments (i. e. changing greater y(r) has no less than (I + 1) roots on get that our solution

the

to

[110].

paper

if

isolated.

r,

turn

we

y(ro) 0, then y'(ro)

earlier,

we

and the

of ro,

root

E(ri)

root

a

on),

so

1)(N

domain

y has

solution

+

monotone

is

c

of the a

a-2)(1

-

of maximum in the

point

a

of minimum in our

that

exists

only

have

can

(a-2,a2)-

positiveness

ro.

is

E

there

Therefore,

ro

the

to

(11.1.10)

satisfying or

ro

has

c

one

to

according

Then,

ro.

2Co(a2

>

solution

our

=

of

of ro nearest

have:

we

E(ri) If

right

the

on

right

the

on

(11.1.4),(11.1.5)

Cauchy problem

of the

of

root

of

roots

no

are

be

be the

47

let

definiteness

y(ri) > 0, hence there exists T E (ri, r2) such that 0 and y(r) > 0 for all r E (ri, T). Then, by (11. 1. 7) y (T) E (a,, a2) (otherwise < 0). hence of T. Therefore, y"(T) < 0, y'(r) < 0 in a right half-neighborhood of =r E (T, r2) such that 0 and y'(r) < 0 for r E (T, =r). Suppose the existence y'(j ) to because if E Then, according equation (0, a,] y(=r) Y(-F) E (a,, a2), then (11.1.4) the facts that of =r y"(=r) < 0 which contradicts y'(r) < 0 in a left half-neighborhood and that y'(=r) 0. But then E(=r) :5 0, and we get a contradiction because E(r2) > 0 > 0

between

and r2.

ri

Then,

=

=

=

where

>

r2

Let

=r.

So,

take

us

Yj and ?/o

V0 =

yo E

V0.

=

inf Y1.

=

First,

(0, a,]

since

second

our an

V0

claim

arbitrary

jyo

G

By V > 0

G(yo)

denote

brcause <

:

y(r) the

due to

0 and

proved. I >

integer

(0, a2)

we

is

0, let

has

no

less

of the

solution

(11.1.7)

consequently

than

a

solution

E(O)

<

(I

+

1) roots}

problem

y(r) 0 for

(11,1.4),(11.1.5) does

these

not

values

with

have

roots

of yo.

if Let

CHAPTER2.

48

Jy0 I

be

taken

77M,

r,'

with

<

0

<

yo

=

y'0

=

Tm,

Yj converging of the

rj-

<

...

r,'

<

smallest

F2n

<

<

There

Lemma 11. 1 .3

all

We will

Proof.

exist

I

be such

ym

that G(al)

=

2

<

for

all

r

therefore,

4

rn

of ym.

small-

(11.1.4),(11.1.5)

Then,

Let

1,2,3,....

=

as

it

is

proved

be

proved

fml+,.

Irmk

-

d is

we

(rm, z) where by (11. 1. 7) for of

(of course, notation).

m z

that

Tm k+,1:5a+b(Tkm+1)2

i

inequality

the

(k=1,...'l)

d

r

first Let

> 0.

the

C(T;'k'+,)

>

(z, zi)

E

depend

zi

from

can

E

z

(a,, a2)

interval

(rm,k for

A;+,)

-pn

which

2

independent

0 is

(z, Tkm+,)

E

zi

point

ym(r)l

dr

>

any

the

because

ym(*fk+,)

that

that

prove

C

where and

second

d where

First,

0.

the

definiteness

C=

independent the

to

the

that

prove

Cauchy problem

a, b > 0 such

the

1

G(")

to

respect

of extrernurn

constants

only

prove

Suppose

for

with

want

m

by analogy.

G(d)

uniformly

< rm <

1-fmk -1 -rml:5a+b(-f'k+1)2 k for

y-

of the

points

...

V0. Now we

to

solution

bounded

are

be the

T-1+1

<

above,

from

sequence

a

I roots

est

STATIONARYPROBLEMS

has

one

is such

we

G(ym(T-k+,))

We have

m.

C,

0 <

<

ym (.1)

that

m but

on

of

I ym(r) I E (a,, a2)

omit

this

dr >

C2(Tmk+,)-l

index

Co for

:'

C,

some

and

G(y- (zi))

the

simplicity

for

0;

>

=

of

Hence

Y12 (r)

2(N

!

y

I

-

12

(r)

r z

if for

r

G

all

(rm,k -). Thus, (II.I.11) such Tlk+,) r E (r', k

holds.

ym (r)

that

rm k

and Lemma 11.1.3

Summing

is

proved.

jTmj+j lm=1,2,3....

is

values

of the

functions

of the

interval

(a-,,

a

ym

a,),

Irmk

bounded their

at

there

-

Tmk+1

from

TMIT, Hence,

> d.

In view

<

a

we

of this

b(T;'A';+,)

+

d2

have

fact

dr2

and

ym(r) :5 -C3 < (11. 1. 11) we get

2

0

inequalities

the

-

by (11.1.4)

Finally,

rm k-

1

>

C,

By points

constant

=

we

obtain

i

I + 1

k

11.1.3,

(T;ml+l )2+ C2-

sequence.

a

of Lemma

statement

:5 C,

first

exists

1

the

1, 2,...,

C

1,

estimate

(11.1.9)

of extremum

lie

the

>

0 such

m=

that

1, 2,3,....

and

from the

since

the

outside

0

EXISTENCE OF SOLUTIONS. A VARIATIONALMETHOD

12.2.

the

Thus,

function

y(ro)

than

than

I roots.

the

on

At the

the

time,

same

yo and

parameter

definition

in view of the

imply

0

==

less

no

of solutions

dependence

continuous if

V has

the

theorem

the

on

that

y(ro)

V cannot

have

fact

the function

of -go that

49

=A

0

more

I roots. Let

that

prove

us

-9(r)

lim

Suppose

0.

=

this

is

the

not

Then,

case.

either

r-00

solution

the Zn

be

only

negative

for

can

number y E

first a-,

=

0) U(O, a,).

Thus,

contradiction

implies

of the m

y(r)

lim

=

y-

by

implies,

(11.1.4),(11.1.5)

0, and Theorem

H.1.3

theorem

the

than

more

on

parameter

I roots.

completely

is

domain

sufficiently

the

on

have

cannot

in the

for

negative

is the

since

case,

extrema

is

energy

problem

solutions

second

have

E(r)

energy

of the

negativeness

values

that

the

should

it

'g(r)

solution

the

of

which

asymptote

an

In the

r.

of extrema

sequence

a

has

P(r)

energy

1,

has

it

or

function

argument to

case

of solutions

large

sufficiently

this

in

But the

too.

the

of the

V is equal

solution

dependence for

that

values

r

of this

Hence,

a,.

=

large

of the

of r,

y

or

large

graph

the

case,

sufficiently

values

continuous yo,

y

of roots

(a-,,

large

the

In

sufficiently

for

V is monotone

+00-

-4

This

proved.0

r-oo

Existence

2.2 In this

section

the

to

we

shall

of radial

existence

f(02)

case

of solutions. consider

of the

jolp-1,

p > 1.

=

application

an

solutions

AO

A variational

wO

=

consider

we

101P-10,

-

Our result

the

on

Theorem

solution

We note

ul

that

right-hand

a

side

similar

sense

to

Theorem

tinuously

r(.)

ul(r),

'(rV)lr=r(,,)

=

point

a

N > 3 be

0,

where

r

=

takes

of the

equation

of

Loo

-

we

present

H be

Let

real-valued

Then, S,

jxj,

with

place

problem N

E R

(11.2.1)

,

(11.2.2)

a

solution

and,

precisely

1 roots

for

1

any on

g(o)

kind

(1, N+2). N-2

Then,

half-line

the

for

the

1,2,3,...,

=

(11.2.1),(11.2.2)

problem

the

general

more

and p E

integer,

for

then

if

a

below

real

are

Hilbert

functional differentiable

continuously

0.

vo E

attention

a r

> 0.

with

functions

g(O)

Theorem

11.2.1.

the in

a

101P`0.

which

a

the

proving

our

0.

radial

positive

result

11.2.2

be

>

similar

differentiable > 0

Jr

critical

Lo

has

=

Two results

Let

Let

to

We restrict

following.

is the

(11.2.1),(11.2.2)

problem radial

H.2.1

the x

=

existence

methods

(11.0.1),(11.0.3).

problem So,

of variational

method

the

Y(h)

space

in

j(v) -0.

when

proving

with

a

H, and S

function

functional

Ih=r(vo)vo

used

=

on

J(r(v)v)

norm =

f

S such

11

-

11,

h E H that

considered

be

J :

for

I IhI I any on

a

con-

1}.

=

v

E

S has

S a

Proof

is clear.

By conditions

Further, such

consider

-y(s)

that

E S for

d

0

arbitrary

an

(-1,

E

s

of all

consisting

Thus,

E L.

w

Remark

by

sults

S.I.

J(r(vo)vo),

H.2.3

The second

Theorem

of H1 with

Lq

is

E

the

compact

for

In

0.

=

is

a

Let

product

;

N-2

function

continuous

For any g E

1)

into

H

r(vo)-j'(0)

+

space

H

>=

>

subspace

the <

I

X(h)

L of the

(vo)v,

h=r

for

W>=0

I

any

E H. 0

w

simplification,

sufficient

for

goals,

our

of

re-

fh

=

for

jxj

as

at

Clearly,

point

the

set

Thus,

0

x

C07,

=

Then,

h(jx 1)}

the

h E

we

H,'

0 and

there

of all

vanishing

exists

that

-->

except element

oo.

C000

from

into

unique

a

each

Ix I

as

functions

radial

subspace

of H,1

everywhere

accept

can

be the

embedding

RN and continuous

in

any

h

:

arbitrary

an

oo.

---+

H'

E

of H'.

norm

everywhere

0

--->

following.

H,'

also,

h almost

Proof.

any

a

and the

4(jxj)

of the

is

is the

2N

2 < q <

with

vo,

for

N > 3 and

addition,

Sketth

H,1.

need

we

scalar

H,' coZnciding

of H,1 in

H.2.4

J.,=0vo

[r(-y(s))]

therefore

(-1,

Then,

0.

[75,76].

from

result

H.2.2

0

obviously

is

,=0

to

>= 0

w

Theorem

Pohozaev

d

1,

7'(s)

kind

-y'(0)

and

vo

=

0.

map -Y from

X(r (vo) vo), -y'(0)

<

orthogonal

vectors

<

7(0)

(vo)

=

0

differentiable

J'(r(vo)vo),

=<

of the

vectors

of all

d

-J(rvo)1,=,(v) dr

>=

1),

r

The set

theorem

continuously

j(-y(s))j.,=0

=

of the

J'(r(vo)vo),vo

<

x

STATIONARYPROBLEMS

CHAPTER2.

50

is dense

Qoo 00

11g1ji,

DN

=

f rN-1 [g2 (r)

+

)2]dr

(g'(r)

0

where

the

C07,

be

a

DN > 0 depends

constant

of the linear

Co7, equipped

space

with

converging

sequence

to

only

on

the

norm

h in

H,.

Hence the

N.

11 111, obviously

Let

-

Then

H,'

space

h E

for

H,1

is

any

r

a

completion

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

and >

0 there

C

exists

00

h (r)

(s)

lim n-oo

and is

for

any

continuous

a, b on

:

0 <

the

a

<

half-line

b

gn r

>

as

0.

n

Further,

-4

(11.2-3)

ds

oo

since

in

H'(a,

the

b);

sequence

in

particular

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

is

EXISTENCE OF SOLUTIONS. A VARIATIONAL METHOD

2.2.

converging coincides and

H,,

in

lim

=

easily

h(jxj)

therefore

h,

H,'

G

g,,

.n-oo

with

(11.2.3)

have

we

and

h(jxj)

=

thus,

as

everywhere

almost

of the

element

an

51

H,1,

space

RN These

in

facts

.

that

imply

00

h(r)

=

I

-

k(s)ds,

0,

>

r

r

(+oo)

hence

Also,

0.

=

obviously

we

have 2

00

00

1 (r)j

h(s)

ds

I

<

S

N-1

2

00

ds

(h'(s))'ds

(N

SN-1

-

2)-12' Jjhjjjr

2-N 2

r

r

(11.2.4) I h,,In=1,2,3....

Let

f 01.

hn belongs

Each

by (H. 2.4)

c(R)

0

--+

r>R

R

as

<

bounded

a

the

of functions

sequence

of H'

embedding C

where

for

Hence

+oo.

--+

be

L, by Cc(R)

to

I hn(r)l

max

Hr'

C

>

L, Further, are independent

RN \

in

for

into

c(R)

0 and

continuous

of

0

r

>

n

and

R> 0

00

Jhnj'Lq(jX:

DN

=

jxj>Rj)

I

r

N-1

(R)jjh,,jjj2,r'

:5 DNCq-2rq-2

Ih.,,(r)lldr

R

Since

here

for

large

ciently

Remark

Sobolev

spaces

Now,

we

functions:

u(jxj)

> 0 and a] I

of functions

from

Clearly

u

<

E

Hr'

Consider

a u

for the

with

(I

E

u

is

by

u(jxj)

0 <

a

Lions

P.L.

the

We consider

H.2.1.

b) H,(0,

<

b.

Let

+

W02)

from

Hr'

and

the

b)

=

than is

in

space

H,(b)

Lq-E3

in

theorems

of

[56].

following

spaces

the

satisfying

suffi-

6 for

compact

embedding

of

case

obtained

of functions b

I hnln=1,2,3....

particular

a

symmetries

(here 0 as I x H,(a, b).

x

any

...,

is smaller

side

the sequence

Theorem

Hr(a, b) or jxj :

space

as

1, 2, 3, H.2.4

proving

to

turn

jxj Hr' satisfying 0

=

Theorem

H.2.5

the =

n

right-hand

6 > 0 the

given

arbitrary

an

R

of

condition

H,(b) of functions and Hr(+ 00) H,. =

functional

f12 (IV012

JW

P + I

IOIP+l}dx.

RN

It

is

clear

that

Consider

J is

continuously a, b:

arbitrary AO

=

wO

-

differentiable 0 <

a

101"_1 0,

<

in

b < +oo.

1XI

E

H,. First,

(a, b) (jxj

let

<

prove

us

b if

a

=

0),

that

the

problem

(11.2.5)

52

CHAPTER2.

0(a) has

positive

a

and

=

J,'(rv)

We

b).

be use

Then,

imply for

0

(to

0

solution.

snH,(a,

G

v

0(b)

=

interpreted

as

Theorem

11.2.2.

conditions

the

STATIONARYPROBLEMS

0(b)

0 if

=

S

Let

a

jh

=

of Theorem

=

0),

(11.2.6)

H,'

G

11.2.2

r

11hill

:

r(v)

=

1}

=

0 and

>

case

our

f

VU12 + WU2) dx

(I

a
r(v)

=

f

JujP+1dx

a
and,

-d

since

lr=r(v)

J(rv)

Ir

=

0, functions

I

(IV012

0

r(v)v

=:

W02)dx

+

r(v)

is

10

M

Then

E

function

H,(a,

get

we

J(O) therefore,

the

embedding there

functional

theorem

C,

exists

0 such

>

b)

0 0

(12

=

(11.2.7)

1

all

0

J ,} this

lim

be

a

)110112 1

for

to

lim

11.2.4

some

0

M.

In

C

0

>

addition,

by

since

independent

of

0

the

E

M,

(11.2.9)

our

is

this

j(ao)

0,

positive. reason

=

the

left-hand

by (11.2.8)

a2(i 2

1 P + 1

We can also

accept

11011,

that

M.

on

<

we

In

view

10,,_}

Let that

there

lim

110,111.

of

be

a

exists

By

have:

IOIP+'dx

==

7

0.

RN

(11.2.7)

of

0 : we

-

weakly compact.

J

M-CO

side

Therefore,

functional

it is

Suppose

RN

for

the

H,.

in

I 0,,_ IP+'dx

lim

But for

on

(11.2-8)

M;

E

> 0

for

110,111.

by (11.2.7),(11.2.9)

and

M-00

latter

0

with

H,1; hence,

in

M-00

Theorem

the

(11.2.7)}.

satisfies

below

! C,

sequence

is bounded

weakly converging Then, 11011, :5

110.,111.

Hence,

We set

that

minimizing

sequence

subsequence M-00

(11.2.7)

E M.

Let

(11.2.8),

0

C`jj0jjP+1

<

11011, for

0 and

from

110112

b).

vEsnH,(a,

of

J is bounded

and

identity

a
smooth

a

the

IOIP+ldx.

a
Obviously,

satisfy

is smaller

0 and there

exists

than a

E

the

(0, 1)

right-hand such

that

get:

)110112 1

,

_

liM m-oo

1 2

P + 1

)II0,111

2 =

inf

J(v),

VEM

side

ao

and E M.

EXISTENCE OF SOLUTIONS. A VARIATIONALMETHOD

2.2.

i.e.

acontradictior.

weget

lim

Hence,

53

J(O)

E Mand

101

Further, 0

that

accept

J considered function

(r)

v

H,(a,

space

b).

VEM

-

v

(I 1)

+

wv

x

J(O

dt

0 in

=

is

critical

a

point

tv) 1,=o

+

0 by 101,

changing

Hence,

0

11.2.2,

Taking

Coc (a, b) satisfying

E

J(O).

=

to Theorem

According

the

on

J(101)

and

M, too,

E

> 0.

for

we can

functional

of the

arbitrary

an

neighborhood

a

J(v).

inf

=

M-00

of zero,

radial

obtain

we

equality

the

+00

I

DN

d

d

fr N-1( Tr v(r) Tr 0(r)

(r) 0(r)

(r) OP(r)) I

v

-

dr

(11.2.10)

0,

=

0

0

hence, with

is

generalized

a

closure

its

(11.2.10)

and

for

a

in the

implies

in

the

<

IxI

sense

solution

impossible

that

b

because

Consider

N

E R

case

:

:

a

I fVv(x)

Vo(x)

b}. 0' (r)

Hence,

(11.2.5)

,

some

0(b)

classical

O(Ixl) > (a, b) which

E

r

0

=

0 is

2.1).

Section

see

contained =

in the

particular,

in

0 for

=

0(a)

0, then

>

a

domain

equality

the

wv(x)O(x)

+

If

equation

<

theorem,

Then,

0.

=

any bounded

satisfies

=

uniqueness

a

<

a

O(Ixl) < JxJ 0(r)

otherwise

of the

the

E R

the function

Ix (because

(11.2.5) in JxJ < b}.

equation N

Ix

set

domain <

of

v(x)o"(x)}dx

-

=

(11.2.11)

0

RN

follows

also

from the

v(x)

condition here

limit

the

function

arbitrary

v

solution

over

Co'

E

v

E

Co'

If b

0

equation

satisfies

point

point

Then,

(11.1.4)

prove

a

continuously x

=

since

to

We take

0.

JxJ

the

<

prove an

as

C-

we

in the

-

<

JxJ

of

fact

in

<

Also, be

the

Taking

zero.

converging

to

(11.2.11) 0

for

a

an

generalized

is

a

is

also

since

the

b which

cannot

equal

radial

a

function

to

in

zero

a

section.

(11.2.5)

this

equation

R > 0 if E

it

and

Hence,

equation

an

0

b.

11.2.4.

JuIP-'u,

previous

O(Ixl)

have

JxJ

of

equality

achieve

we

by Theorem

and satisfies

this

properties

domain

wu

=

arbitrary

W21(BR(O)), with ri L,,(BR(O)) 0

b

solution

the

0

=

g(u)

H1,

C000 satisfying

E

v

neighborhood

open

domain

in

011,,1-0.

functions

these

space

the

(11.2.5)

diff-, rentiable

suffices

it

<

that

an

satisfying in

with

non-radial from

x

of the

support

a

then

+oo,

Let

Clearly,

Jv,,J

equation

domain

is twice

b.

=

b and for

sense

with

of the us

the

in

function.

JxJ

for

arguments >

sequence

a

elliptic

of

above

0 for

=

arbitrary b

=

is at

small +oo

Lq,,(BR(O))

and

where

classical, any

i.

point

that

e. x

JxJ

:

neighborhood R qo

b if

<

of the b

' N-2

it

.

<

+0o.

Hence,

2N qo IOIP-10 E E W,2,(BR(O)) Therefore, O(qo) p(N-2) P where (on this subject see [36]). Thus, by the embedding theorem 0 (E L,,(BR(O)), 2N r1N ' if p E ( 4 and I is if > o(ri) q, arbitrary N+2) q, p N-2 N-2 N-2ri N-21 p(N-2)-4 C(BR(O)) if p E (1, N42 ) .In the two latter cases 0 E Wr2(BR(O)) with arbitrary differentiable in BR(O), therefore it is a classical large r > 1, hence, 0 is continuously =

solution

of

our

=

=

-

*

-

-

equation.

,

=

-

CHAPTER2.

54

Consider also

; 2,

<

P

then

L, (BR(O))

E

that

< r2

r,

sequences

the

with

point

fixed

point

of the

required

arbitrary

is

2N

=

N-2

with

it

< :

large

thus,

it is

0(q2)

for

hence,

it

because

n

to

converge

this

map has

a

no

embedding

theorems

by embedding

theorems

the

1, therefore

q >

no

when these all

must

then

2

number

a

case

that

by

:,

We

increasing

or

6 > 0 and

verified

<

infinite n

in the

some

cases

q2 P

all

all,

qo.

r2

=

get either 2

simply

classical

a

r3

of

bounded,

remaining

that,

=

>

q,

if

show

< L

6 for

-

2

be

can

arbitrary

arbitrary

an

we

rn

that

solution

of

0 < a,

< a2

(11.2.5)

equation

and

proved.

is

statement

So,

L. First 2

-

and

In the

.

if

on.

>

rn

monotone

but qo

so

> qj,

satisfying

and r,,

2

can

one

(p(r2)

and

be that

differentiable,

fix

us

:L

<

cannot

L,(BR(O))

to

Let

it

than

continuously

the

< rno-1

...

map

greater

belongs

W(r3) > q2, fqn}n=0,1,2....

f qn}n=0,1,2....

sequence

q2

::--

verify

easily

can

By analogy, with

and

....

<

infinite

are

fixed

it is

q3

=

One

N-2

,

2

=

1rnjn=1,2,3

sequences

such

E

N+).

4

p(?P(qo)). Lq2PRM)

q-,

0

(N

E

p

case

that

observe

"'

0

the

STATIONARYPROBLEMS

positive

integer

1.

Let

<

<

< al

...

be

oo

be U1(jXj),U2(jXj), ul+,(Ixl) the above- constructed radial solutions of the problem with a (11.2.5),(11.2.6) ak-1 and b the above minimization where they satisfy ak (i.e. respectively, problems), and U2 < O)U4 < 07 because exist if u is a U1 > 07U3 > 0,... (solutions U2, U41 the solution of then -u is a negative We positive problem (11.2-5),(11.2.6), solution). denote by u(jxj) the function which is equal to Uk(IXI) in the domain ak-1 < JXJ :!, ak, and

points

let

ao

0 and

=

a1+1

Let

+oo.

=

....

=

=

...

k

1, 2,

=

Then,

ak.

of aA;,

1+ 1. Let R be the

...'

according

i.e.,

minimizing

R1, ai

0

parameters

Lemma 11.2.6

There

1, 2, 3,

converging

n

<

of

=

J(u)

(il

=

J is bounded for this

sequence

values

ing

functional

the

of all these

set

(11.2.8)

to

...

T' ) Jjujj,'

2

exists

subsequence

a

to

some

a

suffices

It

k=0,...'1-1, 1,2,37

...

It

of

<

a

the

(a,,...,

al)

exist

C, C,

=

)

and because

suffices

to

the

to a

n

I

prove

< -

C1,

other

prove

that for

there

all

n.

estimates

We shall

n

1+ I

of the parameters

values

all

Let

be

the

R with =

(n

+ oo

sequence

a

n

R' where 0

E

and for

E R

correspond-

=

1, 2, 3,

(a n'1 ...'an)

=

< a,

a

< a2

E

1

<

<

...

can

-4

Suppose

+0.

that

110nill

too.

Let

<

Vn(X)

Q2 for =

it

such

> 0

only

that

prove

be obtained

that

n

ak+1 a' > C 1

C,

n ak >

-

-

(n

> 0

-

by analogy.

that

J(U)

inf

b

n

a1

R.

set

the set

on

<

any

u

+oo Proof.

as

...

the

on

J considered

a' 0 < a' 1 < a' < 2

=

all values

for

p+1

from below

functional

for

functions

-

is

some

anOn(bnjxj)

UEHO'(b)nm,

u:oo

right.

Then,

not

where

there

On (here

minimizers an

>

0

--+

are

n

+00

exists =

chosen

a

sequence

1,2,3,...). for

the

Then functions

bn

--+

+0 such

JO.jp+j vn to

:5

C2)

satisfy

EXISTENCE OF SOLUTIONS. A VARIATIONAL METHOD

2.2.

(11.2.8)

condition

1 1 VnJJ21

with

a

b

0 and

=

C3, C4

some

0

>

H,' satisfying

E

theorems,

embedding

using

we

get

for

all

C3 IVVnl2,2 therefore

<

b2nIVOn for

Then,

1.

==

55

0

independent

(11.2.8),

of

+1 1 12>Cap n- 10njP+1 4 P 2-

Hence,

n.

since

10,,Ip+l

earlier

as

! C5

> 0

have

we

2

C6b,,P,-'

0 < an <

--+

+0

as

n

oo.

--*

Therefore,

J(Vn) as

n

(2

=

because

oo

--+

11 VnJJ21

p + 1

2

N+

-

2-N+P4

C7 JVV"'12 2 < C8bn

< -

P41

particular,

In

> 0.

inf vEH01 (i)nm, i.

e.

contradiction.

a

Let of

positive

implies

this

J(v)

Cgbn2-N+p4'

< -

0

-->

that

0,

<

n

return

us

v:Ao

JVO 12 2

I

solutions,

proof

the

to

one

of Theorem

passing

prove,

can

As when

11.2.1. to

existence

if necessary,

subsequence

a

the

proving

that

the

in H,' to a function weakly converges j0n}n=1,2,3.... 0(jxj) which satisfies with and b and k a ak equation 0, 1,-, 1, (11.2.8) (11.2.5) for ak+,, 1. Also, since 0 for k ak < Ix I < ak+l 0, 1, 2,..., 1); in addition (k 0(ak) the function where is solution for of r a r jxj, equation 0(r), (11.1.4) ak, there exist left 0'(ak a derivative on the on the right 0'(ak + 0) and a derivative 0) which sequence

condition

=

=

=

=

=

=

=

are

nonequal Now,

to to

zero

prove

by

the

theorem.

uniqueness 11.2.1,

Theorem

suffices

it

k

as

a

by

function

of ak; at

the

the

and

r

are

E

let

of

r

s

=

equal

to

has

0 the

lim ,3-+o

[ak-1,

c(s))

a(s)

>

theorem

function

c(s)

function

point

which for

implicit

the

and

0 and

satisfies

a

unique

0(r)

c'(s)

C'

to

function

and

r

c(s)

has

lim

c'(s),

S__o

T(o, P) P(s) >

5,

r)

0 be

condition

=

(11.2.8)

((-so,

derivatives

values with

for

of

r a

a

=

E

and

=

.5

the

T(a, 0,

[c(s),

ak+11,

0

such =

E

(aA:-,, + <

ak). s'o(r).

so

w

neighborhood we mean

s,

r)

=

that

the and

that

the

on

where

c(s)

+

that

0(r) 181

and

Let

ak-,,b

:

where

right

01(ak

suppose r

small

0]),

on

respectively.

#w(s, r)

the

C'

as

small

sufficiently

a

=

0

w(s, r)

Denote

0)

-

and >

sufficiently

any

c(s) in ([0, so)) and

root

belongs

for

1,2,...,l

=

-

Then,

0'(aA;

that

show

to

Let us take an 1,2,...,1. arbitrary 0) for all k for Let also the definiteness 0'(ak 0) 0'(ak + 0). Fix 0(r) E Co (ak-1, aA;+,) such that 0(ak) > 0.

left

aw(s, r) a,

0

>

0

function with

a

=

STATIONARYPROBLEMS

CHAPTER2.

56

c(s),

b

P(s), by

Since

UT(a(s), 0(s), 0]; H01(ak-1, ak+,))

Clearly

respectively.

ak+,,

=

UT(a(s),

and

-)

s,

C'((-si,

E

-)

s,

E

for

C'

some

Q0, si); Ho'(ak-1, >

si

ak+l))

0.

definition C(S)

d

rN-1

da

(&2

2

UT2)

+

ITIP+11 dr

P + I

ak-1

ak+1

1

d =

0-0e(s)

-

do

N-1

r

(U 2

-

2

1

WIU2)

+

J-TJP+1

_

P + 1

dr

0,

C(s)

easily

we

derive

using

an

integration

by parts

ak+1

I

d

ds

1

N-1

r

(&2

2

WUF2)

+

JTJP+'] dr

P + 1

ak-1

aA,+,

1

d

ds

r

N-1

(U 2

2

+

WU72)

P + I

S=+0

J!UJP+'] dr

S=-0

ak-1

ak+1

f

=

r

N-1

[0/01

000

+

_

101 P-'Oo]dr

=

a

N-10 (ak)[O'(ak k

0)

-

0'(ak

-

+

0)1-

ak-1

Hence, or

0'(ak

if

negative)

-

such

0) : 0'(ak

+

0),

then

there

exists

sufficiently

a

small

(either

positive

IL 2.1 is

proved.0

s

that

ak+1r 1

N-1

(&2

2

+

VT2)

P + I

JTJP+1

dr

<

a1_1

ak+1

I

<

r

[2 (012 1

N-1

+

W02)

101P+1I

P + I

dr

aA,-,

which

is

2.3

The concentration-

a

contradiction.

Thus,

0'(ak

0)

-

=

0'(aA:

+ 0),

and Theorem

method

compactness

of P.L.

Lions The concentrationof

applications

unbounded a

problem.

compactness to

various

domains; Here

we

method

equations

in

particular,

only

touch

proposed

(differential, in many

upon

this

by

P.L.

Lions

(see [57,58])

integro-differential, cases

subject

it

allows

to

and illustrate

integral, prove

the

the

has

etc.)

solvability

method

with

a

lot in

of an

METHOD OF RLIONS ION- COMPACTNESS CONCENTRA7

2.3THE

example

of

presented of the

problems

simplest

of the

one

below

is

of

solitary

to;

shall

we

the

use

result

problem

the

Weconsider

waves.

functional

of the

minimization

applicable

it

stability

the

investigate

to

57

2 IVU12

IUIP+1Idx

1

E(U)

+ I

RN

under

the

JU122

restriction

I ,x

The result

want

we

Theorem

for a

11.3-1

arbitrary

an

minimizing

in H" and

(1,

1 +

-I-)N

JU122

=

Al

(11.3-1)

limit

arbitrary.

and A > 0 be

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

sequence

arbitrary

its

set

we

H',

E

u

jV such that

C

fixed;

0 is

>

following.

is the

Let p E

A

JE(u)/

Inf

=

prove

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

sequence

compact

to

A where

=

the

of

fUn(*

sequence

point

is

+

(11.3.1) Yn)}n=1,2,3....

the

of

solution

a

Then L >

problem

the

and

-00

there

exists

relatively

is

problem

minimization

(11.3.1). Remark and any

A

>

If p

11.3.2 0

has

one

+

-I-,N

then

0.

To

see

> I

IA

<

I,\

this,

I U(or, .) 122

1 IVU(j,

1

-oo E(u(u, becomes negative -))

E(u(u,

Remark more

general

(27ror2)

X) 12 dx-

11.3.3 kind.

as

for

Really, For

a

>

Remark

providing

the

M N

1 +

e

T

1

(p+l)+N

2

07

P.L.

+0 when p

-4

sufficiently Lions

example,

he

0 and

f (x, u)

11.3.4

As P.L.

relative

is

large

[57,58]

in

c(x)u

p + 1

I I-I--

something Lions

compactness

the

=

like

>

'

x

and,

N

1 u (1, x) jP+1

dx.

for

E

p

(1,

1 +

N

0.

problems

considered

f (x, u),

=

1 +

> cr

investigated

U

c(x)

(1,

RN

-Au +

where

For P E

function

2

N

RN

Hence,

A > 0.

any

the

=A and

(o,, x)) =20r2

E (u

consider

VA

u(c, X) We have

for

-oo

=

of the

essentially

problem N

E R

0,

k(x) ju IP-'u his

with

publications,

noted

in

of any

minimizing

sequence

k(x) the

>

0.

principal

up to

relation

translations

as

58

CHAPTER2.

in Theorem

11.3.1

I,\

is

I,, + I,\-,,

<

for

(0, A).

a

E

a

sequence

STATIONARYPROBLEMS

Below

obtain

we

this

relation

for

problem.

our

Lemma 11.3.5

jUnjn=1,2,3....

Let

be

in H'

bounded

and

satisfying

the

condition

1 U 122 where the

A

0 is

>

following

fixed.

Then,

three

exists

a

fUnk}k=1,2,3....

subsequence

one

of

there

fYk}k=1,2,3....

exists

C

R'

that

such

for

any

> 0

c

there

for which

R> 0

U2k(X +Yk)dx>A-c, BR(O) fx (ii) (vanishing) =

k=1,2,3,...

BR +(0)

yl,

(here

satisfying

properties:

(i) (compactness) exists

there

A,

=

RN:

E

jxj

R});

<

lim

Z

sup yERN

k-oo

U2n,:(x)dx

0

=

y+BR(0)

for

R

all

(dichotomy)

(iii) bounded

0;

>

there

exist

[V and satisfying

in

I

u""

(UA;

-

the

U2)1

+

k

(0, A)

fUk1}k=1,2,3....

and sequences

and

JU2k}k=1,2,3,...

following: --+0

k--+oo

as

for

q

I (Uk1)2dX

lim k-oo

G

a

-

a

I (Uk2)2dX

lim

=

k-00

RN

2N

2
_

(A

_

j-

te)

=

2

0;

RN

(SUPP Ul;k

dist

I

liminf k-.

SUPP

I I VUn) 12

_

U2)k

-+

I VU1k 12

+oo

_

as

k

I VU212 I k

-->

oo;

dX > 0.

RN

With from

Part

the I of

Proof

[57].

of Lemma 11.3.5

We introduce

Qn(0

we,

the

=

actually,

concentration

SUP YERN

f

the

repeat functions

proof of

of Lemma 111. 1

measures

U2 (x)dx.

y+Bt (0)

jQn(t)Jn=1,2,3....

Then, functions quence

that

on

R+

is

and

k-oo

Qn, (t)

=

sequence

liM t-

f Qnk } k=1,2,3.... lim

a

+00

and

Q(t) for

a

Qn(t) function

nondecreasing,

of

A.

=

By

the

nonnegative, classical

Q(t) nonnegative

any t > 0.

result, and

uniformly there

nondecreasing

bounded

exist on

a

subse-

R+ such

OF RLIONS METHOD CONCENTRATION-COMPACTNESS

2.3THE Let

a

+00

t

for

the

Let

us

place

Q(t).

lim

=

Obviously

[0, A].

E

a

fQnk(t)}k=1,2,3,.-

sequence

If

If

a

a

0, then

=

(ii)

vanishing

the

clearly

A, then

=

59

the

takes

(i)

compactness

occurs.

briefly

arbitrary.

be

these

prove

Then,

claims.

two

let

First,

a

0 and

=

let

c

0 and

>

R

0

>

have

we

Qn,(R)

I

sup yERN

=

u',, (x)

dx

<

c

y+BR(O)

for

large

sufficiently

all

obviously

there

where

are

Mk

such

0, and the first

k >

exist

sequences

Ck

integers,

such

positive

that

I

claim

+01 Rk

--+

dx

0,

have for

n

+oo

-->

Qn- (Rk)

that

2, (x)

U

proved.

is

k

as

>

Q,,,, (Rk),

=

Second,

k

--+

=

1, 2, 3,

a

all

Then,

A.

=

fMklk=1,2,3

and

oo

for

6k

let

>_

....

...

yj,+BRk(0)

Then, taking

arbitrary

an

>

c

we

I

n_(x Un

k such

all

y,,,)dx

+

A

>

that

<

el,

all

and for

c

m

> mk-:

c.

-

BR, (0)

Now,

to

R > Rk

that

so

the

Consider

place k

in

this

the

case

for

m

all =

k

0,

1,

p(x)

0( ) .

C

>

=-

to

prove

take

to

sequences

Rk

the

that

=

W be

a,

=

A

...

=

(infinitely and O(x)

respectively.

=

exists

Rk,

=

=

k

'Ek,

a

=

k,,,

number

lQnk(4m)

and

...

Rk_+j

get required

< 1

there

m-1

R,

set

we

cut-off

=

m

Q(m)1:5

-

we

Jxj

0 for

and W,,

(iii)

dichotomy 01

Ck >

17

-

El

=

Ck

takes -+

0

as

> 0

Q(4m)l ...

=

such

'Ek,

=

=

that

m-'

<

Ckm+l

M7 Ck-+l

1, 2, 3,

2A

and,

M_' +

differentiable) =- 0, W(x) We set

M

functions: 1 for sup

Jxj

> 2.

0 <

0,

0

Let

0A

and

:5

I I u. I 1 1. Then, clearly

RN

+

-

each

Q(M);

Y)Unk(X) 12

_

02R(X + Y) I VUnA,(X)

12]

dX

<

1, O(x) W,, denote

there

that

[IV(OR(X

for

a

sequences.

n

0 such

large

sufficiently

a

+oo and

I Qn, (4Rk)

and

integer

Rk_+1

172,3,..., Let

We have exist

'Ek

a

-

Then,

k,,,.

Q(+oo)

since

suffices

it

...'

proved.

is

there

1, 2, 3,

=

(0, A).

E

a

positive

any

>

m

claim

Clearly,

IQ,,,,(m) for

all

that

I Q,,, (Rk) Indeed,

for

second

case.

such

oc,

--+

relation

this

get

-2

CR

)

Take Yk

Mk-

M

exists

CHAPTER2.

60

STATIONARYPROBLEMS

and

I

[IV( OR(X

2

Y)Unk(X)l

+

_

W2R (X

Y) I VUnk W12 ]dx

+

-2


RN

for

N

all

y E R

k

,

1, 2, 3,

=

and R > 1.

...

Q'Ilk (t) (Rk)

Let

I

=

R' be such that

Yk E 2

(X)

Unk

dX.

Yk+BRk(o)

Then,

Ulk

setting

ORk(*

=

f (u 1)2

lim

dx

k

k-oo

Yk)UnA,

+

ce

-

f

lim

=

k-oo

RN

Finally,

let

Wk

-

(Uk

W4Rk (' +

=

Yk)

therefore

oo,

--+

jUjq q we

Ci(lUlq

< -

-

0.

=

Then,

WkUnA:.

we

2

Unk(X)dX

Qnk(4RA;)

_<

Qnk(Rk)

-

0

---+

jX-YkjE[RA:,4RAjj

Sobolev

the

have

inequality

2

2N

2 < q <

2

(q

y-

for

> 2

N

=

1, 2),

(11.3.2)

obtain

we

1

turn

we

for

quence

the

(Uk

+

U2)k

with

q

for

any

P E

(1,

the

sequence

=

p + 1.

1 +

(11.3.1).

Let

a

I,\ E

the

uEH':

I

the

we

:

(p I I VUn12 1,,=1,2,3....

exponent

the

use

2

+

1)

any

A

> 0

and

a

E

E(u)

inf -EHI:

0 -

P +

11

RN

JU12= 2 ,

IuIP+'dx

-

is

bounded,

E(012u)

<

H'.

OI(a),

se-

Indeed,

embedding

N belongs

(0, A)

I

minimizing

Sobolev

ce

E(u)

a

in

(1, A]. Then,

inf JU12=0C, 2

JUI 22=Cj

prove

be

boundedness

its

in H'.

for

inf uEH1:

2'

N

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

Further,

sequence

and 0 E

Let

we

bounded.

L +

[A,2 A)

Io"

0

first,

is bounded

<

2N

2 < q <

oo,

---+

11.3.1.

inequality,

Hence,

jUnjn=1,2,3....

Lemma 11.3.6

At is

In this

1). N

k

as

Theorem

JE(Un)ln=1,2,3....

(11. 3.2)

0

--+

(iii).0

proving

to

problem

sequence

Proof.

-

dichotomy

the

get

Now

the

=

Ujq+N- IIVUI I-N)22

+

2

by

JUnk and

and U2k

fxERN:

k

o2Rk(X + Yk) I VUnk 12 ] dx

2

RN

RN

as

[jVul k I

+ Uk 2))2 dx <

1

(Unk

get

we

to

(0, 2)

therefore

23THE CONCENTRATION-COMPACTNESS METHOD OF RLIONS since

it

as

is shown

in Remark

I,

11.3.2

I(A) Consider that

has

it

Suppose

J,8kU k2122

=

Ilk

A

the

which

opposite.

JU12=Ce 2

k-oo

which

contradicts

Then,

according

in

the

0 and

k-oo

by >

El

have

(iii)

show

that

=

Hence,

oo.

--+

We shall

cannot

0 be such

k

as

a).

-

(dichotomy) a and Iceku 112 k 2 > E(u') k + E(U2)k + E(Unk )

flk

we

I(A

Lemma 11.3.5.

show that

0

---

+

us

1 and

=

[E(akUlk)

lim

!

given Let

yk'

-yk,

Remark

0

-+

+ E p k U2)] k

cannot

k

as

according

+

A

for

because

IA

suffices

it

to

liminfE(u.,,,) k-00

Take

0.

<

Y E

aim,

then

0

arbitrary

an

R

0.

>

R'

jUjp+1+N-I (p+1) L2(y+B (0)) 2

+

(Y

L2

any

this

For

occur. oo

--+

to which

inequality,

(IUIPL+2(y+BR(O))

C(R)

<

lpP+1 '++,'

11-3.2

Sobolev

the

to

(ii)

vanishing

jVUjE(p+I)-N 2(y+BR(0))j 2

X

'R

L

00

N

fZ,},=1,2,3,...

sequence

Pk

where

Jun,

(ii)

case

I IUI'L+p+,(y+BR(O)) a

lim

I(a)

Hence,

< 0.

Lemma 11.3.6.

the

that

>

k-oo

yk/

+

show that

us

I(a):5

a

ak

=

E (Unk)

lim

=

contradicts

prove

Let

ak

k-oo

Let

a

+

Let

lim

+ E (pkU2)k

IA

A

I(a)

funk I k=1,2,3.... (i) (compactness).

Then

a.

-

E(akUlk)

==

a

property

occur.

=

subsequence

a

the

AI(a)

<

E(u)

inf UEHI:

61

be such

C R

RN C

that

U f z,

+

BR(O)}

and each

point

'=1 x

N

is contained

E R

the

above

Sobolev

in at

most

inequality,

I balls

where

get (Ek

we

I

>

SUP

=

0 is

Junk

r

fixed

a

integer.

IPL 2(zr+BR(O))

applying

Then,

1

0

-+

k

as

oo):

--+

00

Iu-'IPP+1

+1

E Iu,'IP+I'LP+ (Zr+BR(0))

<-

<

r=1 00

Jun 1,12L2(z,+BR(O))

C(R)Ck

z2 (p+l)-N

N

(p+l) 12+NL2(zr+BR(O))

IU k

+

2

IVUIkIL2(zr+BR(0))j

X

r=1 00

lu2

Clek r=1

so

that

indeed

By quence

fUn'k(* H1.

IiM k-00

IlUn,'(*

strongly

=

(ii)

vanishing

above

and

a

(11.3.2)

to

this

lim inf E (un,

in H1

as

Iluolli, k

---

0

k

as

oo,

--+

uo

This

-

the

easily

(uo),

E

H" strongly

also

converges

but

the

such

E (uo)

then

of the

k

N

C R

sequence

yields

fUn}k=1,2,3....

subsequence

lYk}k=1,2,3.... some

> E k

therefore oo.

a

subsequence

k-oo

=

exists

sequence

converges

A and

+Yk)JII

-->

impossible.

is

there

arguments,

jUnk}k=1,2,3,... + Yk)}k=1,2,3.... JU0122

1121

C1II5kIJUnk

-<

-'r+BR(O)

By inequality

Hence,

uo

the

the

I VUnk 12 ]dx

+

nk

that

L2 and

in to

uo =

k-oo

statement

E (un,

in

Lp+i), hence, in

k

converges

of Theorem

se-

sequence

weakly

strongly

lim

fUn' (* +Yk)}k=1,2,3.... k

the

11.3.1.0

to

CHAPTER2.

62

In this

section,

author

which

we

suitable

that

state

if

eigenvalue AU,

=

U(O)

eigenfunctions

the

of one-dimensional

(11.0.1),(11.0.3)

to

by

obtained

results

be bases

can

functions".

in

precisely,

More

problem:

(0, 1),

E

x

=

of solutions

recent

"arbitrary

L2 containing

f(U2)U

+

of

similar

segment

a

nonlinear

_U

consider

systems

cases

on as

following

the

of systems

example)

an

in certain

problems

of functions

spaces

consider

(with

briefly

eigenvalue

nonlinear

we

properties

On basis

2.4

STATIONARYPROBLEMS

U(1)

u

u(x),

=

(11.4.1)

(11.4.2)

0,

=

U2(X)dx

(11.4.3)

0

where

again

spectral

a

all

it

correspond

to

first,

separable

an

eigenvalue. introduce

Hilbert

only

a

given sufficiently

with

which

a

partially

are

then

1)

known.

and the

twice we

call

We note

coefficient

the

to

function

problem.

of this

(up

product

scalar

a

=

eigenfunction eigenfunction

one

u(x) is (11.4.1)-(11.4.3),

u

problem

and A is

function,

smooth

A E R and the

definitions

some

space

is

corresponding

that

we

f

(A, u), where [0, 1], satisfies

the

priori

a

pair on

u(x)

and clear

is not

At

a

differentiable

eigenvalue

A the that

real,

are

If

parameter.

continuously

real

quantities

Let

can

H be

corresponding

a

norm

1 2

Definition

for

h E H there

arbitrary

an

Ih,,}n=0,1,2....

A system

11.4.1.

exists

unique

a

called

C H is

of

sequence

basis

a

of the

real

coefficients

called

linearly

space

H

if

janjn=0,1,2,...

"0

such

E anhn

that

=

h in the

of the

sense

H.

space

n=O

Definition the

equality

Ihn}n=0,1,2....

A system

11.4.2.

anhn

0,

=

where

an

are

E H is

real

coefficients,

takes

independent

place

in

H

if

only for

=0

0

=

ao

=

a,

=

=

...

an

In accordance

Definition

11.4.3.

=....

with

the

[5,6]

papers

We call

a

basis

we

introduce

Ihn}n=0,1,2...

the

of

the

following space

two

H

a

definitions.

Riesz

basis

of

00

this

if the

space

series

E anhn n=O

co

when

F_ n=O

a

2< n

00.

with

real

coefficients

an

converges

in H when and

only

ONBASIS PROPERTIES OF SYSTE, MSOF SOLUTIONS

2.4.

Definition

H.4.4.

quadratically

called

f enjn=0,1,2....

Two systems

close

if E Jjhn

H

in

-

f hn}n=0,1,2...

C H and

en

11

is

directly

2

<

63 C H

are

00.

n=O

proof

Our of N.K.

of the

Bary Theorem.

f hnln=0,1,2

in H.

close

establish

any

Theorem

f (r)

orthonormal

basis

of

function

integer

(b) 112(01 1)-

the

also

with

a

proving

II

norm

following

11.

-

p(x)

> 0

p(x p(ax) Afunctionalp(-)

y)

+

3.

space

in the a

theorem

Riesz

a

the

the

to

basis.

system

quadrat-

Bases

bases. of

system

Now

we can

eigenfunctions

result,

We call

for

any

p(x) jalp(x),

a

addition,

Ao

fUn}n=0,1,2....

we

shall

the

prove

proved

real-valued

is

functional

<

Bary

A,

<

basis

of

Let

X be

p defined

on

a

eigen-

is

An

the

For real

the

pair <

...

con-

7

that

this

and

Bary theorem.

[311.

in

a

(An Un);

pair

a

Un; such

(0, 1),

E R

u

Then

unique

<

...

(real)

this

;

space

aim,

Banach

X admissible

we

space

if the

satisfied:

are

on

Un; in

Gelfand

has

interval

of

function

[0, oo).

E

r

eigenfunction

the

in

function

the

of

(11-4. * (11-4. 3)

problem roots

differentiable

function

E

x

+

X;

p(y)-7 E R.

a

X is called

lower

apoint

semicontinuousat

xo E

X ifliminfp(x)

>

X-XO

Xxo) Gelfand tionals

theorem.

defined

in X.

Let

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

be

a

sequence

of admissible

continuous

If

AX)

=

SUPPn(x)

<

+00

n

for

system

a

in H.

Bary

for

H and

basis

literature

basis

of

close

Riesz

a

continuously

a

corresponding

a n

of I.M.

:5

=

of

this

conditions

2.

H is

of eigenjunctions

theorem

a

An and 1

be

the

> 0

precisely

system

Before need

of the

being

basis

Riesz

is

nondecreasing

n

eigenvalue

coefficient

the

f(U2 )u

continuous

a

any

an

a

f hn}n=0,1,2,...

of

property

Let

Un possesses

1.

following

the

on

quadratically

and

called

also

are

the

on

11.4.5.

be

(a) for

up to

based

(11.4.1)-(11.4.3).

problem

sisting

system

result

our

and let

the

orthonormal

to

C H be

independent

linearly

Then

....

Clearly,

of the

section

fen}n=0,1,2....

Let

C H be

....

fen}n=0,1,2

ically

of this

result

Bary.

any

x

E

X,

then

the

fitnetional

p(x)

is

admissible

and continuous

in X.

junc-

CIIAPTER 2.

64

Now

Gelfand

turn

we

theorem.

is based

It

Lemma 11.4.6

each

of

point

on

Let p be

above

following

the

functional

p(x) all

for least

with

the

X lower

in

semicontinuous

at

that

M11x11

:!

of the

Proof

result.

M> 0 such

exists

We begin

theorems.

admissible

an

there

Then,

X.

the

proving

to

STATIONARYPROBLEMS

(11.4.4)

E X.

x

Proof

of Lemma 11.2.6.

in

ball

one

B, (a),

r

We first

0,

>

a

X, then

E

that

if the

functional

(11.4.4)

holds.

Indeed,

prove

bounded

P is

if

p(x)

sup

at <

00

xEBr(a)

for

a

ball

(a),

B,

then

p(x all

for

E

x

B,

(a).

Therefore,

for

bounded

p is

e.

(11.4.4) (11.4.4)

functional, Let

the

By

B,,(xi)

p(x)

r-lp(x

<

C

setting

a)

-

hence,

and

p(a)

B1(0),

y E

=

+

=

r-'(x

-

a),

r

of

an

>

0,

we

get

Cr-',

<

due

y

to

the

properties

admissible

holds. is invalid.

now

lower

B1(0)

on

<

any

p(y) i.

a)

-

Then, the

of

semicontinuity

there

exists

functional

x,

E

p there

B1(0)

a

p(x,) B,,(xi)

that

such

exists

ball

> 1.

with

Then, by analogy there B1(0) such that p(x) > I for all x E B,,(xi). that such exists 2 the lower semicontinuity, there X2 E B, (xi) P(X2) > and, again by is a ball B12 (X2) : B, (X2) C B, (xi) such that p(x) > 2 for any x E B12 (X2) Continue this process. We get a sequence of balls such that f B,. (X,,,)}n=1,2,3.... Bn,, (x +,) C 0 as we can BIn (xn) and p(x) > n for any x E B, (xn); in addition, accept that rn there is > the oo. construction n a n for Then, unique xo E nB,,,(xn). By p(xo) C

-

--+

---+

n>1

any

integer

> 0.

easily

It functional we

n

This

follows

contradiction

from

Indeed,

lemma.0

that

Lemma 11.4.6

in X is continuous.

the

proves

let

admissible

an

Then,

xo E X.

since

lower

p(x)

sernicontinuous

:5

p(x

-

xo) +p(xo),

have

p(x) On the

other

hand,

-

P(xo)

Xx

:5-

-

x0)

M1Ix

<

by the lower semicontinuity

for

any

-

e

X01 I>

0 there

exists

that

P(xo) for

all

x:

jjx

-

xoll

<

6.

Hence,

for

jjx

x:

1p(x)

p(x)

-

-

-

<

xoll

P(xo) I

6

<

< 6,

minf6;,EM-1}

we

have

6

>

0 such

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

and the

continuity

of p is

proved.

prove

the

Now we shall

this

theorem

lower

admissible.

is

theorem.

In view

of p.

sernicontinuity

N > 0 such

Gelfand

Let

of the

xo

of the

6

is clear

above

E X and

functional

the

suffices

it

arguments 0 be

>

c

that

arbitrary.

p from

show

to

We take

the

number

a

that

P(XO) Choose

It

65

0 such

>

continuity

P(XO)

-

Corollary

PN(Xo) -PN(X)l

that

pN).

of

P(X)

Then,

< PN

11.4.7

PN(XO)

-

(XO)

for

2

x

x

6

+

If.(X)}n=1,2,3....

Let

be

a

xoI I

-

II

xo

-

(X)

SUP Pn

-

2

2

IIx

for

< -

any

<

<

8

(this

6

<

possible

6

PN (X)

-

2

of real-valued

sequence

because

have

we

:5 PN (XO) +

is

<

6-

E:'

lin-

continuous

00

functionals

ear

If E If (x) I"

X.

in

<

(p

oo

1) for

>

all

X,

E

x

there

then

exists

n=1

M> 0 such

that 00

E If

WI,

-<

M'IIxIjP

n=1

for

all

C: X.

x

Y (N

N

Proof.

Let

PN(X)

E Ifn(X)I"

=

n=1

the

Gelfand

p(x)

11.4.6

Now

[6].

theorem

Let

<

MIIxII.E1

we

present

h E H.

functional

the

a

proof

1, 2, 3, ...)

=

(x).

sup pn

=

By

n

p is admissible

of the

p(x)

and

and continuous.

Bary theorem.

In

fact,

Thus

the

repeat

we

by

Lemma

proof

from

We have 00

h

00

E anen7

E

n=O

n=O

converges

in H.

a

2<

00.

n

00

Let

us

E

that

prove

an

[en

hn]

Indeed,

n=O

2

-+p

E

an(en

hn)

M+P

n=m+l

as

m --+

hnI 12.

oo

any fixed

Then,

-

n=m+l

Ilen

-

hnI 12

0

,

n=m+l

0"

with

respect

coefficients

We have n

E

a2

to p > 0

by

the

of the

convergence

series

en n=O

Clearly,

continuity.

case.

uniformly

M+P

E

<

the

only

a,

to

corresponding

we can

assume,

are

prove

linear that

functionals if

passing

to

f hl}1=1,2,3.... a'

coefficient a

in

n

--+

0

H.

C H and

Let lim 1

as

subsequence

I

--+

oo.

us

also

I I hI I

=

their

0, then for

00

Suppose

if necessary,

show

that

this

an'

is

!

not co

the >

0.

STATIONARYPROBLEMS

CHAPTER2.

66

g'= (a,,)-'h'=

Then,

E bkek

n

+ en

0 in H

--*

I

as

But

oo.

--+

E bke-k

then

H as 1

g'

clearly

and

oo

--+

g'

--+

in

kon

kon

bA;ek for

bk,

coefficients

real

some

hence

k96n

1:

+

en

bk ek

0

=

H,

in

kon

i.

get

we

e.

Thus

contradiction.

a

indeed

coefficients

an

linear

continuous

are

func-

in H.

tionals

00

Let

E an(en

F

hn)

-

and

F

Uf.

=

The operator

U is

linear

and

is

it

n=O

everywhere

determined

BR(O)

If

=

E H

11f1l

:

H.

in

R}

<

Let

us

and

f

that

prove

BR(O).

E

it

completely

is

First,

remark

we

Let

continuous.

according

that

to

00

Corollary

there

11.4.7

M> 0 such

exists

E

that

a

2

M211fJ12

<

n

-

for

f

all

Then,

E H.

n=O 00

take

an

arbitrary

There

0.

>

e

exists

number

a

N> 0

large

so

F,

that

I I en

hn

-

<

n=N+l

,E, therefore

)

N

F

Ean(en

-

hn

-

n=1

2

an(en-hn) 2

00

E n=N+l

00

E

<

1:

2

an

n=N+l

12

lien-hnj

<

n=N+l

00

EM211fJ12

2 an <

1EM2 R2.

<

n=N+l

arbitrariness

of the

In view

approximated

arbitrary

of

e

0 for

>

f

any

by functions

closely

E

BR(O)

from the

element

the

F

family

compact

=

Uf

can

be

of

of functions

N

the

E an(e-n

kind

hn),

-

i.

U(BR(O))

e.

is

relatively

a

compact

subset

of the

space

n=O

complete

H and the

of the

continuity

operator

U is

proved.

A

E

0

Af.

00

0

Let

f

=

-

F,

F, anhn-

then

Set

U where

-

E is

the

identity.

n=O

A is

Then

a

bounded

linear

in H and

operator

Af can

only the

have

independent.

Then,

and the

solution

A-'

inverse

trivial since

the linear U is

operator

H;

in

in

f

solution

addition,

=

(11.4-5) the

homogeneous Aen

=

u

E H and

v

=

A-'u

E anAen =E anhn n=O

where

00

E n=O

2

an <

()0.

E

a

2 n

<

oo.

00

in H.

Conversely,

A has

a

bounded

Then,

u

=

Av

n=O

00

n=O

operator

the trivial

co

ane-n n=O

00

the

linearly

is

hn-

00

Let

I hn}n=0,1,2.... (11.4.5) has only

system

problem

continuous,

obviously

equation

=o

0 because

completely

But the

if

u

E anhn, n=O

00

then

A-lu

E n=O

anen5

hence

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

As it is indicated

IIA.8

Remark

Ih,,}n=0,1,2....

basis

in H there

m

0 such

>

2

00

E an hn

2

lifil

=

of the

proof

in the

exists

n=O

67 for

Bary theorem,

any

Riesz

that

M2" E a2

>

n

n=O

"0

for

1: anhn

f

all

Therefore,

E H.

have

we

bounded

a

one-to-one

from

operator

n=O 00

12 which

H onto

E anhn

f

maps

H into

E

the

coefficients

of

sequence

a

n=O

(ao,

a,,...,

there

an)

)

...

E

theorem

12. By the Banach

the

inverse

2

<

E

m2

n=O

we

positive

other

property

(resp. [39]).

Bary)

Now

of

we

E

2

an

the

m

2 n

of the

of

is that

11.4.5. use

this

In

generality

In

=

keeps

it

2

of the

of the

(ao,

a,,...,

the

property

fact

we

that

follow

E

...

be

to

12. a

subject,

AnRiesz see

equation

(11.4.1)

theory

of ODEs

paper

our

Consider

0.

=

7

this

qualitative

proof

f(O)

that

an

(on

elements

of the

spite

methods

part

a

of its

we assume

a

n=O

reindexing

we

(a).

statement

loss

basis

Theorem

prove

by quadratures,

00

E

M2

<

Mindependent

<

arbitrary

an

hn

a

n=O

(or Bary)

Riesz after

shall

the

proving

a

basis

be solved

Without

0 <

constants

2

00

1: n=O

for

a

n=O

00

can

e.

get the estimate

2 M

with

i.

00

anhn

So,

bounded,

also

map is

that

M> 0 such

exists

the

[114].

following

Cauchy problem _U

"

+

Au 2)U

u(0) where

a

> 0 is

a

Lemma 11.4.9 that x

has

the

solution

precisely Proof.

of n

For

For any the

roots

solutions

integer

AU,

u'(0)

>- 0 and

a

>

a

=

by u(a, A, x) n

E

x

its

>

(0, 1),

(11.4.6)

0,

(11.4.7)

solutions.

0 there

(11.4.7) regarded 0; in u(a, A, 1) problem (11.4.6),(11.4.7)

(11.4-6),

problem in

0,

=

Denote

parameter.

=

(0, 1)

and

of the

=

X

+ AU2

-

F (U2)

a

unique

a

function

of

addition

A >

(T(n

as

the

place:

[U/ ]2

exists

0,

following

A E R such the +

argument

1))2.

identity

takes

CHAPTER2.

68

f f (p)dp,

F(s)

where

STATIONARYPROBLEMS

therefore

0

[u'(a,

A,

X

for

arbitrary

an

Let

us

At the

G(A, u)

same

=

Au'

-

If

F (U2)

strictly

particular,

tinued

the

for

+oo.

a

these

for

onto

>

A, 0)]2

conditions,

then

X

of A the

segment

[0, 1]

particular,

In

u

E

implies

u" +

c(A, x)u

\

[0, A]

[0, 1].

E

x

follows

it

large

so

from

the

small

positive

the

function

that

F(U2 (a, A, 0)).

-

1 u (a, A, x) as

a

<

A for

function

Therefore,

of

max

XE10111

u(a, 'X, x)

the function

that

any

and

u(a, A, x) 2. 1.

(11.4.8)

sufficiently

and

> 0

by (11.4-8)

in Section

this

(0, 1]

A[u(a, A, 0)]2

+

a'

=

Then,

0.

>

a

E

x

solution

as

A, x))

and for

exists

for

[u'(a,

=

all

0 for

increases

2

0 and

>

A > 0 there

any

values

these

whole

>

F(u'(a,

-

(11.4.6),(11.4.7)

n

u(a, A, x)

for

A > 0 satisfies

a

In

integer

time,

A, x)

problem

of the

that

G(A, A)

Au'(a,

+

arbitrary

an

theorem

comparison A.

solution fix

x)]'

all x

[0, 1].

E

x

be

can

con-

Ju(a, A, x) I

satisfies

the

0

--+

equa-

tion

where

the

A > 0 is of the

problem

values

A

A, be the

most

> n

has

0

A

than

more

above

arguments

by

above

arguments.

the in

(0, 1)

as

dependence

a

> 0

has

to the

roots

continuous

of values

set

(11.4.6),(11.4.7)

problem

According

A,(a)

E

(0, 1),

with respect to large uniformly theorem the solution by the comparison

Hence,

(11.4.6),(11.4.7)

x

arbitrary

> 0 is

large.

0,

n

(0, 1)

in

roots

for

E

x

[0, 1]

if

u(a, A, x)

sufficiently

large

> 0.

Let of the

c(A, x)

function

sufficiently

=

function

theorem

such that

at

least

the

set

The

(n

for each of them the +

1)

roots

as

A,, is nonempty.

corresponding

of the

argument u' X (a,

a

A,, (a)

Let

x

of =

x

because

u(a, A, x) E (0, 1).

A,,.

inf

u(a, A.,'(a),

solution

A, xo) :

solution

function

x)

otherwise,

Then has

at

due to the

0, there (a, A, xo) values must exist A < Xn (a) belonging to A,. as a By analogy u (a, A,, (a), x) regarded function of the argument at least in (0, 1) and u(a, A,'(a), x has n roots 0 because 1) in the opposite the solutions, case to A E An sufficiently close to An(a), corresponding must have at most n roots in (0, 1). Let us prove the uniqueness of the above value A A,,(a), for which the solution u(a, An(a), x) of the problem has precisely in the interval n roots (11.4.6),(11.4.7) the condition x E (0, 0. that there exists 1) and satisfies Suppose u(a, An(a), 1) A' 54 A,,(a) these conditions. satisfying Using the autonomy of equation (11.4-6) and and

since

0 if

u

=

=

=

=

its can

invariance

easily

with prove

that

respect

to

the

changes

of variables

x

---+

c

-

x

and

u

--+

-u,

one

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

(a) for odd with

respect

(b)

the

point

2.

1);

is

the

xo

is

strictly

on

any

(c) u(a, A, x) minimal

ul(x)

exists

the

minimal

value

of the

Ul(X)=h U2(X) byU2(X), the identities

for

x

all

from

2(n+1)

U2(X)

right

a

E

x

(0,

such

x

that

x,

each

that

also

for

point

1

x

f Ul(X)U2(X)

[f(U2(X))

=

where

x,

result

U2(X)

if

2(n+1)

ul(x),

for

the

obtained

[0, Y],

segment

be the

1

=

written

subtracting the

over

Y

let

clear

is

T > 0

or

a

and

it

Let

and

achieves

An(a)

>

0.

(11.4-6),

.

the

and

(11.4.8),

of

view

) Multiplying equation for U2(X), by ul(x),

integrating

to

Section

[0, 1]

E

A'

ul(x)=

that

such

2(n+1)

and

solution;

u(a, An(a), x)

definiteness

the

of the

( 0,

+ 2xi

x

of

respect

also

see

is

u(x)

this

with

increasing

half-neighborhood E

[0, 1];

C

of

2

of two solutions

monotonically

is

'+ 2(n+1) written

equation, another

C

in

1-

"+'2

x0

b]

solution

[XO, X2] and even [0, 1] (on this subject,

Then,

x

+

x0

solution

on

u(a,A',x).

=

b,

this

u.

(a)-(c) (11.4.6),(11.4.7), Let

-

arbitrary

an

extremum

and

any

function

the

argument

same

one

of

and

=

that

2xi) for

properties

point

u(a, An(a), x) ul(x)> U2(X) in

-

+

x

root

problem

of the

[xo

u(a, A,

=

of

point

on

of

< X2

x,

[xi, xo] b, xo + b]

[xo

segment

roots

(11.4.6),

of equation

arbitrary

unique

a

segment

solution

an

nearest

monotone

from the

maximum at

xo

on

two

positive

It follows

u(a, A', x)

point

there

arbitrary

an

arbitrary

(11.4.6)

solution

of

xo

the

to

between

equation this

root

any

69

we

get:

7

0 >

f(U2(X)) 2

_

1

-

An (a) + A'] dx.

(11.4.9)

0

by

since

But

inequality

suppositionUl(X)

our

positive,

is

i.

Lemma 11.4.9

We 11.4.9.

is

keep

Let

we

e.

An(a)

The property

get

(r(n

!

U2(X)

> a

+

for

E

x

(0, Y)

the

,

right-hand

side

of this

contradiction.

1))2

follows

from the

theorem.

comparison

Thus,

proved.0 the

A An(a) for the value of the parameter x)= Un(a, x). By Lemma 11.4.9 these definitions

notation

u,,(a,An(a),

from

Lemma correct

are

1

An (a)

and

0 for

>

any

a

>

0 and

integer

n

Let

> 0.

also

an

f Un' (a, x)

(a)

dx.

0

Lemma 11.4.10 continuous

on

Proof.

contrary, the

U2(X) each

i.

e.

properties increases

of them.

a,

that

>

An (a,)

(a)-(c) on

a2

integer

any

half-line

the

Let

For

<

0-

By (11.4.8).,

segment we

We shall

An (a2).

from the the

function

the

> 0

An(a)

is

nondecreasing

and

> 0.

a

>

n

Let

proof

[0, have

that

prove ul

(x)

=

of Lemma 1-

2(n+1)

u',(xi)

]

and >

x

un

11.4.9, =

U2(X2)

An(al) (a,, x) each

! and

An(a2)U2 (X)

of the

Suppose =

functions

Un

the

(a2 x). By ul(x) and ,

1is the point of maximum of 2(n+1) for any y > 0 for which there exist

(0,

X1) X2 E x

STATIONARYPROBLEMS

CHAPTER2.

70

-(n+jj)-

(0,

E

we

satisfying

2(n+l)

I

I

(xi)

ul

Proceeding

U2

as

(X2)

==

Therefore,

y.

(x)

ul

(11.4.9)

inequality

deriving

when

and

> U2

taking I

(X)

for

all I

-

x

=

2(n+l)

get 2

2(X))

U1(X)U2(X)[f(U

0

f(U2(X)) 2

_

1

n(aj)

-

+

An(a2)]dx,

0

which

obviously

is

proved

is

that

Let that

the

there

the

exists

0 such

>

ao

nondecreasing

is

of the

continuity that

definiteness

the

by analogy). for

each

Then,

1) d(a) 2) u, (a, x)

+0

-+

3)

Un

first

as

one

sufficiently

> ao

a

the

ao+O

as

> un

(a, d(a))

a

inequality

take

easily

verify,

close

to

(ao, x)

E

x

(ao, d(a))

Un

:--:

for

the

An(a). An(ao)

>

(the

place

is

positive.

a

Suppose

the

second

case

0 such

>

it

i.

e.

contrary,

An(ao).

<

Let

be considered

can

(11.4.6)

from

So,

0.

>

An(a)

liM a-ao-O

d(a)

exists

here

half-line

or

follows

it

there

side

(11.4.8)

and

that

that

0;

+

ao

-

An(a)

can

ao

on

function

liM a

for

right-hand

the

because

An(a)

function

prove

us

contradiction

a

(0, d(a)); -A-

and

< -- - u, dx

(a, d(a))

dx un

(ao, d(a)).

of the point x d(a) Un(a, x) < Un(ao, x) in a right half-neighborhood follows from it Then, as above, equality d(a))). (because 0,xx(a, d(a)) < u",xjao, close to ao and for all that Un(a, x) < Un(ao, x) for all a > ao sufficiently (11.4.8) x E (d(a), ; '-+ ). Using the identity similar to (11.4.9) with the integral over the 2(n+l) a contradiction. we get So, the function ' n(a) is continuous, segment [d(a), 2(n+l) Therefore

=

n

n

and Lemma 11.4.10

an(a)

Lemma 11.4.11 a

+0

a

Proof.-

The

(see (11.4.6),(11.4.7)

Lemma

0

as

a

liM a

+0

(0,

all 2(n+l)

and also

addition,

x

us

in

an(a)

x

G

an

(a)

12[

A.

follows

from the

dependence

continuous a

function

continuous

on

the

half-line

+oo.

=

function

parameters

that

prove

such

2(n+l) that

our

lim a-+oo

1

f(U2)U by

an(a)

lim

and from the the

increasing

Further,

[0, 1] (see

the

as

proof

continuity

of solutions

it

proved

is

of

) n(a)

Of

of the

problem

un(a,

earlier,

10),

Lemma IIA.

x)

--->

therefore

0.

(0,

(-=

strictly

a-+oo

11.4.10)

=

a

of the

continuity

+0 uniformly

an(a)

is

0 and

=

on

--*

Let

for

an(a)

liM

0,

>

proved.0

is

-

).

Indeed,

u"

n,

xx

An(a)u supposition

if

(a, xo) is

a

=

we

> 0.

First

+oo.

f (u 2(a, xo))

all,

we

u',x (a, xo) function

on

n

-

observe

0

it

as

the

An (a)Un (a, xo)

that

then

contrary, >

nondecreasing n

the

suppose

But

of

0.

:!'

0

xo

E

n

there

was

half-line >

u",xx (a, x) exists

indicated u

Hence,

E

earlier

[0, +oo); we

get

in

that

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

Un',.,(a, all

x) E(01

x

0 for

>

all

E

x

(xo, 11,

i.

lim

a,,(a)

e.

have

we

So, un,,(a,

contradiction.

a

71

x)

2(n'+I))'

Now,

to

that

prove

+oo

a

suffices

it

+oo,

=

1

u,,(a,

show that

to

2(n+l)

it is proved as (because, above, Un(a, x) is a concave function on the +oo the following segment [0, n+1 T' ]). Suppose that for a sequence ak C Consider the < two cases: place: :5 +oo. Un(ak7 2(n+l) separately following ; '+- ) < +oo. +oo and B. f (+oo) f (+oo) +oo

as

for

< 0

a

of

+oo

--+

x

takes

-+

A.

=

f (+oo)

A. Let we

would

(11.4.8)

get the

functions

Therefore,

An(a) u,(ak, k-oo

Then

+oo.

=

from

Un

(ak x) satisfy +

n,xx

where

9k

ing

standard

to

+oo

-*

Un(ak7 X)

for

all

B.

:!

an(a)

lim +oo

It

for

all

E

x

that

Un

Then,

(r(n

equation

it

is

We also

note ==

Let

the

1))2

e.

we

that

an(a)

is

x

have

E

(0,

)

the

equations

Hence,

2(n+l)

each

of the

; '+-

(0,

in

root

a

But

+oo.

get

by

hand,

one

<

11.4.9,

it

this,

in to

the

inequality

that

there

exist

2(n+l) =

). (a2) xi). Let

Un

of xj.

the

accordfunctions

),

2(n+l)

which

is

theorem,

comparison

the

on

hand,

other

contradiction.

a

So,

Let

monotonically

it

is

the

in

as

proved

0 be

the

the

again

minimal

that

=

that

have i.

n

follows

from

these

minimal

value

x

it

point

Then

contrary.

a2

un,x(al, we

>

this

arguments

in < a2

Suppose

the

Un(a2lx)

for

interval

such

(a2 X) right xj) un,x(a27 Xl)- In An(a2), hence, in view in

7

half-

a

=

<

get

a

contradiction.

half-neighborhood that

indicated

place.

from

> Un

An(al) e.

a,,

takes

the

on

0 < a,

:

un(ai,x)

(a,, x)

Un

(11.4.8) written for x xi, we > (11.4.6), u",x, ,(a,, xi) u",xx(a2;X1)i that u,,(al, x) > un(a21 X) in a right .

any

that

such

a2

show that

us

for

x) :5 un(a21 X) <

(a)-(c)

properties

that

prove

u,,,(aj,

function

increasing

of the

view

0 < a, >

x,

Suppose

a

suffices

To prove

n

proved

; '-+

2(n+l)

theorem,

comparison

on

+

i.

+oo,

E

contradiction).

a

e.

(0,

x

otherwise

01

-=

to

k must

i.

impossible.

equality

of

An(al)

the

X)

respect

numbers

A is

+

prove

; '-+

E (0, (a,, xi)

of

with

on

'-+ ) 2(n+l) e.

neighborhood view

+oo.

=

to

(0,

i. x

<

f (+oo)

of Lemma

opposite, some

case

(0, +oo).

E

a

proof

the

interval

(because

+oo

--

+oo,

=

+oo.

=

remains

half-line the

large

An(ak) A, lim An(ak) k-oo

case

a

sufficiently

f (+oo)

Let

have

we

based

)

2(n+l)

_qk(x)u,,(ak,

uniformly

oo

--

arguments

So,

contradiction.

a

k

as

a

as

'

in the

,

U11

+oo

--

lim

that

un,x(al,

of the

xi)

So, point

un,.'(a2)

>

xi.

X1)

if

An(a2)X2

X2

(n+l)

when

deriving

be the if there

is

inequality

no

such

(11.4.9)

a

E

point with

(xj, in

the

such

2(n+l)

(xi,

1

2(n+l)

integration

that

Un(aj) x)

). Repeating over

the

the

=

Un(a27 X)

procedure

segment

[XI, X21

or

used ,

we

CHAPTER2.

72

STATIONARYPROBLEMS

get: 272

I

0 >

un(al,

X)[f(U2(a,,

x)u,,(a2,

x))

n

f(U2 (a2 X))

_

An (a,)

-

,

n

An (a2)] dx;

+

X1

addition,

in

An(a2).

by

Thus

increasing

the

above

get

we

function

of the

problem

the

interval

integer

any

the

inequalities

following

Proof.

Suppose x

E I

for

all

x

by

the

[X I

i

2(n+2)

21

X

we

,

as

suffices

following < X2

Un+I(X2)

Un

7

(X)) satisfying

has

precisely

+1

of

in

roots

n

function

the

Un;

It

to

Un+I(X)

strictly

be that

un+,(x)

f (u' (x))

>

An for

-

n

exist

two

with

B

increases

E

x

can

and

also

the

integral

Un(X) hence

over

that

occur.

un+,(x) Un+I(X2)

that

!

I,

E I such

xo

A and

such

Un(X2)7

point

a

cases

(11.4.10).

inequality

prove

cannot

2(n+2)

>

(11.4.10)

...

solution

(11.4.9)

deriving

<

should

there

of the

when

it

An+1

case

Un(Xl)i

=

Proceeding

.

n

An

0.

=

+1 (X))

<

The

0 < x,

one

points

Un+l(XI)

1-

X2 <

1

in this

only

exist

...

0.

>

n

2(n+2)

n

theorem

there

(XlIX2);

E

u'+,( f(U2

and

<

11.4.9-11.4.11,

some

then

Therefore,

0.

=

]

2(n+2)

because

E I

A. Let x

'

[0,

=

comparison

(xo)

n

of Lemmas

An >- An+1 for

that

for

u'

In view

monotonically

a

place:

A2

<

An(al)

if

proved.0

(An

Un(x)

coefficient

the

up to

is

is

pair

a

function

the

take

Al

0 <

exists

here

an(a)

that

Lemma 11.4.11

> 0 there

n

equality

strict

proved

0, and

>

a

the

is

it

and such that W-4.1)-(H.4.3) (0, 1) and this pair is unique

addition,

in

So,

argument

For

have

we

contradiction.

a

Lemma 11.4.12

the

arguments,

> =

the

un(x) Un(X2)

if if

segment

get the inequality X2

0 >

i

2

Un+l(X)Un(X)[f(U

n

+1

(X))

_

f(U2 (X))

An+1

-

n

A,,]dx,

+

X1

where,

as

Thus,

we

proof

in the

get

a

of Lemma

contradiction,

11.4.11,

and the

the

A is

case

inequality

strict

place

takes

if

An+,

An

==

impossible.

un+-I(x) < u,(x) for all x E I. Observe that Un+I(X) < Un(X) for some 1 have u'( we would (because otherwise 0). Further, un+,(x) :! ' Un(X) 2(n+2) ) is obvious Let x that E [0, on a then us visually (it picture). prove n+1

B. Let xo E I

for

Un+1

the

equal

<

L2

interval

function

2,, 3,

=

n

all

to

if

Un =

L2

Qn+1 11,

obviously

=

Tk

n+1

For

1.

precisely function

+

-

for

x

arbitrary 1

Un(; T-j) n+1

1). Change the X + ) Un+1 (y k-1 n+1

n

an

k

) (here

has

u,+,

...'

I

Ik

<

one

Un+1 7

=

root on

k

integer

Un( n+1 Tk T (because the

and to un+1

-

x

1, 2,

0)-

In

this

>

;kj lj

k

n+2

n

...'

n+1

Ik by the

segment

(Y

=

+

k n+1

)

for

+ 1 consider

interval, for

each

function x

the

> Y.

k

v(x) Repeat

procedure

this

for

of the

properties

I v (x) I

lies

that

such

k

each

(a)-(c)

under

the

Jv(x)j

<

1, 2,

=

(see

graph

proof

Jun(x)l

all

for

we

e.

get

en(X)

Let an

basis

orthonormal

which

of

plication

u

by

next

interval

an

(0, 1)

J C

of

part

C

0 such

the

Then, u'

that

proving

n

jenjn=0,1,2....

system

(0)

all

0 for

>

(11.4.1)

with

Theorem

IIA.5

equation

of

proved.0

is

=

invariance

In the

graph

is

view

in

function

of the

lUnIL2(0,1)

<

0, 1, 2,.... 1)x], n L2(0, 1)- We also accept

in

-1.

and there

Lemma 11.4.12

+

due to the

possible

is

V2-sin[7r(n

=

IVIL2(o,l)

=

the

Therefore,

E J.

x

Thus,

contradiction.

a

I un (x) I

function

of the

11.4.9),

of Lemma

visually,

is obvious

which

Then,

+ 1.

n

...,

the

Un+11L2(0J) i.

73

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

n

respect

=

to

0, 1, 2,

the

multi-

the

paper

follow

we

is

[115]. There

Lemma 11.4.13

close

particular,

in

0, 15 2,...;

exists

the

>

Jun

that

jUnjn=0,1,2....

systems

-

and

12 < C(n +I)-' for all n are quadratically jen}n=0,1,2.... en

L2(0, 1)-

in

bounded are jUn(X)}n=0,1,2,... of the sequence Consider to x E [0, 1] with respect Un (bn) x) of solutions uniformly this that Let us with the Cauchy problem bn prove 107r(n + 1). (11.4.6),(11.4.7) Let dn bounded. is uniformly Un(bn, 2(n+1) ; '-+ ). Then, obviously it suffices to sequence At

Proof.

first,

we

the

show that

shall

functions

-

=

=

for

number

each

of the

boundedness

show the

n

we

have

An (bn)

(7r (n

>

+

1))2

(11.4.8)

identity

Further,

.

theorem,

comparison

the

By

dn from above.

sequence

yields

the

inequality

(10ir(n which

+

1))'

>

Un(bn7 X))

G(, n(bn),

(r(n

!

dn

large

sufficiently

all

we

would

for

all

get that

in

large

sufficiently

So,

contradiction.

X)

n

-

F(u'(bni

X)),

n

(11.4.11)

that

implies

for

1))'u'(bn7

+

the

numbers

(11.4.11)

Indeed,

n.

if

right-hand

the

numbers

< 20

n

and

for

side all

boundedness

uniform

we

suppose

is

than

greater

satisfying

x

of the

this

that

is

the

not

left-hand

20, jUn(bn,x)j jUn(bn7X)Jn=0'1,2.... =

sequence

then

so,

one

i.

e.

a

is

proved. Let

hn(X)

=

properties

us

show

sin[7r(n (a)-(c)

10

that +

from

jun(bni -)IL2(0,1) 1)x] and observe the

proof -h"

> I

that

of Lemma

=

n

/-tnhn7

for

all

u',x(bn) n

11.4.9, X

G

large

sufficiently

0)

=

u'n,,,(bn) (0, 1),

h' n (0)

1)

=

numbers

and,

hn (1). '

in

view

n.

Set

of the

We have:

CHAPTER2.

74

h,(O) where

un

(11.4.6)

(7r (n

=

for

the

+

1))'.

Therefore,

w,,(x)

functions

"+ -Wn

Wn(O) where

family

the

boundedness

x)

Wn(l)

family

C,

constant

a

Multiplying

this

[0, 1]

segment

independent

> 0

of

applying

and the

place

by

the

by parts,

integration

because

12

An I Wn L2(0,1)

_JW/'xJ2

L2 (0,I)

n

-

-

all fact

for

C2

numbers and

all

independent

0 is n,

we

f

<

n

can

yield

large

be

numbers

proved

aim,

we

first

I u' (0) for

all

numbers

At

first,

integrate boundedness

the

I u'n(O) I

<

e,'

-

n

< -

I hn I L2 (0,1)

and since

sufficiently

all

large

numbers

C2,

> I n.

for

This

n

(11.4.13)

bn

<

(0)

bounded.

?7(n

+

there

Let

I)-',

n

I ul (1)

0 such

e-1

-

that

the

family

of

that

prove

us

>

n

-)

0, 1, 2,....

=

C3

exists

C3 and

u,,,(bn,

functions

the

for

as

show that

n

2W2(X)dx

X

n.

enIL2(0,1)

shall

for

uniformly

is

-

> I

(11.4.13)

using

Jun

equality

In view of this

n.

1

that

fUn(*)In=0,1,2....

functions

For this

of

the

over

0

JUn(bn, *)IL2(0,1)

have

Lemma IIA.11

sufficiently It

the

>

equality

get

0

where

theorem.

comparison

obtained

we

1 xw',,,(x)Wn(x)dx

2

uniform

estimate

the

1

<

of the

(11.4.12)

integrating

n

the

bounded

C1

<

taking

n

by 2xw'(x),

equation

and

uniformly

Pn1

-

0,

=

n

fun (bn, X) ln>o

of functions

J- n with

W,,(l)

=

n

is

equation

(0, 1),

E

x

W,,(O)

=

and

equation

hn(x):

-

y.w,,,

=

f W,,(x)}

of functions

of the

u.,,(b,,,

W.(x)

=

0

=

get from the previous

we

=

h,(1)

=

STATIONARYPROBLEMS

n

(1) 1

that <

(11.4.14)

C3

n.

let

multiply

us

obtained of the

identity

family

(11.4.1)

equation

the

over

segment

f Un (X) ln=0,1,2

of functions

for

written

[0, 1] ....

-

,

u(x) Then, we

1

An

-

C4:5

f[ 0

Uln (X)]2 dx <- An + C4

get

=

Un(x) by Un(x) due

to

the

and

uniform

75

ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS

2.4.

for

a

C4

constant

u(x)

for

written 0 and

(x)

n

and

_

-

n

2A,j

multiply

equation

(11.4.1)

obtained

identity

between

estimate

previous

of the

use

[U/ (0)]2

2

n

us

the

integrate

the

with

12[u' (1)]

let

Further,

n.

transformations

after

We get

1.

+ x)u'

un(x) by 2(1

=

of

independent

0

>

<

I

C5 +

<

I (1

2

X)f(U2 (X))Un(X)Ul(x)dx

+

<

n

n

0

I

:5 C5 +

JF(U2 (x)) I dx, n

0

F(s)

again

where

f f(t)dt.

=

due

since

Hence,

(a)-(c)

properties

the

to

from

the

0

proof

[U/ (0)]2

of Lemma IIA. 9

n

it

a

from

follows

e'(1), get (11.4.14). sign

C6

constant

n

Let

[el (0)]2

=

of functions

=

-

en(x). ,

....

+

tiplying

the

of the

use

a

Gn(X)

f[gn

(X)]2

dx <

C7

0 is

>

constant

11.4.9,

e' n (0)

and,

sign

u, n (1)

(11.4.12),

inequality

uniform

boundedness

X

(1)

of the

as

we

family

(11-4.16)

integrating

C7

-

f[

gn

(X)12

dx

-

2

j

(11.4.16)

and

from 0 to

identity

obtained

the

(11.4.14)

Mul-

functions.

of continuous

sequence

due to

/

(11.4.15)

(0 1),

(=-

0,

we

get

1

xGn(X)g'

n

(x)dx,

0

0

a

sign

=

of

1

0

where

and

by parts,

1

/-In

gn

::7--

bounded

uniformly

integration

(0)

in view

/Lngn(X),

==

(0)

by 2xgn(x)

(11.4.15)

equation

I with

is

due to the

n

of Lemma

have

gn

JGn(X)}n=0,1,2,...

proof and

u'

sign

since

the

2[ln

Then,

we

-9n "(x)

where

Now,

n.

from

n

n

Un(X) gn(x) fUn(X)}n=0,1,2

of

C6

2, n

-

n

(a)-(c) [e/ (1)12

properties

because

have

[U1 (0)]2

2Anj

-

independent

0

>

the

we

n

(1)]2

U/ for

[U/ (1)]2,

=

n

independent

of

n.

Therefore,

applying

inequality

the

I

2ab <

a

2

+ b 2,

we

obtain

that

there

C8

exists

>

0 such

that

[In

fg2 (x)dx

<

n

0

numbers

n.

Hence,

jjgnjj

:5 C

1' 2

7r-'(n

+

I)-',

and Lemma 11.4.13

is

proved.0

C8 for all

CHAPTER2.

76

L2(Oi 1) ger

n

Un(

of the

the

suffices

it

now

show

to

fUnjn=0,1,2,....

of functions

system

expand

0

>

11.4.5,

Theorem

To prove

function

( n+1 t )

u,,

let

aim,

L2(0, 1)

space

independence

linear

the

this

For the

in

STATIONARYPROBLEMS

for

us

each

inte-

Fourier

the

in

in

series:

00

Ea ne-k(') k

n+1

where

ank

coefficients.

real

are

Then

k=O 00

Un(*)

r

=

b,,,e,,,(-)

(11.4.17)

n

M=0

in the k

=

0, 1, 2,

Then,

=

holds

valid

is

each

and,

n

in view of

real

exist

(n+1,

L2

our

Also,

space

=

a

0

points

r

n+1

the

1,

==

obviously

bn0

because

the

where

n+1

(X),

e(n+l)(k+l)-l

(11.4.17)

equality

too,

where

> 0

n

the

the functions

points,

)

r+1 n+1

r

(01

of Lemma 11.4.9

precisely

are

to these

respect

spaces

proof

the

shows that

verification

for all

,

2,

also

Therefore,

n.

bnn-1 functions

0 for

=

U.

and e.

sign everywhere. that

suppose

us

which

L2(0, 1). acceptation, bnn

of the

sense

same

Let

there

the

of the

from

roots

m: (n+l)(k+l)-i place in the space L2

0 if

=

takes

(a)-(c)

to its

respect odd with

are

obviously

properties

the direct

of each

sense

in

of the

are

is odd with

bn,,

nand k

a

=

(11.4.17)

equality of the

0, 1, 2,...,

in the

bn (n+l)(k+l)-l

where

and since

n

k

,

view

un(x)

where

it

in

1, 2,

1)

Indeed,

....

since

function r

L2 (0)

space

the

coefficients

fUn}n=0,1,2,...

system

Cn,

n

=

0, 1, 2,...,

independent.

linearly

is not

equal

all

not

to

such

zero

Then,

that

00

ECnUn

(11.4.18)

0

=

n=O

in the

b1jcj

L2(0, 1). Let (11.4.18)

space

Multiply =

But

0.

contradiction, in the

L2 (07

1)

large

is

radial

solutions

a

proved.

f(02)

method,

for

Independently, existence

jolp-1 =

in

of

a

contained.

presented;

number

N

In

the

the

existence

Thus,

problem

(p

positive

space

M, =A

0 and

that

co

L2 (0) 1) we

as

of the

Theorem

IIA.5

cl-I

In

-

of

view

supposed

cl

=

:

0.

proved,

is

54

0 and cl

(11.4.17)

we

So,

0.

get:

get

we

a

fUn(X)}n=0,1,2....

of functions

system

1)

>

devoted

publications

of

3 and

[96]

the

too.0

remarks.

of the

=

in

independence

linear is

el

above

Additional

There

is

by

proved

is

and the

space

2.5

with

it

as

I > 0 be such

index

the

equality

is

(11.0.1),(11.0.3).

for

the

solution,

[97]

paper

of

based

a

a

solution

f,

function

same

on

methods

refinement with

of

an

I

[71],

a

< p

of the

of the

arbitrary

[71],[96],

papers

paper

existence

of the

questions

In the

In the

considered.

1 < p < 4 the

the

to

< 3

and

N

qualitative

technique given

a

solution =

theory of the

number

variational

proved.

is

3

of

problem

this

by using

positive

existence

proof

a

of

paper

of roots

of

ODEs,

[71] on

is the

77

ADDITIONAL REMARKS.

2.5.

half-line

r

[82]

in

[71] (N

the

the

there

in

bounded

solution

is

i.

satisfies

it

e.

also

applicable

f (0') 0

in

principal achieving

the

W. Strauss

[87]; (non-radial)

mistake

no

nontrivial

=

paper

solutions

that

stating

Till the

101 investigated has

0; sufficient

a

--+

have

reviewed,

Another recent

paper

on

root, for

conditions of roots

case

positive

the

the

half-line

solved

of

is based

a

similar

to are

nonlinearities

for

proved

first

[98];

in

of solutions

of the existence

unsolved

remained

has

aim but

for this

exploited

method.

variational

obtained

by

by

methods

of the

a

the

(11.1.4),(11.1.5)

Cauchy problem

was

theory

qualitative

of

model

is considered

[110]. w-f (0)

> 0

r

of

jxj

an

it

paper,

is

radial

solution

are

obtained

101-1,

=

11.1.2.

supposed with

a

of

results

the

case

f( 02)

One

more

the

an

by using

made

number

that Jim i(Al-00

finite

a

so.

in the

a

of

has

given is not

(02)

Theorem

exists

> 0

f

it

(11-0.1),(11.0.3)

and there a

author

arbitrary

[77,78];

proposed

was

symmetrization

a

the

example

with

In this

=

with

problem

existence

concept

Rabinowitz

the

of the

of

Unfortunately,

solution

a

of P.H. with

the

solutions

radial

positive

on

H1.

from

solutions

oo.

0

>

existence

from results

in the

unique

the

method

existence

of radial

existence

as

number

we

was

of the

r

any

[47].

function

the

directly

now,

half-line

for

two papers

3 and was

(3,5),

p E

ODEapproach

an

completely

proving

jolp-1,

of solutions

the

problem

the

=

:

11.1.2

also

[110].

from

f(02)

3 and

see

his

101P-10.

N

Theorem

latter

0,

=

possesses

A result

0.

of the

r

paper

101P`

=

>

r

for

solutions

methods

[10],

was

[59].

paper

of radial =

half-line

Methods

positive

that

result

f(o')

3 and

=

the

on

ODEapproach,

=

on

N

of this

the

are

solved

point

of the

have

we

an was

solutions

I < P < 5 any

methods

fact,

proof

completely

was

for

that

that

so

with

considered

neighborhood

a

OIP-'

f (0')

3 and

(11.0.1),(11.0.3)

problem

In

of roots

existence

in

follows

roots

N

Shekhter,

way of

nonnegative

g(o)

p

4 < p < 5 the

for

This

in

with

f(02)

paper

Another

+oo

proved

is

5 in the

for

=

problem but

paper

sign

in the

of the

value

B.L.

this

N

the

(11.0.1),(11.0.3).

exploited

problem

by

of ODEs

in

alternative

to

the

on zero

The indicated

on

are

whether 0.

=

derivative

number

we

In the

time.

r

(11.0.1),(11.0.3)

with

result

first

its

sense

a

(11.0.1),(11.0.3) long

proved

been

the

framework

In the

by

there

for

5,

<

< p

point

of the

investigations

proving

its

question

obtained

for

similar

for

the

given

any

of the

remained

with

independently

is

solution I

problem

with

this

possible

for

problem

the

5 the

solution

a

proved

that

made it

radial

a

with

solutions

for

< p <

for

[97],

it is

addition,

In

result

A similar

1 < p < 4.

kind.

of roots

neighborhood [83]: it has

a

paper

hold

of

number

given

bounded

of

3 and

==

general

methods

existence

incomplete:

here

N

more

a

paper

variational

fact,

arbitrary

I

for

3).

prove

in

of

5 in the

and for p ! =

left

proved

> 0 is

nonlinearities

for

In to

Ix I

=

case

-*

is

function

[Lo_f(02)]

<

arbitrary

given

methods

of the

STATIONARYPROBLEMS

CKAPTER2.

78 of ODEs.

theory

qualitative

of

The methods in the is

[2,3,9,71,75,76,82,83,97].

papers

of solutions.

radial

wo 101P-10, g(o) The final [25,54,68]. the positive w > 0), proof

a

functions

of

system

established

fortunately,

(without is

so

a

x

Here

Another

this

>

0

we

the

on

is

Also,

[119]

in

way consists

that

in

exploited in

upper

of a

general

triangular

even

for

the

in

basis

a

analog

attempt

[118].

to

properties

and

all

system

11.4.5 use

the we

of the

elements of functions

of its

on

expansions note

that

where

JUn}n=0,1,2....

of

s

that

the

< so

and

on

the

is considered.

Bary

the

(11.4.17) b,',

(the

diagonal to

theorem. for

example

an

coefficients

principal

probproblem

proved

problem

based

which,

transform

Fourier

eigenvalue

However,

the

in

of the

of Theorem

proof an

is

an

is

systems

boundary-value

[1181 H'(0, 1)

IIA.5

Similar their

is

of

monograph errors

for

first

discussion

[119].

it

paper

nonlin-

in the

was

essential in

a

of the

small

of this

basis

a

[118],

In

nonlinear

a

being

In the

denumerable,

eigenfunctions presented is

property

results

interesting

of Theorem

proof

published

knowledge We only

thorough

approach

the

eigen-

completeness

the

contains

of

direction.

some

more

The first

of

page

111.

best

arising Bary theorem

the

are

that

under

A

proof

the

is considered.

which

approach

[39];

[6].

in

[115-117].

in

[6].

in

in

paper

The

corrections

presented

are

proved

contained

in that

parameter)

over

and

approach

the

results

L2

have

shows

[5]

proved.

is

These

constant.

is (bn )n,m=0,1,2,... insufficient are zero) L2(0) 1)M

from

solutions,

natural

This paper

in

However,

spectral

negative

half-line

aim.

in

of its a

proof)

corresponding

eigenfunctions system

a

is

equation,

operator

problem,

theorem

this

in

note

system

author's

[62] containing author [63] where

same

linear

be corrected.

can

and

lems

[114].

in

the

nonlinear

a

results

no

on

of the

papers

wo- 101"o,

=--

error

properties to the

nonlinearity in the

We also

principal

a

uniqueness

the

(for g(o)

exists).

problem,

almost

there

of positive

proved

is

proved

it

basis

on

for

Makhmudov

by

paper

differs

theorem

2.4

are

a

this

around

questions this

by

(if

contains

Liouville-type there

A.P.

of

of

(without

announced

to

and

and the

perturbations

ear

field

eigenelements

of

[83]

in

as

unique

from Section

Sturm-

monograph

subject

this

on

new

a

the

mention

presented

the result

nonlinear

a

quite

is

always

is

particular,

it is

the

uniqueness

solution

positive

[54]:

in

knowledge

our

concerns

the

on

In

of the

of

chapter)

results

(11.0.1),(11.0.3).

is obtained

solution

only

best

introduced

proof.

this

in

are

uniqueness

result

result

similar

Concerning this

the

-

of

there

problem

of the

solutions

considered

literature,

In the

==

(not

problem

The second

methods

to the

the

to

of the

variant

our

close

are

However,

precisely

containing

paper

no

11.2.1

Theorem

proving

our

be

this from

matrix are

complete

non-

in

Chapter

3

Stability

of solutions

it

chapter,

this

In

noted

is

Sobolev

JxJ

with

respect

the

to

distance

stability

the

study

to

of functions

spaces

the

p in

solitary

of

we

(for

definitions

named

a

solitary

see

u(x, t),

wave

Introduction

where

(x, t)

u

O(w,

=

=

x

x

mathematical

pioneer

by

paper

field.

investigations

in the

like

vanishing

solutions

respect like

the

to

O(x -Lot) can

or

such

that

the

i.

the

"forms

if

Thus, the close

two sufficient

which

we

u(x

sense

of the for

conditions

suppose

t)

-r,

graphs"

as

to

x

be

--+

solution

u

p for

O(x

-

graph

wt)

the oo

of

stability for

one-dimensional,

P.E. Zhidkov: LNM 1756, pp. 79 - 104, 2001 © Springer-Verlag Berlin Heidelberg 2001

our

of usual

as

is

of

is

an

arbitrary

perturbed

---->

with

i.

respect

e.

stable,

to

the

Simultaneously

with

N

=

1.

wave

of functions 'T

"almost

then

for

solution

distance we

close

functions

these

almost

are

In Section

coincide".

KdVE.

spaces

u(x, t)

as

and

clearly

x

soliton-

a

travelling

translation

a

functions

O(x -wt) oo JxJ

p "almost

distance

0, then

of

with

u

=

KdVE are

standard

of the

t >

t > 0 there

of the functions

vanishing

visually:

of distances

for this

and

(x, t)

some

of soliton-

form a

the

numerous

f (u)

if

point

a

waves,

stability

the

of

stability

the

a

0(+oo).

further

the

KdVE

is

xo

=

KdVE with

understood

easily

sense

however,

and of the

graph

a

in the

solution

vanishing

solutions

other

of

of the

in the

to it

stability

be

distance

of the

spaces,

graphs

of its

forms

this

and

oo

-+

each

soliton-like

a

x

sense

Sobolev

Lebesgue e.

to

of the

can

proved

has

standard

oo

p; he called

as

in the

close

be not

x

origin author

the

the

that

E R and

x

0,

=

initiated

which

d for

recall

of solitary

stability

the

to

the

was

paper,

--->

terminology

vanishing other

to each

In this as

distance

This

solution.

[7]

Benjamin

T.B.

devoted

literature

as

of the

all

x

In the

vanishing to

case

-

are or

We also

Section

or

NLSE

Lebesgue

KdVE and

of the

case

as

waves

3.1). wt) in the and u(x, t) e'wto(w, x) for the NLSE, a kink if 0' (w, x) = 0 for if there is a unique xO E R such that solution soliton-like 0' (LO, xo) and the of x function of a of extremurn 0(-oo) argument O(Lo, x) as

NLSE

of p and d

As

waves.

KdVE and

of the

waves

of standard

and it is natural

spaces oo

-4

respect

solitary

usually

distances

to

of solitary

stability

of the

questions

consider

Introduction,

the

in

with

unstable

shall

we

3.1

as

r(t)

=

G R

identical, coincide". each

t

> 0

sufficiently we

consider

p of soliton-like

study

the

NLSE

STABILITY OF SOLUTIONS

CHAPTER3.

80

As it behavior

noted

is

as

x

Section

3.2

p under

assumptions

we

x

In two

stability

cases

prove

solitary

of

stability

of

stability

a

the

I

can

=

and kinks. the

to

respect

In

distance

defining

NLSE nonvanishing

interesting

new

vanishing

solutions

stability

the

of the

waves a

type.

solutions

of soliton-like

with

N

on

type.

stability

We begin

considered. one-dimensional

NLSE with

KdVE with

of soliton-like

the

section,

In this

assumptions solutions

soliton-like

are

of the

of kinks

a

we

Stability

3.1

these

natural

and the

of solutions

x

types:

consider

we

in

two

general

of

3.3

oo.

--+

only

the

prove

In Section as

of

waves

KdVE under

the

2.1,

oo of derivatives

--+

solitary

have

Section

in

of such

as

x

solutions

for

be

will

00

-+

KdVE and

the

NLSE. Let

p(u, v)

=

I Ju(-)

inf TER

v(-

-

r)

-

H1

E

v

u,

and

d(u, v) where

H1 is the real

prove

that First.

we

in the

space

in each

case

I Ju(-)

inf yE[0,21r]

'rER,

first

greatest

the

Cauchy problem +

ut

lower

f (u)ux

remark

for

We also

u(.,t)

let

[0, a) I Ju(-, t) 111

continued

solution >

Let

C for

1), (111.

1.

=

1.

2)

C,

0 be

in

t C-

in

I,

the

(uo

> 0

then

[0,

a

interval

an a

twice

a

constant,

existing

0,

2 E H

(x);

(111.1.2)

H2-solution

from Theorem

holds

1.1.3

continuously

differentiable

differentiable

function

twice

(M.1-1),(X.1-2) if

exists

a

=

0).

that

exists >

in

If there

6 > 0 such

8) (resp. there of time [0, 6), 6

+

1.

R,

x, t C:

continuously

twice

0, if

in

in view

an

interval

case

of the

a

=

solution

the

H2 -solution

a

=

can

be

embedding

> 0

of

time

that

u(-, t)

can

be

of the problem

0). u(., t) be a H2_ [0, a) or I [0, a], considered by analogy).

function of time

0

C

and

such

interval

an

exists

a

differentiable

continuously

1.

easily

can

here.

problem

there

a

One

second.

KdVE:

the

arbitrary

an

(Ill. 1), (111. 2) < C (the IV: Jju(-,t)jjj

problem a

of a

interval

f (-) be

0, bounded >

all

the

Let

of the

[0, a],

be

uo

of

result

f(.)

Let

H'-solution

I

onto

Proof.

a

a

or

<

111. 1.1

=

=

in the

is achieved.

for

uniqueness

following

the

present

be

=

(111.

of the

with

Proposition I

proof

the

problem

the

f (.).

that

(X, 0)

u,vEH',

complex

bound

+ uxxx

U

we

and

case

the

consider

eiy v(--,r)jjj,

-

I

and

=

of H'

=

into

C,

such

that

STABILITY OF SOLITON-LIKE SOLUTIONS

3.1.

lg(-)Ic

:5 Cl for

(1.1.3),

condition

global problem

obviously

Thus

ul

+

a

(., t) 6). El

is

E

u

[-I

since

Cl,

(., t)

is

continuously

a

twice

1 +

Cl ]

and

==

f, has

f

with

H2-solution

a

solution,

a

u

Jul (., t) Ic (111.1.1),(111.1.2)

that

problem

of the

u

of such

0 such

>

be

-

taken

uniqueness b

exists

H2-solution

a

f (.)

for

obviously,

Then

due to the

there

fl(.)

C and let

<

problem

the

(-, t).

ul

JlgJJl with

coinciding that

so

in I and

H' obeying

g E

H2-solution

Also,

[0,

all

function,

differentiable

81

(-, t)

=

(., t)

ul

in the

a

unique

of the

Cl for

< I +

satisfying

t

latter

for

[0,

E

t E I.

interval

b).

+

a

of time

00

Definition

f (-)

111. 1.2 Let

U C2 ((-n,

E

n); R),

so

in accordance

that

Propo-

with

n=l

H2-solution, O(w, -) E

any

and

Hl.

x

this

p(uo(.), problem (111. 1), (111. 1.2) has p(u(., t), 0(w, .)) < e for uo E

1.

By analogy,

consider

R, be

E

0 <

6,

f(JU12)U

U(X,O) satisfy

of

solution

(fl)

ing the

any

are

has

the

t), U(-, t))

with

Now

P.L.

Lions

to

the

KdVE with

stability

we

assume

well-posedness the

sense

of the

of the space

0

E

exists

b

>

=

that

v

E

v

1. 1. 3 and

local

>

H')

for

0

>

u(x, t)

solution

t > 0 and

half-line

entire

[2,4);

the

NLSE:

0,

X,t

E

Then,

0 such

half-line

and

uo

of

this

H'

E

0 and

>

formulating 111. 1. 1 if

in

of the

>

according one

a

proves

suitable

parameter

u(x, t)

solution

method

following to which or

result

E

f (.)

supposes

sense

of

We consider on

connected

2 is

v

satisfy-

the problem for all t > 0

solutions. the

v

V(x,t)

solution

u(x, t) of t

of soliton-like

However, values

any

uniqueness

Proposition

(111.1.1),(111.1.2)

this

concentration-compactness

stability When

call

for

H'-solution

entire

existence

we

that

requirement

other

(111.1.3) soliton-like

=

H'.

of the

0.

R,

f(JU12)U in equation e'wtO(x) be a U(x, t)

let

the

function.

problem

b

uo(x).

the

onto

of the

Jul',

the

for

corresponding

application

investigation f (u)

differentiable

continuously

c (the L2.4).

an

of Theorem

sumptions

<

e

exists

nonlinearity

where

b the

<

Theorem

consider

we

there

0

=

and

1.2

be continued

can

d(u(-,

proved

(JI1.1.3)

>

=

the

1,

=

Section

d(uo, 0)

condition

(111.1-3),(M.1-4) one

c

N

from

NLSE

the

if for

stable

Let

111.1.3

condition

any

KdVE,

the

> 0 there

corresponding

the onto

of

solution

local

unique

a

t > 0.

iUt + AU +

Definition

then

Cauchy problem

the

if for

be continued

can

all

has

soliton-like

a

stable

O(w, .))

H and

the

one

w

solution

2

(111.1.1),(111.1.2)

problem

the

wt),

-

call

we

H2

E

uo

0(w,

let

Then

if

that,

such

of

for

111.1.1

sition

with is

a

the as-

twice

the

(for example, (0, 4), then all

local in the

the

from

arguments

Theorem soliton-like

Let

O(x,t) from ftn}n=1,2,3.... as

n

111.1.4

is

(n) uo

=

Let

the

=

has

solution

a

clearly

and

boundary

> 0 and

w

E(u)

functional

is defined

I V

(the parameter

w

this

boundary-value

belongs

01

two functions

A,

we

problem n --+

n

JUn(*) tn) 122 a

02

the

P(U(n), 01t=0)

0

_

0

Cauchy

of the

Theorem

to

following

for a

A,

(11. 0 (up

family =

1

-3)

11.3.1, with

equation

is

shown

it

some

2.1,

of Section

2.1, 0 that

solution

positive

of the

values

of

y,,)

any

according minimization

our

+

for

parameter

Therefore,

wn(-

that

102 122

10, 122 :

since

different

C R such

in Section

beginning

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

a

sequence

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

sequence

it

the

Further,

with

to

0

translations,

0.

>

=

as

As at

up to

minimizing

any

U(00)

solutions). some

A and

=

our

we

take

--+

A

as

vn n

--+

=

oo

-+

e.

in the on

the

a

(-,tn) .Since JU.(*,tn)12 and E(Un('7 0)) Aun

for

sequence

n

-+

contradiction.

111-1.5

For the

papers

by

stability

the

above

C R such as

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

of solutions

sequence

jYrjn=1,2,3.... 0 p(vn, 0)

oo7 i.

fact,

results

sequence

a

1.1.3.

According

2.3.

otherwise,

unique,

a

with

from the

sequence

--+

0,

>

E(u),

LOU,

=

because

has

(11.1.3)

and

to

Remark In

problem

exists

minimizing

Hence, n

solution

0

--+

in H'

as

oo.

any

a

nontrivial

11.3.1,

there

Return For

positive

no

A,

that

get

Theorem

to

be

has

family

the

to

jul"u

+ 1

must

problem

above

is stable.

problem

satisfies

solution

its

every

E

that

H2-solutions

are

3)

1.

conditions

U" +

the

in Section

(11.

exist

by Theorem

given

inf JU12=A>o 2

uEHI,

the

Un(Xi t)

A > 0 the

any

corresponding

the

there

minimization

IA where

that

from H2such

where

e

for

family

the

Then,

0

P(Un(*i tn)i Olt=tn) > with uo (111.1.1),(111.1.2) lo(.,t)12 2 A > 0. Consider

problem

Suppose

ju(n)}n=1,2,3....

sequence

a

Then,

1) from

1.

stable.

not

[2,4).

E

v

(Iff.

A > 0.

(11.1.3)

R+ and

C

arbitrary

an

where

Kd VE

the

hold.

below

Jul"

=

and

oo

-+

fix

family

the

f(u)

Let

O(x, t) of

us

theorem

of the

proof

solution

Proof.

is

STABILITY OF SOLUTIONS

CHAPTER3.

82

with

proofs

+

lun (-,tn)

12

time,

based

on

1,

111.1.4

Lions the

is

[23]

we

1.1.3

is clear

it

that

therefore in

converges -->

111.1.4

Theorem and P.L.

Theorem

problem,

y.,,)

A

Theorem

Thus, first

tn))j

v,,(.

that

T. Cazenave

E(un(-,

minimization

Since

oo.

according

to

=

problem

of the

get

H' to 0 P (Un

('I tn)

JUn(.,0)12

2

f vn},,=,

2,3....

there as

exists

n

oo.

--+

0

7

as

proved.0 was

and

concentration-

proved by

P.L.

in the

paper

Lions

compactness

[57,58]

[1011. the

method

83

STABILITY OF SOLITON-LIKE SOLUTIONS

3.1.

for

investigated

in these

applying

this

method

Now we consider

the

"Q- criterion"

possibility

ishing

of

jxj

as

"Q-criterion"

name

often

by

P and

stability of the

KdVE

in the

the

close

used

waves.

to

conser-

of the

condition

in the

law Eo

conservation

the

van-

necessary.

the

in which

here

rename

variable,

example

solutions

stability

KdVE)

We also

a

is

solitary

of

literature

physical

of the

case

by Q.

denoted

been

has

(E0

NLSE

law P of the

vation

originates

the

an

of soliton-like

of the

conditions

spatial

with

stability

of the

stability

of the

from

on

particular,

in

the

illustrate

to

problem

to the

sufficient

gives

which

oo

--+

only

wanted

we

considered:

are

depending

coefficients

Here

papers.

kind

general

more

NLSE, admitting

multidimensional

The

essentially

NLSE of the

a

U

that,

recall

the

in

7(u)

KdVE,

the

on

case

=

f f (p)dp

and

0 U

F(u)

f 7(p)dp.

=

0

Theorem

111.1.6

plex argument

f(JU12 )u NLSE (111.

Let

for the

u

differentiable

continuously

be

a

1.

3)

with

(f (u)

N= I

of the

function

differ-

continuously

be twice

com-

'r

entiable

for

KdVE).

the

f f (s)ds

F(r)

also

Let

and let

there

wob 2

0 and F (02)

exist

wo E

R and b > 0

0

f (0)

that

such

-

0, f (b 2)

wo <

>

L,,o

-

wo < 0, 7(b) (0, b) (resp., f (0) the KdVE). As for 0 E (0, b) for

0

G

there

tions

U(x, t) 0(w,

exists

O(wo,

x

E

-

wot) for

it

-

0, F(b)

>

proved

is

U(x, t)

solution

soliton-like

a

wob

-

-

F(b 2)

0,

2

!Ib

-

2

e'woto(wo,

=

as

x) for

these

0

<

condi-

NLSE

the

for

0

<

2202 2

_

there

which

for

oo,

x

under

1,

2.

W002

_

F(02)

0 and

=

Section

in

KdVE) vanishing

the

=

(resp., exists

L2 and 00

d

-P(O(W, -))

I

2 W=,o

O(wo, x) 0' (wo, x)

dx.

00

If the condition

I

dw

satisfied,

0 is

>

then

this

soliton-like

solution

the

[92]

U(x, t)

is

W=WO

stable.

Remark like

solutions

author

For

cases.

complete.

(U(x, t) the

(N

As =

paper =

1,2).

paper

example, we

O(w, x [35], 0

notes, in

the

know,

any

wt)

for

the

is radial

The function

w

> 0

where

point

from

satisfies

a

equation

f(X, JU12)

exists

paper

NLSE.

when N

incomplete

there

in

multidimensional

a

KdVE),

about

0

is

N > 1 and

case

for

-

0 of

proof

his

obtained

is

statement

functions

positive

with

of this

A similar

111.1.7

=

a

solitary

O(x)

> 0

RN, if p E

(11.0.1).

JuIv

>

the

wave

and,

However,

proof

U(x, t) (N

O(x)

the

several

from

[92]

=

to >

as

for

except

according

(0, N'2)

Substituting

I

soliton-

for

2) =

e'wto(w,

is

x) of

results

and p

>

0

WPV(W2X),

84

we

CRAPTER3. find

in the

of the

case

NLSE: AV

and

by analogy

tion

of the

we

for

the

KdVE.

IVIPV

+

V

-

(11.0.1),(11.0.3) 2.-E Wp f v 2 (y)dy

=

Vjj"j_"

0,

=

Hence,

problem

P(U)

have

STABILITY OF SOLUTIONS

by

the

uniqueness

mentioned is

of

fixed

a

positive

a

Additional

in

v(.)

where

2

0

=

radial

remarks

function.

to

Thus,

[92] (for case

N>

of the

what

shall

on

the

result

Proof

f [lh'(x)

=

is

One

for

KdVE is

the

change

the

paper

(0, 4).

can

Here

>

In

'.

A

N

[15].

in

v(x, t)

the

in the

NLSE if p

We

(111. 1.3).

prove

the

if p E

proved

of variables

that

0 if

By analogy,

NLSE.

NLSE

2,

>

from

results

0 is satisfied

>

place

the

can

and

condition.

dw

consider

of the

12 ]dx.

this

takes

the

1)

=

dp(o)

for

> 4

making

12 +wolh(x)

under

instability

solution

a

N

condition

the

We first

wo > 0

u(x, t)

where

is stable

when p

111.1.6.

(for

111.1.6

the

instability

of Theorem

11hl 121

=

show that

and, respectively,

0

=

if necessary, that

we

U(x, t) I u IP

f (u)

KdVE with

follows,

similar

f (0)

According 1) the solution

Theorem

to

Chapter dw

RN

0 < p < -1. N

solu-

=

also

accept

lower

bound

we

greatest

that accept 6- f (O)tu(x, t)

-00

in the

expression

remark u

that

in the

real).

d( U, u)

for

generally

is achieved

and

-r

e-'(-/W+w0t)u(-

form

Differentiating

7

t)

r(t),

+

the

expression

T(t)

7(t)

(we a perturbed solution unique). 0 + h(x, t) where h(x, t) v + iw (v, w are with to and T d(U, u) respect 7, we get at

some

T

=

E R and

E R

We represent

not

are

=

for

=

00

I

V[f(02)

+

202f/(02)]

O'dx

=

X

(111.1.5)

0,

00

00

j

Wof(02 )dx

0.

=

(111.1-6)

00

Further, AE+

WO

dx!

AP

a(s)

where d2

2

=

o(s) f(02).

=

+ W0

_

as

Lemma 111. 1.8

s

--

There

2

[P(O+h)-P(O)]

+0 and

C

exists

+

dX2

0 such

1f(L+v,v)+(L_w,w)}+a(jjhj

12)

2 d2

L+

>

>

that

W 0

(L-

-

w,

[f(02)

w)

+202f/(02)]

CIIWI12 for

all

WE

(111.1.6).

satisfying Proof. of the

Wo

E(u)-E(O)+

operator

Let

w

L_

=

ao

+ wi-

where

corresponding

(L-w,w)

to

=

(0, wi-) the

=

0.

eigenvalue

Then, A,

=

since

0

0,

have

(L_w.L,w_j -) ! A2 JU,_L12 21

we

is the

eigenfunction

H'

STABILITY OF SOLITON-LIKE SOLUTIONS

3.1.

because,

and,

hence,

is

[28]).

see

0

since

is

I

(on

L-

operator

0 is minimal

=

this

subject,

+IW_L0f(02)dX

02f(02 )dx

0.

=

(012

W002 )dx

+

0,

>

get:

we

-00

-00

Jal IW12

C11WI12,

!5

and of

independent

C2

>

sup

I f(02) 1)

0

A,

00

00

f 02f(02 )dx=f

hence

1 2 is positive

L-;

eigenvalue

of the

spectrum

of

spectrum

by (111.1.6):

Then,

a

Since

positive

corresponding

the

of the

point

isolated

an

of the

function,

positive

a

bound

lower

2 is the greatest

where

85

(111-1.8)

therefore

H'.

E

w

CIW112i

:5

(L-w,w)

implies

k

For

independent

0

>

C21 W2 12 with

! of

some

(M

have

w we

X

(L-

I

w)

w,

(1W 1122 )k -+1 + 1

1

W01W122)

+

2f(02 )dx

W

00

Thus,

(L-w, w) ! for k

k >

>

sufficiently

0

0 the

I

00

1

small

independent

and

k+1

2

i-T-1 OW/12

+

W01W12). 2

IIWI12 1

k + 1

of w, because

f W2f(02 )dx

(IW112+WOIW12)_ 2 2

1

expression

k

k

+

for

is not

a

sufficiently

smaller

small

than

-00

00

C2 k + 1

IW12

I

k

2 -

k + I

W2f(02 )dx

(C2 k +

>

k _

k + 1

1

M) IW12

>

2

0.

-

-00

Lemma 111.1.8

In what

is

proved.0

follows,

condition

the

we use

(V, 0) Lemma 111.1.9

(111.1.5)

satisfying

Since

Proof. value

there this

There

2

=

0 and

exists

eigenvalue

an

(111.1.9).

clearly

0'

0'

has

is

an

precisely

eigenfunction is

C

exists

and

minimal.

Lot

gi

that

eigenfunction > >

root,

of

92

(L+v, v) >CIIVI12

L+

A2 is the

0 with

0,

(111.1.9)

0.

such

> 0

one

g1(x)

=

a =

with

second

corresponding

mo'

be

the

for

all

corresponding of L+

eigenvalue

eigenvalue

eigenfunctions

A,

of the

E

v

H'

eigenso

< 0

that

and

operator

CHAPTER3.

86

L+

o,',

normalized =

W

ag,

L2 and let

in

b92

+

01,

+

v

192

+

+

(L+v, v) It

follows

from

the

spectral

C,

of

v)

0

Using

now

Schwartz

the

using

0 <

Hence,

1

to

C1

(III.I.10)

jvj_12 2

(III.

(11.0.1)

equality

i

(L+v,,Vj

_)2(L+0,,0j

conditions

of the

d dw

there

P(O)

=

exists

implies

-L+O".

that

(111.

Combining

(L+v,v)

1

=

.10)-(111.1.

Alk

2

)Alk Now,

let

us

use

+

2

I (L+

>

2

theorem,

2(0,

W

-2(L+O,,,

13),

(L+v-L,

+

(I

(L+0

<

we

v ,

)

(111. 1.5).

of

,

,

A, I a

0,,) v

oi-)

-2AIa

such

! A, k2 +

v-L,

k

1.

(111-1.13)

inequality:

(Allal Ik 1)2 -V"r-Aja2

v

=

+

(1

v

112,2

kgl

192

+

-

v -r) (L+ C2

+

V1

)2

+

into

>

v-L,

0-

this

obtain: 00

JIMI

j

00

(of 12 f(02) X

+

202f/(02)

}dx

j

11m I

=

00

[(0/1

XX

WO(of )2]dX X

-00

CC)

1 0/ [f(02) X

+

202f/(02)]

(kgl

+ vi

-)dx

<

C31kgj

+

V112-

00

111

:5 C4 I k91 +

VJ-

12.

This

inequality

(L+v, v)

together

! C5JV12'2

0 1 ).

-L'

.

C2 I kgl +

Substituting

2(L+O

-

2

:5 -rAja

v

2

that

get the following

vr) (L+

-

oi-)

v-L,

find

we

=

v i

i,

obtain

we

(0, 1) independent

E

r

-)

Alak + (L+0

=

inequality,

0

Thus,

Let

Then,

vj-).

,

92.

gi,

that

)

v ,

-L,

Further,

v.

(L+v

+

vi-

orthogonal Lj-.

E

by (111.1.9)

Therefore,

Hence,

2

Alk

of L2

0 _L,

(see [28])

theorem

independent

0 is

>

where

VI

=

(L+v where

subspace

L_L be the

kgl

=

STABILITY OF SOLUTIONS

with

C5

>

0.

(111.

1.

14) implies:

v-L)

=

(111.1.14) equality,

we

STABILITY OF SOLITON-LIKE SOLUTIONS

3.1.

Proceeding we

further

proof of

end of the

at the

as

and Lemma 111.1.9

CIIvII1,

::

Lemma III.1.10

I lh(.,

Proof.

for

We have

IIU(',tl)

t) Ill

is

e

-

i('Y(t2)+WOt2)

I I h(.,t2)

0(.

Now h

Theorem

prove

Then,

=

by (111.1.7)

111.1.6.

P(O

+

aj(0)I

-

lim

0. 8

+0

AE +

WO

2

all functions condition

Also,

I I,,

JIU(*itl)

h

Let

P(0)

-

-

ao

=

=

-(t2))Ill

T

<

hence

U(*7t2)111)

+

h_L where

a

a,

+

ia2 and

1012+ Ih 122 2

2a,

h

jai I

I lh(-, t) 12

C,

2

and Lemmas 111.1.8

LOO

S

h)

=

AE+-AP>C2(lllmhj-+a20l, 2 where

,

-

-u(*,tl) :5

_

of t; hence

lai(t) Further,

ei(Y(t2)+WOt2)0(.

7-(t2))Ill

functional

the

AP

independent

1

+

proved.El

is

we can

0.

7

1 IU(*,t2)

:5

_

U(* t2)1 11

-

Ilh(-,t2)III

-

t.

e'(^f(t2)+WOt2)0(.

-

JIU(')t2)

-

I IU(i t1) I 1,

of

function

JIU(*)t2)

-

7(t2))Ill

-

III -I Ih(-,tj)

and Lemma III.1.10

0,

>

IIh(-,t2)II1

-

7(tl))Ill

-

I Ilh(.,tl)lll

the

inequality

and t2:

tj

<

By analogy,

C

continuous

a

arbitrary

ei('y(t1)+0j0t1)0(*

-

1 lu(-, ti)

for

8, from the last

proved.0

is

Ilh(.,tl)lll

is

Lemma 111. 1.

get:

(L+v, v)

(0,

87

Hence,

AP ->

(73

(1 lIm

sufficiently <

exists

m(l Ilm

112) -C3IajIlIReh-LIJj-C4aj+a(lIhlj

h

,

+

12

1

+

0 such

+

in H1 with

a201 11

12),

2

1

m>

h-L + a201

small

lh(., 0)12 2

and 111.1.9

12+ IlRehithere

_

that

IlRe h-L 112), 1

C3

corresponding

>

(111.1.16)

0,

coefficients

a,

satisfying

I lRe h-LI 11).

clearly AE +

WO

2

AP <

C411hl 12,1

C4

>

0,

(111.1.17)

88

CHAPTER3.

for

all

small

in H'.

of the

kind:

sufficiently

h

hood of

in H'

zero

06

h

=

Let

H"

E

m-1 be

>

a

6

arbitrary

an

1-8, 2

<

IlIm

sufficiently

f lai(t)l

sup UoEH':

t>O,

jail

:

For

large

I IIrn

+

h

6 > 0 let

+

,

t)

Suppose

+

IlRe h_LIIj

Let

us

prove

a20(',

+

h(.,O)EO6

06 be the

a20111

constant.

h

STABILITY OF SOLUTIONS

t)II1

+

neighbor-

open

<

1

6

-

that

I lRe h-L(., t)l I,}

0

(sufficiently small) C > 0 and with u.,&, t) problem (111-1.3),(111.1.4) sequence hn(', 0) E 06,,, either +0 as n 8,n c or I IIrn hi -n( -, tn)+a2n(tn)O(*7tn oo, such that aln(tn) )112+ 1 2 > h c2for First of some n > IlRe In tn tn) 112 0, 1, 2,3, all, (111. 1.15) implies that for all sufficiently < euntil large n wehave laln(t)l IIImhj_n(*it)+a2n(t)0 t) Ill 2+ E and t IlRe hj-n(., t)112 I < _C2f2 because if lain(t)l + a2n(t)O ( IlIm hj-n(.,t) 1112 + -6 V < h IlRe E2, then by (111.1-15) n(', t) 112I ! 2 +C562 which is a contradiction lain(t)l as

+0.

-->

this

right.

is not

of solutions

a

-->

Then,

there

exist

a

of the

-+

....

=

_

-

1

-

because

tn

>

0,

n

Then,

<

we

have

for

lal,,(tn)l therefore, small

according > 0

c

large

all

(111. 1.16),

to

independent

sufficiently

[AE

of n,

large Wo

+

Thus,

we

arrive

which

the

solution

2 at

taking

we

numbers

+

to

the

place

for

all

exists.

soliton-like

for those

t > 0 for

solutions solution

which

exist

sufficiently

a

Loo'2 'P] Ltn At the

[AE

+

I I Re h_Ln ('; tn) 112

+

IlRe hj-n(*,tn)lll];

large

2

=

02,E2.

>

m-'

and

sufficiently

a

solutions

AP] I

t=0

:5

also

c8i

t >

exist.

t

C7,8,n2

=

yield

1.

0 in the

Hence,

0

-+

as

a

n

--+

proved

are

priori

get

we

oo.

for

all

t for

estimates

0)

(111.

problem

point

(111.1.17)

from

(111-1-18)

relations

<

> 0

time,

relations

of the

the

C6

>

same

wo

and

These

u(x, t) V(-, t) at

these

+

n:

-

contradiction,

u(., t)

there

get

n.

API I t*=tn a

assumption,

our

+a2n(tn)0(',tn)II1

JIU(*it)lll taking

112

numbers

m[IIIm hln(*,tn)

[AE for

to

that

sufficiently

all

<

according

Hence,

IlIm h1n(',tn)+a2n(tn)0(*)tn)

and

c

small.

such

1, 2, 3,...,

=

lain(tn)l

arbitrary

0 is

>

c

3), (111. 1.4) sufficiently

close

of the

d and

sense

according

distance

to Theorem

1.2.4

and

STABILITY OF SOLITON-LIKE SOLUTIONS

3. 1.

proved the

statements,

111.1.6

us

change

u(x, t)

now

consider

the

variables,

of

for

the

(it

global

is

be continued

can

NLSE the

of the

case

f (0)

t)

+ r,

solution

of the

case

that

u(x

form

in the

soliton-like

the

u(., t)

solutions

0). Thus,

t >

onto

of Theorem

statement

proved.

is

Let ate

of these

any

half-line

entire

89

=

0 and wo > 0.

O(wo, x)

=

making

KdVE. We can assume,

under

consideration

r,t)

01 (Wo'.)12

Representing

h(x, t),

+

ff(x,

where

and the

a

t)

parameter

perturbed =

r

appropri-

an

=

solution

O(Wo, is r (t)

x

-

Wot)

chosen

is to

minimize

U,

(.

x

the

get

we

+

_

2

x

+WOIU(.

7-,t)

+

opo'.)12

_

2)

constraint 00

I

f (O(wo, x))O'(wo,

t)dx

x)h(x,

x

(111.1.19)

0.

=

00

Also,

as

in the

AE,

Wo

+

2

NLSE

of the

case

AP

=

El (u(.,

t))

(Lh, h)

>

where the

0

s

-->

NLSE,

of the

case

as

+0 and L we

=

-4 -

-

dX2

+

h E H1

all

[P(u(.,

t))

-

flo(wo,

.))]

!

Loo

f (O(wo, -)).

-

Proceeding

further

as

in

get the estimate

(111.1.19)

satisfying

2

7(1 IhI 11),

+

(Lh,h) for

Wo

El (O(wo, .)) +

-

>

the

and

C911hl 121 condition

00

j

h (x)

0 (Loo, x)

dx

0.

=

00

The end of the

111.1.6

is

proof

Under

III.1.11

NLSE and

KdVE the

d

In

< 0.

the

only these

the

our

of the Here

restrictions,

N,

2

(to

then

we we

2, the problem

< P <

the

those

repeats

case

be the

of Theorem

assumptions

instability

consideration,

Chapter if 0

case

[40,84].

papers

simplify from

in this

NLSE. Theorem

the

for

proved.0

Remark

in

stability

of the

with

respect

KdVE it

is

consider

focus

to

the

proved

in

these

on

the

interpreted problem

as

has

with p > a

the

0 for

radial

f(JU12) w

>

N

IUI,'.

0 has

positive

for

a

1, 2);

cases

takes

[15];

paper

=

=

in both

distances

only

questions

case

(11.0.1),(11.0.3)

111.1.6

above

for

the

in

the

to

solution

addition,

solution

place

NLSE

According positive

0.

of the

if

NLSE,

and,

to

results if and

if p satisfies

According

to

CHAPTER3.

90

111.1.7,

Remark

the a

Let w

wo > 0 and

positive

a

the

prove

)

4

N-2

positive

radial

and

b such

constants

all

x

a

e"OtO(wo, x) a

complex

Section

in

as

of the

(p

4

>

N

for

with

respect

to

We present

N=

only

solution

h

=

ih2

+

for

given

a

(11-0-1),(11.0.3)

problem

of the

IVO(x)l

+

Further,

h,

0

all,

of

First

1.3.

number

there

exist

that

z(t)

=-

0 for

can

easily

the

jY0+h(0)-Y95(0)j
solitary

verify

H' such that

E

ae-blxl

<

one

Then,

2.1.

NLSE.

function

jjhjjj<e,

e""O(x) 1, 2).

of solutions

N> 3

Section

from

notation

10(x)l for

instability

where

proof.

the

use

us

to

4, (N

p E

(incomplete)

formal

=

have

we

d for

distance

STABILITY OF SOLUTIONS

for

any

following

the

U(x, t)

solution

wave

that

C

> 0 there

exists

take

place:

inequalities

JE(0+h)-E(0)j<e

Z0+h(0)>1E-1,

and

(111.1.19) where

YO+h and ZO+h

e""o O(wo, we

hand

x)

side

functions

values

of t

there

are

T > 0 such

exists

where

u(-, t)

of the

NLSE is

this

section,

kinks

are

10(w, x) 1

:5 C,

solutions

of this

natural

only for

various

Here like

E R.

kind

Perez we

KdVE,

"soliton"

Consider considered

in

are

is

the

clear t for

Thus,

0 the

that

YO+h(t)

t > 0 for

place

takes

+ h.

t

all

for

for

Hence,

0.

-

instability

"almost

a

the

for

all

is

a

which

all

these and

> 0

t

IVU12

SUP

right-

=

+00

of the

e'woto(wo,

solution

X)

stability

KdVE

of, kinks

always"

general

type

common

too.

this

[42] It

stable. on

of the

the

-

It

means

main

idea

in the

idea

first

appeared

where

is used

a

for

analysis

semilinear estimates

0' (W, x) :

Introduction,

the

stable

under

they

are

Although of the

we

stability by

in the

paper

wave

equation

from below

that 0 and

X

in the

that

f (u).

function

the

KdVE. We recall

conditions

it is noted

=

consider

of kinks

D.B.

Henry,

is considered.

of the

functionals

follows).

KdVE

Section

is

0

uo

at

be continued T

--4

O(w, wt) satisfying 0+. As O(w, oo)

Wreszinski

in what

the

t

as

the

and

We set

idea,

this

(h, Lh) (see

h(x, t)

x

equations;

and W.F.

exploit

+

prove

of the

there

it

cannot

+oo

---

with

that

0

waves

assumptions the

J.F.

x

such

Further,

1.

Zq5+h(t)

Z.O+h(t)

x)

shall

we

3), (111. 1.4)

H'

an

of kinks

travelling

1.

in

solution

to the

tE[o,T]

Stability

3.2 In

proved.

than

function

that

e"00(wo,

=

(111.

corresponding

z

function of increasing Therefore, inequality (1.2.19) is

determined. the

y and

small

smaller

ZO+h(t)

Hence,

0.

>

arbitrary

not

and

functions

Cauchy problem h

is

function

these

of the

function

a

(1.2.19)

of

decreasing

t)

+ h (x,

choose

can

of the

values

are

2. 1:

Conditions for

a

kink

O(w,

x

providing -

the

wt) satisfying

existence

of kinks

equation

(11.

1.

1)

are

and

STABILITY OF KINKS FOR THE KDVE

3.2.

0(oo)

conditions

the

ing conditions

0

=

it

sufficient

is

and

f f(s)ds,

=

that

necessary

.0

(hereAo)

satisfied

are

exist,

to

91

fj(0)

7(0)

=

-

the

follow-

wO + wo-

and

00

f fi(s)ds):

F1(0)

0-

A:

fi (0-)

B:

F, (0-)

C: F, We also

fl (o+)

0;

F, (o+)

(0)

for

< 0

0;

all

(0-, 0+).

G

require -W

Clearly,

(111.2.1)

condition

provides

I O(W, X) Without

(111.2. 1)

and a

suitable

the we

result

loss

shall

of

(111. 1. 1),(Ill. following.

Theorem

Then,

.

there

of

these

with

Let

the

such

solutions

the

that

of the

of

a

0-

> x

-

0-

Under

wt).

solution

u(00,

infinity

the

>

i

O(w,

kink

on

C1 C2

0+

that

uniqueness

conditions

For this

u(x, t) t)

aim

A-C we

of the

This

need

Cauchy result

(111.2. 1) be valid, f (-) be a twice and a function uo(-) be such that u0(-) O(w, -) E and solution the a (0, a) unique u(x, t) of problem For any O(w, -) E CQ0, a); H2) n C1([0, a); H-'). A-C and

assumptions

-

half-interval

a

1XI)

!5 C1 e-C2

accept

conditions

function

exist

(111.1.2)

stability and

(111.2.1)

< 0.

estimates

we

existence

differentiable

continuously H2

generality

1.2)

111.2.1

the

A0)

I Ox'(W, X) 1

+

show the the

on

problem is the

0 I

-

+

u(-, t)

-

quantity

2IU (X, t) 2

IM-1 0)

=

F, (u (x, t))

-

x

dx

00

does the

that

of

not

above

depend

I Ju(-, t) this

-

solution

and

exists

0(w, -)111

The Proof 1.1.3

i.

t,

on

solution

onto

a

of this

Proposition

<

e.

the

on

a

C for

all

t E

[0,

half-interval theorem

I(.) is [0, a),

functional

half-interval

can

[0, a), a

+

then

6),

be made

6

a

conservation

a

>

there >

law.

0, and there exists

a

In

exists

(unique)

addition, continuation

0.

by analogy

with

the

proof

if

C > 0 such

of Theorem

STABILITY OF SOLUTIONS

CHAPTER3.

92

Remark

111.2.1,

111.2.2

suffices

it

analogous

nition

Remark

t))

get

write

to

I.I.I.

careful

a

the

Since

111.2.3

I(u(-,

quantity

To

to

equation

by

IF 1 (U)I

construction

is well-defined.

difference

the

for

A formal

u(x, t)

of solutions

definition

u

C(U

<

verification

0

-

0 )2

-

from

Theorem

and formulate

as

x

the

oo,

--+

independence

of its

defi-

a

of t is also

obvious.

The result

on

Theorem

111.2.4

stability

the

of kinks

KdVE we consider

the

for

here

follow-

is the

ing.

(111.2. 1) u(-) such

A-C and For

any

p'(u, q

=

distance

uO inf

f I u' (x)

for

any

1.

and

for

Remark the

again

111.2.5

stability

Proof

One of the

of Theorem

a(s)

-

Lot)

easily

o(s) h(x, t) =

+

the

prove

as

s

where

existence

kink

the

of

0p,

_

the

KdVE.

0

_,r) 121, 2

X

is

stable

with

6 > 0 such

that

if uo(.)

corresponding can

Pq(U(',

t), O(W, -))

We first

111.2.4.

_

<

the

onto

O(Lo, -)

half-line

entire

E

problem

the

t > 0

place.

takes

E

-

the

to

respect

u(., t) of

solution

be continued

that

the

stability

from

prove

the

following

estimate:

CP2(U, 0)

I(0)

q

+0 and C

-*

T

=

T(t)

>

Theorem

111.2.4

(P2(U' 0)),

-a

is

chosen

(111.2.2)

.

independent

0 is

C- R is

for

of

Let

u.

Pq(U, 0)

to

u(x, t)

=

be minimal

O(W, (one

x

-

can

T). Then,

of such

AI 00

=

I(U)

-

1(0)

00

Ih,2

2

kink

form.

J(U) where

corresponding

qlu(x)

+

111.2.1

see

conditions

Let also

set

exists

the

easily

can

be the

T)122

_

Then,

0 there

then

inequality

any t > 0 the

1}.

+

>

c

<

X

X

1w I

+

6, Pq(UO('), O(W, ')) 1), (111. 1.2) from Theorem

(111.

01 p,

_

X

H2 and

T

O(w, x -wt) OP, -) E H1 we

-

differentiable.

continuously

and

I If (s) I

-1E[0-,0+1 e. pq, i-

be twice

satisfied

7-ER

max

=

f (.)

Let

that

0)

q

where

be

+

[w

-

f (0)]h

2jdx-

-1 2

-00

I

If (0

+

Oh)

-

f (0)

1h

2

dx

1 =

-

2

Ij(h)

+

a(P2 (U, 0)), q

-00

(111.2-3) "0

where

O(x, t)

0

E

(0, 1),

a(s)

=

o(s)

as

s

--*

+0 and

II(h)

f -00

L

=

d2 dX2

+

hLhdx

with

STABILITY OF KINKS FOR THE KDVE

3.2.

Now

we

(for details, set [a, +oo)

exists.

a

h

0'

is

ILO'

=

of

b

A2 if A2 exists

=

g)

+ g where

and

of the

2

differentiating

pq

(U,

with

in the

a

=

We obviously

0.

=

b

0'(x)

Since

with

A2

respect

to

the

positive

is

case.

have

bjg 122'

::f

=

L with

operator

opposite

the

LO'

0 and

>

eigenvalue

second

=

operators

L coincides

operator

> 0.

-

Ij(h) Further,

of the

spectrum

differential

ordinary

symmetric

maxjf (0+), f (0-)} the first eigenfunction (positive) 0. A, Hence, the eigenvalue w

=

minimal We take

Let

theory

The continuous

where

corresponding if it

[28)).

see

function

0, the

spectral

the

use

93

(111.2.4) the

at

-r

of

point

minimum,

find

we

00

I

[q

Lo

-

f (0)]O'hdx

+

0.

=

00

Substituting

h

po'

=

equality,

this

+ g into

P '5

which

together

(111.2.4)

with

last,

we

>

in the

(111.2.5).

the

continuity

of Lemma 111. 1.8.

The last

t.

Take

and 0

an

<

CT

C31 I h 112,1

all

u

arbitrary

11u(-,t)

arbitrary

c

VC such

<

proof

of the

part of

H'

:

p,

6 > 0 for

(111. 2.6)

due to

t,

E

we

get

yields

Then,

0.

>

of Theorem

0(w,-)jjj

-

all

t

>

Theorem

we

0 for

111.2.1

finite

any

u(-, t)

is

0.

global

111.2.4

(111.2.5)

0,

is

easily

above

As in

Lemma

jjh(-,t)jjj

of

as

a

there

arguments

(111.2.3)

from

follows

clear.

continuity

of the

in view

>

(111.2.2)

the

2

III.1.10,

function

80

exist

(u, 0)

<

which have

6o.

AI

<

61

Let

C62

<

61.

which

the

and since

E

Let

(0, 8o) u(-,O)

81, therefore,

solution due to

half-interval in time

[O,a)

<

and Theorem

of 0

>

is

that

<

11h(-,t)JI,

that

of the

continuity

of

Thus,

Take

c.

<

6.

h (-,

Then,

t)

by

the

latter

statement

C, of the

proved.0

an

in

(111.2.7)

<

exists.

<

(111.2.6)

be such

V81

existence

111.2.4

be such

in view

u(-, t) (111.2.7)

of the

-CTjjhj 121

< AI <

IM-10111 in

,

that

I Jh(-, t) 111 for

,

C3

Now inequality

Cjjhjjj for

C2

get the estimate

proof

and

C1 19 12

C21h 12,2

(Lh, h) as

get

gives

Ij(h) At

easily

we

solution

u(-,t),

the

solution

of

CHAPTER3.

94

Stability ing as I x I

3.3

In this

section,

are

consider

(111.1.3)

space-homogeneous

noted

in

begin

with

the

u(.)

Then,

one

can

adding

27rm,

m

=

to

absolutely

ao +

a(x)

=

lu(x) I

u'(x) we

have

R,

that

(ao for

+ a(-))w(-)

example

a

w(.),

absolutely

u(-)

a(x) E

consider

X1 and here

all

x

u(x) : is the

(ao

=

all

for

any

E R and

all

0, and kinks. a

such

4D(t)

as

type.

We

example,

if

parameter.

valid,

for

with

fixed

a

a(.)

function

real-valued

finite

any

a

is

already

we

interesting

new

(this

As

one-

solutions

interval

and

unique

a(x))e'lc+w(x)] R;

E

x

+

in

t E

R).

E

X' to

up

if 0

there

finite

addition,

due to the

< ao +

< cl

relation

small,

(111-3.2) a(.)

interval,

such with

The result

on

< C2 < +00 for

W'(.)

then

real-valued

exist

(111.3.1)

that

(111-3-1)

a(x))w'(x)]e'1c+w(x)1,

sufficiently

is

E R.

x

E R

x

of the

These

oo.

--+

ao > 0 is

unique

in

+

i(ao

+

I I a(.) I I, u(.)

in

0 for

a

particular,

In

.

if

continuous 0 for

>

L2

exist

function

complex-valued

if for

ao +

E occurs

,

x

it, such that

[a(x)

=

of

constant

nonzero

to

> 0

ao >

where

all

for

of solutions

as

stability

a

continuous

0, 1, 2,...

stability

where

,

if(a2)t 0

there

u(x) and that

prove

a

that

the

on

if(a2)t 0

aoe 4)(t) u(x) 54 0

verify

w(.),

function

a

aoe

=

=

in X'

close

NLSE nonvanish-

1) non-vanishing

=

X' and

E

easily

N

shall

we

solutions

R, u(-)

E

c

sufficiently

is

and

Introduction,

results

two

(with -1)(t)

ones

studying

Let

of the

oc)

-->

shall

we

NLSE

dimensional

of solutions

STABILITY OF SOLUTIONS

functions

(ao

that

some

the

E

c

+

L2

-

stability

of solutions

x

E

Conversely,

a(-) a(.))w'(-)

E R takes

all

E

X1 and E

place,

L27 then

4b(t)

we

following.

differentiable f(JU12 )u be a twice continuously juncbe real-valued and a 0 for < f function (.) u, f'(ao 2) E9be Let such that 0. 0 H' and Ila(.)Ill some ao > < bo > < 6o. if a(.) (-= la(-)Ic 2 if(aDt in the following is stable Then, the solution aoe of the NLSE (111.1.3) 4D(t) sense: 6 E (0, for an arbitrary bo) there exists 8 E (0, 6o) such that, if uo(-) E X1, uo(x) : 0 and uo(x) (ao + !T(x))ewW where u(x) luo(x) I ao E H1, I IZT(.) 111 < 8 and IW'(') 12 < 6, then the corresponding X'-solution u(x, t) of the Cauchy problem the entire onto given by Theorem L2.10 can be continued (III.1.3),(M.1.4) half-line t > 0, u(x, t) : - 0 for all x E R, t > 0 and for any fixed t > 0 the functions a(x, t) in the representation andw(x,t) Theorem

tion

of

the

111.3.1

complex

Let N

=

1,

argument

=

=

=

u(x, t) satisfy

thefollowing:

a(-,t)

E

=

H',

(ao

+

Ila(.,t)lll

a(x,

-

t))e'U(-5')+-(x,t)1 <

c

and

IWX'(*it)12

<

C-

AS JXJ STABILITY OF SOLUTIONS OF THE NLSE NONVANISHING

3.3.

Proof.

smoothness

I

=

k this

the

X'-solution be such

T1, T2,

...,

of the

existence

TI,

I

any

Under

d'

the

which,

of both

T, ! T2 Tj

assumptions

exists

any

integer

coincides

of course,

positive

Let

half-interval

maximal

Tk

>

...

there

for

solutions.

be the

sufficient

a

1.2.10,

Clearly,

k, [0, TI)

...,

suppose

0.

>

T2

TA;

=

T

>

0.

inequality

f(JU12)U 12

dx1

we

Theorem

to

existence

Clearly, above

If

V-solution

1, 2,

=

XL.

(111.1.3),(111.1.4).

of the

V-solution.

Due to the

I

for

E

according

local

unique

a

intervals

that,

of the

Lemma 111.3.2

Proof.

has all

in

then, problem

of the

problem

uo(-)

and

integer

f(JU12)U,

X'-solution

1, 2,...,

with

k > 1 be

function

of the

(local)

unique

a

first

Let

oo95

--+

MIJIU1111-1)

_<

d'

CII(IIJU1111)l

+

U

dxl

1 21

have

we

Ci(max Jjju(-,s)jjjj_j)

111uJI11:5

+

SE[O,t]

I

C2(max 11jull1j) SE[041

t

jjju(-,s)jjjjds

0

which

by step

step

the

implies

of Lemma 111.3.2.0

statement

00

f (JU12

I(u)

Let

2)2

a

-

dx.

0

-00

Lemma 111-3.3 oo

for

(111.

uo E

some

3), (111. 1-4)

1.

function

is

entiable

let

given

Then,

I(u(.,

be valid

u(x, t) of

t))

<

for

00

the

all

I(uo)

and let

<

Cauchy problem

[0, to)

t E

and this

by

X'

E

uo

f(JU12 )u

and

Theorem

be

a

four

X'-solution

corresponding 1.2.10

we

times

u(x, t)

formally

continuously

differ-

Cauchy problem

of the

have: b

(u)

12i[( IU12

lim

-

dt

[0, to).

111.3.1

X1 -solution

corresponding

t E

For the

(111.1.3),(111.1.4)

of Theorem

assumptions

in t.

At first

function.

dI

for

exist

continuous

Proof.

the

Let

X1 and the

a,b-+oo

-

2) (U -g

ao

U;u X

-

I-x=b

)]

X=-a

+ 2i

J(U2g.2

_

X

U2;U2) )dx X

-a

Then,

according

right-hand rlosed

side

subset

continuously for

uo E

f(juj2)u

to

the

of the

of

[0, to),

latter

differentiable

converging

equality

t))

I u(-,

in t.

X' by the passage to

result,

standard

to

<

uniform

is oo

for

Now the the

limit

non-smoothed

I(u(.,

since

all

t E

statement over

ones.0

a

0))

with

[0, to)

<

respect and

oo

to

the

and

t from

function

of Lemma 111.3.3

sequence

the

of smoothed

can

limit an

the

in

arbitrary

I(u(-,

t))

is

be obtained

functions

uo

and

CHAPTER3.

96

Lemma 111.3.4

the

the

Let

corresponding

of Theorem

assumptions

X'-solution

of

the

STABILITY OF SOLUTIONS

(111.

problem

be

111.3.1

valid,

3), (M. 1.4)

1.

for

X'

E

uo

exists

and

[0, to)

t E

Let 00

faO) JU12 2 2

M(u)

JUX12

2

U(JU12)

_

+

dx,

+ D

00

where

3

-f(a2)a2+

D

U(a 02)

0

2

U(s)

and

d P* If 1(uo) f fW

1

=

2

the

then

< oo,

quantity

0

t))

M(u(.,

determined

is

Proof.

One

for

easily

can

all

verify

M(uo)

quantity

function,

(111.1.3),(111.1.4). indeed M(u(-, t))

problem hence

differentiable

f(JU12 )u

by

Let

be the

in the

as

E

uo

a

exists

and

oo

C(u)(Jul

u(x, t)

be the

differentiation

X1

one

of t.

easily

can

)2,

be

four

a

contin-

times

X'-solution

of the

shows

that

dM(u(-,t))

For

a

twice

continuously

obtain

the

of smoothed

sequences

over

ao

-

corresponding

independent

E

uo

limit

t.

X3, f(JU12 )u

formal

I(uo)

that such

for

t E

Then,

in view of Lemmas 111.3.3

M(u)

in the

following

uo E

<

of

and because

0,

_

dt

of

statement

functions

uo

and

of Lemma 111.3.3.0

half-interval

(III.1.3),(ITI.1.4)

+ D <

and

to the

passage

X1 be such

maximal

the

is determined

proof

1

JU12

and let

f(JU12)u

function

Lemma 111.3.4

I(uo)

2

Let first

Then,

independent

and

since

2

is determined.

differentiable

uously

that,

+: W)

_U(JU12) the

[0, to)

t E

<

[0, to)

and

oo

the

that

and

uo(x)

0 0,

corresponding is

and 111.3.4

for t E

these

for

nonzero

[0, to)

E R.

x

Let

X'-solution values

we can

of t and

rewrite

[0, to)

also

of the

problem x

E R.

the functional

form:

M(U(') t))

M(U(') t))

=:

-

M( )

=

MI + M2 + M3)

00

M, (u (., t))

where

f I a 2 (X, t)

2

2a

-

2(X, t)a 02f/ (a 02) } dx,

M2(u (-, t))

_C0

co

f

1 =

2

L02 (X, t) (ao +

a

X

(x, t))2 dx,

M3(u(.,t))

=

t) 112)

ci(jja(-,

-00

and

lim S_+o

for

fixed

any

w(x, t)

is

By of the

0

S

t

as

absolutely

(here

(x, t)

functions continuous

Lemma 111. 3.3

condition

u

of the

=

of

(ao

x

+a

an

11 a (., t) I I I is a theorem f(a 02) M1(u)

0

>

arbitrary

0, there

ClIal 12I

t)]

function

continuous <

,

where

a(-, t) belongs finite interval).

function

the

in

(x, t))ei[f(a2)t+( (X

,

exists

a

E

H1.

to

of t E C

>

ao+a(x,t) H'

and

[0, to).

0 such

the

Also,

>

0 and

function

in view

that

(111.3.3)

STABILITY OF SOLUTIONS OF THE NLSE NONVANISHINGAS JXJ

3.3. Take

the

from any

s

(0, bo) and let H(s) Then, there (111.3.3).

arbitrary

an

estimate

(0, 61]. By

E

E

6

above

the

arguments,

exists

61

there

exists

H(jja(-,t)jj&5 if

Ila(-,0)jjj

for

such

82 and JW,1 (*) 0) 12

<

At

I I a(-, t) I I,

initial

data

last,

in view

of the

equality

Ilw.,(ao

+

2 we

IW.,' (7 t) 12

get

;

<

c

jjju(-,t)jjjj

these

C, for all

<

Corollary

t E

c

=

2

for

data

[O,to).

C

all

Thus,

=

H(s)

such

constant >

CS2 for 2

that

812 of

-

a

(., t) Iwe get

<

M3(u) 62

62 where

there

sufficiently

0 is

0 such

>

111-3.1

that

proved.0

is

Then,

be valid.

111.3.1

Theorem

>

C,

exists

and Theorem

+oo,

of

assumptions

(0, 61)

continuity

Mj(u)

0)12 by (111.3.2)

to

0 is the

that

such

E

>

[0, to)

t E

-

4

the

using

M(u)

62 and

<

initial

the

Let

111.3.5

<

a)1'

I I a (., 0) 11 1

if

for

Therefore,

small.

81

<

82

<

C

where

(0, c)

E

M(u)

62. Then,

<

a(s')

Cs' +

=

oo97

--+

the

(0, 8o) and (an arbiao E H', if uo E X1, luo(-) I (0, 60) trary where WOis a realuo(x) (ao + ii(x))e"M aoll, < 6o (consequently 11 luo(-)l and interval in continuous valued function W'(-) E L2) and for absolutely any finite X'-solution the corresponding u(x, t) of the problem (111. 1. 3), (111. 1-4) given by Theand 6 L 10 2. < orem IWx' ( 0) 12 < 6, then for any t > 0 and xo E R there 11 a(., 0) 11 1 exists -y(xo, t) such that 7

solution

4P(t) large)

is

d

stable

also

> 0 there

in

following

the

for

sense:

6 E

exists

such

E

6

any

that

-

=

-

=

Ju(-, t)e"I Proof

Remark for

aoe-

f(s) ia2t5

=:

The

111.3.6

0,

differentiable

and let

0,

to

IH1(xo-d,xo+d) proved

the

of

IE-

Theorem

Theorem

theorem,

this

<

111.3.1.0

111-3.1

valid,

are

the

function

0- and 0+

(a) 0-, 0+ > 0; (b) _Zj + f(02 ) (c) _0 + f(02 )

stability of the

be such

that

+

202 ff(02

=

0;

of kinks

complex the

<

for

the

NLSE. Let

argument

following

u,

f (-)

conditions

be are

f(JU12 )u a

+

U(02

_U02 + 2

+

satisfied:

0;

U(02+ )

where

U(s)

2

f f (r)dr; 0

be

real-valued

S

(d) _ZU02

example,

OD(t)

solutions

corresponding

the

for

stable.

are

Now we consider

ously

from

4)(t)i

assumptions

According

-s.

ao >

,

follows

immediately

-

a

continu-

function

98

CHAPTER3.

ZU' S2 + T

U(S2)

<

As it is shown in Section

2.

(e)

_

ei! 7to(x)

of the

implies

the

[O(x)

lim X__00

The

g(. -,r)

=

speaking,

generally

ro

-

lim

[O(x)

9(-

7-)

-

is

0+]e'x

-

0'(x)e-'

lim

=

X__00

we

shall

need

it

in what

g(-) 0(-) llg(T) 0(.)Ill 0(T) + 0(- -'r)

function

that

E

-

-

-

-9(x, t)

kink

a

(b)

condition

=

be such

the

0(.)

-

such

g(x)

function

U(02 ) for all s E (0-, Oj (c)- (e) provide the existence of 0. As it is well-known, 0(oo) that

> 0

technical; 0- < 0+.

is that

minimizing

number

have

a

with

X_+00

definiteness Let

c

=

(a)

assumption the

for

of

0-]e-"

-

+

-

1, conditions

(111.1.3)

NLSE

existence

U02 -i

_

STABILITY OF SOLUTIONS

-

X_+00

follows.

H1.

-

We also

We denote

(since 0(0(-) E Hl).

-

0'(x)e'

lim

=

T)

-

suppose

by

0(-)

-

Of course,

0.

=

a

To

real

E

H',

as

earlier,

we

non-unique.

X'-solution

u(x, t) of the NLSE (111.1.3) be such that lu(-, t) I 0(.) E H' for some t > 0. We set v(x, t) for g(x) u(x To, t), where To is taken lu(x, t) 1, and a(x,t) As if is lv(x,t)l then O(x). earlier, Ila(.,t)lll sufficiently small, 0 < cl value :5 O(x) + a(x, t) :5 C2 < oo for this t and respectively there exists function a real-valued continuous in finite inan absolutely w(x,t), arbitrary terval and unique the term 27rm, m to it, such that up to adding 1, 2,..., Since v(-, t) E X' and vx'(x, t) v(x, t) (O(x) +a(x, t))e'1'_0t+'0(x,t)1. [0'(x) +a'(x, t) + we have i(O(x) + a(x,t))w.,(x,t)]e'l'-t+w(x,t)), E L2 if Ila(.,t)lll is sufficiently small. if u(x, t) : 0 for some t > 0 and all x E R, where By analogy, u(x, t) is a Let

a

-

=

=

-

=

-

=

=

=

X

X'-solution

of the

Theorem

of

tion

the

111.3.7

complex

corresponding for

if

X1, luo(-)l

uo G

any

ing X1 -solution onto

the

<

Proof.

c

>

let

the

of

the

0(-) E H1, Ila(., O)l 11 u(x, t) of the problem (Iff. 0)

t >

<

Consider

the

and

for

NLSE a

(111.

1.

Lo(x, t).

introduce

continuously

twice

exists

(a)-(e) 3) is

differentiable be valid. stable

sufficiently

small

in

6

funcThen,

the >

the

following

0 such

that

ILO,1(*) 0) 12 < 6, then the correspondis global 3), (111.1.4) (it can be continued t > 0 one has I U(-, t) fixed 0(-) E H1,

6 and

<

1.

any

a

we can

assumptions

0 there

-

and

c

e'w_t0(x)

small

half-line

entire

Ila(.,t)lll

V(x, t)

sufficiently

and

u

=

then

1, f(JU12 )u be

Let N =

argument

kink

sense:

(M.1.3),(IIIJA),

problem

6-

functional

00

J2

M(u)

1UX(X)12

_

U(JU(X)12)

+

WIU(X)12 2

+

D,

dx,

00

where

U702 -i

D,

-

determinedfor

Lemma 111.3.8 an

interval

+

U(02 ). -

afunction

of

time

Let

[0, to)

u(-) u(x,t)

One E

and let

can

easily

verify

X' it is sufficient be

a

X'-solution

lu(-, 0) 1

-

0(-)

that

for

the

M(u(.)) to be that lu(.)I-O(-) E L2in problem (111-14,(111.1.4) Then, lu(-, t) I 0(-) E L2 for quantity

and necessary

of E

the

L2.

-

STABILITY OF SOL UTIONS OF THE NLSE NONVANISHINGAS IX

3.3.

all

[0, to)

t E

Then,

u(x,t)

and

by

lemma

of the

converging

uo

For

uo E

X' nonequal

to

M(UO)

M(O)

AM:--

over

-

For

[0, to) and,

E

in

M(V)

-*::--

-

function,

uo E

(Ill.1.3),(IIIAA). M(u(., quantity

we

of smoothed

get the

t))

statement

f( JUI 2)U

functions

0

ones.

luo(-)l

that

and such

zero

X'-solution

a

sequence

a

the

0, hence,

=

dt

[0, to).

problem

of the

dM(u(-,t))

that

non-smoothed

the

to

X4_solution

limit

to the

passage

a

t

differentiable

continuously

times

of t E

independent

and

determined

five

a

shows

calculation

direct

the

be

corresponding

be the

for all

determined

is

oo99

--+

t.

f(JU12)u

Let first

Proof.

X' and

t))

M(u(.,

quantity

of

independent

addition,

is

the

that

so

I

M(O)

-

0(-)

H'

E

have

we

(111-3.4)

M1 + M2 + M31

=

where 00

00

M,

Ila

2

2 X

(0

+

f(02)

_

))a 2}dx,

202f/(02

-

M2

I

2

W12 (0

a)2 dx,

+

X

-00

-00

00

ja

I

M3

2

0

where

=

21f(02)

O(x, t)

+

H1 satisfying

f ((0

-

+

Oa)2)

There

C,

exists

0 such

>

202f/(02) The

Let

.

q+}

minjq_;

eigenfunction

q

L.

all

9 E

I I Ju(-

(see [28]).

Further,

was

ro

+

Oa)2 )dx (111.3.5)

CIJgJ2 2 for

real-valued

all

).

as

a

=

eigenvalue

corresponding

and isolated that

theorem

point

where

By condition

from

and Lemma 111.3.9

defined

-0 + f(02) + q(x) have 0. < we q (b) the half-line [b, +00) where b equation (111.1.3) that 0'(x) is an

&) o

-

L fills

it follows

the

(111.3.6)

0.

is

A0

=

0; in addition,

eigenvalue

M, (g)

=

of the

(g, Lg)

>

(R (T))

1

'r='ro

of minimum

of the

function

"0

2

I

00

a(x,t)o'(x)[q

opera-

C1 Ig 1 22 for

proved.0

Therefore,

2

=

-w"

202 fl(02

+

spectral

the

(111.3.6),

t) 0

Oa)2f/((o

Mi(g)

that

Ao is the smallest

sign, from

=

=

operator

L with

operator

it follows

number

-r,

the

of

H' satisfying

The -

0

f(02 )

+

spectrum

is of constant

Hence,

tor

>

LW

operator

_Zj

=

of the

0'(x)

since

the

Consider

continuous

+

condition

the

(91 01) Proof.

2(0

-

(0, 1).

E

Lemma 111-3.9 g E

202fl(02)

ay +

f(02)

+

202f/(02

)]dx.

R('r)

STABILITY OF SOLUTIONS

CHAPTER3.

100

We take

jZj

sup

f (0'(x))

-

20'(x)f'(O'(x))

-

I

+ 1 and

xER

10112 1 01(V

K

f(02)

+

+

202ff(02))

12

00

f

+

f(02)

202fl(02))dX

+

-00

g E

H' sat,sfying

C2Ig 12,2

MI(g)

Lemma 111.3.10 the

C2

where

Cj(1

=

K) -2, for

+

all

real-valued

condition

W

I

g(x)o'(x)(q

Zj

-

+

(02(x))

f

+

202(X)f1(02(X)))dX

(111.3.7)

0.

=

00

Proof. g

=

ao'+

Represent

p)

(0',

o where

arbitrary

an

0.

=

function

Then M,

(g)

CIJW12.2

M, (W)

=

(111.3.7)

H1 satisfying

9 E

in the

form

from condition

We get

(111.3.7): 00

a

j

0/2rq

_

f(02)

Zj +

202f/(02

+

+j

)]dx

00

W01[q

_

ZU +

f(02)

+

202f/(02

)Idx

=

0,

00

-00

hence, co

f 001[-q I 1101 1 2 a

=--

101 12

f(02)

+

+

202f1(02)]dX <

oo

f 012[-q

+

f(02)

+

202f1(02)]dX

-00

<

I ot [-t7

I W1 2 10112

ay +

_

f(02)

+

202f/(02)]

12

=

co

f 012[q-

-

U+

f(02)

+

KJW12-

202f1(02)]dx

-0"

In addition

1912

Lemma 111.3.10

IC11 10'12

<

9 E

Proof.

inequality

There

with

:5

3.

(1

+

K)1 012.

M(g) -2 C1 JW122 Thus1

> -

C3

>

0 such

C311gJ12 for

MI(g)

that

C2Ig 122

from Lemma 111.3.10

implies

the

k > 0:

arbitrary

00

M, (9)

C2 JgJ2 2, and

:

k

2(1+k)

_lg,12+ '2

C2

(1+k)

jgJ2

f

k

2+T(I+k)

q

-00

Hence,

M, (9)

all

real-

7).

M, (g)

inequality an

exists

(Iff.

H1 satisfying The

1 012

proved.0

is

Lemma 111.3.11

valued

+

k

JgX1122

+

2C2

2(1

-

+

kVlgl2

k)

2'

(X)g2(X )dx.

following

STABILITY OF SOLUTIONS OF THE NLSE NONVANISHINGAS IXI

3.3.

taking

Thus,

There

Lemma 111.3.12

L(s)

lim

that

0 such

8-+o

k

small

sufficiently

a

exists

For

Lemma 111.3.13

-

follows

Proof

Let

and

IW'(',

of

la(., t) 111

[O,to).

time

is

a

u(x, t) of Then, a(.,t)

X'-solution

corresponding

tl7t2

[O,to)

E

that

and

argument

lu(., t) I

that

110('

0(.)

(111.3.4)

that

=

conditions

the

E

(111.3.5).0

and

0(-) lv(-, 0) 1 ), (111. 14) exist

H'

for

all

t

f [JU(X)12

-

02(X)]2

H1 and

E

-

1 .3

-

110(*

JU(*,tl)l Ill

that

JU('Itl)l

-

-

+

in

half-

a

[O,to)

E

and

To prove

=

J(u(.,

dx.

Clearly,

if

second

claim

holds.

t))

is

Ill

:5

tl) IU(*)t2)1

JU(*it2)1

-

consider

determined

for

X',

then

any

follows

claim

expression

the t

11

H1 for

E

The first

0.

second,

the

-I U(')t2)1

T(t2))

11 JU(

JU(*7 t2)1 111

-

prove

Hl.

I Ill

JU(')tl)

T(t2))

+ to

E

X1 satisfying

E

0 such

>

< 6-

a(-, 0) problem (Iff. 0(-) Iv(-,t)l

-

11 JU(*7t2)1

-

0) 12

6

00

J(U)

>

s

inequalities

'r(tl))

suffices

it

liM

that

+

-

t2-tl

fact

from the

110(*

=-:

JU(*) tl) I 111

-

Lemma 111.3.8

and

0.11

>

t.

formal

the

from

I I a(-, tl) I I I -I I a(., t2)112

T(t2))

of

function

continuous

uo(-)

the

=

It follows

Proof.

+

the

small

sufficiently

a

and relations

X1 be such

E

uo

exists

nonzero

Lemma 111.3.12

from

Lemma 111.3-14

there

any

-

110(*

of

C3

some

llgll2L(Ilglll).

0

>

c

any

M(0) I < c for IM(v(., 0)) luo(-) I 0(-) E Hl, I la(., 0) 111 < 6

that

interval

with

L(s)

function

nondecreasing

a

C311glll

-f

is obvious.El

Proof

the

get Mj(g)

we

IM3(g)1:5

0 and

==

0,

>

00101

--+

E

[0, to)

and

-0.

in

continuous If

f(JU12)U

is

the

u(x,t)

function

smooth

sufficiently we

have

as

function

differentiable

continuously

a

a

X'-solution

sponding is

t, then

and

uo

G

in the

proof

of Lemma 111.3.3

of t E

[0, to)

and

the

for

corre-

J(u(.,t))

that

00

dJ(u(-,

t)) =

dt

2i

I

[(U2-g2

UXU2) + 200x(uU., 2

_

X

-

u.,U)Idx.

-00

For

an

a(., 0) sequence

over

a

of

X'-solution

Let a

us

u(.,t)

Xl-solution

arbitrary

condition

G

H'

get

we

of smoothed

prove

Theorem of

the

the

of the same

problem

property

and

(111.1.3),(111.1.4) relation

the

satisfying

by passing

to

the

limit

solutions.E]

111.3.7.

problem

Let

[0, to)

(111.1.3),(111.1.4)

be the

maximal

satisfying

interval

lu(.,O)l

of the existence -

0(.)

E

H',

CHAPTER3.

102

I la(., 0) 111 be sufficiently (111.3.5) and Lemmas

small

H(s)

and let

111.3.11

and e

there

Lemma 111.3.12 that

E

lg(.)Ic

<

and

apply

(0, 61)

2

AM<

111.3.14

and

IWx' ('7 0)12

82

.

also

such

i.

in

t

these

Remark the

case

the

theorem

0

0-

u(x, t)

solution

can

x

x(t)

=

have

Remark of

all,

w(x, t) easily

a

exists X2

for

any interval

constants, and

(X2

speaking, solution the

sufficiently

of

At the

Take

2

is

arbitrary

an

(0, e)

G

(0, 6,)

E

s

such

that

by Lemmas 111.3.11

[0, to)

I la(-, 0) 111

if

and

62 and

<

M1

M3)

each

solution

kink

the

111.3.7

proved

0-

< 0

=

(O(x)

t

exists

u(x, t)

of the

0 in the

=

to

in the

0+,

<

impossible

is

point

there

right-hand

problem

above

then

sense,

of this

side

where

method

our

represent

[106]

paper

of

arbitrary

an

a(x, t))e'r7'+W(x,')1

+

0

>

proved.0

is

first

was

Thus

0.

at

0

-

also

proving

perturbed

because

perturbed

a

has

representation

root

a

we

is close

time,

same

to

smallness

the x

want

to as

of a

Theorems

comment

IW.,' (*; t) 12 it

constant,

in the

for

a

can

be

Corollary

and

111.3.1

fixed

not

for

any

as

that

one

and

> 0

C

First

mean

unbounded

even

111.3.5,

does

t

111-3.7.

d

can

>

0

IWx'(*i t)12 < 8, lL0(X1,t) W(X2) t)l < c any x, in X21 < 2d so that the function w(x, t) is close to a constant of the 2d to but the as + +oo 0, d, xo d) length tending IW,' (* t) 12 (xo the function for intervals d, x, + d) w(x, t) is close to, can be different (xl if the difference is + Thus, roughly d) X2 sufficiently large. d), JX1 X21 8

>

0 such

JXI

which

that,

if

then

for

-

-

--+

-

i

-

-

-

each

in

case

close

sufficiently

unperturbed

solution

that

function

verify.

there and

as

Now

111.3-16

observe

we

the

If

it

the

but

root

no

-

lW 1(')O)J2 [0, to) for

E

to

u(x, t)

form

all

E H'.

a(., t)

if

the

in

there

E

+ t

considered.

is

t

61.

<

Then,

all

for

c

C4 (AM

'.!

111.3.7

work since

not

82.

<

C3,52 for 2

!

which

to

f

Theorem

Theorem

0+

<

does

solution

global.

are

111.3-15

=

according

E

solutions

H(s) jig(.)Ill

that

that

that

e.

(111.3.4),

due to

Then,

(111-3.8)

satisfying

JW, *;0)12 1 la(-, t) 111 <

[O,to) as Ila(-,0)llj I I lu(., t) I ill :5 ?7 for all close sufficiently (111.1-3),(111.1.4)

uniformly

H'

E

JWI (*; t) 12

0 <

0 such

>

and

(111.3.8)

Observe

s'L(s).

-

H(lla(-,t)llj).

:

111.3-13, 62

<

inequality

<

g(-)

all

Lemma

Q_62 if Ila(-,0)jlj 4

81

exists

for

C3s'

=

and 111.3.12

AM(t) By

STABILITY OF SOLUTIONS

can

solution be shifted

far

from

each

of

Theorem to

the

111.3.1

in any above from

other.

one

or

unperturbed another

111.3.7

one

at

the t

bounded

interval

in

such

two

=

of

structure

0 is close

but

intervals

to

phases

a

the of the

if these

perturbed of

structure

perturbed

intervals

are

103

ADDITIONAL REMARKS

3.4.

3.4

Additional

Partially

results

and in the

[7]

presented

variable

3.1

proved

rigorously

for

waves

However,

proved

particular

in several

only

insufficient

knowledge

investigate

the

stability,

physical

of soliton-

problem

was

a

a

was

results

literature). statement

[61]).

example,

result

rigorously

is

related

partially

are

result

similar

(11.0.1),(11.0.3)

of

uniqueness

this

for

such

case

stationary the

stability

a

(see,

difficulties

situations; need

we

d(., -)

physical

fact,

In

literature

the

the

about

distance

many other

like in the

1.

=

multidimensional

in the

noted,

already

it is

as

proof

N

case

in

spatial

the

in

Probably

earlier

rigorous

a

NLSE is used

multidimensional

the

without

the

for

"Q-criterion".

[92] (it appeared

paper

periodic

NLSE the

condition

the

one-dimensional

the

only

studied

We have

oo

-->

in the

solitary

of

stability

the

on

Ix I

as

called

[43,88]

Benjamin

T.B.

t100].

in

sufficient

a

by

paper

waves

of the

case

independently,

and,

the

to

travelling

of

In the

have obtained

we

vanishing

solutions

first

[21,23]

in

In Section

like

in

corrections

monographs

in the

contained

also

are

[14], some small [81 the stability

KdVE is considered.

of the

x

chapter

in this

In

Also,

introduced

first

t95].

review

made.

are

remarks

to

our

(in particular, of the

solution

positive

to

latter

problem). approach

Another

vanishing

currently

case,

"Q-criterion"

and

for

solution

instability

is

to

Several

papers

is not

ff.

is

waves

the

waves

that some-

with

deal

we

of

uniqueness

of the

problem

exam-

be noted

when

case

simplest

a

for

solitary

of

P.L.

inequality

[15]

again

when

case

for

we

dw

KdVE and

the

difficulties

meet

the

have

in

the

in

>

is the

chapter

in this

upon

0

the

provides

instability.

4p

inequality

opposite

[40,841

for

<

dw

NLSE

the

of

stability

multidimensional

0,

though We

case.

literature.

(see,

used,

for

to the

NLSE with

the

two results

to

the

NLSE in this

[22,81]),

example,

oevoted

are

known of

touch

the

in

multidimensional As it

do not

papers

this

readers

presented

to the

again

almost

In the

proved

these

from

solitary

in

occur

speaking,

we

roughly

that,

soliton-like

a

stability

of the

which

example,

for

related,

problem

Another We remind

method

those

should

it

by

solutions.

stationary

refer

the

to

in

problems,

difficult

However,

problem

to

similar

method

of this more

nonlocal).

solutions

method

compactness

application

essentially

to

sometimes

method

difficulties

meet

proofs

this

apply

to

attempts times

a

(and

concentrationan

applications

finds

it

the

on

considered

have

multidimensional

ple,

the

we

based

is

oo

-->

Although

Lions.

the

JxJ

as

of soliton-like

stability

of the

investigation

the

4-o

author,

depending

coefficients there

the concentration-

of the

investigation

non-vanishing direction.

in which

are

as

almost

JxJ

-+

stability on

no oo.

the

results In

compactness

of

solitary

spatial on

Section

waves

variable. the

stability 3.3

we

of have

CHAPTER3.

104

Finally, of

solitary

of the the

we waves

stability

original

mention

for the in abstract

equations.

the

papers

equations spaces

[12,40,84,93]

under with

where

consideration further

STABILITY OF SOLUTIONS the

problem

is reduced

applications

of the

of the

to abstract

obtained

stability

problems results

to

Chapter

4

Invariant

Invariant

measures

when

case

of

space

dynamical

a

(see

dimensional

Of course,

there

is

Numerous measures

in the

by

nonlinear

partial

a

tigations

for

subject these

objects

wave

equations

variant

allows

well-known

for

we

mass

points

points

by

tions)

that

refer on

(mainly boundary Poisson.

t,,,

by computer

conditions

and it

of trajectories one

has

a

proof

paper

generated

(partial) [73].

each

of this

Later,

oo.

of

bounded

case,

constructing

generated

is

these

inves-

literature

cdlled a

For

on

speaking, system a

a

the

(nonnegative)

description

this

partial

observed

becomes

arbitrary

recurrence

many

"soliton"

Fermi-Pasta-Ulam

system

invariant

the

phenomenon phenomenon a

close

the

measure

for

initial

with

stability

mass

was

equations

at

found

suitable

An analo-

according a

of

simula-

the

to

trajectories

in

chain

neighboring

(by computer of

in-

differential

phenomenon.

is called

bounded

considered

with

interacts

First,

of nonlinear a

of this

authors

they

similar for

consider.

Second, nonlinear

a

these

point

mass

dynamical

P.E. Zhidkov: LNM 1756, pp. 105 - 136, 2001 © Springer-Verlag Berlin Heidelberg 2001

by

we

of the Fermi-Pasta-Ulam

Briefly,

simulations) was

or

mechanics

statistical

invariance).

of the

waves.

Roughly --+

this

in

illustrate

measures

constructing

system

where

law.

of time

again

If

the

configuration

the

property

circle

nonlinear

a

a

of nonlinear to

of invariant

for

explanation

an

theory

the

[55,65,67]

dynamical

readers

moments

some

a

exist

Some other

two results.

our

Another

dimensional.

system

we

finite-

remarks.

without

give

to

in the

detail

in

(in [55],

measure

equation

gous

used

are

dynamical

a

N.N.

invaria

measure.

existence

chapter,

In this

applications

some

when

case

phase

by

known,

is infinite-

measures

the

result

normalized

invariant

proving

when the

well

is

system

invariant

to

equation.

in Additional

mention

we

dimensional

differential

is indicated Now

devoted

are

NLSE and KdVE with

the

a

dynamical

a

it

as

the

In

systems.

nonnegative

a

Liouville

whether

interest

papers

of

By analogy,

of

space

dynamical

of

(more precisely, space) the classical

existence

possesses

phase

infinite-

theory

metric

[72]).

system

recent

such

the

also,

natural

a

the

in

compact

a

states

and,

when the

occurs

is

Krylov

[13,52]

tool

is finite-dimensional

system

Hamiltonian

situation

basic

system

and N.M.

measure

too.

the

are

dynamical

a

Bogoliubov ant

measures

corresponding

to

CHAPTER4.

106

dynamical

Now we

present

satisfying

(see below)

theorem

recurrence

Mfor

)

the

of

explains

be

t

this

there

are

following.

the

separable

metric

homeomorphism

a

literature,

In the

system.

choose

we

complete

a

fixed

any

dynamical

a

and

concept

Let Mbe

IV.0.1

Mi

x

Poincar6

definition

of this

Definition g

a

definitions

R

:

the

partially.

phenomenon

different

then

system,

INVARIANT MEASURES

and let

space

of

a

function

the

space

M into

phase

space

M.

itself

properties:

x for 1) g(O, x) any x E M; g(t + -r, x) for 2) g(t, g(T, x)) =

Then, Borel

measure

Borel

set

defined

system

This

is

theorem

t

system

then

R,

it

=

called

is

E M.

the

p(Q)

M and

space

E

sufficient

for

applicable.

is

Also,

section.

invariant

for

phase

all

X

with

y(g(Q, t)) for invariant

an

If

is

ft

a

arbitrary

an

for

measure

the

g.

definition

recurrence

the

E R and

T

dynamical

a

g

on

Q C Mand

dynamical

next

function

t,

any

=

the

call

we

Proposition

goals.

our

For

provides

IV.1.1

particular,

In

definitions

of Borel the

with

the

it

Poincar6

and measures,

sets

of

correctness

our

the

see

definition

of

an

measure.

h(t, -) be a dynamical system with a phase space M. Then, we a trajectory h(t, x), where x E Mis fixed and t runs over the whole real line stable to Poisson R, is positively according if there exists a sequence (Or negatively) to such that oo +oo as n x (resp. tending ftnln=1,2,3.... h(tn, x) h(-t,,, x) x) IV.0.2

Definition

Let

say that

---

in

M as

n

positively

-+

and

In this we

negatively

our

ing dynam;cal

which this are

is

a

dynamical stable

space

We call

to

a

if

Poisson

it

is

of constructed of

KdVE

or

the

nonnegative

invariant

of

trajectories

measures,

correspond-

NLSE, following

Borel

from

measure

y in

the

phase

a

the

metric

oo.

theorem

M, and let

Then,

system.

according

<

by the

to

Poisson.

properties

recurrence

generated

y(M)

recurrence

metric

on

to

--+

according

stable

is

applications

about

theorem. if

-+

trajectory

a

according

attention

recurrence

Poincar6

stable

systems,

Mbounded

space

that

speaking

chapter,

concentrate

Poincar6

We say

oo.

Poisson.

almost

h be

Let y be

all

a

a

dynamical

nonnegative

points

of

system, bounded

M (in

the

invariant

sense

of

space

measure

the

measure

of

for

P)

ONGAUSSIANMEASURESIN HILBERT SPACES

4.1.

This

result

(see,

for

proof

known.

example,

We refer

In this

section,

used

are

review

the

in

for

example Let

algebra

in the

X be

[41].

infinite

an

O,X

A be

conditions

Hilbert

for

its

the

to

further facts

basic

some

of

set

a

[27,85]

subsets

some

and

dimensional

be used

to

which

measures

monographs in infinite-

theory recall

spaces

of Gaussian

measures

briefly

we

and

set

two

of

measure

Here

literature

numerous

properties

readers

theory

general

the

book

following

if the 1.

from

those

We refer

advanced

an

information

only

consider

the

to

in

measures

sections.

next

[32]

paper

All the

mainly

we

readers

[72]).

monograph

the

On Gaussian

4.1

for

well

is

107

spaces.

contained

is

theory.

of this

of X.

the

A

We call

an

satisfied:

are

A;

E

AUBandA\BbelongtoAifA,BEA.

2.

00

We call

algebra

an

Aa

sigma-algebra

nA,

if A=

A for any An E A (n

E

1, 2, 3,

=

n=1

such

that

A,

D

A2

D

(n

=

D

...

An

D

if A is

is known that

It

....

00

any An E A

for

for

that

too,

contained

v(A)

sigma-algebra

nonnegative

+

v(B)

defined

for

A.

on

function

A,

any

If y is

A;

containing

E A.

It

known,

is

n=1

sigma-algebra

unique

a

nAn

A and

E

n=

exists

then

this

Mcontaining

sigma-algebra

is called

A and

the

minimal

A.

containing

A

U An

has

one

A there

algebra

any

in any

sigma-algebra

1, 2, 3, ...)

sigma-algebra,

a

00

defined

v

B E A

algebra

an

satisfying

An B

additive

nonnegative

a

on

A and

0 is called

=

measure

on

V(A

satisfying

additive

an

A such

that

U

B)

measure

Y(An)

0

n An

07

lim n-oo

for

An

any

then

the

the

1, 2, 3, ...)

=

additive

on

A,

which

D

A2

D

...

D

An

D

and

...

n=1

additive

A,

algebra

an

sigma-algebra

minimal

for

countably

called

y is

measure

countably

is

(n

A

E

then

A such

M containing

the

algebra

exists

its

that

this

on

there

A.

If

unique

'U

onto

countably

is

measure

( nAn)

measure

a

extension

0"

additive

sigma-algebra

this

on

the

in

that/,

sense

:--

arbitrary

An

(n0=1U )

E

0

p

i

An

M (n

1, 2,3,...)

=

satisfying

Ai

D

A2

D

y(An)

lim

for

ail

n-oo

n=1

:)

...

An

D

...

and

that

"0

E p(An)

=

for

any

An

E

M (n

=

1, 2, 3, ...)

such

that

Ai

n

Aj

if

n=1

: I.Let

containing it

should

is

not

an

Mbe all

a

and

be remarked

algebra

complete

separable

open

so

closed

that that

in a

metric

space.

Then,

subsets

of Mis

called

the

general

the

of all

open

careful

definition

set

of the

the

Borel

minimal

Borel

and

sigma-algebra

sigma-algebra closed

sigma-algebra

in

subsets

M;

of M

needs

an

CHAPTER4.

108

additional

A

consideration.

sigma-algebra

Borel in the

M,

space

then

for

Borel

a

in the

measure

y(A)

y(K)

sup

=

the

is taken

is taken

supremum

all

ov, r

open

Now we present of the

definition

Proposition M

N be

)

of

subsets

Sketch

a

of the

space

Now

with

real

that

(Cx, a.)

with

the

a

integer,

components >

0 for

called

a

of

in

dynamical

system.

the

any

B

Borel

Gaussian

remind

that

Rn)

of the

closed

in

.

a

this

measure

one

can

V/(-2ir)n

centered

follows,

easily

verify

if

(B)

det

(nondegenerate)

In what

Gaussian a

a

in Mand in

all

Borel

A is not

finite-dimensional

nonnegative

in

n

Rn

C

measure

we

that,

exp

_2

(B-l(x in

measure

-

for

a

n

x n

matrix

a), (x

Rn; obviously

-

a))

w

means

in

Rn

1

w(Rn)

=

We Call

1.

need for

=

of the

properties

some x

(x-1,

...

)Xn)

xixjw(dx)

=

measure

Rn,

E

(B)i,j

=

bij

A

I (Ax, Rn

x)w(dx)

=

Tr

(AB),

w.

Let

a

=

0.

.

matrix

x n

matrix

a

0

space

defined of

form

contained

sigma-algebra.

Borel

definiteness

subsets

N obviously

space

of the

Rn

that

Borel

0.

=

I so

W transforms

positively

The Borel

and W

spaces

Then,

and

sets,

definition

positive

the

Then,

Ar.

W(A)

symmetric

a

metric

correctness

sigma-algebras

Borel

measures

B be

0 from

x

=

the

N.

space

subsets

and

open

separable

the

in view of the

E Rn and

(we

that

particular,

N.

infirnum

in A and the

providing,

M onto

space

o into

all

consider

the

density

PW is

by

contradiction

briefly

we

be

> 1

n

is

the

a

complete

N be

subsets

Mi, containing

But this M 1.

of

Suppose

A E M.

Mtransforming

on

measure

y(0),

K contained

sets

Let Mand Ar denote

Proof.

Let

of

Borel

Minto

space

sigma-algebra

Let

M and

Let

Borel

a

ODA

statement measure

homeomorphism

a

N, respectively. of the

containing

A.

invariant

an

IV.1.1

the

closed

following

the

of

all

over

0

sets

space

inf

=

KCA

where

defined

measure

M. If y is

AE M

set

any

nonnegative

additive

countably

is called

INVARIANT MEASURES

Then,

ONGAUSSIANMEASURESIN HILBERT SPACES

4.1.

109

n

where

(C)

Tr

is the

ci,j

of

trace

J(Ax,

a

x)2w(dx)

C

matrix

x n

n

[Tr (AB)]2

=

(Cij)ij=1,2,...,n.

=

By analogy

(AB)2

+ 2Tr

Rn

for

symmetric

a

Lemma IV.1.2

that

such

AB

=

A.

matrix

x n

n

Let

a

following

The

0 and let

=

A be

result

is in fact

positively

a

defined

taken

[27].

from

symmetric

n x n

matrix

Then,

BA.

w(fx

E R

n

(Ax, x) ! 1})

:

< Tr

(AB)

(AB)

1)

and

w

where

c

defined

(f

x

>

0 is

n

arbitrary

and

(AB) I

Tr

-

pi,

)fLn

...

c-\/-Tr

<

eigenvalues

are

> 1

2

2c-

-

max

Yn

n

of the

positively

symmetric

A.B.

matrix

Proof.

I (Ax, x)

E R

We have

w

(jx

E Rn

J(Ax,

(Ax, x) ! 1}) :5

:

x)w(dx)

Tr

=

(AB).

Rn

By analogy

(I

W

x

[(Ax, x)

E Rn

-

c2Tr

(Ax, x)

(AB) I

Tr

-

(AB)]2 w(dx) (AB) Tr

=

Tr

2C-2

Tr

Rn

Now, finishing on

invariant

directly

related of

system

the

measures

but used

measures

differential

finite-

x(t) (x1(t),---,x.,,(t)) ( oi(x),---, on(x))

=

be the

transforming time

R

=

o(x) h(t, x)

:

corresponding any

t E R and

t, of the above

system

ordinary

(AB) < -

<

2C-2 maxYn-0 n

case,

we

present

differential

in what follows.

a

result

equations

Consider

not

following

the

equations:

0(x),

=

where

[(AB)2] (AB)

dimensional

of autonomous

systems

to Gaussian

ordinary

of the

discussion

for

c-\/Tr

>

Rn

)

function xo

Rn is

a

an

with

unknown

continuously

("dynamical

E Rn into

supplied

Rn is

)

l

the

solution

the

initial

vector-function

differentiable

from R x system") x(t), taken at the data x(O) xo. =

and

map.

Rn into moment

Let

Rn of

Theorem R'

IV.1.3

the

For

R.

into

M(xi,

Let

Borel

Xn)

---I

be

continuously

a

is

for

be invariant

bounded

open

determined

for

Q and

for

and bounded

any t

all

t E

n

the

in

from

that

sense

interval

an

(-T, T),

it

OW(4pi(X)l

all

x

=

(xi,

Proof. QC R

n

Xn)

We here

=

sufficient

=

0

that

and necessary

n E R

be

interval

an

I(t)

[72].

monograph

the

follow

(-T, T)

and let

v(h(t, Q)) v(Q) for any small that h(t, Q) so (-T, T)

is

Oxi

for

from

function

M(x)dx

h(t, x)

function

the

domain

differentiable

in Rn

measure

V(Q)

to

INVARIANT MEASURES

CHAPTER4.

110

I

=

Take

the

from

M(x)dx,

arbitrary

an

formulation

bounded

open

theorem.

Let

of the

set

(-T, T),

t E

h(t,Q)

and t, t +

r

E

(- T, T). Then,

I(t

+

M(x',,...,

-r)

xn)dx'l

...

dxn.

h(t+T,Q)

h(T, h(t, Q)) from h(t, Q) x' O(T,x) (01(7,,x), (x,,...,x',) is and one-to-one + h(t h(t, Q) T, Q)

Clearly, e.

the

=

=

=

I(t

+

f

....

M(7k(T, x))

h(t + On (ri X))

into

map

smooth

-r,

Q)

where

with

the

a(X1

det

is

a

for

diffeomorphism, a

inverse.

X

fixed

the

map

Hence,

dx.

Xn)

T

i.

(IV.

1.

h(t,Q)

Further,

we

M(V, where

have

-r,

x))

=

M(xi

+,r

obviously

(20-1 ) a-r

+

'('r),

-

Xn

+

T

T=O

( aT)T=O ao

( a7kn ) T=O+ aT

(P(X).

Therefore, n

M(X')

=

M(X)

+

T

Wi(X)

axi

+

O(T).

(IV.1.2)

1)

4.1.

ONGAUSSIANMEASURESIN HILBERT SPACES

Also,

one

can

easily

verify

that

( axjaxi )

=

6ij,

6ij

where

I if

=

i

=

j and

otherwise,

0

and

0xi aXj

d d-r

'9 oi (X) '9Xk

Hence, a X'i axj

8ij

=

ax,

k=1

'awi(x)

+

(i, j

axj

k

+

axj

o(r),

n).

i, j

n.

yields

This

(

det

therefore,

'9(x,, a(X1,

(IV.1.1)

due to

-

n

Xn1) -Xn)

+

r

(IV.1.2),

and

n

I(t

+

7-)

M(X)

+

am

7

+

axi

j=1

h(Wr,0)

mawi

Now, of this

we

immediately

presentation,

we

Definition ble

Hilbert

,n

IV.1.4

> 0

implies

infinite-

follow

this

[27].

bounded

linear

a

basis

Gaussian

First,

H.

Ben

Let

=

In several

the

following a

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

Anen,

n

=

places

definition.

in

operator

sequence

a

IV.1.3.11

measures.

need

self-adjoint

possess

in

we

dx

dx.

of Theorem

statement

dimensional

operator

orthonormal

an

the

book

the

Let B be

H and let

space

forming

ments

study

to

turn

axi

i=1

h(t,Q)

relation

I

'9(M(x)W'(x))

1:

1'(t)

o(-r)

+

axi

and thus

The latter

o(-r),

+

aXi

real

separa-

of eigenelewhere

1,2,3,...,

00

for

all

Then,

n.

B is

called

an

of

operator

iff E An

class

trace

<

00-

n=1

Definition a

self-adjoint

IV-1.5

operator

We call

in H.

FCRn such

a

set

Let with

a

real

separable

cylindrical

Hilbert

fen}n=1,2,3.... if

there

and B

space

forming

exist

integer

an n

H

:

-->

H be

orthonormal

> I

and

a

basis

Borel

set

that

=

For

a

eigenelements

MC H

M

cylindrical

H be

fixed subsets

operator of H.

fx

E H:

B from

One

can

[(x, el)H, Definition

easily

verify

...

)

(Xi 6n)HI

IV.1.5 that

we

A is

E

F}.

denote an

(IV.1.3) by A

algebra.

the

set

of all

112

Definition

IV.1.6

orthonormal

an

INVARIANT MEASURES

CHAPTER4.

-

unbounded)

basis

defined

H be

Let

the

Ben

rule

self-adjoint

linear

a

Anen;

where

domain

of

=

Hilbert

separable

real

a

B be

H and

in

by

again

1enJn=1,2,3_. in H (generally

space,

operator

An

0

>

are

(so

numbers

some

be

00

that

1:

x

E H

Xnen

the

belongs

to

call

the

definition

the

of

if

B

if

only

and

n=1 00

AnX2

oo).

<

n

Then,

we

(but

additive

in

general

not

ME A

of

additive!)

countably

n=1

measure

defined

w

on

A by the

algebra

the

for

rule:

the

(IV.

kind

1.

3)

n

n

w(M)

=

0'

(27r)-2

i

21

E \,-I

2

X2

dxl

e

dXn

...

F

the

Gaussian

centered

Proposition

Then,

Hilbert

space

algebra

of all cylindrical

H.

Proof.

Let B be

IV.1.7

suffices

It

to

the

is

sets

of

B.

operator class

trace

sigma-algebra

acting

Min the

in

H

space

separable

real

a

the

containing

sigma-algebra.

Borel

that

prove

correlation

operator

an

minimal

the

the

in H with

measure

arbitrary

an

B,(a)

ball

closed

with

E H and

a

00

r

>

0

belongs

M. This

to

from the

follows

relation

B,

(a)=

nMn where cylindrical n=1

sets

Mn are defined Mn

=

Ix

(x

E H:

principal

The

follows:

as

-

result

el)'

a,

H

+...

theory

of the

(x

+

a, en

-

of Gaussian

)2H < r2},

n

1, 2, 3,

=

-

in Hilbert

measures

El

...

is the

spaces

following. Theorem

countably

IV.1.8

additive

on

Gaussian

centered

The

measure

A if and only

algebra

the

if

B is

w an

from Definition of

operator

trace

IV.1.6

is

class.

00

Proof.

First,

E -n

let

Suppose

+00-

that

the

measure

w

countably

is

addi-

n=1

Consider

tive.

the

first

case

A

SUP

::--:

An

<

oo.

Let

Pn be the

---,

en}

and

let

RnJ

<

orthogonal

projector

n n

in

the

H onto

Consider

subspace

cylindrical

we

the

second

get

w(Mn)

B Rn-O,VrIFn

contradicts

_(O) n

the

x

span

Jel,

1. 2

of the

J11PnX11H 2

G H:

of Lemma IV.1.2

statement

!

=

R,,

=

(PnBPn)

Tr

Ai.

sets

Mn

By

H,,

Therefore,

arbitrary

countable

since

large

additivity

Rn

-

w(Mn) ---

radiuses of the

*

+oo

aVRnJ,

! 1-2Aa-2. as

n

-->

oo,

w.

>

0-

Taking in

w(B Rn-

satisfying measure

I

here

H there III/Rn

n

(0))

a

=

exist 2

20, balls

which

ON GAUSSIANMEASURESIN HILBERT SPACES

4.1.

Let

An

sup

now

and let

+oo

=

0 < p,

:5 Y2

<

113

be

:5 /Jn

...

subsequence

a

of

n

the

IAnjn=1,2,3....

sequence

cylindrical

B

operator

satisfying

lim

pn

+oo

=

j9.}n=1,2,3,....

each

For

integer

consider

> I

n

the

set

Mn where

of the

n-oo

eigenelements

corresponding

with

eigenvalues

of

I(Xi9i)HI

E H:

i

< n,

1,2,...'an}i

=

We have

integer.

0 is

>

an

jX

=

an

ng-I

w(Mn)

(27r)-

=

integer

an

X?

dxl...

2

-n,u

We take

nAan

2

dz

2

e

_nAan

I

0

>

an

L2

(27r)

dXan <_

e

that

large

SO

right-hand

the

side

is smaller

here

2-n-1

than

00

each

for

and that

n

an

+oo

--+

as

n

Then,

oo.

--*

the

on

U Mn

hand

one

H and

=

n=1

w(H)

=

(because

1

H is

cylindrical

a

00

the

other

w( U Mn)

hand

of the

part

E w(Mn)

<

n=1

first

(IV.1.3)

kind

of the

set

with

F

=

R'),

but

on

00

i.

2

e.

get

we

contradiction,

a

and

the

n=1

theorem

proved.

is

00

Now,

E An

let

Let

+oo.

<

us

that

prove

the

measure

is

w

countably

additive

n=1 on

the

there

exists

which

cylindrical

A of

algebra

compact

a

aim,

that

C H such

If,

set

For this

sets.

let

w(M)

first

us

<

for

c

that

prove

for

any

cylindrical

any

>

C

0

Mfor

set

0.

Mn K,

00

bn

Let

be such

> 0

bn

that

--*

+oo

as

n

--+

oo

E bnAn

A

and

<

Take

+00-

n=1

arbitrary

e

cylindrical

and

> 0

R

by

the

inequality

first

Lemma IV.1.2

from

for

kind

Mof the

sets

Then,

0.

>

jX

M

=

E H

:

[(x, el)H,

(X, en)HI

....

F},

E

n

bix?

where

>

R2 for

:5

R21),

any

x

=

(xi, ...,x,,,)

E F

(i.

e.

mnE

=

0 where E

=

Ix

E H:

00

E bn (x, en) H2

w(M)

we

have

for

K, the closure

<

AR-2

and

obviously

the

set

E is

P in H of the

set

E with

be

of

cylindrical

relatively

n=1

compact the

in H.

above

Now,

Taking

compact

A,

let

R>

we

get

set.

A2

D

D

D

...

An

D

...

a

sequence

in

sets

H and

00

let

nAn

0.

Then,

by

known property

the

of Borel

measures

for

any

6

>

0 there

n=1

exist

closed

cylindrical

sets

Cn

C

An (n

2,3,...)

of the

kind

(IV.1.3)

such

that

n

w(An \ Cn)

< c2

-n-2 .

Let

also

Dn

n Ck. Then, Dn

are

closed

k=1 n

w(An \ Dn)

<

w(U (Ak k=1

\ Ck))

6

<

2

cylindrical

sets

and

114

CHAPTER4. also

Let

above-

KL be the 2

let

En

Dn

=

constructed

Clearly,

n Ki. 2

En

compact

corresponding

set

compact

are

any

An

0,

=

nEn

have

we

n=1

a

number

all

n

i.

n-Oo

Proposition be

countably

IV.1.9

additive.

The

0 for all

=

0. But hence,

there

w(An)

Thus,

> no.

n

w(En)

:!

+

centered

Gaussian

w(B,(a))

for

taken

is

statement

0

>

measure

any

from

w

nMnwhereMn=fxEH:

the

>

r

lim n-oo

where

ak

that

such

no > 0

Taking

exists <

for

c

B,(a)

<

> no + 1

n

we

<

r

2}. Therefore,

(k=no+l +00

2

Ak k=no +1

(a, eA;)H.

=

Fix

6

IV.1.6

Again,

]2

+00

w(Mn).

<

0.

[109].

paper

[(x-a,el)H]'+---+[(x-a,en)H

n=1

=

e

from Definition

E H and

a

00

w(B,(a))

and

r;

O,n

=

Then,

of

w(An \ En)

and

n=1

Let the

of this

Proof

E,,

w(An)

lim

e.

that

such

no > 0

> no,

An

C

instead

2

to

00

sincen

addition,

In

n.

H, En

in

sets

00

for

INVARIANT MEASURES

and

i6

2

a2k

<

4

get n

n

w(Mn)

]10' k2

(27r)-2

=

'\-3k 2

i 2

k=1

k

dxl

...

dx,

>

Fn n

n

J1

n-no

C(27r)

2

f

Ak

k=no+l

where

C is

)2+ + (Yn r2 T} nfy

a,

---

---

+ -

positive

a

(Yn an )2

-

E R

n

Further,

valid,

we

'\

2

k

k

dzn,,+,

k=no+l -

dzn,

...

Fn'

ly (yi, ---'yn) E Rn : (y, (Yno +1) iYn) E Rn-no : (Yno+1 ano+1 )2+ < Z:2 } (this is valid because + (Yno ly E Rn : (y, a, )2+ 4 ano )2 < r2 < a F a + + C (Yno+l )2 )2 (Yn } n)no+1 4

)2

an

independent

constant

<

r2

I

Fn'

and

Y

of n, Fn

=

=

=

-

...

-

-

n

=

-

-

according

since

fZ is

E

(Zno+l)

...

choice

the

to

)

Zn

)

:

-

embedding

of no the

zn2,,+l

+

+

...

zn2

:5

r

16

1

C

F,

n

get: n

n

w(Mn):' -

C(27r)-

n-no

2

Ak

2

k=no +1

e

Z2

n0+1

Now we

above

use

the

inequality

first we

inequality

+...+Z2

n

< -

from Lemma IV.1.2.

E

!

C(I

6 r

k

dZno+l

...

dz,,.

Y6

According

+00

EAk) A;=no+l

k

r2

have:

W(Mn)

A-1. 2

k=no+l

>

C1,

to

this

result

and the

ON GAUSSIANMEASURESIN HILBERT SPACES

4.1.

C,

where

Let

positive

is

a

w

be the

Let

subsets

of the

F C R

n

Gaussian

(finite-

measures

and fix

take

us

Mn is

integer

(IV.1.3)

with

sigma-algebra

a

fixed

our

the

as

M" of all

set

arbitrary

and

n

a

jw"}"=1,2,3....

and consider

> 0

n

We define

IV.1.6.

measures)

Gaussian

dimensional

kind

Definition

from

measure

arbitrary

an

H of the

space

.Clearly,

0

constant.

centered

of Borel

sequence

follows.

115

Borel

sets

setting

and,

n

n

wn(M)

(27r)-'T

=

][I

n

L

Ai

2

i=1

obviously

we

Mn.

get

procedure

this

Repeating

Gaussian

dimensional Now

we

sigma-algebra

Borel

Hn

that

subset

I el,

span

=

each

Min .

-

-)

A n Hn

sigma-algebra Borel

M1 is

sigma-algebra M, obviously

subsets

because

by

all

and

open

Now,

separable

closed

let

the

and

only

SO,

(n

=

open

1, 27 3,...)

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

C

:

and closed

be

weakly

real-valued

arbitrary

Lemma IV.1.10

the

Then,

c

--+

>

as

Borel

with

get

measures

Borel =

easily

can

we

the

1,

the

to

is

a

M,

set

M. But

then,

contradiction

containing

in H.

in

complete

a

We recall

1, 2,3,....

=

that

that

a

measures n

Borel

verify

sigma-algebra

converging

I W(x)vn(dx)

measure

v

in

Mif

W(x)v(dx)

M

n

H,

of

minimal

vn(M)

=

one

a

M}

A E

some

Consider

coinciding

subsets

nonnegative

v(M)

that

Then,

whole

A E Msuch

exists

supposition.

is the

A n Hn is

show that

A n Hn for

the

onto

if

lim

an

of finite-

A E M(we recall

Hn),

n

there

in Mand not

is called

n-oo

for

=

the

be considered

Wn can

Msuch

space

sequence

Then,

sigma-algebra

Borel

sets.

v, vn

metric

that

the

definition

sigma-algebra

sequence

our

extended

wn(A to

A n Hn E M1. n

that

all

=

suffices

it

Mn by

n

contains

get

we

the

on

naturally

wn(A)

C H

Mn7 M1 :

in H contained

a

0,

>

be

opposite.

fC

=

A of H such

of all

.3ince

the

n

and M1 C n

dx,,,

...

defined

wn

n

can

this,

To prove

M'

Clearly,

Mn.

Wn

rule:

the

by

H

Suppose

of Hn if A E M.

integer

measure

1).

en

dxj

jWn}n=1,2,3,.-

measures

show that

e

measure

all

for

1

2

F

additive

countably

a

Y>'-1.?

-1

Let

of

sequence

M

bounded

the

functional

continuous

measure

w

from Definition

jWn} weakly

measures

0 in M.

be

IV.1.6

converges

to

the

countably

measure

additive. w

in

H

as

oo.

Proof.

First

0 there

exists

of a

all,

one

compact

can

set

prove

If,

as

in the

C H such

proof

that

of Theorem

w(If,)

> I

-

e

IV.1.8 and

that

1,Vn(K,)

for > 1

any -

c

CLIAPTER 4.

116

integer

all

for

functional

let

Further,

0.

>

n

W defined

take

us

j

lim

W(x)w(dx).

W(x)w,,(dx)

H

Take also

arbitrary

an

>

E

one

easily

can

that

verify

there

6

exists

=

8(e)

>

0

that

such

IWW W(Y) I any

ifn

Let

=

if,

H,,,

n

Ix

satisfying

and y E H

If,

E

x

n

f

p(x)w,(dx)

so

that

f

-

I W(x) 1,

M= sup

<

6. for

integer

any

o(x)wn(dx)

> 0

n

<eM,

Kn

H

where

IH

Obviously,

1, 2,3,....

=

y

-

(IV.1.5)

<

-

for

(IV.1.4)

H

Then,

0.

bounded

continuous

that

in H and prove

n-oo

real-valued

arbitrary

an

INVARIANT MEASURES

we

have

XEH

I W(X)Wn(dx)

liminf n-oo

W(x)wn(dx) Kn

H

Let of

c

us

now

> 0, such

show the

of C > 0,

existence

I

n-oo

W(x)w(dx)

in view of the

(IV. 1.4),

i.

Y E H:

Y

Then,

K,

C

the

e.

=

Kn,

arbitrariness

of

statement

of the

E

Hn7

sufficiently

all

I all

the

by

w,,

Hilbert the

large

sufficiently

product

rule:

0

I w

on

<

CC-

n

Hn,

space

for

a

W and

independent

n.

I

Y2

E

(IV.1-6)

0, relations

Hn,

I

K.,,,,

Let

IY21H

numbers

-

which

Further, U)n in

measure

Ix

(IV.

1.

7)

<

and

(IV.1.7)

together

=

8(c),2

dist

(y,,K,,)

<

2

Hence,

n.

W(x)w(dx)

<

Me

K,,,c

cylindrical M=

W(X)Wn(dX)

lemma.

large

W(x)w(dx)

numbers

of the

>

c

H

for

only

K,,

Y1 + Y27 yi

for

I

-

H

yield

depending

that

liminf

Clearly,

(IV.1-6)

<Me.

is the

it is clear

Hn and the

orthogonal

that

the

Gaussian

complement

HnL

:

[(Xi en+I)Hi

...

)

(X) 6n+k)H]

direct

w

measure

w'n defined

in H of the

set

E

is the

measure

E

F},

subspace

in

Hn,

ON GAUSSIANMEASURESIN HILBERT SPACES

4.1.

where

R' is

F C

Borel

a

wj-(M)

set, n+k

n+k

(2-7r)-t

=

n

11

f

A' z

i=n+l

Here,

Wn(Hn)

particular,

in

Kn, njx

E H:

I

X

Y E

Hnj.

...

dXn+k-

I

Wn(dXn)

Hn let

E

Xn

K-L, (Xn)

also

n

f:

by (IV.1.5)

Then,

f

XnEKn,

Kn,c

dx,,+,

For

1.

=

n

W(x)w(dx)

\-IX2

+1

=n

e

F

w'(Hn)

=

+ Y,

Xn

=

117

W(Xn + Xn_L)W_L(dxj_)

n

n

X-LC=K-L, n, n

a(n, c)

f

+

V(Xn)Wn(dXn)i

Kn,c

I a (n, c)

where

Now, on

(D(x)

let

bounded

C2 c for

Y.

where

be

Q is

an

arbitrary

the

and

Borel

subset

n.

functional

continuous

p(Q)

in H bounded

4D(x)w(dx),

of H.

Yn(Q) ! p(Q) for

liminf

and

c

quantities

4D(X)Wn(dx) bounded

Lemma IV.1.11

of

nonnegative

Consider

(Q)

independent

> 0

real-valued

a

of H.

subsets

C2

some

arbitrary

an

open

bounded

set

Q C H.

n-oo

(K)

liM SUP [In

:5

p(K) for

an

arbitrary

closed

bounded

K C H.

set

n-oo

Proof us

prove,

Fix

an

satisfying

example, >

0 and

c

following

the

0,(u)

0 <

2.

0,(u) 0, (u)

Here the

arbitrary

1.

3.

Phen,

is standard.

for

according

=

=

0 for

to

statement

take

a

u

for

all

any

u

u

(the

real-valued

E

IV.1.10,

'b_00

=

0,ju)

satisfying

the

be

0,(u)

functional

f

0, (u)

(u, aQ) we

>

the

[109]. Let by analogy).

paper

proved continuous

in

H and

c.

have:

0,(u)y.,,(du)

over

above

a

0e(U)(D(U)Wn(dU)

lim

=

n-oo

4D (u)

(du)

w

=

Q

the limit

from can

X;

dist

lim

>

n-oo

here

second

Q;

E Q and

Lemma

lim inf Yn (Q)

Taking

taken

variant

properties:

< I

1 if

its

present

we

first

fV),(u)dy(u). 0

sequence

properties

c,

-*

with

+0 and c

=

E,

an

arbitrary

we

get

sequence

liminf n-oo

tln( I)

of functionals

!

P(Q)-O

CHAPTER4.

118

An invariant

4.2 In this

section,

[107].

paper

A)

L2 (0,

construct

of two

invariant

an

space

L2(0, A)

samples

of the

0, the

A>

Let

L2 (0, A)

shall

we

for

measure

INVARIANT MEASURES

the

measure

be real

NLSE for

the

let

X be the

and

NLSE;

follow

direct

L2 (0, A) equipped

space

we

with

the

product the

scalar

product

(7-VI W2) X where

(ui,vi),

wi

as

a

Let

us

of

system

consider

the

equations

for

*1+ t

U

*2t

UIX

2

+

X

_

_

X

i

following real

i

the

for

imaginary

corresponding

of the

parts

jjujjx

norm

NLSE written

real

in the

unknown

form

function:

f(X, (UI)2

+

(U2)2)U2

=

0,

xE(O,A),

tER,

(IV.2.1)

f(X, (Ul)2

+

(U2)2)Ul

=

0,

xE(O,A),

tER,

(IV.2.2)

=

u'(x, the

(V1 V2) L2 (0,A)

and the

1,2,

=:

problem

and

u'(0, t)

Formally

+

I

ui,viEL2(0,A),

'I

(u, U)2X.

(Ul U2) L2 (0,A)

:--

i

u'(A, t)

to)

(IV.2.1)-(IV.2.4)

problem

u'o

=

=

E

0,

(IV.2.3)

L2 (0, A).

equivalent

is

R,

t E

(IV.2.4) the

to

integral

equation

t

u(t)

A(t

=

-

to)uo

+

j

B(t

-

JU(s)j2)u(s)]ds,

s)[f(.'

(IV.2-5)

to

(to

(Ul (t), U2(t)) as X), interpreted

u(t)

where be

=

A(t) -

In this

section,

(f)

Let

possessing

a

is the Uo

cos(tD) sin(tD)

our

f (x, s) be

a

on

real-valued

partial

continuous

all

(x,.5)

E

[0, A]

One may define

H'-solutions

of the

x

and

the

values

in

f

f,,(x, +

If.,'(x,

is the

a

functional

space

s) s) I

and let

<

-

cos(tD) sin(tD)

following.

function

continuous

5) 1

sin(tD) cos(tD)

B(t)

function

derivative

if(x, for

with

(Ul,U2), 0 0

sin(tD) cos(tD)

hypothesis

function

unknown

=

of (x, s) there

exist

E

[0, A] C

>

x

[0, +oo)

0 such

that

C

[0, +c*). X-solutions

NLSE from

of the

Definition

problem 1.2.2.

(IV.2.1)-(IV.2.4)

by analogy

with

FOR THE NLSE ANINVARIANT MEASURE

4.2.

C(I; X)nC'(I; tion (IV.2.5),

H

H

Proof

(a) for (IV.2.1)-(IV.2.4);

the

at

h((ul,

0

U2), t) 0 (ul (-,

of

time

equa(IV. 1)- (IV. 2.4) satisfies is (IV.2.5) C(I;X) of equation -2 H the class x n (0, A) of C(I; X) C'(I; T1, T2 > 0).

problem

exists

be the

global

to))

t +

the

Then,

d

I JU(., t) 112X

dt

let

w

D`

0

0

D-1

of

u(., t) of

X-solution

any

the

function above

X-solution

dynamical

a

)

for

0

=

be the

arbitrary

an

Gaussian

centered

the

in

with

measure

S is

Since

X.

space

additive.

countably

is

t E R

,

problem h is

problem

the

(u' 0 U20 ) E X and (IV.2.1)-(IV.2.4)

pair

taken

with

system

the

of

f F(x, (UI)2

(D(u',u2)

also

Let

S

operator

class,

trace

+

(IV.2.1)-

problem

the

correlation

the

operator

an

of

A

w

1.2.313

Proposition

section.

transforming

function

to.

of

(f)

unique

a

to), u2(.,

t +

t +

of this

hypothesis

the

E

proof

the

result

main

E

2.

X;

space

(C) (IV-2-4); (d)

the

any uo E X there

let

with

by analogy

u(-, t)

solution

arbitrary

an

u(-,t)

1)- (IV. 2.4) + T2] where

2.

T1, to

-

Under

IV.2.2

moment

phase

[to

present

can

X-solution

the

into

=

be made

Theorem

(b)

I

can

(0, A)) of

the

any solution

(IV.

problem

the

Now we

x

(f)

hypothesis

the

H-2

conversely,

and

(0, A)) (here

-2

(0, A)

-2

of

solution

a

Under

IV.2.1

Proposition

119

the

(U2)2 )dx

measure

where

F(s)

in the

space

0

2

f f (x, p)dp (the functional

4

obviously

is

real-valued

and continuous

X

0

bounded

of X).

subsets

bounded

on

e(1(U1'U2) w(dul

Y(Q)

(here

Q is

measure

for

forms ues

the of

T

dynamical

the

Proof

of Theorem

depending

0

(IV.2.5) IA,,, en}n=0,1,2....

has

D and let

L2(0,A)

onto

be

Xn the

=

a

of X)

subset

h with

system

into

itself

only

unique

on

and

eigenvalues

span(eo,...,

en)

and

Let

in X

measure

dU2)

well-defined

in X and

phase

X.

a

space

right-hand

the

I

of

=

the

corresponding

side for

contraction

(here

solution the

Xn.

is

JJuoJJx

local and

subspace

is

the

The map from

JV.2.2.

C(I; X)

space >

equation tor

Borel

arbitrary

an

Borel

the

Then,

[to

-

class

it

is

of

(IV.2.5)

+

T]).

C(I;X).

eigenfunctions

Pn be the orthogonal

projector

Xn & Xn.

Consider

also

Xn

=

trans-

small

sufficiently T,to

invariant

an

val-

Therefore, Further, of the in the

the

let opera-

space

following

120

CHAPTER4.

problem

approximating 1

2

Un

+ Un,xx + Pn[f(X,

t

2

Un

_U

t

U

n

Let

pn

in the

space

Then,

the

the

Pn

0

0

Pn

)

1 n

,

X-P n[f(X,

gi

eigenelements

of the

(U2)2)U2]

n

0,

t E

R,

(IV.2.6)

(Ul)2

+

(U2)2)Ul]

n

0,

t E

R,

(IV.2.7)

n

(eO, 0),

=

is

(0, eO),

=

Clearly,

that

the

Xn is

the

onto

-

-

orthonormal

S.

nU2(X). 0

n

has a unique local solution (IV.2.6)-(IV.2.8) (as it is well known, in a finite-dimensional and

n

projector

92

an

operator

n

U2(X,to)=p

orthogonal

f9n}n=1,2,3,...

system

+

PnUI(X), 0

=

be the

(Ul)2 n

X

(X, to)

Let also

X.

(IV.2.1)-(IV.2.4):

problem

the

INVARIANT MEASURES

92n+1

7

-

basis

for

linear

=

(en) 0), the

in

X

space

integer

n

with

Xn 92n+2

(U1(X't)'U2(X'

=

any two

space

equipped

subspace

positive

any

un(x, t)

(IV.2.8) Xn (D Xn

=

(0, en)

n, the

n

norms

7

consisting

....

of

problem

t))

E C (I;

are

equivalent,

Xn)

X). In for 0 these solutions. 112 t) X for any n and for any uo Therefore, X the problem 1, 2, 3, (Ul0 (.), U20 has a unique global solution (IV.2.6)-(IV.2.8) Un(*) t) E C(R; Xn). it is clear the above solutions that Further, Un(*,t) of the problem (IV.2.6)the equations satisfy (IV.2.8) 1, 2, 3, (n we

mean

addition,

direct

the

space

shows

verification =

the

-dt-JlUn(*,

that

of the

norm

space

=

...

=

Un(', t)

=

A(t

_

to)pnUo

f

+

t

B(t

-

S)pn V(.' JUn(*, S)12 )Un(*) s)]ds.

(IV.2.9)

to

Hence,

u(., t)

(IV.2.5)

from

(IV.2.9)

and

one

has for

those

of t for

values

which

the

solution

exists:

JU(', t)

Un

-

(', t) I IX

:5

C1JJUo

pnUoJJX

_

+

I

C2

t

I JUn(*, S)

-

u(., s) I lxds+

to t

J

+C3

I JU(., 3)

pnU(.,

_

s) I Jxds.

(IV.2:10)

to

the

Here the

constants

solution

right-hand respect

u(-, t) side

to t E

C1, C2, C3 do

of this

[to,

for

exist

to +

t E

[to,

to +

T]

obviously

inequality

TI,

depend

not

therefore

we

on

where tends

get from

the

initial

value

T > 0.

Then,

to

as

zero

n

the -4

(IV.2.10)

inequality

to and t.

uo,

third

+00

term

in the

uniformly

with

by the Gronwell's

lemma that

lim n-oo

By analogy,

if the

max

tE[to,to+Tl

u(-, t)

solution lim n-oo

exists

max

tE[to-T,to]

JJU(',t)-Un(',t)JJX=Oon

a

segment

JJU(',t)-Un(',t)JJX=O-

[to

-

T, to],

Let

T >

0, then

ANINVARIANT MEASURE FOR THE NLSE

4.2.

121

Hence, lim for

all

fact

I

segments

implies,

tEI

of

the

it is easy

then

for

for

verify

to

any fixed

T2]

+

(c)

that,

if

u(-, t).

solution

and, hence,

IV.2.2

This

global

the

X.

u(., t)

function

a

of the

existence

function

t E R this

(IV.2.11)

0

=

of Theorem

uo E

any

t) I Ix

u,,(.,

-

of the

statement

(IV.2-5)

equation

Further,

(IV-2.5),

T1, to

-

particular,

in

solvability

[to

=

I Ju(-, t)

max

n-oo

is

G

C(R; X) of the

solution

a

satisfies

equation

following

equation:

T

u(.,,r)

A(T

=

t)u(-,

-

+IB(,r

t)

_

S) [f (.' I U(.' 5) 12) U(.' s)]ds,

R,

E

r

t

which, any

the

as

fixed

earlier,

for

the

map uo

t

transformation

(t

uo

fixed

any

has

t

u(., t) u(., t) as

is

--+

--+

Therefore,

X-solution.

from

map from

a

global

unique

a

one-to-one

X.

X into

X follows

X into

for

The

continuity

from

the

of

estimate

to)

>

t

U

('i t)

Vt)

-

X

C1

!

U

('; to)

V

-

(*) to)

X

+

C2

)rI

lu(.,,.s)

v(-, s) I Jxds,

-

to

u(., t)

where estimate

v(., t)

and

for t

<

Lemma IV.2.3

For

arbitrary

two

are

Thus,

to.

any

and T

> 0

c

all

numbers

n

=

T,to+T]

1, 2, 3,

(IV.2.6)-(IV.2.8),

problem

for

and

...

taken

I JUn(*, to)

(here

u,,(.,

to)

Proof

=

pnuo

follows

v,,(.,

and

from the

to)

=

estimate

-

same

(c)

of Theorem

8

exists

value

>

and

IV.2.2

0 such

a

similar

proved.

are

that

<

solutions

Vn(*) to) I IX

-

(IV.2.5),

equation

Vn(*l t) I IX

two

any

the

with

of and

0 there

>

I JUn(i t)

max

tE[to

for

solutions

(a),(b)

the statements

u,,

(., t)

satisfying

n,

t) of

and Vn (*) the

the

condition

<

pnVo). (t > to) t

I jUn(',

t)

-

V.(-, t) I IX

!5 C1

I JU.(',

to)

-

Vn(*i to) I IX

+

C2II

JUn(', 8)

-

Vn(-, s) I Jxds,

to

that

results

from

By hn(UO) t)

Un(')

t +

to)

where

equation we

(IV.2.9),

denote

Un('i t)

the

is the

and

function solution

an

analogous mapping

of the

for

estimate

any

problem

uo

E

t <

X and

(IV.2.6)-(IV.2.8).

to.0

t

E

R into

It

is clear

CHAPTER4.

122

that n

the

the

let

..

.

correlation

According

.

Borel

measures

Also,

Xn the

S

S*

=

Section

from

f2 C X is

arbitrary

an

Lemma IV.2.4

phase

space

Yn is

is

wn

can

functional

continuous

a

in

as

Xn'

e(1'(u)dWn(U) set)

Borel

defined.

well

are

invariant

an

Let

the

rewrite

us

n

for the dynamical

measure

hn with the

system

(W.2.6)-(W.2.8)

system

the

for

coefficients

bi where

aj,

n

E ai(t)ej

and U2n

E bi(t)ej.

i=O

Then,

ai(to)

(Un1(*7tO)iei)L2(0,A)i

=

(t)

(ao (t),

=

an

...'

VbEn(a, b),

=

(t)

a

get

we

i=O

a(t)

where

U2)

well-defined

be considered

Xn.

Proof.

uln

measures

Wn with

measure

Wn is

measure

each

For

.

measures

Yn(n) (where

4.1,

(D (Ul,

(D(u)

since

the

Xn'

Xn

space

Gaussian

centered

in

> 0

phase

the

with

system

space

Since

result

the

to

Borel

in the

S.

in X.

following

dynamical

a

consider

us

operator

in Xn

the

hn is

function

1, 2, 3,

=

INVARIANT MEASURES

-VaEn(a,

=

bi(to)

(t)),

b(t)

b),

(IV.2.13)

(U2 (-,to),ej)L2(0,A)i

=

(i==1,2,...,n),

n

(bo (t),

=

(IV.2.12)

bn (t)),

...'

un

(IV.2.14) b)

(Ul,

=

U2n ) and En (a,

n

A

f I! [(UI,X)2

E(Un)

2

(U2,J2]

+

n

F(x, (Ul)2

-

n

+

n

(U2)2))

Then,

dx.

n

according

The-

to

0

the

IV.1.3,

orem

dynamical

(W.2.12)-(W.2.14)

system

with

system possesses

Borel

a

phase

the

invariant

n

y' (A)

ii

-(n+l)

(27r)

=

n

A,

'=O

A C R2(n+l)

where

IV.1.1,

there

space

R2(n+l)

is

and

A C R2(n+l)

is

an

natural

a

Borel

arbitrary

Borel

Q of the

subsets to

Borel

a

e

set

ji':

measure

En(a,b)

n

db,

da

Further,

set.

space

according

between Xn defined

0 C Xn if

an

by

A of the

subsets

the

element

Proposition

to

Borel

rule:

to

9 when and

only

when

(a, b)

E A where

ul

Un

(Ul,

=

(ao,

...,a

correspond measure

n);

,

to

it,.

b

each

These

=

(bo,...,

other

bn).

In

in this

arguments

addition,

sense,

easily

then

aiej,

imply

if two sets

jUn(Q) the

=

A C

y' (A) by

statement

n

set

U2 ) ben

n

n

U2

E biei

and

i=O

'=0

a

Borel

a

n

longs

by the

generated

A

correspondence

one-to-one

corresponds

f

R2(n+l)

space

R2(n+l) the

and

definition

of Lemma IV.2.4.0

1

C X"

of the

ANINVARIANT MEASURE FOR THE NLSE

4.2.

According the

to

verges

measure

w as

n

Lemma IV.2.5

p(Q)

Proof.

arbitrary

the

IV.1.10,

Lemma

to

of Borel

sequence

measures

weakly

Wn

con-

oo.

--*

t)) for

tt(h(fl,

=

123

any

bounded

open

Q C X and

set

for

any

t E R.

Fix

according

Then,

set.

is open

4) is bounded

4.1, the

proved

too,

and

Q,

c

=

For any A C

Lemma

ly

IV.2.3,

for

B6(x)

x

and for

n

K,

of the

x).

y

-

B6,(xi)

be

the

since

Further,

< e.

h(K, t)

=

h(Q, t)

functional

from

of results

obviously,

Section

according

is

>

a

that

for

to

compact

a

set,

11hn(u,t)

covering

of the

According

0.

any

has

one

finite

a

Then,

aQ,)}

Then,

1,2,3,...

=

IV.2.1,

A and let

set

6 > 0 such

exists

any

B6, (xi),...,

Let

IV.2.2,

a.Q); dist(Ki,

yEB

E K there

x

61

<

C Q.

inf EA,

any

X1 Ily-xllx

E

and

B)

y(Q \ K)

that

boundary

minf dist(K,

0.

>

bounded

open

of Theorem

X and in view

space

of Theorem

aA be the

X, let

dist(A,

again

of the

e

arbitrary

an

(a),(b),(c)

arbitrary

an

K C Q such

(a) and (b) h(f2, t).

a

(where

fix

us

set

Q C X be

statements

subsets

compact

a

statements

K,

Let

bounded

on

exists

proved

to(...

let

E R and

t

the

to

bounded,

and

there

an

u,

E

v

to

B6(x)

-h,,,(v,t)llx compact

0'

<

K

set

3

by

I

balls

these

and

let

U B6,(xi).

B

by (IV.2.11)

Then,

and

by construction,

we

get

i=1

the

existence

for

all

(Q)

a

number

Further,

> no.

n

y

of

:5 p (B) +

inf yn

< lim

n-oo

(because Hence,

p(Q)

yn(B) due

>

=

to

p(Qj).

open

an

Remark be

unbounded,

the

lim

=

R

arbitrary

proximation

p(Q)

Xn) =

proved

(B)

+

E

=

yn(hn(B of

tt(Qj),

c

n

Xn' t))

0,

>

(hn (B, t))

(c)

u

E B and

we

+

c

(f2j)

< y

+

c

hn(B n Xn' t) C hn (B, t)). By analogy ft(Q) < jz(,Qj).

and

have

and Lemma IV.2.5

statement

2

all

and IV.2.4

lim inf yn

<

,

for

c'

is

of Theorem

proved.0

IV.2.2,

we

have

for

an

arbi-

Q C X:

set

p(Q) For

to

n

(hn (u, t); X21)

n-00

arbitrariness

Thus,

According trary

(B

/In

the

dist

get by Lemmas IV.1.11

we

c

that

0 such

no >

+oo

y(BR(0)

Borel

of the

set

e.

Q)

A by open sets

it

=

may

the

happen

y(BR(0)

lim R-+oo

A C X we obtain

Generally,

IV.2.6 i.

set

n

the

containing invariant that

Thus,

measure =

h(Q, t)) IL(A)

equality A.

I.L(X)

n

p

+oo.

=

=

y(h(Q, t)).

jz(h(A,

Theorem

IV.2.2

t)) by is

the

proved.

given by Theorem IV.2.2 However,

according

to

ap0

can

the

CHAPTER4.

124

(c)

statement

dynamical a

ball,

that

get

we

according

stable

Remark

physical

two

second

all

and the

Poisson

to

IV.2.7

+oo for

<

of the

points

space

(in

of all

these

(f)

hypothesis

The

f (x, s)

nonlinearities:

a

X

the In this

section,

spatial

consider

we

variable

for

corollary,

a

w)

measure

in the

space

for

example,

valid,

is

where

e-"

=

invariant

of

following

the

for the

0 in

>

a

are

X.

uo(x

where

We shall

A)

uo(X),

=

construct

laws

+

E,

>

n

x

A, t)

the

norm

R,

E

infinite

an

f (u)

and A

space

X, t E

Uo

=

1.1.5

for

(IV.

Hper(A) by

3.

1)

(IV.3.2)

R,

Here

we

follow

the

associated

with

dynamical

systems

generated

Throughout simplicity.

for

this

conserva-

by the

section

Our first

[111].

paper

measures

spaces.

n

to

respect

(IV-3-3)

is fixed.

> 0

phase

suitable

with

(X),

of invariant

series

on

in the

(X, to)

periodic

u:

==

x,tER,

u(x, t),

=

by Theorem

3, given

(IV.3.1)-(IV.3.3)

problem denote

+

for

measures

Cauchy problem

KdVE with

standard

the

U

shall

we

related

result

following.

is the

Theorem the

Then, rule:

IV.3.1

function

(Uo' t)

hn

(IV.3.1)-(1V.3.3)

Hpn,r(A).

The

n

correlation

Let hn

=

integer

from

U(.,

given

t +

> 2

n

>

2, A > 0, T

Hpner(A) to),

u(., t)

where

Eo,

...'

E,,,-,

>

Hpn ,JA)

into

by Theorem L1.5,

functionals

dynamical Let

the

IV.2.2

of

view

in

As

0.

>

of such

any

KdVE

u(x

this

on

of the

is dense

points

ut+uux+uxxx=O,

the

r

sense

f (x, s)

and

1+s

series

An infinite

tion

the

of Theorem "

X and

c

phase

new

a

By analogy,

R > 0.

of the

set

for

bounded

4D is

any

any

set

balls

case.

4.3

the

0 <

almost

for

invariant

an

any of these

functional

the +oo

<

R > 0 is

with

choose

can

Since

h.

jt(BR(0)) jL(B,(a))

0 <

IV.1.9, that

obtain

we

system

BR(O)

ball

any

Therefore,

dynamical

Proposition we

h.

system

of the

space

theorem,

of this

INVARIANT MEASURES

is

from

is a

t E R and to E R be

0,

defined the

Hpner(A)

dynamical

Theorem

for

uo

-solution

system

L 1. 5

any

are

E

of the

with

arbitrary.

Hpner(A) the

phase

space

laws

conservation

by

problem

for

system. be

integer

operator

and Wn be the

S

=

(I

+

Gaussian

centered

Dn-1)(Dn

+

I)-1.

Since

measure

clearly

in

S is

Hpn,-r'(A) an

operator

with of

4.3.

ANINFINITE

trace

class

SERIES OF INVARIANT MEASURESFOR THE KDVE 125

Hp',,-,'(A),

in

the

countably

w' is

measure

Let also

additive.

for

Hpn,,,(A)

E

u

A

Jn (u)

En (u)

=

1 f[U(n)]2

1 -

U2 }dx

+

x

2

I

En(u)

=

(S_1U) U)HP1_(A)

-

2

=

0

A

JJCnU [U(n-1)]2 x

2

u

2

(n-2))Idx.

qn(Ui

_

U

x

0

For

arbitrary

an

Borel

Hpne-r'(A)

QC

set

we

e-Jn(u)dWn(U)'

,p(Q) where

QC

Hpn,-'(A)

Theorem

IV. 3. 1.

orem

For

Since

due

it

be taken

can

applicable. Wn)

for

So,

Poisson

form

A

where

k

0

=

Ao

of <

Hpn,,,(A) Hpn,-,r(A)

in

following

A,

onto

set

Hpn,-' (A) phase

onto

<

in

Hpn ;' (A).

Hpn,-r'(A)

from IV.3.2.

A2k-1

=

orthogonal

an

A2k

<

I eo,

-

-

7

is

measure

according

the

proof

to

of the

basis

e2m} L'

M

to

Pm' the

of the

Cos

space

corresponding

the

Pm be the and

27rkx

2

+ An2k)

orthonormal

Let

complement

(and

1.1.5

the

stable

points

e2k(X)

D with ....

-

all

Theorem

of

sense

h n-1

system

Recurrence the

The-

1}.

-

dynamical

(in

by

given

wh e re

Let

A

operator

span

=

is

the

Theorem

27rkx sin

n

Poincar

addition,

in

follows

J6mJm=0,1,2....

Lm

of

space

hn-1

< + 00,

0,

Borel

nonnegative

system

(Rd)

=

set

Theorem

of the <

k

the

A _) n2k-1)

...

,n

therefore

2

subspace the

:5 d,

invariant

-

Theii.,

A2

0 <

and,

Poisson

proving

to

dynamical

0

>

following.

is the

section

well-defined

a

for

the

to

eigenfunctions the

of

is

,n

d

an

space,

points

e2k-l(X)

=

Rd is

of this

the

Ek(u)

:

immediately

IV-3-1

1, 2,3,....

consisting

all

dense

a

Now we turn

eo(x)

new

according

Theorem

latter).

a

large

result

> 3

n

measure

IV.3.1

almost

stable

are

G

U

Theorem

to

integer

any

sufficiently

any

Rd

The main

set.

invariant

an

For

Borel

a

IV.3.2

and it is

measure

is

.r

A

Hpn,,(A)

eigenvalues

orthogonal

projector

be the

orthogonal

subspace

LIn.

proj

Consider

in ector

the

problem: u't

+

Pm [U'U'l

x

+ Um xx x

=

0,

x

E

(0, A),

t E

R,

(W-3.4)

CHAPTER4.

126

(X, to)

U,

Clearly,

for

C'Qto of

any

T, to

-

Hpn,,(A)).

Since

HP",(A)

it

for

T > 0

E

uo

T]; L,,,)

+

it

some

be

can

has

easily

is

all

t, and,

(it

global

known fact In

hence, that

in

this

k

as

--*

that

mk

oo.

Then

u(.,t)

where

IV.3.3

Let

for

k

as

t

any

Hpn,,r(A)

proving

Lemma IV.3.4

R(n, d)

0, such

>

-+

1julln

umk

space

(IV. 3.4)

this

statement

For

any

that

if

u

E

E

(., t)

we

E

norm

have:

Lmh

two

and uO

=

consider

a

---

--+

let

and with

initial

k

0

Hpn,,r(A),

in

umk(-,t)

and

the

u

Hpn, ,,(A)

in

strongly

(IV.3.1)-(1V.3.3)

mk

sequence

a

strongly

uo oo

the

and t.

x

and

k

k

as

(We apply here norms are equivalent).

in

Hpnr(A)

u(., t)

=

nonnegative

R).

E

any

E

uo

m

(.,to)

t

n any uo E HPe r(A)

for

differentiable

--+

with

Hpn r(A)

line

problem

ofthe

-solution

0,

=

t)

the

is

data

um". 0 number

integer

of lemmas.

d

and

n

>

there

0

R

exists

and

d,

<

En (u)

...'

<

d,

R.

<

Consider

Proof.

linear

"

uo

oo,

E R

Eo (u) then

real

integer,

um'; Before

entire

infinitely

> 2 be

n

of equation

Lmk -solution

the

+oo

--

the

is

obviously

is

by

-dt-Ej(um)

=

u'(x,

solution

generated

(IV.3.4),(IV-3.5)

problem

the

is

1Um(*,tO)jL2(0,A)

=

of the

local

in L,,,

-dt-Eo(u')

that

verified

onto

(IV-3-5)

classical

unique

a

finite-dimensional

a

solution

Proposition be such

solution

be continued

can

addition,

the

(x).

uo

(the topology

1UM(*7t)1L2(0,A) for

P"'

=

INVARIANT MEASURES

the

(n

law

conservation

>

2):

A

2

Eo(u)

I(DnU)2 2

+ En(U)

X

1

+

2

u

2+ cnu(D'-'

X

U)2

-

Dn-2U)

q,,,(u,...'

X

dx >

0

1

211UG)11n where

n

mates

for

Ej(u)

(s)

is

have

1U1L2(0,A))1'_1

2

p

function

!Eo(u) 2

,

in view where

of the

-

and

continuous

functionals

the

we

a

2

+

?7n(JjU(-)jjn-1)) increasing

!Eo(u) 2

E2(u),...'

known

on

inequality

+

[0, +00). En-1 (u).

lUlLp(o,A)

Repeat For

1+1

'EO(U)

2

2

IJUI12 1

-

?71(1U1L2(0,A))(jJUJj12*

+

I)-

esti-

functional

JU12L2 (0,A) (jDxujL2(0,A) P

p > 2:

El (u) +

the

these

+

ANINFINITE

4.3.

We get

SERIES OF INVARIANT MEASURESFOR THE KDVE 127

by step

step

from

the

obtained

I I u 111 for

all

There

Lemma IV.3.5

(R, s)

on

[0, +oo)

E

satisfying

0

(d),...,

:5 C,

and Lemma IV.3.4

R,

t E

estimates:

proved.

is

n

functions

exist

[0, +oo),

C,,, (d)

u

(R, s),

-y,,

such

that

(R, 0)

-y,,,

nondecreasing

monotonically

in

=-

and

8

defined

0,

continuous

following:

the

d

Tt En(umk(*7t)),<

(R,

< -y,,,

for

all

t

1JU011n-1

E <

Proof. of LUmk in

get

all

R, and all Let

Lg

all

3,4,5,...,

n

above -gg.,

=

k

all

1, 2,3,...,

=

For

g.,.,,.

-

+

11Pmk(UmkUTk)111) X

E

uo

Hpn,-,,'(A),

for

which

UMk. 0

sequences

Of UMk in the t

place

(see [50]),

zero

k

:5i j
i+,;' '1-2

R,

t)DxjUMk(*it)11L2(0,A)

-L

Pn [Dx Uml'('7

max (,

E

uo

integrand

Hpn,-,'(A),

from

the

under

since

for

expression

substitution

the d dt

En(umk (-, t))

we

have:

we

A

d

Tt En(umll(-,

t

))

=

j

a

I

at

2

(DnUm,, )2

cnumk (Dxn-lUmk)2_

+

X

0

-qn(umk,

+'CnPm-L, n-2

+

E i=O

Dn-2Umk

UmkUMk)(D X

n

+ i=O

dX -k= Pt Ik

n-IUMk)2

+

X

Dn X -2Umk)

aq (umk,..., -

0 (DiUMk)

(U-k

=

U-")

IAl (_j)nDx2nUmkp-Lk

M

(UmkUmk)+ X

0

( -1)'-'2cnD

Um11)Xn-IUMk)

n-1 X

X

-L

7nk(UM'UxMk)+

P-L

A

D'

X

J

P-Lk (UMh UTk) M

X

CnpMk (umkumk)(D n-lUMk)2+ I -L

X

X

0

+2Cn( n-2

)u

n-lUmk)p.Lk

UMkDX

aq,, (umk, ...,D n-2Umk)

a(Di

X

UMxk

M

ID

n-2

E Cn-2 i %

X

n-2-i

D

X

I

um D'+'u

Mk

X

i=O

+

A

p-Lk [D' ( umkumk)] ,,,,

X

dx

X

Cn

=

PM (UmkUMk)(D n-lUmk)2+ J-

X

k

0

+2cnPmL,(um-'Dx n-lUMk) n-2

+

E i=O

ID

n-3

E Cni-2 t

X

Dxn-2-iUmk

D'+1u

Mk

X

i=O

aqn(u"',..., Dn-2Umk) i -PM-L , [DX( UMkUM91 a(DXi UMk X

X

I

dx.

I

+

X

CHAPTER4.

128

These

equalities

imply the

INVARIANT MEASURES

of Lemma IV.3.5.0

statement

T, to + T] there [to and nondecreasing exists a function on #,(s) continuous [0, +00) such that for any the quantities bounded by Pn(IIUOlln) in t E I are uo E Hp',,(A) Ilu-(.,t)ll,, uniformly and m 1, 2, 3,... of the problem (IV. 3.4), (IV. 3.5)). (here u- (-, t) are solutions Lemma IV.3.6

For any

integer

> 2

n

and any segment

I

=

-

=

Proof

max

I IDxnum(.,

0,A)

and

On(s)

>

2).

functions

are

fix

I

IIUM(*7t)112

<

proved.

Lemma IV.3.7

For

the

0 and

>

t) Ill :5 C(R)

all

for

I I UOI I

let

t,

we

IIUM(*it)lln

C2(R),...,

to):

On(IIUM('7tO)IIn-1),

-

nondecreasing

and

continuous

>

En (um (-, to))

-

On(IIUM(*)t)lln-1)

-

R

(t

estimate

n

on

the

half-line

Then

< R.

,

t

aking

by step:

get step

C,,,(R),

:5

0

any

E

uo

Hp', ,,(A),

T > 0 and

integer

n

> 2

the

following

place

11 UMk(.' t) -U(',t)lln-1--->O

max

tE[to-T,to+T]

Proof.

IV.3.1,

I

arbitrary

an

from

1 E,, (um (-, t))

2

I um (-,

is

> 2

to) IL2(0,A)

Um(.'

earlier

as

and Lemma IV.3.6

takes

-

Indeed,

that

account

IDx

n

L

where

into

t)122(

follows

(-, t) I In-1)

u'

an

tEI

an(s) [0, + oo) (n

of Lemma IV.3-5

in view

we

First,

let

Using

H'.

E

uo

as

Lemmas IV.3.4

k--+oo. and

IV.3.6

and

Theorem

get: A

I d

I D'-'(u",

2 dt

(-, t)

X

_

U

(.' t)) 12 dx

=

0

A

21 Dx

'-'

(UMk (.' t)

-

U

(.' t))

n xDX [(Umk

(.' t))2

_

(U(., t))2 ]dx+

0 A

+

2

I

P,, L, [ Dn- (U (.' t))]D'[(umk(.,

t))2 ]dx

1

X

X

<

0

I

C1(IIUOIIn)IIPMk(U(*,t))IIn where

Ilp'Lk(U(*It))Iln

0

as

+ k

--+

oo

C2(IIU(*)tO)IIn)IIUM1'(,t)

for

any

t and

2

_

I IP1mk (U(* t)) I In 7

analogy I d 2

Tt

Umk(.'

t)

_

U

U(., t) 1 In-17

(.' t) 12L2(0,A)

<

C for

all

t.

By

SERIES OF INVARIANT MEASURESFOR THE KDVE 129

ANINFINITE

4.3.

C1(JJU0JJn)JJPm-LA:(U(*)t))JJn

<

+

C2(11 Uoll.)IlUmk(.

1

t)

_

U(.,t)112

n-1,

Hence, t

(.,t)112_1

um

JIUMk(.,to)_U(.,to)112

<

1 11 UMk(."9)_U(.,'S)JJ2

n-I+C3

n-lds+a

..

k

to

where

am,

+0

--+

obviously

also

lemma,

k

as

valid.

For t

Lemma IV.3.7

Lemma IV.3.8

any uo E

uo

estimates

take

Hpner(A)

and

IV.3.6,

JE (UMk (.' t))

(R,

7n

0

umk

k, P-L

liM for

L

all

s

E

n-

suffices

Mk

k-oo

(-, s)

it

P

max,


to

0

x

as

k

-->

oo

by

X


X

X

Di

x

I Tm-,, [D'umk (-, s)

-D'u(-,s)

(D'

X

mk

Di

x

umk (.'

X

respect

to

+

X

and

t, and Lemma IV.3.8

proof

is

llp

Jim k-oo

8))JL-2(0,A)+

s

E

[to, t]

positive

and

(UMk(.'S)UMk(."5))JJ1

<

IV.3.1.

X

Ulk

x

X

Theorem

Wehave:

I Pmj-, [D'

JPmJ-,,[D'u(-,s)

=

X

k

wherei+j: 2n-2.

D.,j Umk(., -5)] 1 L2 (0,A)

Diu(-,-9A1L2(0,A)

in the

with

-5)] 1 L2(0,A)

umk


Lemma IV 3.7

as

<

)111)dsl,

uniformly

-L

Now just

S)

umk

x

oo

any

n(UMk (.' to)) I

(Umkumk.

k1iM PMk (UMk (.' for

0-

=

that

prove

X

and

bounded

are

1

[D'umk (-, s)

[to,t]

Gronwell's

the

J,962-2

+JJPJ-

integer

is

t

to

where

inequality

since

-E

n

I

Hn, ,,(A) the latter place and, due to

En(Umk (', to))]

i

Lemmas IV.3.5

<

E

and any t E R

[En (Um' (' t))-

lim

By

arbitrary

proved.El

k-oo

Proof.

an

to similar

<

is

For

For

oo.

--+

(.' S)

x

Diu X

Dju(-,,5)11L2(0,A)

ml,

(-,S)--->

X

0

By analogy

t)Umk (.' t))

0

X

proved.El of Theorem

IUMk (.' t)

-

1.1.5,

one

U(., t) I In

__

can

0

prove

that

for

any

t

0

CHAPTER4.

130 k

as

Hpn, ,,(A),

if uo C-

oo

--+

Corollary

IV.3.9

[E" (um (-, t

lim

Let

M-00

Proof IV.3.3.

the

repeats

be integer.

> 3

to

such

> 0

Theorem

prove

Proposition

Then,

Lemma IV.3.8

IV.3.10

IV.3.2,

For

any

>

n

for

Hpn,,-,'(A)

any uo E

and

t

any

in

view

of the

need

three

Proposition

proved

any

we

shall

also

2,

uo E

Hp'e,(A),

>

e

statements.

0 and t E R there

exists

that

I 1Um(-' 0 for

proved.0

is

0.

==

of

proof

IV.3.3

D

Below,

8

n

Proposition

(um (., to]

E"

-

and

INVARIANT MEASURES

m

and

1,2,3,...

=

UN., 0 11.

-

arbitrary

an

<

ul(-,t)

Lm-solution

(IV.3.4)

of equation

satisfying

jjUm(*itO) (here

um (x,

t)

there

are

the

is

the

and ul

m

of

solution

Suppose

Proof.

the

is

um,

exist

sequences

mk

uT'

Clearly,

accept

we can

as

k

in

Hpn,,(A)

-->

then

But

oo.

as

k

-+

Proposition

n(s)

function

oo,

umk

Lmk

that

the

all

m

Proof

=

e.

IV.3.11

1, 2,3,...

(.' t)

Let

get

n

ml,

-

t)jjn

for

that

8

any

>

0

be

En (Um (*

7

t)

and

__+

[0, +oo)

:5 77n

Hpn,-,'(A).

Lemmas IV.3.5

with

Then, on

to)) 1

0

==

ul). k

as

uo

and, u

--+

Then,

there

such

oo,

IV.3.6.13

=

also,

uT"

and

that

that

for

IV.3.10

any

such

(I I Um(*, to) I I

t

1)

+oo

--+

--+

U(.,t)

proved.0

E R there

that

n-

is

Mk-

m=

mA;

(-, t) andUMk(.,t)

Proposition

integer.

i

uo

-+

> 6,

contradiction.

> 3

11,,

uo

is unbounded

UMk (. 1

IV.3.3 a

with

Pm,

-

(W.3.4),(W-3.5)

and continuous

and uo E from

UMk(.'

-

sequence

we

nondecreasing

follows

7

by Proposition i.

I I uml'

where

problem

of the

I En (Um (') t)) for

0 such

>

c

(W.3.4),(W.3.5)

problem

of the

and

solution

is the

exists

Jjumj(-,to)-um(-,to)jj,<6

and

UM;: I where

8

Lm satisfying

E

solution

a

there

Then

contrary.

<

(IV. 3.4), (IV. 3.5)).

problem

jJUMj(*it)-Um(*7t)jjn :c (here

UMj(*itO)jjn

-

exists

a

4.3.

ANINFINITE

set.

Then

for

to

(here

K

uo E

We first

mo

such

for

any

all

E

uo

Lemma

B,( U)

IV.3.5,

and

> 0

e

Hpn,-,,'(A)

K c 0

--

as

K there

;U E

be

compact

a

uniformly

oo

m -->

exist

r

with

0 and

>

a

that

I E,,, (u' (-, t)) for

let

En (u' (-, to)) t)) Pnuo).

that

prove

and

integer

-

(.,to)

u-

be

> 3

n

En (u'

any t E R

Proof. number

Let

IV.3.12

Proposition respect

SERIES OF INVARIANT MEASURESFOR THE KDVE 131

ju

E

Hn,,,,-,'(A)

have

for

any

=

we

1ju

:

to)) I

I

U11,,_1

-

0 and

>

r

En (um (*

-

<

rl

<

c

and

all

m

of

In view

> mo.

B,(U):

uo E

t

IPm(D'u'(.,s)xD ,;um(*)-9))IL2(0,A)+

-y,,(R,,)max

En (um (* to)) I ., 5

En (u7n

I

X


to

+jjPmL(um(-,s)um(-,s)jjj)dsj. X

Let

integral

the

estimate

us

JPmL[D'um(-,s) 1 D'

X

um (.,

s)

D1Xum (.,

x

s)

+ I D'Um (-, X

:Um(-,s) (IV.3-1)-(IV.3.3),

I Pm-L [D' ff(-,

respectively, of

right-hand

in the

such c

for

we

all

uo

that

get

first

the

that

E

B,

for

(m, r)

( U)

and all

any

>

c

-y.,,

there

.9

(R,

exist

as c

all

0 <

bounded the and

use, a

r

for

and

< ro

all

for

0

<

example,

number

mo >

r

m

< 1

of the 0 such

> mo,

and

if

m

uo =

E

and

a

ro

>

(., s)

u'

B, ( U).

1, 2,3,...

Egorov theorem)

way the

<

In

by that

x

D'XIum

r

mo

on

than

such

(-, 3)] 1 L2 (0,A)

that +

the

addition, any

c

>

0 (m, r)

function This

0 there

implies exist

r

E

i

-

En (um (,) to)) I

<

c

is

(with (0, ro)

that

En (Um (. t))

0

>

'E

Lemma IV.3.6.

for

exists

Pm (um uxm)

term

number

a

terms

depend

is smaller

inequality

latter

0 and

X

and third

and do not

oo

> 0 and t E R there

+11PM'(UM(-'s)UXM(-'s))111) for

m --+

similar

in

second

the

IV.3.1,

zero

of the

I Pm' [D'

max

0:5"35--l i+3:pd2n-2

UoEB,(U)

to

side

Estimating

m.

0 and

sup

=

right-hand

(IV.3.4),(IV.3.5)

problems

the

+

U.

arbitrary

for

IV.3.10

in the

term

of

+

i

X

tend

inequality

of this

side

uo

=

-9) 1 L2 (0,A)

D1X-9(.,

x

S) I L2 (0,A)

D' Um(.,

x

Di U(-, s)] 1 L2 (0,A)

x

and Theorem

IV.3.3

by Proposition

Further,

uo.

with

Propositon

Dx%-, s)

-

solutions

the

are

(-, s)

D'i!m X

s)

X

and

In view

s)

<

X

-

D1XV(.,

x

+

where

s)

We have:

inequality.

of the

Djum(-,8)1jL2(0,A)

x

X

:5

side

right-hand

the

from

(IV.3.6)

CHAPTER4.

132

all

for

> mo

m

Fix

these

place

takes

if

uo E K if

Hpn,-,'(A)

wi

of

(I

+

Bjuj).

A '-'),

+

point

B, (ul)

-,

where

i

finite

covering

a

any i the

for

(IV.3.6)

jek}

K

ball

a

of the

K

set

(IV.3.6)

relation

is also

the

is

S of the

change

of

valid

for

all

basis

in

above-indicated

kind

(i.

I in the

definition).

by

n

orthonormal

n

-

Then,

0, 1, 2,....

=

set

proved.0

is

operator the

compact

obviously

From here

with

of the

be

that

such

IV.3.12

of the

functions

A' )-'(l

-

.

Then,

3.

>

n

eigenvectors

above-defined =

B, (ul),

Proposition

integer

consisting

also

Let

mi

and

each

to

be numbers

and uo E

arbitrary

the

are

el,

ml,...,

> rni i

an

r.

Let

property.

> maxmi,

m

Fix

e.

m

<

0 and compare

>

f-

and let

balls

V1 1,,_1

-

the above

possessing by

I luo

if

arbitrary

an

INVARIANT MEASURES

eigenvalues

are

wi

of

S. Consider the

subspaces

the

in

dimensional

finite-

Lk

Gaussian

Hpn,-'(A)

C

measures

Wk

Wk(Q)

(27r)

=

2A;+1II

Wi

Q

Then,

W1, is

the

a

measures

sian

I [(Ui eO)n-1)

Lk

Borel Wk

the

IV.1.10

E

u

7

be considered

fWk}k=1,2....

Wn For

Borel

a

7

.

any

the

functional

of this

subsets

J,,

weakly

the

space,

(IV.3.4),

(IV. 3.5)

and any

uo E

taken

tl,

at

Hpn,-,'(A)

L

so

of time

moment

e

I,

into

according

Lemma IV.3.13

lim Proof.

(ZO (t)

7

Let

us

(t))

where

Z2m

and the

h,,, (.,

rewrite u-

(x, t)

also

set

HPn,;1(A),

sets.

Section

by

and

4.1,

Lemma

infinite-dimensional

phase

t) (-,

Gaus-

rule

any

t +

to)

h,,, (u, t)

-

t).

i(t)

a

=

(IV.3.4),(IV.3.5) + Z2m (t) zo (t) eo (x) + =

JV-,H(z(t)),

-

-

u,

fixed

bounded Let

the

system

L..

into

L..

(IV. 3.4), (IV.3.5) t transforms

bounded

closed

(Q))

it,,,

-

problem

h,,, (P.,,

system =

of the

a

be

by

t transforms

h,,, (., t) for =

W2'-'

fixed

on

HPn,,,1(A).

in

generated

L,,,

space

for

and bounded

well-defined

are

_

Obviously

(p,,, (h, (Q, t))

the

/,n

to.

Let t E R and Q C

M-00

the

Borel

are

from

results

to

in

Hpne-,' (A)

in

u-

the

to

we

solution

t +

&2ki

...

F C R2k+1

and

to

converges

ltk

with

function the

maps into

Fj

measures

continuous

system the

that

Idzo

e_j'(u)dWk(U)-

measures

h, (u, t) be the dynamical

jei}i=o,1,2'...'2k

rule

Z? E W` i

According

k.

Hpn,,(A)

Q C

obviously

is

the

_0

e

E

Borel

Ak (f2)

Since

2

(U, e2k)n-1] as

set

by

F

in Lk for

measure

can

sequence

measure

...

1

f

I -

i=O

where

defined

vectors

over

2k

2k -

spanned

set.

also

Then

0.

in

the

e2m

(X).

coordinates

Then,

we

z(t) get

(IV.3.7)

SERIES OF INVARIANT MEASURESFOR THE KDVE 133

ANINFINITE

4.3.

(to)

z

H(z) matrix (i. 1,2,...,m)and where

Let

::::::

+

J*

0 for

det(

that

prove

Lebesgue

the

IV.1.3,

=

J is

(1 (jrk)2n-2) +

A

)

azo,3

f

orm(Q)

k,1

indexes 1 for

ij=5_,_2m

measure

(2m + 1) x (2m + 1) (k -(J)2k,2k-I

=:

A

of the

values

2aut

(IV.3.8)

2m,

0,

skew-symmetric

a

_2irk

=

other

all

i

and

+

---

(J)k,l

us

Theorem

-2 2me2m) -A (J)2k-1,2k

Ej(zoeo

e.

(uo, ei),,-,,

=

dzo

all

0,1,...,2m.

according

Indeed,

t.

dZ2,,,

...

=

is

invariant

an

to mea-

n

for

sure

dynamical

the

(IV.3.7),(IV.3.8).

(hm (Q, t))

I

=

h-

arbitrary

an

this

Borel

immediately Let

generated

L,,

space

by

the

problem

Therefore,

orm

for

phase

the

with

system

arguments,

dZ2,rn

...

dzo

...

dZ2m

(O,t)

V =-:

that

of the

In view

of the

continuity

17,

function

1.

closed

arbitrary

an

Vdzo

dZ2m

...

9 C R2m+1

set

implies

take

us

dzo

bounded

Hp'e-r'(A).

QC

set

In view of the

above

get:

we

ym (hn

(Pmu) -E,, (hm (u,t))

eEn

(9, t))

dyn (u).

Further, ym (Q)

therefore,

according

integrand

in the to

KC

Hpne-r'(A)

proof

m

of Theorem

side

equality

E Q.

u

p(Q \ K)

<

Take

c, the

[ttm (K

lim

n

Q)

an

a

I dym (u),

IV.3.6,

Lemma is

arbitrary

we

bounded

function

>

c

of which

existence

By Proposition

IV.1.8.

and

IV.3.11 of this

0 and

>

that

such

jn(P u)-En(hm(u,t))

(Q, t))

Proposition

to

right-hand

integer

respect

pm (hm

-

0 and

can

be

obtain

the

that

uniformly a

with

compact

proved

as

set

in the

IV.3.12, -

tt,,

(h,,, (K

n

Q, t))]

=

0,

M-00

hence,

by Proposition

IV.3.11,

we

lim sup

get the relation

[ftm (Q)

-

ym (hm

(Q, t))]

<

C,

c,

M-00

which,

in view of the

Corollary

IV.3.14

arbitrariness

For

any

lim M-00

of

c

>

bounded

I tt,, (Q)

-

0, yields

open

p,,,

(h

the

set

statement

0 c

(Q, t)) I

Hpn,-'(A) =

0.

of Lemma IV.3.13.0

and

for

any

t E R

134

CHAPTER4. Lemma IV.3.15

,,n(Q)

,n

=

By Theorem

IV.3.1

set,

B)

Clearly, E

v

that

,n

inf

=

and

(0 \ K) < c. f2j. (Q, t) JJU Vjjn-j

these

u

-

balls.

any

also

Let

P

=

aA is

and

B1,

E

v

of

Proposition

r'(A),

for

sufficiently

all

,,n(f2)

,n

<

(B)

+

c

Further,

m.

<

by

(B)

lim inf y,,,, M-00

Therefore,

in view

/.,n(Q,)

+

and Lemma IV.3.15

us

(Q).

unbounded)

proved.

is

is

exists

aQ,)}, a

set

compact

a

where

Hpne-,'(A).

A C

set

a

compact

B,(u)

ball

that

covering

of the

compact

set

K by I

where

0, and

>

U Bi.

B

i=1

h n-1

--4

(U, t)

Hpn,-,,' (A)

in

as

m -4

oo

=

and

lim inf ym (h

of

c

>

Corollary

(B, t))

+

IV.3.14 e

<

tz'(f2j)

+

e.

0

<

Hence,

Theorem

prove set.

open

a

Qa

c

c

,,n(gj)

Let

bounded

3

M-00

arbitrariness

of the

< ,n

a

a

<

Lemma IV.1.11

Un(Q) By analogy

of

such

r

finite

a

h,, (u, t)

IV.3.3

is exists

E K there

u

radius

ffli)

dist(v,

:

t)

there

get that

we

large

any

Bi be

...,

Qi

h'-'(Q,

boundary

a

positive

a

hm(B,t) for

Then

hn-1

=:

(h,, (u, t); h,,, (v, t))

Let

m.

Qq

in view of nE H

u

a

IV.3.10

r}

<

v

B, (u) and all

Since for

Let

-

dist E

E R.

t

(K, t). Then, If, minf dist(K, ffl); dist(KI,

K,

Let

=

By Proposition

Hpn,,-r'(A)

v

and

set

open

Then,

0.

>

c

uEA, vEB > 0.

a

for all

bounded

a

Lemma IV.3.4

arbitrary

an

and K, C h n-1

too,

dist(A,

Take

too.

K C Q such

be

(Q, t)).

n-1

Hpn,,-,'(A),

in

set

Q C

(h

Proof.

Hpn,,-,'(A)

Let

INVARIANT MEASURES

=

,n(Qj)'

n

IV.3.2.

First,

let

Q C

Hpn,,-,'(A)

be

an

open

(generally

We set

f2k

=fuEQ:

11h

n-1

(U, t) I 1 _j

+

jjujj.,,-j

<

kj,

00

where

k

>

0.

Then

U Qk

Q

,

and

each

set

f2k

is

open

and

k=1

bounded;

in

addi-

00

tion,

hn-1

(n, t)

U

h n-1

(Qk, t)

and

/,,n(gk)

.

,n

(h

n-1

k=1

(Qk' t)) by

Lemma IV.3.15.

Therefore,

,,n (h

n-1

(n, t))

=

liM k-00

Un (h

n-1

(Qk' t))

=

liM k-oo

ttn

(f2k)

=

,n

(f2).

135

ADDITIONAL REMARKS

4.4.

Let

hn-1

(A, t)

now

can

last

is

a

Eo,...,E,,_1

all,

of

trajectories based

not

[16,20,53,66]).

Here

Concerning Some of them

[105]

in

an

with

the

sociated

invariance In

Poincar6

consider

there

equation.

is

nonlinear are

papers

for

with

quite

different.

In

the

and A

for

finite

for

any

f(X, S) :5 C(l p E (0, 2) for N

=

result

this

the 1 in

our

ball

The obtained

B C X.

+

S

d2)

A > 0.

for

is obtained

problem

all

for

by an

x,

>

J.

=

paper

initial-boundary with

Bourgain

arbitrary

>

A.

initial

[16,17] This

seem

to

data

this

allowed

AJuJP

for

in these

exploited

two

explicitly

more

this

a

author

AluIPu

and

to

space

be

p >

question

sufficient

and

nonzero

I_L(B) < -C(l+s di.)

00

:

0 if A < 0 and remains

(IV.2.1)-(IV.2.4)

open

with

L2_ The required

well-posedness of this

0

p >

0 <

such that

like

of

case

where

and obtain

implies:

problem

the

with

following:

important

the

=

IV.2.2

(0, 1)

E

from

the

equa-

completely

constructed

are

case

the

are

0, d2

proved

wave

be not

connected

is

value

who

abstract

an

[109].

f(X, JU12)U

an

However,

for

nonlinear

from Theorem

d,

[4]

paper

cubic

others,

investigate

f(JU12)

in this

case

as

conditions

0 and

For

s.

of the

superlinear

this

C

exist

Unfortunately,

well-posedness

we

(as-

measures

proved.

measures

measures

to those

equations.

KdVE and E in the

not

Methods

such

[107],

paper y similar

measure

a

proof

from

invariant

is

of the for

besides

construction to

con-

is obtained

direction

of the

measures

invariant

author,

example

to their

NLSE. Similar

a

nonlinearity.

the

nonlinearities

B C X if there

for any ball

about

the

[11],

related

In

constants.

are

conditions

weak

a

are

[4,11,18,19,24,30,55,65,67,86,107-

in

of the

[108],

(see

for

differential

in this

case

result

a

details and

abstract

an

superlinear

NLSE with

[67]

In

equation

problem

from

properties

NLSE which

devoted

partial

is constructed

measure

wave

Another

Y bounded

an

on

recurrence

of papers

results

law El in the

important

paper.

reestablishes

carefully

of

in detail.

by nonlinear

follows

easily

sense

some

in this

satisfactory

The

continuity

the

theorem

number

a

is constructed

measure

conservation

invariant

Unfortunately,

tion.

a

[30],

an

(A, t))

El.

the

recurrence

One of the first

earlier.

energy

our

(h

n-1

from outside.

KdVE and

the

investigations

these

generated

systems

invariant

in

proved.

n

=

boundedness

explain

by

for example, NLSE) are considered, of these the invariance In [4,24,55], 109,111-1131.

the

(A)

IV.3.4,

Lemma

to

generated

of the

case

open

completely

is

approaches

measures,

indicated

are

from

and from their

of the

invariant

where

sets

A

set

follow

systems

do not

we

dynamical

for

struction

equality by

IV.3.2

are

application

the

on

there

dynamical

of

Hpn,,-,' (A).

,n

remarks

that

note

we

set.

The

IV.3.2

Theorem

Additional

First of

Borel

of the

IV.1.1,

By Proposition

arbitrary

Hpn,-'(A)

in

Thus,

.r

4.4

an

of the space

of Theorem

Hpn,-' (A).

from

be

by approximations

statements

functionals

set

subset

Borel

be obtained

two

the

Hpn,,-,'(A)

A C

now

paper

in to

a sense

construct

of

an

invariant

to

show its

the

In the

IV.3.2

of the

sinh-Gordon

the

existence

in the

spatial

above

variable

the

\ is

a

ing question

may be

conservation

laws

conservation

law Eo.

make invariant

from

a

similar

space

difficulties

main

above

in the

in the

the

in the

t

constructing

[113],

paper

'

-

UXX + V(X)

U2+ U'X t

V(x)

assumed

-

phase this

the

that

to

(or

at

that.

In

our

from

least

prove

initial

H-2-',

opinion,

this

is

of the

corresponding

measures

0).

>

6 one

the

to

Cauchy problem

for

the

NLSE written

(U2)2)U2

=

0,

x,t

E

R,

(IV.4.1)

f(X, (U')'

+

(U2)2)Ul

=

0,

X,t

E

R,

(IV.4.2)

for

a

X

=

(_

above-

the

system

L2(R)

results

supercritical

0 on

u',0

=

of

L2(R)

the

considered.

is

+

the

potential

V(X)

Yi

described,

in

this

generated

by

the

L2(R)

where

f (x, s)

similar

increases

boundedness

case

R,

E

=:

as

IP.

In

to

V is

IxI

satisfy

---

the

those that

oo

*

this

introduced this

is

that

the

<

+oo.

E v,,,

condition

in

function

rapidly

so

it

paper,

n

paper

we

problem is

of the

A Is

(IV.4-3)

1, 2,

=

conditions on

and

d2 d2

i

x

satisfies

oo

--

operator

like

to)

hypothesis

IxI

as

dynamical

some

in the

f

function

+oo

of the

hypotheses

paper

H-1

should

we

KdVE with

the

an

law.

+

function

The main

v,,

space

obtained

real-valued

a

IV.2.2.

eigenvalues Under

is

tends

measure

measure,

for

and IV.3.2

conservation

(U')'

u'(x,

positive,

required problem

invariant

following

the

U, + f (X,

V(X)U'

-

X

Theorem

like

nth

the

we can

form

real

I U

where

a

lowest

consider

IV.2.2

to the

follow-

the

However,

yet.

in Theorems

that

the

with

example,

For

answered

is not

space

con-

problem

the

connection,

associated

corresponds

know any results

way of

In this

NLSE)?

the

evolution

Sobolev

is

of Theorem

for

measures

measures

construct

to

result

law.

conservation

Finally,

space

that

to

do not

we

Hn-'

corresponding

of the

Unfortunately,

observe

hypothesize

we can

well-posedness

data

question

One could

to the

0,

[112].

paper

conservation

NLSE =

invariant

our

phase

the

on

\IU12U

KdVE (or

the

case,

it.

on

in the

for

In this

cubic

Ux +

there

are

Eo, El, E2

measure

Therefore, the

posed:

comments

some

presented

is

constant,

analogous

in this

function,

unknown

of invariant

usual

A result

higher

a

and

nonlinearity

power

with

of the

A result

iUt + where

associated

derivative

sequence

for

the

(0, 5) (see [18]). [65]. paper

p E

in the

measure,

second

infinite

an

for

sense

equation.

of

NLSE with

presented

invariant

an

square

the

on

periodic

[67],

the

for

structed

in the

NLSE is also

cubic

paper

one-dimensional

the

boundedness

for

containing

law

for

measure

direction

is

INVARIANT MEASURES

C11APTER 4.

136

the

real

measure

construct

an

invariant

(IV.4.1)-(IV.4.3) space.

under

In

on

addition,

consideration

the in are

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