Boundary value problems in time for wave equations on RN

BOUNDARY

VALUE

PROBLEMS IN

TIME

FOR WAVE

EQUATIONS

ON R

N M.W. SMILEYandA.M. FINK

Department

of Mathematics

Iowa StateUniversity

Ames,

Iowa50011 U.S.A.

(Recetved December 5, 1989)

ABSTRACT. Let

Lx

denote thelinearoperatorassociated with the radially symmetric form of thewaveoperator

0t

A -b

A

togetherwiththeside conditionsof decaytozero as r

Ilxll

--

-boo

and T-periodicity in time. Thus

L,w

wtt-

(wr,.

-b

Y--w,.)

-b

AT,

when there are N space

variables.

For

,

R,

T

>

0 let

DT,

R

(0, T)

x

(R, -boo)

and

L(D)

denote theweighted

L

space withweightfunction

exp(r).

Itisshown that

Lx

isaFredholmoperatorfrom

dom(Lx)

C

L2(D)

onto

L(D)

withnon-negativeindexdependingon

.

If

[2rj/T]

:z

< A

<

[27r(j

-b

1)/T]

then the index is 2j

+

1.

In

addition it is shown that

Lx

has abounded partialinverse

Kx" L(D)

--H(D)IqL(D),

withallspacesweighted bythefunction

exp(r).

Thisprovidesakeyingredient for the analysis ofnonlinearproblemsviathe method ofalternativeproblems.

KEY

WORDS

AND

PHRASES. Waveequation, radial symmetry,boundaryvalueproblem, eigenvalueproblem,Hilbertspace,weightedSobolevspace,Fredholmoperator, Laplace

transform,

Bessel functions.

1980AMS SUBJECTCLASSIFICATION CODE. 35B10,35G15,35L05

1. PROBLEM STATEMENTAND NOTATION.

In

this paper we are interested in discussing the existence of solutionsfor certain types of

boundaryvalueproblemsin time associated with linearwaveequationsonspatialdomainsof the form

I]x[I

>

R,

with

IIx]]

(Z

x,

2") 1/2.

In

dimensions N 1,2or3wewillallow

R

0. Oneof ourprimarygoalsisthedevelopmentofalineartheorythatcanbeusedin studiesofnonlinear

problemsof thesametype

(cf.

[1]).

The most prominent caseofinterest isthat of time-periodic solutions. Althoughthis case is of natural interest, it has onlybeen in the last ten years that

contributions towards an understanding ofthisproblem have been made.

Many

of the results

onthe nonlinearproblem have shown nonexistence ofsolutions

[2],[31,[4],[5].

These worksare

complemented by theexistenceresults

[6],[7]

when g 1, and

[8]

when g 3.

In

this work

wewillextend thelinear theory of

[9],

which played a crucialrole inthe analysis of

[8],

toall

dimensionsNandthus set aside therestrictionof

N

1or3.

Let

n

{z

E

1N

[[x[[

> R}.

For

agiven real number

A

e

[t wewill beinterested inthe set ofsolutionsoftheproblem

u.- Au

+

)u

f,

u(t,z)

--,

o

a

Ilzll-,/,

(t,x)

(0,T)

x

fir

(1.1)

e

(0, T)

(1.2)

whichalso satisfycertainboundaryconditions in time.

In

particular these will beeitheraperiodic

626 M.W. SMILEY AND A.M. FINK

u(t

/

T,x)

u(t,x),

(t,x)

E

I

x

R,

(1.3a)

(0,1

u,(O,)

0,

lu(T,x)

+

2ut(T,x)

0, x(

R.

(1.3b)

In

thecaseof the periodicproblem

(1.1)-(1.2)

should besatisfiedfor

In

(1.1)

thefunction

f e

L

((0,T)

x

R)

isassumed to be radially symmetricin space,sothat

f(t,x)

F(t,r)

where r

Ilxll.

Weseek solutions

u(t,x)

U(t,r)

also havingthis symmetry. It is well-known in this case that Au

Urr

+

-Ur.

Thus the standard changesofvariables

w(t,r)

r’-u(t,x),

h(t,r)

r--f(t,x)

transforms

(1.1)

into

(g-

1)(g-

3)

wtt-w,-+

A+

w=h,

(t

r)(0 T) x(R,+o)

(1.4)

4r

Since solutionsaretobe at leastsquare-integrableweobserve that w,h L2

((0, T)

(R,

+c))

if andonlyif u,

f

e

L

((0, T)

R).

In

fact thesolutionswefindwill be weaksolutions inthesense describedbelow.

In

particularthedecaycondition

(1.2)

will beinterpretedas

w(t,.)

L2(R,

+),

(a.e.)

e (0, T),

(1.5)

Despite theapparantweaknessofthisdecaycondition it will turn out that infact

w(t,

r)

decays exponentiallytozero. Inthecaseof the periodicconditionswecanwrite

(1.3a)

intheform

w(t

+

T, r)

w(t, r),

(a.e.)

(t,

r) e

Itx

(R,

(1.6a)

whiletheseparatedconditions which become

(lw(O,r)

(2wt(O,r)

0

r(

(R, +cx)

(1.6b)

,w(T,r)

+

2wt(T,r)

O,

haveonly aformal meaning.

Ourmainresult canbebriefly stated asfollows. Let

Lx

be the

(unbounded)

linear operator

associated with the boundary value problem

(1.4)-(1.6).

Then

Lx

is aFredholm operator

(cf.

[10])

with finite non-negative index depending on A.

A

description of the dependence ofthe

indexon

A,

aswell asthespacesinvolved,requiresmorenotation.

For

R

>

0let

D

denoteeither

(R, +c)

or

(0,

T)

(R,

+c).

Let

>

0 anddefine

L(D)

to be weightedHilbert space offunctionh

e L2(D)

forwhich h.exp(ir) issquare-integrableoverD.

For

anintegerm

>_

0weset

(cf.

[11])

H(D)

{h

_

L(D)

Dah

_

L(D)

for

0

<

lal

<

m}.

The derivatives are to be understood as distributional derivatives relative to

C(R,

+cx)

or

DT,

Ras definedbelow,respectively. Thenormusedin

L(D)

will be

IlhllL](O)=

Ihle2Srdl

(d#=dr

or drdt

resp.),

D

andin

Hn(D)

we use

Similarly we define

LT’oo(D

as the set of functions h

L(D)

for which

IDhlexpOSr)

is

boundedon

D,

for al0

_<

I’1

<

m. When m 0wemore brieflywrite

L(D).

Theclass oftest functions tobe usedindefiningsolutionsof

(1.4)-(1.6)

is

DT,

n

{q0

C

(R

(R,

+co))

q0 satisfies

(1.6)

and

qo(t,

.)

C(R, +co)}.

Let

qN(r)

(N

1)(N

3)/4r

2.

Wesaythat w L

((0, T)

x

(R,

+co))

isaweaksolution of

(1.4)-(1.6)

if

T+oo T+oo

/ /

w[9,t-rr+(A+qN(r))qo]drdt=/

/

hcpdrd,,

o R 0 R

v0

e

r,.

(.7)

Theoperator

L

x definedformally aboveisnowtobe understoodas aclosed linearoperator in

L

2

((0, T)

x

(R, +co)),

with

dom(L)

C

HJ

((0, T)

x

(R, +co)).

The distributional definition above

(cf.

[12])

allows for an analysis based on separation of variables. Itiswellknown

(cf.

[131)

that theeigenvalueproblems

"

+

82

0, 0

<

< T,

(1.8)

((0)

(T),

’(0)

’(T),

or

(1.9a)

a(0)

a2’(0)

0,

(T)

+

fl2’(T)

0

(1.95)

admit countably infinite sequences of eigenvalues

{8,}

and eigenfunctions

{(,(t)}.

The

eigen-functionsare knowntobecomplete inthesensethat linear combinationsare dense in

L(O,

T).

Allowing for thedifferent casesin

(1.9)

we assume

{8,}

is enumeratedso that the sequence is

nondecreasing and includes repetitioninthe periodiccase.

In

termsof thesenotationsourmain

resultcanbe statedasfollows.

THEOREM.

//’8

2

<

A

_<

8_1.1

leor somej and0

<

6

<

(A-

8)

’/ _{then theoperator}

Lx

_is aEredholm operator onto

L

((0, T)

x

(n,

+co))

with indexequalto thedimension ofthelinear spacespanned by

{,(t)},<,.

Moreover,

if the boundary conditions

(1.9)

are eitherperiodic, Dirichlet, or

Neumann,

then thereis a boundedpartiaJinverse

g

L

((0, T)

(R,

+co))

H

((0, T)

(R,

+co))

such that

Lw

h if and onlyifw v

+

g;h, wherev

kez(Lx).

In

addition

range(gx)

C

ker(L)

+/- _{with the}

orthogonMityin

g

((0, T)

x

(R,

+co)).

We looselydescribethe arguments leadingtothe above result. Let

0(t,

r)

,(t)(r)

forsome

C(R, +co).

Then(1.7)implies

+oo

/

w,

[-"+ (-8

+

A

+ qN(r))1

dr

/

h,dr

R R

where

T T

tOn(F

C-I

/W(",

,$)n(l)d,,

hn(v)

C’I

/

o o

for a constant

boundaryvalueproblem

(1.10)

(1.11)

Thus it follows that

w,(r)

is a solution of the generalized

ii

w,

+

pn(r)w,

-h,,

e

(e, +oo),

(z.x2)

628 M.W. SMILEY AND A.M. FINK

where

p,(r)

02

A-(N- 1)(N- 3)/4r

2.

Onthe otherhand,if solutionsoftheseproblemscan

be found for every nthen

w(t, r)

w,,(r),(t)

should satisfy

(1.4)-(1.6)

inthesenseof

(1.7).

In

this way we may showexistence of solutions.

However

we are interestedin uniqueness and

continuousdependenceaswellasexistenceofsolutions. Thuswemustcharacterize allsolutions

of

(.2)-(1.13).

n

particularweneedtoshow

(1.13)

isapropertywhichguaranteesuniqueness.

When

₀

>_

A

this can be done. When

02n

< A

an additional constraint is needed to obtain a

particular 1-dimensionalaffinespace of solutions. Verification of theseremarksrequires several steps.

First we show existence and uniqueness ofsolutions toa morerestrictive problemin which

(1.13)

isreplacedbythe requirement w

.

L(R,

+o)

forsome6

>

0.

At

thesametime,apriori

estimatesof solutionsaregiveninthe

L](R,

+oo)

norm. Theseestimates are crucialforsubsequent developments, and are strongerthan would be suggested by

(1.13).

Tobridge the gap between

the competingdesiresfor strongestimatesononehand and weak uniquenessconditions onthe other handweconsideranintermediateproblemin which

(1.13)

isreplacedbyw

e

L]_e(R,

+oo)

forsomee, 0

_<

e

_<

6. We then showapriori estimatescanbeobtained whene 0, and that

uniquenessisimpliedwhen e 6.

The outline of the arguments forthis are asfollows.

In

the nextsectionexistence and uniqueness

is established withe

<

6.

Some

apriori estimates,which require

R

large, arealso given. Then

inthe third section some specialsolutionsare considered. Thesesolutions are shown tosatisfy

a priori estimates with arbitrary

R

but require e

>

0. Finally in Section 4 the results of the previoussections arecombined toobtain acharacterizationofsolutionsandaprioriestimates.

Throughout the developments we have outlined, it is essential to show that the constants

involved in the apriori estimates areindependent ofn. This allows forapassage to the limit

argument. Thisargument is giveninSection 5 andisused to show that solutionsof

(1.4)-(1.6)

exist aslimitsofsumsof the component functionsdescribedabove.

2.

A

CHARACTERIZATION OF SOLUTIONS.

It

iscrucial forourpurposesthatwecompletelycharacterizesolutionsof thegeneralized bound-aryvalueproblems

(1.12)-(1.13).

Towards this

goal

weconsider theproblem

w"

+

p(r)w

h, r

_

(R, +oo),

(2.1)

w

_

L_,(R,

+oo),

(2.2)

where

p(r)

b-

q(r)

with b

e

I

and

q(r)

r-2

(a

2-

1/4).

In

_particular we want toinclude the case b b,,

8

A

>

-A (i.e.

bounded

below)

anda

(N

2)/2.

With these choices p p,, and q qv are the functions of the previous section.

In

general we assume $

>

0 and ifb

<

0 thenalso$

<

Ibl

/,

R

>

O,

0

<_

<_

,

and h

oz.

L(R,+o).

Wheng 1 or 3so that

p(r)

b is constant, andwith

R

0, ananalysis ofproblem

(2.1)-(2.2)

has been given

in

[9].

The analysis uses theclassicalPaley-Weiner Theorem

[14] (cf.

[15])

which characterizes functionsh

e

L2(0,

+o)

in termsoftheirLaplacetransforms

g(s)

{h(r)}.

We

summarizea

numberof theresultsobtainedin

[9]

inthe statementfollowing.

i)

If b

2

>_

0 and0

<

<

-boo then, foreveryh E

L(0,

-boo),

there isaunique solution

o/’(2.1)-(2.2).

ii)

Ifb

-2

<

0 and 0

<

<

then, for every h C

L(0,+oo),

(2.1)-(2.2)

has a

1-dimensional affmespaceof solutions. All solutions have the form

w(r)

a

exp(--flr)

-b

t(r)

wherea

n

isarbitrary and

t(r)

-’

{[H(s)-

H()]/(s

32)}

isuniquelydeterminedbyh.

Furthermore the function

t,

in either case, belongs to

H(O,

+oo)

and satisfies the a priori

estimates

IIllL0,+oo)

<

CllhllL0,/oo),

[b[/2llllLO,/oo)

<

CllhllLO,+oo),

whereC

C(5)

if b

>

O,

orC

C(5,

5)

if b

<

O,

isa constant independent of h and such

at

C

+oo

as --,₀

+,

_andas

-

if b

<

O.

REMARK:

The above result shows that uniqueness is obtained by only requiring w

L2(0,

+oo),

and that the uniquely determinedsolutionsactuallybelongto

L(0,

+oo)

where$

>

0 is such that hq

L(0,

+oo).

Ifweonlyknow that h

e

L2(0,

+oo)

then thereisnoguaranteethat

solutions exist

(cf.

[9]).

Belowweneed to consider

(2.1)-(2.2)

wheregenerally wemust have

R

>

0.

We

will comment

onthe exceptionalcases

R

>

0subsequently. If 0

<

<

then

exp(-/r)

E

L(R,

+oo)

forany

R

>

0. Let

.A/’R

denote the 1-dimensionalsubspace of

L(R,

+oo)

spanned by

exp(-flr),

andlet

Q

L(R,

+oo)

(2.5)

denote theorthogonalprojection onto

A/’R.

An

easy calculationshows

Qh(r)

cexp(-flr),

where

c

2(

)exp(R)Ha(

2t/)

with

Hn(s)

{h(r

+

R)}.

Clearly wehave the

orthogonal

decomposition

L(R,

+oo)

A/’

+

A/’,

with

Qw

0for allw

A/’-.

COROLLARY 2.2.

Let/

denote anon-negativerea/number, and let

p(r)

b,

R

>_

O, ande /n

(2.1)-(2.2).

Then thereisacontinuous linearsolution operator in

L(R,

+oo)

for

(2.1)-(2.2).

i d=dodi=$

o

oowigs:

i)

Ifb

=/2

_>

0 and 0

<

<

+oo

then

L(R,

+oo)

L(R,

+oo)

is

wheret is the unique solution of theproblem. Thusw satisfies

(2.1)-(2.2)

if andonlyif

w t

=/cah.

ii)

If b

_/2 <

0 and0

<

g

<

_then/Ca

L(R,

+oo)

Af

isdefined by/Cah where

630 M.W. SMILEY AND A.M. FINK

Furthermore wehaveineithercase]Cbh

e

L’(R,

+c)fq

H(R,

+cx)

and

II.bhllL(R,+cx)

_

CIIhlIL(R,+cx),

Ibll/211bhllL(R,+cx)

_

C]]hL(R,+)

(2.6)

IIbhllL,’,R,+

<

CIIhlILn,+,

(2.7)

fora constt C having these

ter

(2.3)-(2.4).

In

pic there is aconstt

Ko

independent of

R

su

that

IIEII

5

Ko

with the norm being the lineoperatornormin

L(R,+).

PROOF:

Givenafunctionwdefinedon

(R, +)

weset

wR(r)

w(r

+

R)

to obtn the trlatedeedon

(0, +).

Ifw

e

L(R,

+)

L(R,

+)

then

l:0,+

-’:,+,

r0,+

-’7,+-

(2.s)

Suppose

that w satisfies

(2.1)-(2.2).

Then

w

satisfies

(2.1)-(2.2)

on

(0, +)

with h replaced by

h.

Hence eitherwR

R

isiquely deteined

(b

0),

orh the fo

wR(r)

a

exp(--r)

+

a(r)

descMbed inThrem 2.1. It follows that these istruefor

w(r)

wR(r

R).

Let

(r)

a(r- R).

Then by

(2.3)

d

(2.8)

Repeating this gument using the other nos involved shows that satisfies the estimates

(2.3)-(2.4)

on

(R, +),

d whenb 0 this veMfies

(2.6)-(2.7).

If b

<

0 thenwend toshow the

seestimatesapplyfor

(I-

Q).

Butsince

Q

is orthogonM projectionin

L(R,

+)

(2.6)

follows immediately. Toobtn

(2.7)

westobsee thatif

cexp(-r)

Q(r)

then

agn

by orthogonty

e-(-8)R

Hence

bythe trigle inequMity

(2.7)

nowfollows from these inequMities dthe estimatesMreadyestablishedfor

.

We

now considerthe variable coefficient problem originally stated. Assuming

q(r)

0 as r

+c

onemight conjecture thatthisproblemwill admit uniqueor_{multiple solutions, according}

to the value of

b,

inthesamewayasdescribedinTheorem 2.1. That istheproblemsshould be asymptotically equivalentinthesenseof solution sets.

THEOREM

2.3. Let denote a_non-negativerea/number, and let 0

_<

e

<

in

(2.2).

Let

p(r)

b

q(r)

in

(2.1)

with b

e

I

and

q(r)

(a

2

i)

/r2

_{Then thereisanumber}

Ro

>

₀

such that when

R

R0

in

(2.1)-(2.2)

thefollowing

i)

/fb

2

>

0 and 0

<

d

<

+cx

then,/’or

everyh E

L(Ro,

+x),

thereisaunique solution

ii)

Ifb

_#2 <

0and0

<

6

< #

then,

for every h E

L(R0,

+oo),

problem

(2.1)-(2.2)

hasa 1-dimensiona/ane spaceof’so/utions. Moreoverthereisasolution of

(2.1)-(2.2)

such that allsolutionshave the[orm

w(r)

aw0(r)+

t(r)

wherea Itand

w0(r)

isasolution

ofthecorrespondinghomogeneous problemwba’ch satisfies

(2.2).

In

eithercase the solution satisfies thefoflowingestimatesfora constantChaving thesame

dependenceon6, and

IllllL(R0,+oo)

_<

Cllh]lh(Ro,+o)

IbI’/2IIIIL(Ro,+)

<

CIIhllL]gRo,+o),

(2.9)

,.,

<

CI}h}L:

,+).

(2.10)

PROOF" Firstwesume 0. Forconvenience set

la

2 _i

l"

Let

Eb

be the line operator deedin

Coroll

2.2,d suppose b

fl

0.

Let

K0

denoteaboundfortheoperator

noof

Eb

in

L(R,

+).

We cse

K0

isindependent ofR. Choo

>

(Koco)

a/*.

By

the

eoroay

we knoww

L(,

+)

satisfies

(2.1)-(2.2)

if d

oy

if w

Eb

[qw

+

h].

Let

T

L(,

+)

L(,

+)

bethe bonded

e

operator definedby Tw

b

[qw

+

h].

We

have

IITw-

TVllL](Ro,+oo)

<_ IlIQIIIIqllL--(Ro,+oo)llw

K0-21lw-

VIIL](Ro,+

It

follows that

T

isacontractionby theoiceof

.

Hencethereisauniquely determinedfixed point q

L](,

+).

EquivMently, when 0 d

R

,

problem

(2.1)-(2.2)

h aique

solution@

e

L(,

+).

If b

-f12

<

0 then w

L](,+)is

a lution of

(2.1)-(2.2)

if d only if

w(r)

a

exp(-#r)

+

(r),

where

,[qw

+

h].

Nowfor aeda

e R,

dee

T

o o

by

Tw

,[qw

+

hi,

where

w(r)

a

exp(-r)

+

w

(r).

Then beforewefind

HT

w, Tv,

I1(o,+o)

<

KocoRo-llw,

for all wl, vl E

Aft0.

Therefore

T

has a unique fixed point wl E

Aft0,

and all solutions of

(2.1)-(2.2)

belonging to

L](R0,

+oo)

havethe form

w(r)

a

exp(-r)

+

wl

(r).

Forconvenience ofnotation let

e(r)

exp(-r).

Since the operator K:bq hasnorm lessthan

oneitfollows that

w

(I- ICbq)-llCa[qae

+

hi.

Thusw, dependslinearly and continuouslyon

botha

R

and h

L(R0,

+).

In

particularwemaywrite

w

(r)

aw2(r)

+

(r),

where

w2

(I-

K:q)

-1K:qe,

(2.11)

Hence

w(r)

cr

(exp(-#r)

+

wz(r))

+

(r),

wheredearly

wo(r)

exp(-#r)

+

w2(r)

isasolution of thecorresponding homogeneous problemwhichsatisfiesw0

L(R0,

+oo),

and

t

isasolution

of

(2.1)

whichisuniquely determinedbyh.

It remainstoverify theestimates

(2.9)-(2.10).

First assumeb

>_

0. Thenthe bound

K0

isalso independent of b.

Hence

both/Cband

(I-ICq)

-

are normbounded independent ofbasoperators

632 M.W. SMILEY AND A.M. FINK

supposeb

<

0. Then the bound

K0

dependsonthedifference

-.

Itfollows that

inherits thisdependence. Asbefore

t,

asgiven by

(2.11),

satisfiesthefirstinequalityin

(2.9)

with

the constant

C

nowdependingon

t.

Since

t

:b[qt

+ hi

we c usethe samebt-strap gument above toobtn the

remMng

estimatesin

(2.9)-(2.10).

2

Now suppose0

<

<

.

Then$

-

>

0 d h

e

L(,+)

C

L$(,+).

Hence by the gument jt given, with $ replacing

,

there is a ique solution of

(2.1)

satisfying

(R0,

+)

d thereforew in this w

L(R,+),

when

b

0dR

R0.

But

L$

ce. If b

<

0onlythe existenceof h bn clMmed. Thechacterizationoflutionsfollows before.

REMARK"

The gument above fls when

.

In ts

ce wewould have 0 d hence

b

wouldnotbewell-defined.

COROLLARY 2.4. Le$

p(r)

b-q(r) where b

e

d

q(r)

(a

2

)/r

.

Assme 0

<

.

Thenconclusio

i)

d

ii)

ofhe previous heoremeid with replaced byy

R >

O. additionestimates

(2.9)-(2.10)

remn

id for

Ro

scientlylge.

PROOF" Thisfollowsimmeately by continuationof theiquesolution to

(0,

+),

which ispossiblesince p

C(0,

+).

3. SPECIAL SOLUTIONS.

In

this section weconsider somespeciM solutionsobtMned from the iationofpetem forma by ceMn oices of the fr constts. Letting

p(r)

b-

q(r),

where b d

-

differentiMequation

(2.1)

becomes

q(r)=(a

)/r

2withaE

,

-i _=h.

(a.

w"

+

b

r

This isanonhomogenusBessel equation. All solutionshave the fo

(=

-(

e

(+

+

(

(,

(a.l

R R

inwNeh c,c ebitryeonstts,

w(r)

d

w(r)

elinely independentsolutionsof the corresponding homogenous problem,d

W()

is theirWronski. Itfollows immediately from Abel’s foule that

W()

is

here e

t

distinct ees to eoider. Let u

1-

If b for me

>

0 thenwe

functions of he 1st d 2rid kind respectively. Nrom known results

(el. [1])

it follows that Sinceweknow the uniquesolution

N

of

(2.1)-(2.2)

tends toero r

+,

wNle

(

.

w(r)

d

w(r)

e ymptotieNly cosine d sine functions, we hueristieNly gue that the

choices

+

f

/

l/2J,()h()d,

r

s/2y(s)h(s)ds

c

=

R R

should yield

.

Withtheseoices

(3.2)

becomes +

w(r)=

/

Gb,(r,s)h(s)ds,

where

(.)

1/2

Then

(3.2)

becomes

w(r)=

/

Gb,(r,s)h(s)ds,

where

(3.6)

R

Gb,u(r,s)

(

-(rs)ll2l(s)K(r)’

R

<

s

<

r,

(3.7)

-()’/()K(),

<

<

+.

Ultimatelywendestimateson

I111

d

Ibll/llwll

inthe

L_t(R,

+)

normfor thefunctio

w(r)

defined in

(3.3)-(3.7).

For

convenience we sume that

b0

1 is afixed number. Then

(hl/2

/2

)

both

w]

d

bllw]

ebondedbym

k0

bl

wJ.

Thusit

sces

toboundthislt qutiW. N

generN

wehavebyH61der’s inequiW

+

I()1

Ilhll:(R,+)

f

IGb,(r,s)l

2e-2s’ds,

R

Gb,(r,s)

extended the zerofunctionon

R 5

s

<

rin c b 0. Setting

b,(r,s)

with

m=

(b

I,

Ibll/)

Gb,u(r, s)

wethusobtn

m=

(b12,

lbl

/)

IIWIIL_,(R,+)

C(b,R,,,)IIhIIL(R,+),

suming

(3.8)

IOb,(r,s)12e-2[s(’-)+t]dsdr

C(b,R,,s,u).

(3.9)

O

goN

istoshow hat he eonstt

(b,R,,e,u)

isfiNted hthe following Wpesof de-pendenceonthe

vous

pee:

634 M.W. SMILEY AND A.M. FINK

(3.10ii)

If b

_>

0 then

C(b,R,

6,,v)

C(R,

6,

e,v)

is independent of b, and if b

<

0 then

C(b,R,$,e,v)

C

(]b[

1/2_-6,

R,

_6,

e,v)

_{dependsonbonly through thedifference}

]b]

1/

>

0 and tendsto

+

as

]b[

1/.

then then

C(b,R,

8,e

v)

C(b,v,e, v)

is independent of

R,

and if v

>

(3.10iii)

If 0

_<

v

_<

C(b,R,5,,v)

+oo

as

R

0

+.

Thereareseveralcasesthat must beconsidered. Firstsuppose that b

3

2

>

_0sothat

Gb,(r, s)

isgiven by

(3.4).

Wedefinetworelatedfunctions

Fr, Qu

R

+

-

Rfor v

>

0bytheformulas

Q,,(x)

Y,,(x)

From

the well-known asymptotics

(cf.

[1])

for

J,,Y

itfollows that

F(x)

O(1), Q(x)

O(1)

Consequently

then

Q

e

L(+).

Alsowehave

()+1/2

F(x)

boundedon

+,

and

L(l

+)

andif0

<

v

_<

Using thesefunctions

(3.4)

takes the

()

1/2-

Q(x)

bounded on

I

+ provided that 0

<

v

<

_3"

form

(recall

b

3

2

>

O,

_v

> O)

hl/2_and

b,v

1/2,, _{then both}

Fu

Suppose

that 0

<

b

< b0

sothat0

<

fl

<

0 =o0 b, If0<v<

and

Q

areboundedon

R+.

Hence

thereisaconstant

C

C(v),

which isindependent ofb,

R,

t and

,

such that

10,(,,)l<C,

o<<5,

0<b<bo,

O<r<s<-I-oo

(3.11)

then

Q,,

isnot bounded on

I

+ but

Q(3x)/x-1/2

is bounded independent If however v

>

1/2

of/ _<

b₀ for x

>

R

>

0. Hencefor any

R >

0 thereis a constant

Cn

Cn(v),

which is

independent ofb, and e, and whichgrowswithout boundas

R

0

+,

such that

1

Ib,u(r,s)l

<

CR

8u+1/2

(v >

,

0

<

b

<

bo,

0

<

R

<

r

<

s

<

oo)

(3.12)

Suppose

that b

> b0.

Then

1,,v(r,s)=

Gb,v(r,s)=

[(flr)V+1/2Fv(lr)(fls)1/2_VQv(fls)

-(3s)V+1/2F,(3s)(3r)1/2-Q,,(3r)].

Since

()+1/2

F(x)

and

()1/2-

Q(x)

areboth

boundedon

+

when 0

<

v

<

,

thereis aconstant C

C(v),

which is independent ofb,

R,

and e, such that

(

1

)

10b,(,,)l

_<

c

0

<

v

<

,

b

>

b0, 0

<

r

<

s

<

+oo

(3.13)

When v

>

weonlyhave

()

1/2-v

Q,(x)

bounded on

[R, +oo)

for

R

>

0. Hencefor any

R

>

0 thereisaconstant

C

C.(v),

whichisindependent of

b,

ande,and growingwithout bound

Nextwesuppose that

-b0 <

b

_f12

<

0.

In

thiscase

b,v

bo/2Gb,v

with

Gb,v

givenin

(3.7).

Wenowalso restrict6 to lie in the range 0

<

6

<

.

As before wedefine tworelatedfunctions

Fv(z)

e

()-v

Iv(x),

Qv(z)

e

()v

Kv(x)

for u

>

0. From the known asymptoticswe

Fv(x)

O(1), Qv(x)

o(1)asx-

0+ and

Fv(x)

0

__(x-v-1/2)’

Qv(x)

0

(xv-1/2)

as obtLin

x

+o.

In

terms ofthesefunctions

(3.7)

becomes

ab .(, )

-()’/*

()"

-.(-’)F.(#)Q.(#),

[

_(rs),/2

(_)v

e_#(._,)Fv(flr)Qv(fls)

R

<

s

<

r

< +oo,

R<r<s<+oo.

then

Qv

E

L(R+).

Hence there is a Notice that as before

Fv

E

LO(R

+),

and if 0

<

t,

<

constant

C

C(u),

which isindependent ofb,

R,

6 and e, such that

-b0

<b<0,

0<s<r<+oo)

Cre-(*-)

(0 <

_u

<

Ib,v(r,

s)l <

(3.15)

Cse

-/(’-*)

(O<u<

1/2,

-bo

_{<b<O, O<r<s<}

Ifv>l

then

Qv

is no longer bounded on

R

+,

but

Qv(x)/xv-1/2

is bounded independent of ha/2

<

₀ for x

>

R >

0. Thus for any

R >

0 there is a constant

CR

CR(v),

which is

independent ofb,6 ande,and growingwithout boundas

R

0

+,

such that

CnsV+1/2e_(,_o)

-bo

<b<O,

O<R<s<r<+oo)

(r, >

,

(3.16)

(r,>

,

-b0<b<0_

O<R<r<s<+oo)._

The remaining cases, b 0orv 0,areinfactlimitingcasesofthoseconsideredabove.

For

brevitywe simply state that

(3.11)

and

(3.12)

applywhen b 0, and

(3.11), (3.13),

and

(3.15)

applywhen r, 0.

LEMMA

3.1. Let

bo >

1 bea,given constant,and lete,6,

R >

0 with

<

6.

Suppose

that b

>

-bo, a

>

-1/2,

and

p(r)

b-

(a

2

1/2)/r

2.

Ifb

<

0 a/so assumethat 6

<

IbiS/.

T

thesolutionsof

(2.1)-(2.2),

given by

(3.3)

with

Gb,ll

given by

(3.4)

when b

>

0 and

(3.5)

when b

O,

orgiven by

(3.6)-(3.7)

when b

<

0,a//satisfy

max

(bo/2,

Ibl

/2)

IlwllL$_,(,+oo)

<

C(b,R,,,

Il)llhllL$(,+o)

(3.17)

where

C(b,R,

6,,

Il)

is

,

constant of thetypedescribed in

(3.10).

PROOF: First consider the case b

>

O. According to estimates

(3.11)-(3.14)

we have

Ib,lal(r,s)l

2

<

_Cs

v,

_{fora constant C}

independent of b, 6, and

,

and powerp with 0

<_

p

<

max(2,21a

+

1).

Using the inequality

(r

+

s)

v

< 2V(r

v

+

_s

v)

_wehave

4-o0 4-00

/

spc-2(s-r)d3-

/(s+r)pc-2Sd3

<_Cl__c2rp,

o

for constantsCl,C2

>

0dependingon6 and p but independent ofb,

R

and

.

Sincerp

exp(-2er)

isintegrableon

R

+ for p

_>

0,

>

0itfollows that theintegralin

(3.9)

isboundedbyaconstant then the boundon

(,lal

isvalidfor 0

<

r

<

s

<

+.

C(R,,,Ia[)

independent of b. If

I,1 _<

Therefore we may let

R

-

0+ in

(3.9)

without loss of convergence in the integral, and take

636 M.W. SMILEY AND A.M. FINK

When b

_/2

<

0, and 0

<

6

<

/3, it follows from

(3.15)-(3.16)

that

CrPe

-2(r-’),

s

<

r, and

Ib,l,l(r,s)l

2

<

_CsPe

-2(’-r),

_r

<

_{s, for a constant C independent}

ofb,6 ande, andapowerpasabove.

Hence

+oo

R

<_

C

r1e-2(-5)(r-")ds

+

C

sPe-2(+)(’-")ds

<_

_Cl

+

_c2rI

R

wherecl,c2

>

0areconstantsdependingon6,p, andthe

difference/3

6,but independent of e,

R.

Againitfollows that the integralin

(3.9)

isboundedbyaconstant

then

C

C(/3

6,

R,

6,e,

lal)

which in thiscasedepends alsoonthedifference

the integralremainsconvergent with

R

0 andC

C(/3

6, 6,e,

la])

canbe taken independent ofR. Estimate

(3.17)

nowfollows from

(3.8)-(3.9).

4. A PRIORI ESTIMATES.

The purposeofthis section istoestablishaprioriestimatesoncertainsolutions of

(2.1)-(2.2).

Throughout

we assume that

p(r)

b- q(r)in

(2.1)

with b E

R

and

q(r)

r-2

(0

.2-

1/4)

_for

a>-.l

Alsowe assume 6

>

0and if b

<

0 then 6

<

[b[

1/2.

Itiscrucialfor laterdevelopments that the constants involved in these estimates be independent of b, for b

--

+oo.

It will be

convenient for our purposes to assume b

>

-b0

for somefixed

b0

>

1. We begin withsome technical lemmas showing that it issufficient to establish

L(R,

+oo)

estimates on solutions in

orderto obtain

HI

(R, +oo)

estimates. The first isavariantofawell-knowninterpolationestimate.

LEMMA

4.1. Let₇

>-

1 be given andsuppose6

>_

O,

R

>_

O. Then/’ora11w

H(R,

+o)

Shereis aconstant

Co

independen of7,6,

R

such hat

][w’IIL(R,+o.)

_<

Co

[max(l,62)llwllL(R,+)4-"y-IIIw"IIL(R,+oo)]

(4.1)

PROOF:

From

[11,

p.

70]

weknowthat foranye0

>

0thereisaconstant

K,

independent ofe0 and

R,

such that forany

f

H2(R,

+o)

+oo +oo

Taking

f(r)

exp(6r)w(r)

ande0

7-2e

forsomee

>

0yields

I1’11

’

+

2(,o,,o’)

+

llwll

_<

K-r-

I1o11

+

4K-’

(11o"11

where allnorms are

L(R,

+oo)

norms, and the pairing

(w, w’)is

theinnerproductin

L’(R,

+oo).

For

anyr/>0wehave

21(w w’)]

<_

rl-1]lw]12+rl]lw’]]

2,

sothat

26(w, w’)

Since₇-1

_<

_{1 it follows that}

(1

6

lK’)llw’ll

_<

(K-r

-’

+

4K

’

+ 6.-)llwll +

4K-r-ll,"ll

.

Wenowmake thechoicer/=

(36)

-1 and require that

16K62e <

-

to obtain

1

72

llw’ll

<

(K

-a/4K /

S-)llwll

/

4K-llw"ll

LEMMA

4.2. Let

bo >_

1 be given.

Suppose

that

w(r)

isasolution

o[(2.1)-(2.2),

and that

(a&/,

lal

’/*)

Ilwll:n,+)

_<

Cllhll:n,+)

fo o=tt m

i, other constt

C

C(,a,,e,

I1),

hvingthese type ofdependence on theindicated

peers

the firs const,such that

IIw’ll(n,+

5

C(;

+ Iq(R)l)llhll{n,+).

(4.2)

PROOF" From

(2.1),

heMinkowskiinequiy, d he sumedestimatewehave

a

I1"11(,+

Ilhll{,+

+

m(0,1l)+

a

I111,+

{

(

(#/

/,)

a*-m

ko

,11

+C

+

m

IIlln,+

Hencefor &

eons

C

1

+

C

(1

+

[q(a)[)

d wih m

o

[[

in

(4.1)

ifollows

[[w’[[#(,+)

C0

[m(1,)m

k

,Ibl

)Ilwllgl(,+)

C0

(Cm(1,52) +

C])IIhlIL](R,+).

This veeries

(4.2).

According to Thereto 2.3 dits corolla, problem

(2.1)-(2.2)

h a

que

solution

(r)

when 0 e

<

d b 0. Sin

(),

e

by

(3.3)

d

(3.4)

or

(3.5)

spetiely,

solutionof

(2.1)

that h bn shown tosatisfy

(2.2)

wehave

6(r)

w(r).

When b

<

0there

is no longeraique mlutionof theproblem, but wehave two picul solutions,

(r)

d

w(r)

ven

by

(3.6)-(3.7),

satisfyingestimates

(2.9)

d

(3.17)

respectively. Noticethat

(3.17)

holds for

R

>

0 while

(2.9)

holdsonly for su6cientlylge. It iso pueosesubsequently toshow thata

uque

reprentativelution be selectedbymesof projectionoperators, md that this solution satides bothof theseestimates.

Ts

in tm leMs to

HI

estimateson mlutions. Wefirstshow that

6(r)

satisfiesthesineestimate

(3.17)

thefunction

w(r)

defined

i

(3.6)-(3.7).

LEMMA

4.3. Le 1 befixed, dle,5,R>0 withe

<

.

Supposetha b-bo,

,-,

dp()

b

(,

)/i(2.).

fb<0oum+2e<=lbl

n

Th

thefunction of

CoroHy

2.4 satisfies

(b/,

Ibl

n)

IlellL_.,+)

C(b,R,,e,

I’I)IIhlIL{R,+)

(4.3)

0 0

C(b,R,,e,

I’1)

*h *hpopeidd

(3.0).

PROOF: As

pointedoutabovethisfollows iediatelybyuniqueness when b 0.

Su

pose

b=-

<

0 d 0

<

<

< +2 <

.

Since b

-

it su6ces toshow

(4.3)

holds

with

(b&/, Ibll/*)

1.

t

()

b by

(3.6)-(3.7).

Th th

that

e()

()+

/*K,(#).

Fo

oei

t

thth

I1.

t

>

0 b the number

td

toexistby Thereto2.3. Thenamust satisfy

(note

thatw

e

L(R,

+))

+

638 M.W. SMILEY AND A.M. FINK

By

writing2,

(- )

+

(g

+

)

and using the Schwarz inequalitywefindthat

kll

<

I1

wllLL.Ro,+oo)ll(’)l/2K

((’))

Since

>

+

2 and

K=(r)

is monotonically decreasingwe have

K(r)

< K=(vr),

where

7

+

2 for convenience. Using

((z)

e

()=

K=(z),

tr

>

0, as definedpreviously, and noting that

K(z)

isapositive functionitfollows that

R0

Clearly 7r

>_

($

+

2e)R0 >

0 sothat

(7r)

-2, _and

IQ.(,=)I

both boundedat the lower limit. Also 7-

-

>

0 and

Q,

hasat most polynomialgrowth at infinity; thus theintegralon

theright isconvergent. Let

C(Ro,g,,a)

denoteitsvalue. Then from

(2.9)

and

(3.17)it

follows that

1-1

<

c(,,,o)116-WllL_,(Ro,+oo)

foraconstant

C(b,

R,

,

,

tr)

asdescribedin

(3.10).

Nowchangingvariables z

r

inthe integral definingk and using

/<

,1/2 shows immediately that k₀ isboundedawayfromzeroindependent of

Ibl/.

Thua

I1

-<

CIIhllL:,+oo)

for constant

C

asabove. It nowfollows from this,

(3.17),

and thetriangleinequality that

t

satisfies

(4.3).

We

now introduce the projection operators

Q",

which will be used in determining unique

solutions. Let 0

<

<

and suppose that

>

0. Let

H"(R,

+oo)

{w

E

L(R,

+oo)

w(k)

Ls(R,

+oo),O

<

k

<_ m}

bethe weighted Sobolev spaceasdefined in Section 1. Since

>

wehave

r/2Ks(r)

H’S"(R

+oo),

R

>

0.

We

define

Q’

tobe theorthogonalprojection

in

H"(R,

+oo)

onto the 1-dimensional subspace spanned by

r’/2K(r).

Ofcourse

Q’

also dependson

R,

and tr. If0

<

tr

<

1/2

and m 0 then

Q"

is also well-defined for

R

0.

In

general, however,wemusthave

R

>

0. Computationally,wehave

Qw(r)

ar/2K(r)

where

.

II(.)/K.

((.))i1-

i

))).r(n,+oo)

.r(,+oo)

(w,

(.) K.

((.

(4.4)

THEOREM 4.4. Let

b0 >

1 beagiven constant, andin

(2.1)

let

p(r)

b-

q(r)

where If

Il

<

1/2

let

R

O,

otherwiseassume

R

>

O.

>

-o

=

()

(

1/4)/

,it

>

-.

Suppose

that in

(2.2).

i)

/fb

_>

0 and0

<

< +oo

then foreveryh

L(R,

+oo)

thereisaunique solutionw* of

(2.1)-(2.2).

ii)

/fb

<

0 and0

<

<

[b[

’/2 _{thenforevery h C}

L(R,

+oo)

_problem

(2.1)-(2.2)

_hasa

Moreoverin eithercase thereisaconstantC

C(b,

R,

,

[a[),

having the properties describedin

(3.10),

but beingindependent of e, such that thefollowingest/mateshold:

max

(bo/2

Il

’/2)

II’llq(,,+)

<

Ollll,.:(,,+oo)

I1,,,*11,,(,+oo)

< 6’(1

+

Iq(R)l)

Ilhll;(,+oo),

R

>

0,

(4.5)

(4.6)

IIL(R0,+oo)

Thus from

(2.9)

and

(4.3)

weobtain

(4.9)

thenwe canallow

R

0and take C independent of for aconstant C

C(R,,,

[hi).

If

[hi

<

R. Wenow apply

Lemma

4.2 toobtain

(4.6).

If b

-

<

0 and 0

<

e

<

then all solution of

(2.1)-(2.2)

have the form

w(r)

(r)+

arl/UKa(r)

asdescribed inTheorem2.3. Againwewrite arather than

[a[.

Since

<

3

all of If

u(r)

is

these solutionsbelongto

L](R,

+oo)

with

R

>

0 ifa

>

,

and with

R

0if 0

<

a

_<

.

asolutionof

(2.1)-(2.2)

with e then

v(r)

(r)

u(r)

satisfies

(2.1)

with h identicallyzero.

Thus

v(r)

crl/2g(r)

+ c2r/2I(flr)

forsome c,c

_

It.

But

then

v(r)- clr/2g(r)

belongs

to

L2(R,

+oo)

which impliesc2 0. Thus

u(r)

t(r)-

clr/2g(r).

Thisshows all

solutionsof

(2.1)-(2.2),

with e

,

have the form

w(r)

(r)

+

ar/2ga(flr).

Now

set w*

(I-

Q)t.

Clearly w* is well-defined and satisfies

(2.1)-(2.2)

when e

.

Moreover

[[w*[[

_<

[[t[[

inthe

L(R,

+oo)

normsince

Q

isanorthogonalprojection.

By

choosing

e rain

(,

("))

wehave 0 <e

<

5

<

i

+

2

<

ft.

Hencewe canrepeat theargument above leadingto

(4.9),

with replacing*. Then using

{1*11

<-

I111

yields

(4.9).

Theestimate

(4.6)

thenfollowsfrom

Lemma

4..

To

obtain

(4.7)

weuse

(4.5)

and thefact that

*(r)

-

0asr /toestimate

<

]

I*’()ld

<

,-e-’"llw"llz,(n,+oo)

_<

Ce-’"

(1

+ Iq(R)l)IlhllL(n,+oo).

I*(")1

Toobtain the two remainingestimateswebegin by multiplying

(2.1)

by

w’

where for convenience

wewritew inplaceofw*. Integratingfromr to

+oo,

and using the limits

w(r),

w’(r)

0as

I*(,’)1 <

c-’"

(I

+

Iq(R)i)Ilhll.(a,+),

,-

>

R

>

O,

(4.7)

I"(,)1,

Ibl’/=lw*(,)l

<

C-’"

(

+

Iq(R)l)Ilhll,,,,(,+o),

,

>

R

>

0.

(4.8)

PROOF" Suppose

that b

>

0. Let

t

denote the unique solution of Corollary 2.4i, and

let w be the solution givenby

(3.3)

and

(3.4)

or

(3.5)

respectively. Sincew satisfies

(2.1)-(2.2)

with0

<

e

<

wehave w.

Suppose

that

u(r)

is asolutionof

(2.1)-(2.2)

with e t;. Then

v(r)

(r)-

u(r)is

asolutionof the correspondinghomogeneousequationwhichsatisfies

(2.2)

with e

.

From the form that vmust take

(cf.

Section

3)

it follows that visidenticallyzero.

Therefore

(2.1)-(2.2),

with e ti, hasaunique solution w"

t

w where w isgivenby

(3.3)

and

(3.4)

or

(3.5)

respectively.

Toverify

(4.5)

welet

R0

>

0 bethe number guaranteed to existby Theorem 2.3, and choose

640 M.W. SMILEY AND A.M. FINK

r

--

+oo,

weobtain

(with q(s)=

(a

s

1/4)/s)

I’()1

+

bl()l

-2

()’()a

+

q(s)w(s)w’(s)ds

By

the previousestimate it sufficesto consider b

> b0.

Bounding the firstintegral onthe right

wehave

+oo +oo

[2/h(s)w’(s)dsl<2/

[h(s)[eSSlwt(s)lesse-2SSds

L](R,+)

+

IIL](R,+oo)

<

Ce

-’( +

Iq(R)l)II

foraconstant C

C(b,R,,,o)

as describedabove. Similarly forthesecondintegralwefind

12

/

q(s)w(s)w’(s)dsl <

CIq(r)l

e-s

(

+

Iq(R)l)

Estimate

(4.8)

nowfollows.

Inthe preceeding theoremitwasshown that thereisauniquelydetermined functionw

satis-fying

(2.1)-(2.2).

When b

<

0 the uniquenesspropertywasobtainedby addingtheconstraint

Ow=O

(4.10)

with m 0. Since

Q’

iswell-definedforallm

>_

0when

R

>

0 it isclear that uniquenesscanbe obtainedby addingtheconstraint

(4.10)

forany choiceofm

<

2.

By

the natureof the apriori

estimates

(4.5)-(4.8)

wehaveestablished it turnsout thatthecasem 1 also allows apassage

tothelimittothe originalproblem

(1.4)-(1.6).

Thuswealso record the following.

THEOREM

4.5. UnderthehypothesesofTheorem4.4, withthe exceptionthat

R

>

0is nowrequiredun/ess

I1

12,

allconclusionsof Theorem4.4 remain va/idwith

Q,

replacing

Qo.

PROOF"

When b

>_

0 the results are identical. When b

<

0 we set

Ibla/

and w**

Qw*,

where w* isthe unique solutionobtainedviathe projection

Q.

Since

Q

isan

orthogonalprojectionin

H(R,

+oo)

the inequality

(4.6)

follows immediately. Since

1511/2

<

"o

the firstinequality

(4.5)

then follows from

(4.6).

Havingverified

(4.5)-(4.6)

forw**wemayrepeat thesameargumentasbefore to obtain

(4.7)-(4.8).

5. PASSAGE TO

THE

LIMIT.

Our

goalin this section isto proveexistenceofsolutionsandtocharacterizethe set ofsolutions

ofproblem

(1.4)-(1.6).

As

in Section 1 we set

qN(r)

(N-

1)(N- 3)/4r

2.

We begin by consideringthefinite dimensionalapproximatingproblem

BOUNDARY VALUE PROBLEMS IN TIME FOR WAE EQUATIONS ON RN 641

Win(t,

.)

_

L2(R,

E

(0,T)

(5.2)

w(

+

T,)=

W..(, ),

(,

)

a

x

(,

+oo),

or

(.a)

0

Win(0,

r

2Wm(0,

r O,

e (R, +),

(5.3b)

fl,

W(T,r)+fl

ot

(T,r)

0, r fi

(R,+),

where

R >

0 d

H(t,r)

h,(r),(t).

The fctions

h,,,

were deed in Section 1.

We ek a solution

Wm(t,r)

w,(r)(,(t)

having thesefo. Leta

a

(g-

2)/2

sothat

(N- 1)(N

3)/4r

(a

)/r

qN(r)

inthe previous ctions. Reclingthat

,,

0 and that

,

satisfiestheboundyconditions

(1.9)

we eletothe componentwise

equations

w

+

(bn

qN(r))w,, -h,,,

w,

_

L2(R,

+oo),

r>R,

inwhichb,

02,-A.

Thereareactuallytwocasestoconsider:

i) A

<

rnin

8

and

ii)

₀

< A

<

0+,

forsomej. Howevertheargumentsare similarsofor brevityweonlyconsiderthemoredelicate

second case. Also, sincewe areinterested in the limit of

Wm

as rn

+oo

we assume rn

>

j.

,Thenforsome

b0 >

1wehave

-bo < bn <

bj <

0forn

<

j, and

bn >

0forn

>

j. Set

n

Ibn

11/2

for n

<

j. Accordingto Theorem4.4allsolutionsof

(5.4)-(5.5)

areeither uniquelydetermined

whenn

>

j,orgivenby

w,(r)

w,(r)

+ a,r’/2K(#,,r),

a,

,

(5.6)

whenn

<

j. Henceallsolutionsof

(5.1)-(5.3)

have the form

W.,(t,r)

Z

w,(r)(,,(t)

+

Z

a"rl/2K(#"r)"(t)

(5.7)

-_<-, -_<

Let

QO..

L(R,

+oo)

--}

span{rl/2K=(#,,r)}

be the orthogonal projection definedin

(4.4),

with m 0

and#

/,. Then 0

Q#.w,,

0 where

w,,

is the function appearing in

(5.6)

and definedinTheorem4.4.

Clearlythe functions

,(t,)

l/g=(z.),(t)

(, <

j)

(5.S)

satisfy

(5.1)-(5.3)

with

g,,,

identically zero, and also

(1.4)-(1.6)

with h identically zero.

Let

Lx

denote the

(closed unbounded)

linear operatorassociated with problem

(1.4)-(1.6),

so that

Lxw

wtt-wrr

+(A

+

qN(r))w

for smooth functionswsatisfying

(1.5)-(1.6).

Then

vn

_

ker(Lx)

for n

<

j. We define anorthogonalprojection

Px

L

((0, T)

(R,

+oo))

span{v,,},,</

by

pw

E

.’/g(.).(t),

where

r

(.9)

,

C2.,()/’/=

g=(,),

,,()

f

(t, ).(t)dt.

o

Then

W,,(t,r)

(I- Px)W,,(t,r)

isthefirst summand appearingonthefightin

(5.7),

andisa

uniquelydetermined solutionof

(5.1)-(5.3).

Moreoversincethefunctions

n(t)

areothogonalin

L2(O,T)

wehave

w(r)k(t)w,(r),(t)e2Srdrd

,,11

(z

o

642 M.W. SMILEY AND A.M. FINK

Hence

from

(4.5)

itfollows that

(R,+)

.(m

<

c(6,

,

,

N,

)llhll

,

_Ls((O,T)x(R,+oo))"

Here

c(6,

flj -6,

N,R)

is used to denote aconstant depending on theindicated quantities, but independent ofrn and h.

In

general c --,

+oo

as 6-,₀+ andas 6

flj-,

and alsoc

+

as

R0+ifN>3.

Frombasic Hilbert spacetheory

(cf.

[18])

it followsthat thereisauniquefunction

w*(t,

r)

lim

Win(t,

r),

belonging to

L

((0, T)

x

(R, +)),

withthelimitmeant inthe senseof

/,

((0, T)

x

(R,

+oo))

convergence. Alsowemusthave

[[W*IIL$((O,T)(R,+oo)

<

C(6,j

6,

N,R)llhllL$((O,T)(R,+oo)),

(,5.10)

and accordingto

(4.5)

ifN

<

3we cantake

R

0andc

c(6,//i

6,

N).

Let

/)T.a

bethe setof testfunctionsdefinedin sectionone. Then

T +oo T

/

/

Wn

[9tt-rr+(A+qN(r))]drdt=

/ /

Hmqodrdt,

o R o R

I

6.

T,R.

(5.11)

Since qa 6

VT,

R implies ,t qarr

+

(A

+

qN(r))99

L

2

((0,

T)

_x

(R, +oo))

_{we may pass to the}

limit in

(5.11)

toobtain

T+oo T

/ /

w*[tt-tPrr+(,+qN(r))cp]drdt=/

/

hcpdrdt,

o R 0 R

V

_

VT,

R.

(5.12)

Hence w"

is asolution of

(1.4)-(1.6),

and depends continuously on h according to

(5.10).

If

w 6/,2

((0,

T)

x

(R, +oo))is

anothersolution then theargument leadingto

(1.12)-(1.13)

shows

w,(r),

as defined by

(1.11),

must satisfy

(5.4)-(5.5).

Hence

w,(r)- w,(r)

a,rl/2K,(,r)

for n_<j and

somea.

6 R. Ifn >j then

w,(r)-w,’(r)

0. Thereforew

w*+v

where

v 6

span{v,}

with

vn

asdefinedin

(5.8).

In

particularthisshows

ker(Lx)

span{v,},<j. We

have verified thefollowingresult.

forsomej and define

.

18

A]

1/2

THEOREM

5.1.

Suppose

that 8_i

<

A

< 8j+

n

<

j. Then fora/16 6

(0,fli)

and anyh

L

((0,

T)

x

(R,

+oo))

thereisafinite dimensional affme space ofsolutionsof

(1.4)-(1.6).

Allsolutionshave the formw w*

+

v, where w* isthe

solutionuniquely determined by

Pxw*

0andv

ker(Lx).

Furthermore thesolutionw*satisfies

(5.10),

and ifN

<

3 then

R

0 isallowed.

For

certainchoices of theboundaryconditions

(1.6)

it ispossible to show that solutions have

more regularity than stated above.

For

instance when the boundary conditions areperiodic, Dirichlet,or

Neumann

wehave

T

,(t)t(t)d

8,cn,

n

k,

O,

n#k.

o

Thus forafunctionof theform

Wm(t,r)

E

w,(r),(t)

itfollow that

--Wm

_{L((O,T)(R,+c))}

With these choicesof the boundary conditions

(1.6)

inmind we again consider thesequence

2

ofproblems

(5.4)-(5.5).

Asbefore weassume

0

<

X

<_

0,i+x"

By

applying Theorem4.5wefind that allsolutionshave the form

(5.7),

with

w

uniquely determinedbytheconstraint

Q.w

0 when n

_<

j.

Here

Q,

denotes the projection defined in

(4.4)

with m 1 and

f,. We

nowdefine

P

g

((0,

T)

(R, /c))

span{v,},,<

as in

(5.9),

butwith

Q

replacing

Q.

Thenasbefore

W

(I-

P)W,

isthefirstsummandonthe rightin

(5.7).

THEOREM5.2.

Suppose

that theboundaryconditions

(1.6)

aresuch that theassociated

eigenvalue problem

(1.8)-(1.9)

has eigenfunctions which satisfy

(5.13).

Assume that

+

o

omj .d dn

I.-

1/,

,

<_

j. T.

o

e (0,,)

i

((0,T)

x

(R, +))

there is a ite-dimensionM

ne

space ofsolutions of

(1.4)-(1.6).

All

oo

o.

to

H

((o,

T)

x

(R,

+))

v

t

om

"

+

,

"

solutionsatisfying

Pw*

0 dv

ker(L).

Fthermore w* satisfies theaprioestimates

wherec

c(,

-,

N, R)

isa consttwith thesepropeies in

(5.10).

PROOF: Theexistenced iquenessofw* followsintheseway before. Toverify

(5.15)

it remMnsto showthat

w,

w

exist d satisfy appropriateestimates. Fimtweconsider

w.

Since

9

b.

+

X

itfollows from

(5.14),

appliedto

W

(I-

P)W,

d

(4.5)

that

wherec

c(,

I, N,

R,

)

nowalsodependson

I,

but isindependen ofm. hisshows that

a

convergesin

((O,T)

x

(R, +)),

dastdd gument verifies that

in hsense ofdistributions. Essentially the se gument applies

o

w

lira

,

with

(4.6)

udinplaceof

(4.g).

o

verify

(g.16)

we

agNn

eorthogonty, destimate

(4.7),

toobtMn

T

lW(t,)lat

c.lw:()l

0

nm

c

-

(

+

IqN(R)I)

IIh.

Silestimatesapplyfor

W

d

W.

Thesecombine to

ve

T

}

IW(,.)l’

+

I

+

c(

+

Iqu(R)l)e-llhllLo,T)n,+))

where once

agMn

c

c(6,

j

-6,

N,R,)

isindependent ofm. Nowfrom thisestimate d the Sobolevembedng

H(0,T)

C(0,

T)it

follows that

(a.e)

(t,r)

(0, T)

x

(R, +)

Iw(t, )l _<

clIW,(’,)IIH’(O,T)

<

-"

(

/

IqN(/)I)IlhllL((0,,)(,+)).

Since

W,(t,

r)

-

w(t,

r) (a.e) (t, r)

(5

(0,

T)

(R, +c),

thesame estimateholdsfor

w(t, r),

which

644 M.W. SMILEY AND A.M. FINK

REMARK"

When A

<

min0, analogous arguments apply and show there is a unique

solution w* E

H

((0, T)

x

(R, +oo))

of

(1.4)-(1.6)

whichalsosatisfies

(5.15)-(5.16).

COROLLARY

5.3. The

(dosed unbounded)

linear operator

Lxw

wtt-wrr+(A

+

qN(r))w,

where

qN(r)

(N- 1)(N

3)/4r2,

associated with

(1.4)-(1.6)

is a Eredholm operator from

dom(Lx)

C

L

2

((0, T)

_x

(R, +oo))

_onto

L

((0, T)

_x

(R, +oo))

_{with non-negative index. The}

pa-rameter

>

0 and theindexof

L

dependon

A

in thefollowingway.

H" A

<

min/2 then

>

0 is otherwiseup.restrictedand theindex iszero. If

19

< A

<

t9i+

then 0

<

<

13j

IX

-/gj[

1/2 _and

the indexispositive andequaltothe dimension of thelinear spacespanned by

{,,(t)},,<j.

/n ad-dition

Lx

hasa ou=a d(partial)inverse

Kx

L]

((0, T)

x

(R,

+oo))

H2

((0, T)

(R, +))

L

((0,

T)

(, +oo))

a rm a y

the correspondence h

--

w*, where w*isdescribed inTheorem

5.2.

REFERENCES

1.

SMILEY,

M. W. Complex-valuedtime-periodic solutions and breather of nonlinear

waveequations, toappear inDifferential andIntegralEquations.

2.

CORON, J.

M. Priodem/n/male pourunecordevibrantedelongueur in/in/e, C. R. Acad. Sci. ParisSet

A

294

(1982),

127-129.

3.

LEVINE,

H. A. Minimalperiods forsolutionsofsemi//near wavequationsin exterior domainsand forsolutionsof the equationsof nonlinearelasticity, to appear in

J.

ofMath. Anal.

&

Appl.

4.

STRAUSS,

W. A. Stable and unstablestatesofnonlinearwaveequations,in Nonlinear Partial DifferentialEquations,

Contemporary

Mathematic Vol. 17,

Amer.

Math.

Soc.,

Providence,

R.L,

1983.

5.

VUILLERMOT,

P.

Nonexistence ofspatiallylocalizedfree vibrations forcertain

semi-linearwaveequationson

R

2,

C.

R.

Acad. Sci. Paris, Set. 1 9

(1985),

395-398. 6.

WEINSTEIN,

A. Periodicnonlinearwaves on ahalf-line, Comm. inMath. Phys. 99

( 9S),

3S-38S.

7.

WEINSTEIN,

A.

Erratum,

Periodic nonlinearwaves onahalf-line,

Comm.

inMath. Phys. 10._._7

(1986),

p. 177.

8.

SMILEY,

M.

W.

Breathers and forced oscillations ofnonlinearwaveequationson

R

3,

Journal/’firdie reineundangewandteMathematik 398

(1989),

25-35.

9.

SMILEY,

M. W. Time-periodicsolutions ofwaveequationson

11

and l

,

Math. MethodsinAppl. Sci. 10

(1988),

457-475.

10.

KATO,

T.

Perturbation Theory forLinear

Operators,

2nded., Springer-Verlag, New

York,

1976.

11.

ADAMS,

R. A. Sobolev

Spaces,

Academic

Press,

New York, 1975.

12.

LIONS,

J. L. and

MAGENES,

E. NonhomoreneousBoundary Value Problems and Applications, Vol.

I,

Springer-Verlag,

New

York,

1972.

13.

HARTMAN,

P. Ordinary Differential Equations, John Wiley

&

Sons, Inc.,

1964.

14.

PALEY,

R. and

WIENER,

N.

FourierTransformsin theComplexDomain,A.M.S. ColloquiumPublications, Vol. 19,Providence,

R.L,

1934.

15.

RUDIN,

W. RealandComplexAnalysis, 2ndedition, McGraw-Hi11, New York, 1974. 16.

ABRAMOWITZ,

M.and

STEGUM, L(eds.),

Handbook ofMathematicalFunctions,

Dover Pub., New York,

1965.

17.

WATSON,

G. A

Treatiseon theTheoryofBessel Functions, CambridgeUniversity

Press,

1922.