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SOME PARTIAL DIFFERENTIAL EQUATIONS
WITH DELAY

Dissertation for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
DAVID RICHARD HALE
1977

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LIBRA RY I“
Michigan Stag-3 '
University

     
   
   

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thesis entitled . .
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presented by

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has been accepted towards fulfillment
of the requirements for

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ABSTRACT

SOME PARTIAL DIFFERENTIAL EQUATIONS WITH DELAY

BY
David Richard Hale

This thesis is primarily concerned with some partial

differential equations with delay. for example

t
(—§%)(x.t) = cAu(x.t) - j g(t-S)Au(s)ds

-Q
where A denotes the Laplacian. In the first chapter,
several physical models are discussed. which lead to

equations of this type.

In the second chapter, a more general equation

t
u)(t) = cAu(t) - f g(t-s)Au(s)ds

(.51.
dt
where A is an infinitesimal generator and is considered

at first. This equation is formally transformed into a

second equation.

This second equation is shown to have a unique solution
for appropriate initial conditions. A semigroup is defined

using these solutions. The infinitesimal generator for this

David Richard Hale

semigroup is found, and its spectrum computed. In the
last section it is shown that the solution of the modified
equation gives solutions of the original equation when

A = A and the solution is in a weak sense.

The third chapter discusses existence and uniqueness
for the nonhomogeneous linear equation and the perturbed

equation.

The last chapter involves a saddle point property

for the perturbed equation.

SOME PARTIAL DIFFERENTIAL EQUATIONS WITH DELAY

BY

David Richard Hale

A DISSERTATION

Smeitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1977

$107075

To my parents

ii

ACKNOWLEDGEMENTS

I would like to thank my advisor. Dr. Shui-Nee Chow,
for suggesting the topic for this thesis. His constant

help and encouragement were also important to me.

Also I would like to thank Michigan State University
for giving me employment. and Mrs. Milligan for typing

this thesis.

iii

TABLE OF CONTENTS

CHAPTER I: INTRODUCTION
CHAPTER II: EXISTENCE, UNIQUENESS AND THE
INFINITESIMAL GENERATOR
§1. Introduction
92. The modified equation
§3. Existence and uniqueness
§4 The infinitesimal generator

§5. Decomposing W’ into subspaces invariant
under a

§6. The operator 4 and solutions of (2.1)

CHAPTER III: NONHOMDGENEOUS LINEAR PROBLEMS AND
PERTURBED NONLINEAR PROBLEMS

§1. Introduction
§2. The nonhomogeneous linear equation

§3. The perturbed linear equation

CHAPTER IV: THE SADDLE POINT PROPERTY
§1. The spectrum of the semigroup
§2. The saddle point property

BIBLIOGRAPHY

iv

Page

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42

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49

53
53
66

75

CHAPTER I
INTRODUCTION

Functional differential equations have been well
studied by various writers. Such equations arise in phys-
ical models where the rate of change of a system depends
not only on the present state of the system, but also on the
past state or "history" of the system. A natural general-
ization of functional differential equations, which are
ordinary differential equations where the derivative depends
on the history of the system [11]. are equations involving
partial derivatives and the history of the system. These
equations occur as models in some physical and biological
problems. In this introduction models of 1. Gene frequency,
2. Heat conduction, 3. The "dangling spider," and 4. Visco-

elasticity will be considered.

1. Gene frequency.

Fleming [9] has used the equation

g2.=.§32.+ up - 6

to describe the frequency of a selectively—neutral gene. in
a habitat consisting of a number of colonies of animals

arranged in a row. Here p(t,x) is the frequency of the

gene type at time t and position x. a is a positive
constant depending on the rate of reproduction and death,
and B is a positive constant taking into account other
variables. The migration from colony to colony is

p(t.x + hi + p(t.x - hi - 2p(t.X)
2h

2
which is approximated by 3L2-. Note that this model

5x2
assumes that the rate of change of the gene frequency
depends on the migration at the present moment only. To

relax this assumption a more realistic model might be

a 32 t 32
3% = —-§- + I g(t-s) (—-§-) (8.x)ds + up -6.
OX -m OX

where g(s) 4 O as s 4 ~,.g(s) 2_O. This integral in-
volves the past history, and thus takes into account the
rate of migration in past times. Since g(s) 4 O as
s a w,, the system is of fading memory type. That is,
although the value of p in the distance past does affect
the integral, it does so less than the value of p in the
immediate past.

This model also assumes that the colonies are arranged in a
row. It is more valid to consider colonies arranged in a
plane. By reasoning similar to that in the preceding model,
this can be represented by

t
gE.= AP + I g(t-s)Ap(s,x)ds + up - 6.

-co

where all the variables have the same meaning as before,

except x is a two vector.

2. Heat conduction.

Jace W; Nunziato [21] has studied a similar equation in
regard to heat conduction in materials with memory. Let M
be a homogeneous heat conductor with memory. For each x 6 B,
there are three response functionals: the free energy E,
the entropy N, and the heat flux Q. Each of these depend
on the temperature T and the temperature gradient VT. both
at the present time and the past history.

To be more specific, let TT(s) = T(s-t) and VTt(S)
= vT(s-t), considered as elements of a function space. Let
H = {g:(—e,0) -o R such that J: g(s)h2(s)ds < co}. (Identify-
ing functions equal except on sets of measure 0.) h(s) is
a positive function decreasing to O. T(x,t) is the tempera—
ture at position x, time t. T(t) is the same function
considered as a function from the real numbers into a Hilbert
space H. FOr each t, T(-,t) E H. H is a Hilbert space.
Let < , >» be the inner product in H. Then Q (similarly
E and N) is a function of

(T,VT,Tt.VTt) 6 R x H x R3 x H3.

E is assumed twice Fréchet differentiable, N and Q each

once Frebhet differentiable.

Define G = E + TN, where G the internal energy.

Then G and Q must satisfy the energy balance equation
G = - vQ + r

where r is the heat supply from the body's surroundings.
The second law of thermodynamics gives the Clausius-

Duhem inequality [3]
N12; - v(.-I]:. Q).

To Obtain constitutive equations these equations are

linearized. This gives after considerable computation

O
Q -K(0)v'1‘ -j K’Is)vr(t-s)ds.
O
In the same way

G

Go + g(orr + I: a'(s)T(t—s)ds.

Here K is the heat conduction relaxation function and Q
is the energy temperature relaxation function.
Combining these equations and using the energy balance

equation results in

a(o)T(x,t) + I.<z'(s)T(x,t-s)ds
O

= K(O)V2T(x,t) + I.2K'(s)v2T(x,t-s)ds + r(x,t).
0

If a’(s) = O (in other words if the energy tempera-

ture relaxation function is constant) then the equation becomes
0 I
T(x,t) = aAT(x,t) + j‘ g(s)AT(x,t-s)ds + r(x,t)
0

an equation similar to the previous one. Paul Davis has

studied this equation [7], [8].

3. The "dangling spider."

Another example of a model where such an equation
appears is the "dangling spider," studied by B. Coleman and
D. Owen [55] among others. A mass M hangs from a ceiling
by a massless but strechable filament of length 2. The
forces acting on the ball are the tension T in the fila-
ment and a body force F in the z—direction. It can in
many cases be assumed F = - §%%§lm, where h is indepen-
dent of time, that is, F is a potential.

Assume T at time t is given by a function g of
the history of 2 up to t: T(t) = g(zt). Using Newton's
second law, the equation of motion for the ball is
M2= F(z) -g(zt) .

If

T(t) = G(O)zt(0) + I G'(s)zt(-s)ds

(this form appears in the linear theory of viscoelasticity
[4 ]) where G is the relaxation function, and 6(0) the

equilibrium modulus, then the equation becomes
0
M°z°= F(z) - G(O)zt(0) - Le’mntt-sms.

For F, the simplest nontrivial case, and also a case
with physical importance, is when F = gM,, the gravitational

case. In more general cases, F does depend on 2.

4. Viscoelasticity.

Another example is linear viscoelasticity of the Boltz—
Mann type, considered by Dafermos [6] and Coleman [4].
Consider a one dimensional homogeneous body with density
p > 0, stress at position x at time t = o(x,t), displace—
ment from position x at time t = u(x,t), satisfying the
constitutive equation o(x,t) =cux(x,t) - jtg(t-r)ux(x,r)dr

where g(s) _>_o. g’m go. and “I g(§)d§ > o.
O

The body has endpoints b and d,, which remain fixed,
so u(b,t) = u(d,t) = O, for all t 6 R.

The state of the body at time t is described by the
displacement u(x,t), the momentum. v(x,t), and the history
of displacement ut(x,s).

The equation of motion is

t
pii(x,t) =cuxx(x,t) —I g(t-T)uxx(x,'r)d'r.

In the following chapters, a generalization of the
equations which appear in the first two models will be studied.
Chapter 2 is concerned with the homogeneous linear equation.
Existence and uniqueness is shown for a related equation. A
semigroup is constructed using solutions to this equation,
and its infinitesimal generator is computed. The spectrum
is of the infinitesimal is computed. In the last section of
chapter two, the solution of the modified equation is shown

to give solutions of the original equation in a weak sense.

The third chapter is concerned with the nonhomogeneous
equation and the perturbed linear equation. BaSic exist-
ence and uniqueness theorems are proven for these.

In the fourth chapter a saddle point pr0perty is
established for the perturbed equation. This involves more

accurate determination of the spectrum of the semigroup.

CHAPTER II

EXISTENCE, UNIQUENESS AND THE
INFINITESIMAL GENERATOR

§l. Introduction.
In this chapter we consider some partial differential
equations with delay, for example

0
(2.1) 4%%(x,t) = cAu(x,t) - I g(t-s)Au(x,s)ds, t 2_O

-co

where u is a scalar function of xeRn and tER. "A"
denotes the Laplacian with respect to x and g is a

scalar function defined on (-<=,O] and c is a positive real
number.

Let 0 'be an open, bounded, connected subset of R9
with boundary an. Let F2 be the closure of {2. Let an
be locally Lipschitz, that is, if x 66 0, there is a neigh-
borhood Uk of x in R? such that Ux n at} is the graph
of a Lipschitz function. Together with (2.1) we impose the

following initial-boundary conditions:

U(X,t) 0 X650: tZO

w(x,t) X E Q o t < 0

u(x,t)

where w(x,t) is a given function.

The solutions to (2.1) found here will be solutions in
the sense of distributions, or a "weak solution" [15], with
the solution u(x,t) defined on a Sobolev space [1].

Let Cam) be the infinitely differentiable functions

into R with compact support in n, and define

 

In] 1 = algrad ulz. Let 1131(1) be the completion of c;
H
0

in the norm l°l 1' Then Hg(CD is a Hilbert space [15].

Ho

The Laplacian is defined on C'

0
closed self-adjoint operator on a dense subset of H3(n). [15]

and can be extended to a

new the solution u(x,t) will be considered as an
element in a Hilbert space. Let a >'0 be a positive number
such that I. ezatlg'(t)|2dt < a.
0

Let x = L at[-ua,0;Hc1)(Q) ], be the Hilbert space of
2,e

functions from (-.,o] into 33“» with inner product
0
<f..g>x = f ezat<£(t).g(t)> 1 dt
-9 no

where < , > 1 is the inner product in 113(0) .
H
O
The space H3(0) is used to correspond to the boundary

condition

u(x,t) = O, x 6 an, t‘2;0.

Elements in GENO) clearly satisfy this condition, so
the condition u(°,t) 6 33(CD corresponds to u(x,t) = 0,

by the trace theorem [15]. Also let V = X x Hg(fv.

10

W will be used as the space from which the initial condition
will be taken. If (w,p) e w, w e x, p e Hé(0), then
u(x,t) = w(x,t) for t < O, and u(x,0) = p(x). Since
w 6 X does not imply w is continuous, both w(x,t) for
t < 0 and w(x,0) must be specified.

The first question concerning (2.1) and the related
initial boundary condition is naturally, "Does a solution
exist and is the solution unique?" To Show (2.1) has a
solution in the sense of distribution under the previously
mentioned restrictions, we will use semigroup theory.

A semigroup is a one parameter family of continuous

linear operators T(t) on a Banach space E such that [13]

(l) T(t+s) =T(t)T(s) for t,s_2_0.

(2) For any fixed x, T(t)x is a continuous function in t

for t > 0, in norm on the space E.
(3) T(O) = I, the identity operator on E.

The infinitesimal generator of the semigroup T(t) is

defined by

A(x) = lim 335%54L5 . x e D(A).
1:4 0*

where the domain of A,D(AL is given by

D(A) = {xI lim. IJE%§;:§ exists].

x-OO+

11

It can be shown that A is a closed linear operator
'with dense domain and that the semigroup is uinquely
determined by the infinitesimal generator [13].

In the next sections a semigroup will be found using
solutions of an equation related to (2.1). The infinitesimal
generator of this semigroup will be found and its spectrum
computed. Then using a special case of the general equation,

it will be shown.weak solutions of (2.1) are obtained.

§2. The modified equation.

Let E be a Banach space and A the infinitesimal
generator of a semigroup on E. The semigroup generated
by A will be denoted by eAt. Let a 20, set
K = L at(-a»,O;E). The Banach space E in applications
can b: ghosen to correspond to boundary conditions, for
instance H$(Q) ‘will correspond to the boundary condition
u(x) = 0, x 6 am.

Now consider the equation
Bu _ t
(2.1) at — cAu - I.“ g(t-s)Au(s)ds
with g differentiable

I. [g(s) [2 e238 ds < o, and
0

I. [90(8) I2 e238 d8 < a
O

12

for some a > 0. Suppose this equation has a unique solu-
tion for any initial condition. (w,p), w E K, p E E. Then

the variation of constants fbrmula [12] gives

t S
u(t) = eCAtp + %-I eCA(t-S)( I g(s—s')Au(s')ds')ds
O -03

for t > 0. Integration by parts gives

At 1 It g(t-s)u(s)ds

(2.2) u(t)==ec p + E

O t
.. {—5 eCAt< I g(—s’)u<s’)ds‘) - -'};( I eCA‘t'S’uiswsmm)
-m 0

t s
_ é-I eCA(t’S)( I g’(s-s’)u(s’)ds')ds.
0 .00

This is the equation that will be studied in the next

two sections.

§3. Existence and uniqueness.
First, it will be shown that equation (2.2) has a unique
solution for (w,p), w E K, P E E.

Lemma 2.1. Let w E L at(-e-.O:E). Then
0 2,e

G(t) = I g(t-s)w(s)ds is a continuous function of t for

1:20.

Proof: For any t1,t2 2.0, [G(t1)-G(t2)I

O
_<_ LI (g(tl-s)-g<t2-s))w_(s)dsI _<_ (I I9(‘v:1-:=.2)-g(1:2-8)I2e

O
(In ezast(s)I: ds)‘.

-2asd8)§

13

'2a3(g(t1-s)-g(t2-s))2as 4 o as t -+t

. 0
Since I...» e 1 2 ,

G(t) is continuous.

Ihggram 2.1. Let E be a Banach space, let A be the
infinitesimal generator of a semigroup on E and let

g[0,w) 4 R satisfy

(i) I [g(t)]2e2atdt < a:

0
(ii) g'(t) exists for O th < w: and

... 2
(111) I [g'(t)] dt < m.
0
Then the equation

t
(2.2) u(t) = eCAtp + %-I g(t-s)u(s)ds

o t
- ea“: (13' [I 'g(-s)u(s)ds- %I eth's)Iog‘(s—s')
-. 0 —.

t
u(s')ds')ds - %N:I eCA(t’s)u(s)ds]9(O)
O

has a unique solution on [0,t1] where t1 > 0 is

arbitrary.

 

Proof: Let C[O,t1:13] be the Banach space of con-
tinuous functions from [0,t1] to E, 0 < t1 < c with

norm

IuI = max Ie-ktUWIIE

C te[o.t1]

14

where k is some positive number. Define

LM(W:P)V] (t) = e p + -

C

CAt 1 Io g(t-s)w(s)ds

-u
t

+ %’I 9(t-s)v(s)ds — %-eCAt ID g(—s)w(s)ds
0

"C
t O
- %-I eCA(t-s)( I g(s-s’)W(S’)dS)dS

O
t s
_ % I eCA(t-S)I g(S-S ’)V(8 ’)d8’
0 O
1 t cA(t-s)
_ ENii e v(s)ds]g(0).
0

It will be shown M(w,p) is a contraction for k chosen
correctly.

Note that eCAt

O
(p4-I g(-s)w(s)ds) is continuous in t
t —Q
and I g(t-s)v(s)ds is continuous (since both 9 and v
0
are continuous). By Lemma 2.1. % Io g(t-s)w(s)ds is contin-
"Q
uous. For the other terms in M(w,p) continuity is shown
t cA(t-s)
using the theorem that I e f(s)ds is continuous if
0
f is continuous [12]. Hence, M(w,p): C[O,t:E] 4 C[0,t:E].

Next, we have for any v(t),r(t) in C[0,t:E]

sup Ie-kt[M(w,p)v(t) - M(w,p)r(t)]

t€[0,t1]
t
g sup e-kt 21; I Ig(t—s) Ilv(s) - r(s) Ids
t€[0,t1] o
+ sup - t _ 3
te[0,t1]e kt % I IeCMt 8’ I (I Ig(s-S') IIVIS')-v(s ’) Ids ’)ds
0 O

15

t
”kt 1 cA(t-s)

.... 0) ) - d

+ t€[s()u,ptl]e Clg( I LIe IIVIS V(S)I s

- t k ..
< sup e kt-c1;-Ie s ekslg(t- s)IIv(s) -r(s) Ids

-kt 1 t cA(t-s) S , , t I
+ s — I I '8 I ( )-r( ) d )d
te[011,1?t1]e c I) Ie I LIST s “V s s | s s
k 1 t k k
SUP Ig(O) Ie- t E I e 86- slV<s) -r(s) Ids.

tE[O.t1] 0

Now, IeCAtI g Rec"t for some R > O and some a. [13]

Using this
kt l t ' k
sup e' —I else— sIg(t-s) I|v(s) -r(s) Ids
tE[O,t1] ‘3 o
s
- kt 1 leCA(t-s) l I I I
+ — ( (s-s) v(s )-r(s ) ds )ds
..fouptf I:Ie lIolg II I
-kt l I: k
+ Ig(O) Ie E I e SIv(s) -r(s) Ids
t6 £I 't F1 0
1 —kt t ks
3.239 (I6 IV-rchS) “$.21 Ig(s )l
R -kt t as 8 ksI o
+ E- e I e (I Max Ig(s) Ie Iv-rchs )ds
0 0
kt l t k
+ sup (Ig(O) Ie' E- I e sIv—rchs)

t1€[0.t] o

Mang(s) I,s€[o,t ] Mang(s) Is€[0,t ] at
( kc 1 + cIk+c) 1 Re 1+T1€I9I°’ I%)IV‘rIc°

 

 

 

 

16

Choosing k large enough so that

at

R e 1) + Ig(O)I'3-_',_'<k

max Ig(s)I(%—+m

sE[O,t1]
makes M(w,p) a contraction. Therefore M(w,p) has a unique

fixed point. Since t1 can be any positive number, this

gives a unique solution to (2.2), for all t > 0.

Theorem 2.2. Let wEK, and peE, K and E
the same Banach spaces as in Theorem 2.1. Let u(t) be

the solution of (2.2) with initial value (w,p). Then

Iu(t)IE g C(t)I(W.p)IKxE where C(t) depends on t but

not on (wyp).
Proof: By (2.2) after taking norms

0
Iu(t)IE g ReatIpIE + %-I Ig(t-s)IIw(s)IEds

t
+ %,I Ig(t_s)IIu(s)IEds + ReatGIwIK

O
R t t I 8 I
+ E-I eaI '3 ( I Ig'(s-s')IIu(8')IEdS )ds
0 O

t
+ EISJELL I eait’s) (u(s))ds
0

V\

Reo‘ttlplE + lelK] +-,1;-G(t)

t

l
+ — Max (g(s) Iu(s)l ds
° [0.t] II, E

17

t
_]_'__e(a+a)tG_Iu(s')e
v’Za o

 

nIm

I
-(a+a)s ds'

4.
E
O
O

t
I eaIt-S)Iu(s)IEds.
O
at

Now fix t1, 0 < t1 < w . Then on [0,t1], eOLt S_e 1.

50 using Gronwall's inequality on the above equation,

IIu(t1)IIE < RthIHpHE + GHWHK] + %-G(t)

 

t
1 1 R 1 (“+aIt1 -(a+a)t
+ (exp [Max g(s))-—+— ____e e
IO [0,t1 c c/Za
u(t —t)
+ W e 1 dth IplE

_<_ c<t) l(w.p) leE -

Now define T(t) for t 2_0 by
(2.3) we (mp) = (ut.u(t)>

where u is the solution of (2.2) for the given (w,p)

and

ut(s) w(t-s) t < -s

ut(s) u(t-s) t > -s

Theorem 2.3. T(t) defined by (2.3) is a strongly

continuous semigroup of linear Operators on KarE-

18

Proof: For each t,T(t) is linear since equation

 

(2.2) is linear. T(t) is bounded by Theorem 2.2.
T(t)(w,p) is continuous by Theorem 2.1.

So to show T(t) is a strongly continuous semigroup,
only T(t1)T(t2) = T(tld-tz) must be checked. Let w E K,
p E B, then

cA(t1+t2) 1 1
u(t1+t2)==e p+~— I g(t1+t2-s)u(s)ds

cA(t +t -s) o
-%-e 1 2 g —s)u(s)ds
"Q

t
-3? I eCA(t—s) (If: g’(s-s ’)u(s ’)ds ’)ds
0 C

t1+t2 ecA(t1+ tZ-s)

go(o)I u(s)ds

cAt 1 t2
p + E I g(tZ—s)u(s)ds

cA(t -S)
_.% e 2 IO- g(-s)u(s)ds

t cA(t -s)
"'% I29 2 (I g'IS-s')u(s‘)ds')ds
t CA(t -8) t +t
" i [I 2 e 2 u<s>ds]g(on +§ I: 29(t1+t2-s)u(s)ds
t +t cA(t +t -s)
+ ' 21:: It: 2 e 1 2 (Ij.9'(S-S')u(s')ds ’)ds
t +t cAt t2
__ AI 1+ ZeCA(t1 +122 "5)u(s)dsg(0) _ e 1 lI-Z .g(t2 'S)U(S)ds
c
t

2

19

cAt1 1 t

= e u(t2) + E I.1 g(tl-s)ut2(s)ds

cAt

O
.. i e 1 I”. g(-s)ut2(8)ds

t cA(t -s) '
_ % I01 e 1' (Ii. 9 (s-s”)ut2(s')ds’)ds

--(I e u

c (s)ds)g(O)

0 t2

making the change of variables -t2 +5 4 s, and t2 +s ’ 4 s ’.
So T(t1+t2)(w,p) = T(t1)(T(t2)(w.p)). and so T(t) is a

semigroup .

§4. The infinitesimal generator.

Now the infinitesimal generator of T(t) will be found.

Theorem 2.4. Let T(t) be the semigroup given in

Theorem 2.3. Then the infinitesimal generator of T(t) is

. O
(2.4) «(w,p) = (w,A(cp - I g(-s)w(s)ds))

where the domain D(A) of a is given by

D(a) = {(w,pl Iw.w E K, p 6 E. limw(t) = p, and
o t40

(cp -I g(-s)w(s)ds) e D(A)}.

~-
Proof: Let «(w,p) be the infinitesimal generator of

T(t). Then for (mp) GEM)

u -w
= lim(—E——-— 9.13%.;2)

t I

6! (WP) = lim Tit) (“3:13) 'iWoP)

t40 t40

20

 

w -—w
By definition lim tt = w where w is the weak deriva-
t40
tive of w [15]. For w to exist, lim w(t) must exist. [20]
t4O
Since
cAt l t
lim u(t) = lim [e p + EI g(t-s)u(s)ds
1240+ t40 ...,
1 At 0 1 t A t
_ E. J I 9(-s)u(s)ds _ E- < I ec I ‘s’u(s)ds>g(m
...co 0

- 21? I eCAIt-s)( Is 9‘(s-s’)u<s')ds')]ds=p. lim u(t)
t"0

= lim w(t) =p .
t"O

For the second component Of %(T(t) (mp) - (MM).

0
-};~(u(t) -p) = (eCAttp + - %I 9(-8)W<8)581

0 o
— (p+- F]; I g(t-s)w(s)ds) + -<-]:'-I g(t-s)w(s)ds

t t

+ % I g(t-s)u(s)ds — 213 < I eCA‘t's’utsidsmm)
O O
t o

.. 21::I eCA(t'S)( I g'(s-s I)w(s ')d8 ')d8
0 _m

1 t A t s 1

- EI e‘3 I '8)( I (s-s'IW(S’)dS')d8) q:-

o o

o 0 e
= %[eCA(t'3) (p -% I g(—s)w(s)ds) - (p ”2‘:- I g(-s)W(8)ds)]

o t o
EIEII [g(t-s)-g(-s) ]W(8)ds ”cl-:- I eCAIt'SI (I g'Is-s 'IWIS'IGS'I

21

fl

(g(t'3)"g(0))u(s)ds]%

[I

(HP
0

[1 - eCMt'S) 19(0)p dshfi

+
OIH

(I

rrOrf

[I ecA‘t's’ gconp [ Manda-,1;

+
OIH
O

t _ 8 ’
+ - g (I eCA(t S)(I 9 (S-s’)u(s’)ds’)ds)%
0 0
We will denote these terms by
1

o o
(i) E (eCAt(p-;1: f g(-s)W(s)dS) - (p-c-ls I g(-S)w(s)ds)
-. -cu

0
(ii) Eli-f [g(t-s)-g(-s>]w<s)ds

t
— g I eCA(t-S)jp g'(s-s’Tw(s')ds

O a
t
(iii) 51- [f0 (g(t-s) -g(0))U(s)ds]%
t -
(iv) (1: (f [l-eCA‘t 5)]g(0)p ds)%

0

t -
(v) f eCA(t 8’ g(0)[p-u(s)ds]%
O

t ..
(vi) 2% [fo eCA‘t 3’ (j: 9'(S-8’)u(8')ds')ds]%
each of these terms will be considered separately.

As t e 0. term (vi)

t _ ; o o
31; IIO e‘mt “([2 g <s-s')u<s )ds )dsl

IA
“I
w

t Rem-VJ"t g'(s)2 ds)i(ft u(s)2 ds)i
O 0

ll
OIH
(D

(It g’(s)2ds)i(j‘t u(s)2ds)i » 0.
O 0

22

Fer term (v). since u(s) is continuous for 3 2:0.

for any 6 > 0, there exist 6 such that

OSs<6=Ip-u(s)l<€.

And so

t -
115% [(J‘O £3th S)[p-u(s)]ds) -g<o>l s t ° $11th same

.3 Reab g(O)€ 4 O as t 4 0. since e is arbitrary.
For the term (vi)

t o -
31%- (Io [l-eCA t 8)]p ds)g(0)

l-'

_ 1 1 t cAs 1 t cAs
—E-ag(o)fo (l-e )p dSIE°5-9(0) [OH-e )p dsl

1
S ‘t' ° c-g(0) -t sup leCAs p-pl-

eggs:
. cAs . . . .
Since e p is a continuous function of s, this a O as

t 4 o,
For the term (iii) _
1 t
IE f0 <9 (t-s) - g (o) )u(s)ds|

° t * 0

«us

max (Ig(s) -g<0)llu(s>|)
s€[0.t]

as s 4 0, since 9 and u are continuous for 3 2:0.

Next. for the term (ii),

[% J‘O [g(t-s) -g(-s)]w(s)ds

— O
- 3;- J‘t eCA(t 8) (JV g'(s-s ')W(8 ’)d8 ')d8]%
0 -.
1 O 1 1 O ' ’ ’ ’
‘ 5f [g‘t'S’ -9<-s>1-ew<s>ds-a I..." (-3 ms )ds

23

$33319 ’(8-8 ’) - 9 (-8 ’)w(s ’)ds ‘)ds %
t

I

o

o
= % I [g(t-s) -g(-s)]% W(s)ds—% If” g'(-s’)w(s’)ds

OIH OIH

(eCA(t'-S) — 1) (I0 g ’(s-s ’)w(s ’)ds ’ds

 

 

 

t
- 31E IO(G(s) -g(0) Ids
1. t cA(t-s)
- —- (e -1)G(s)ds.
ct f0
NOW,
.. .. 0
lim % Io g(t-s)tg( S) W(s)ds = % I g'(-s)w(s)ds
1:40 -a -a
so
lim (l: f0 g(t's’t‘gi‘slwmms-é {0 9'(-s)w(s)ds=0.
t‘O -O -o
Also,
lim'JL'It (G(s) G(O))ds - O
t-vo Ct o '
and
. _l_ t cA(t—s) _
a: ct Io (e -l)G(s)ds—O,
since G is continuous. Therefore
lim ut-p = lim (eCAt[P"]: JO 9(-S)W(s)ds]
t-oo t t+o ° -.
O
- p-% I g(-s)u(s)ds)%

and the existence of either limit implies the existence of

the other. By semigroup theory. the right hand side equals
Mcp - JD 9 (-s)w(s)ds) .
"I

and so (wyp) E D(A) iff wyw 6 K, lim‘w(t) = p, and
t40

O
A(cp-f g(-s)w(s)ds) 6 E.

24

In this case

. O
«(wup) = (w.A(cp-f ‘W(S)g(-s)ds)).

NOW the spectrum.of « will be found. First a few
definitions are needed.

Let R(X.V) = (V-XI)-1, where l is a complex
number. and. V is a linear, not necessarily continuous
operator on E. Let p(V) = {l E CIR(1.V) exists. is
continuous, and domain = E} and let 0(V) = complement in
c of p(V). 0(V) is divided into three subsets.

P0(V)

[l 6 CIR(X.V) does not exist}.
CU(V) = {l 6 CIR(XJV) exists with dense domain,
but is not continuous}.

R0(V)

{l e clR(x;V) exists, but its domain

is not dense}.

p(V) is called the resolvent of V, 0(V) the
spectrum of V and P0(V). co(V) and
Ro(v) the point. continuous. and residual

spectrum,respectively.

we have the following lemma
Lemma 2.2. For Re(l) > -a. define

u(s)ds.

‘5 Mt- )
Mx(u)(t) = £0 e s

25

Then Ml: K 4 K is a bounded linear Operator and
1
ml} 5 YT? '
2mg; Using the Cauchy-Schwarts inequality and

Fubini's theorem, we have

 

 

Q t
I I e2at( I ek(t_s)u(s)ds)2dt}E
..co 0
0 O
_<_ I eZat( [ e)‘(t—S) [u(s) IEdS} 2dt
..co t
0 ° (1+a)(t-8) as 2
SI ( I e e [u(s) IEdS) dt
_m t
O 0 O
5.} ( J‘ e(M-a) (t-s)ds) . ( J‘ e(H-a) (t-s)eZas|u(s) '1: ds) dt
..co 1; t
O O . - 2
-<- xia J. ( Lama) (t snfi‘zaslub) IE ds) dt
0 t 2
= xia I ( I 9””) (t’s)e2“|u(s) IEdt) ds
0 2
l 2as 2 l
= —— e Iu(s) ds = --—-——|u| .
X+a)2J~_m ‘ (x+a)2 K
1
5° “41' 5- x+a

The next theoran describes the spectrum of a in

terms of the spectrum of A.

Theorem 2.5. Let a be the operator in Theorem 2.4. Let

26

——-l‘-——— J -— R(x.A)A

c-§(X) X
p = -‘{R(u.A)v + h; I: g(-8)Mxeu(s)ds

where v E E, u E K. Let

q
ll

1 {llRe k g -a}

02 = {Mu 6 can}
03 = {Me-Sm = 0}.
Then
0(a) = 01 u 02 u 03.
Also, if k 6 PM). then R()..d) = (qu+e>‘tp,p).

Proof: First, 5m exists for Rem) >—a and

'1’; e‘XS g(S)d8| S (I: e(-Re).+-a)28 d8)i

 

if“ 8238 92 (8)d8'i S 1 . G
o flRe x+a)

Now. suppose (d— H) (w.p) = (u,v). Then w — w = u.

$0

= I: eMt-S)u(s)ds + extp,

26

u = —-—2;-,——. Jx = R(1.A)A
c-gU.)

p = .}{R(u,A)v + l; I?” g(-s)M>\un(s)ds

where v e E. u e K. Let

Q
II

1 {llRe l g -a}

Q
ll

(Mn 6 0(A)}

0
II

3 {Ale-6(1) = 0}.

Then

0(a) = 01 U 02 u 03.

l

Also, if x 6 pm). then R(l.d) = (qu+e tp,p).

Proof: First, 30.) exists for Rem) >-a and

'I’ e-XS g(s)ds| S. (I; e(-Re).+-a)2s ds)i
O

 

’IC e238 92 (S)d8li S 1 . G
O flRe x-I-a}

Now. suppose (a- u)(w.p) = (u.v). Then w - w = u.

So

w = J": eMt-s)u(s)ds + extp,
O

27

since lim w(t) = p. Substituting this for w in the
t-‘O
equation gives

0 s , a:
cAp - [ g(-s) (I eMs-s )Au(s ')ds ’)ds - (I e'xsg(s)ds)Ap - lp =v.
-c: O 0

So

0
[c-EU) ]Ap - lp = v + I g(-s) (MxAu) (s)ds .

If c-§(1)%O, dividing by c-fi'fl) gives

0
Ap - ——*———p =-——‘-’-—— + (f g(-s)(MxAu)(s>ds) —L—.

~

c-Em c-E'u) -.. c-gm

If ——2‘——-=u€P(A), then
c-§(M
o
p = (R(u.A)V) Lf-t % J- g(-8)MX(R(u.A)Au) (s)ds.

If -—-E"—— E C(A), then p will not be a continuous
c - g (1)
function of (u,v) where (u.v) E K x E. Also

if Re(1) S-a, M). is not a bounded linear operator, so
again 1 6 0(4) .

If c-;(l) = 0, then the equation for p becomes

0
- xp =v +I g(-s) (MxAu)(s)ds.

SO
v 1 0
P = - T - TI g(-s) (MxAu) (s)ds .
1 O
=-%t§AfgummfiHmu

28

Since A is an infinitesimalgenerator it has dense
domain and so Rum?) is defined here with dense domain, but
is not continuous. So 1 E C 0(a) .

Now suppose u 6 Po(d) , and Rem) >-a. Then if
(A-uI)q = 0 . (CZ-XI) ( enmq) = (o,c(A-u1)q) = (0.0) . so
x 6 P 0(a) .

Finally. for Re A _<_ -d, since functions vanishing
for S g t1.t1 variable are dense in K. it can be shown
1 E C 0(a) . Since this is not actually used in what follows,

details won't be given.

§5. Decomposing W into subspaces invariant under a .

In this section, it will be shown W can be decomposed
into subspaces invariant under a for the case where
A = the Laplacian A . and 83(0) = E. To do this, several
definitions are needed.

Let E be a Banach space and V be a linear operator
on E with domain D(V). Let the null space of V be
denoted by 72(V) and the range of V by MV) . Also let
x be a complex number, and let the generalized eigenspace of
l. denoted by mlW)’ be [xlx E E. (V-lI)kx = O for
some k 2 l. k an integer}.

A point A E P O (V) is said to be normal [10] if
1. mx (V) is finite dimensional.
k
2. MXW) = 72(V-11) for some integer k .

3. E = 771x(V) (+3 72 (V—KI)k

29

The set of all normal eigenvalues of V 'will be
denoted by NO(V).

The decomposition of W’ will depend on the normal
eigenvalues of d. the operator defined in Theorem 2.4.
So it must be shown that a has normal eigenvalues. This
is done using the theory of Banach space valued functions
of a complex variable. i

Let E be a Banach space. A function f:c 4 E is

analytic in an open subset G czc if lim f(p)-f(g)
u-k
Wk
exists at all points A 6 G [24].

A function f has a pole at a point l.€ C if f
is analytic in a deleted neighborhood of l and

lim(u--).)n f(u) exists for some integer n‘z l [24].
ufil

Lemma 2.3. Let A and a be the same Operators as
in Theorem 2.4. Let u E NO(A). and a be as in

Theorem 2.4. Suppose Re A > -a. Then X €1NO(d).

Proof: By Theorem 2.5.

 

kt

R(koa) (“IV) = (qu+e Px'px)

where

P)\ = L): R(u.A)v + { If. g(-s) (MxAR(u.A)u) (s)ds.

30

Now if “1 E N<J(A). then R(u,A) has a pole at
“l [24]. Since u is an analytic function of l and
R(u.A) has a pole at “l as a function of u..R(u,A)
as a function of X has a pole at 11 [24], where Al

is a number such that

Thus, for some n.

n
lvlz

exists. Also. if we let Jl = AR(u.A). then

J1 = AR(u, A) = I + uR(u, A).

and so

A n n
lim (x—xl) Jx = lim (x-xl) [I+pR(L1.A)]
1411 l+11

= 1' 1.. n , ,
x‘.l{!;-( ll) R(u A)

Thus Jl has a pole at Al-

Now

lim pxu - x1)“ = lim ifnumx) (x 41)”
X*Xl X4X1

. n
>‘l(1141)1\n1 (l - ll) JX) (w) (s)ds

and since the limit on the right exists. px has a pole

L11 0
+1?) g(-s)M

at X1 . NOW

31

lim (1 - 11)“ R (1 . 0) (mp)
1+11

. it n n
=11m((1-1)nMw+e(1-1)p.(1-1)p)
1411 1 1 1 1 1 1

and since the right-hand side has a limit. the limit on
the left exists, and so R(X.d) has a pole at 11 . Since
R(l.67) has a pole at )‘l . 11 6 No(a) [24].

Now consider the case where A = the Laplacian A.
and Hé(fl) = E. This is done not only to correspond to the
models in Chapter 1. but also to make E a Hilbert space
and A self—adjoint. These properties will be used in the
following theorems.

For the Laplacian, if 0 is an open, bounded region
with boundary. then A has only normal eigenvalues, which
consist of negative real numbers [16]. Thus by Theorem 2.5.
if d’ is the operator defined in Theorem 2.4. for the
case where E = Hé(0) and A = A and 1 is a complex

number, then 1 E 0(a) if and only if one of the three

following conditions are satisfied:

1. Re 1 < -a

2. -——:§——— 6 0(A)

c-gD.)
3. c-§(1) = 0.

Now for the Laplacian on 113(0) . MA) = N 0(A) . So

for A = A and E = Hé(0). Lemma 2.3 gives if

32

~ 6 0(A)
c-g(l)

then 1 E No(a).

Now define

IuI _1 = sup [ I uv.,IvI l = l}
H HO

for u an infinitely differentiable function with compact

support in 0.. Let H-1 be the completion of this space

in the norm I I _1 . Also define
H

< a,b >H = I ab .
0

Then H is the dual space of H3 using the duality pairing

< , >H" Also, A is a continuous linear operator from
H; to H"1 [ ]. This space is used to define a bilinear
form on W’ and it will be shown later that if u(t) is a
solution of (2.2) with E = Hé(n) and A = A. then for
t go, {1(t) is in H'lm).

Recall the definition of ‘W. In this case,

W = L at(

1 1
”“0071! (0)) XH (0) .
2,e 0 0

Now a symmetric bilinear form1will be defined on W’ by

(2.5) < (mp). (u.v) >4: C<p.v >H
o O
_ I ( I < w(s-s'). Au(s'):fiids')g(-s)ds.
..u: 8

Since A is a continuous linear operator from H; into
H-l. this is defined.

33

This bilinear form is similar to the one defined by
Jack Hale for ordinary delay differential equations [11].
and is used for a shmilar purpose. to decompose W’ into
subspaces which are invariant for (2.1).

First. it will be shown that d’ is "self-adjoint"

with respect to the bilinear form (2.5) .

Lemma 2.4. Let a be the operator defined in

Theorem 2.4 for the case where E = H310) and A = A .

Let < . > be the bilinear form defined by (2.5). Let

4

w and u be elements of L at('°° . O : Hé(fl)). where a
2,e
is as in Theorem 2.4. let p and g be elements of

113(0), 1et (w.p) €D(d) and (u.q) €D(d). Then

< d(W.p).(u.V) > < (W.p).c7(U.V) >-

Proof:
0
< d(W.p). (u.V) >a = < (W.c Ap- Ig(-8)W(8)ds). (u.V) >0

0
< c Ap- I g(—s)Aw (s)ds , v>H

o 0 .
-I (I< w(s—s') , Au (3’) >H (-s)ds')ds

...a: 3

O
< c Ap.V>H - I g(-a) <Aw(s) .v>Hds

0 O .
-I ( I < w(s—s'). Au(s') >Hg(—s)ds')ds .

34

Since the Laplacian is self-adjoint in the < . 2H

inner product. and since

d
a?" <M(s-s ’).u(s ')>H = - <A17v(s-ss').u(:ss')>H
+ <M(s-s ’).13(s ’)>H

we have

(C Ap.v>H = <1>.cAv>H

0 o
_ I.“ f(-S) <M(8).V>Hds = I a g(—s)(W(s),Av>Hds.

Integration gives

<cw(m.u(s)>H - <AW(8).\1(0)>H = -f <m(s-s').u(s')>Hds'
o w

+ I: <Aw(s-s').t°1(8’)>H

where w(o) = lim“w(t) and u(O) = lim‘ u(t). Since (w,p)

tdo tfio
and (u.q) are in D(a). Theorem 2.5 gives w(o) = p and

u(O) = q. This gives

<Ap.u(s)>H - <l‘.w\V(s).q>H =
S 0 I 1 , d ’ O . '
_. IO (M(S‘S )pU(8 )>H S + J: (”(3.8 )pU(S )>H.

Using this.

35

o
<cAp,v>H- I g(-s) < Aw(s) ,v>Hds
o o .
- I (IONS—5') . Au (8') >Hg(-S)dS')ds

..cn S

0
= <p.c Av>H - I g(-s) <Aw(s) ,v>Hds

...m

0 o
+ I 9(-S) <Aw(8) .V>Hds - I g(-s) <p,Au(s) >Hds
O
- I (I <w(s—s’>.m‘1(s’)>Hg(-s)de')de.
..co 5
Making the change of variables s’I = s--sI in the last

integral, we have

0
(g(wop)o (uIV) >d = (PoC AV>H ’I g(-S) <p,AU(S)>HdS

o o .
- I ( I <u(s—s”). ANS”) >dS")g(—S)ds
” s

=< (W.p).67(U.V) >d .

This result is used in later calculations.

Lemma 2.5. Let d’ be the Operator defined by (2.4)
with E = Hé(fl) and A = A and let k be an integer '2 1.

Then a necessary and sufficient condition that the equation
k
(4-11) (W.p) = (11.1!)
has a solution for some given (u,v) is that
(2.6) < (u.v) . (S.q) >4 = 0

k
for all (S.Q) E W(d¥-1I) -

36

Proof: It is easy to show (2.6) is necessary. If

(dF-XI)k(w,p) = (u,v), then using Lemma 2.4,

< d- XI)k(W.p) . (S.q) >a

< (W, P): (d- AI)k(S,Q) >d ((Wop)o (0,0)>a = O

for' (s.q) E W((d¥-XI)k) .

To show the condition (2.6) is sufficient, a
characterization of W((d¥-11)k) is needed.
If (6(— u)k(w,p) = 0, then (2%.- u)kw = 0. So for some
xjfl 6 113(0).

k-l 1t -
J
W "' Z ‘Y. 2'7- t .
j=0 3+1 J.
Also, for O 3.1 g_k
(a411)1(w.p)
. k-l It k-l 1t .
‘ d
((3915-1111 '23 YJ+1 9.7-9. lim (33-11).): v3.” 5357—9))
=0 3 t-vo" J=0
k-i-l elt J
= ( jEO YJ+1+1 j: t ' Yul)
Since
(a-u)(a-u)1<w.p) = (a-11)“1(w.p) .
this gives
k-i-l 1t . k-i-2 it .
(d’XI) Z Y. . §.—;— t3 ' Y. = y. . ?- JIY' \
( j=O 3+1+l 3. 1+1.) ( j=0 J+r+2 3. 1+2/

which gives, using (2.4),

36

Proof: It is easy to show (2.6) is necessary. If

(03-11)k(w,p) = (u,v), then using Lemma 2.4,

< a— u)“(w.p) . (s.q) >d

< (w. p).(a—mk<s.q> >4 <(w.p>. (°'°)>a = o

for (s.q> e 72((d—11)k) .
To show the condition (2.6) is sufficient, a

characterization of fl((d%-1I)k) is needed.

k
If (d-XI) (w.p) 0, then (é%-xl)kw = 0. So for some

1
xj+1 e Hom) .

k-l 1t -
J
W: Z ‘Y- 27—1“. .
j=0 }+1 J.
Also, for O g_i g.k
(d- XI)1(W.p)
. k-l kt k-l kt .
‘ d
((§€-).I)1 Z yjfl fife-t3 . lim (gag—XI); Y3.” 957-9))
'=o ' t-vo" 3=0
k-i-l 1t . .
= ( Z Y. . £31- tJ , Y. .
j=0 3+1+1 3. 1+1 )
Since
(d-lI)(d-XI)1(W.p) = (a-11)1+1<w.p) .
this gives
k-i—l 1t . k—i-2 1t .
- _e___ 3 - . s,_ 3 \
(d XI)( jEO Yj+i+1 j: t ' Y1+1) —( jEO Yj+i+2 3! t 'Yi+2/

which gives, using (2.4),

37

k-i -l els sj
GAYi+ I: g( s) j§0 AYk+i+1-—— ej. d3 )‘Yi +1 = Y1+2

for O < i < k-2, and

c Ayk-( I g(-—s)e)‘sds)Ayk-).'Yk = 0

.00

for i = k-l.

The last equation gives

The equation for i = k-2 is, after some simplifica-

tion,
~ ~I
[C-9(M]Mk_1-g (1) MR - 1 Yk_1 — Yk
or
~I
A“'k.----"1"“Yk1 1 AYk= Yk 3
c- g(l) c-g(l)

Applying (A-uI) to this gives

2 _
(A—uI) Yk_1 - 0

and so Yk-l E w(A-u1)2. But since the Laplacian is self-
adjoint. 72(A-u1)2 = nus-(11> and so vkd e flux-111).

Proceeding by induction, Yk E W(A-uI) for 1 g.1 ka. and

“’00 1

Also (—1)1 1: =3? 30 if (w.p) enm-xnk, then

kl j
wejzjy “=2.

(2.5) j+1 e j!

, and p = y1,. where Yj is

an eigenfunction of the Laplacian, with eigenvalue 1, and

38

. (1)
(_1)J E_ITlAl.= %-. Direct computation shows this is
also sufficient.
Now if ((u,v),(s.Q)>d=O for all (SJ?) 672(d-ll)k.

then by equation (2.5)

o o k-l el(s-s') . j ’
C<V'Yk>H -I°:3(-s)( I j§o<Yj+1 --5.—--(s-S) .Au(s )>dS)
" S

k-l 0 0 I I j
x _ _
= C<V.Yk>H—j2 (Yj+1: I 9(-S) I6 (S s ) 5—3:?)— Au(s’)

> ds = 0

where each Yi is any element in fi(A-—uI). Since the

yi's are arbitrary elements in fl(A-ul). this means

0 0 I
c<v.\(k>H ~<Yk. I 9(-S)( I ens-S)

Au(s') ds')ds >H = o

for all Yk E fl(A-uI). Since A-uI satisfies Fredholm's

alternative [ ], this means

0 1
cv - I g(-s)Mx Au(s) ds 6 R(A—LLI)

and so,

0
cv - I g(—s)MxA(s)ds = (A-uI)L

for some L 6 H3 .

Also, since each Yi is arbitrary,

o o ,
< Y1. Im9(-8)( I eMs-3 )Au (S')dS')ds > = 0
... S

for each Y1 E W(A-uI). In the same way

39

O
I g(—s)M)\1Au(s)ds= (A-—L1I)|II.1

>\

t .¢i+etp+(M£(u)(t).

Then direct calculation shows
k
(d-XI) (pr) = (up‘f) .
k
Thug: (uov) 6 9(4- AI) 0

Theorem 2.6. Let [11.....1n} be a finite set of
eigenvalues for the operator a. d the same operator as in
Theorem 2.4, for the case B = 113(0). and A = the Laplacian.

Let 7):)“ (d) be the generalized eigenspace of a for
1

)‘i and m: 9751(0) 63 ”(Mm-~- ® mxn(m. Also let

m°= {yEWI<y.z>a=O for all 2677:}.

Then 77: and 7R0 are invariant under the map eat, i.e. if
YEW). then eatyem. and if 2677?, then eatz 677:0
for all t > 0.

Let [col,- --,cpm} be a basis for 77: . Then there is

another basis {11:1, - ° - , Im} for 7): such that

<mio¢i>d=l and <COiI¢j>=O if 1;!j.

Define P: W 4 W by

m
(2.7) 92:: Z“. <¢., z> mi.

40

Then P is a projection onto 771, that is, if z 6 W,

pzem, and if 2577),Pz=z. Also for any t_>_O

edtP = Peat

. . . 0
Also I -P 13 a progection onto 773 , and

eatu —p) = (I —1>)e”t .

Proof: First, since the Laplacian has only normal
eigenvalues, each xi is a normal eigenvalue. Thus, each of

the ”‘1 (a) is finite dimensional and so is their direct
i

k.
1

sum. Also each Mk (0) = 72(d-1I) for some ki . Since
i

772x (d) n ml (6) = {O} , Lemma 2.3 and the definition of
i 3

normal eigenvalue give

n
9 Md-xl).

(2-8) W = 771x(d)@772)\(d) @°°-® 771)‘ (d)@( 1
n 1-

1 2

Now assume {col,---,com} is a basis for 7):. If for

every I 6 M. 4'?! O , there is a :0 such that

< $.60 >4 51 O , that is, if < , > is nondegenerate

0
on 771 [14], then the Gram—Schmidt process will give a

basis [I1.--'.\)n} such that <cpi . *1 >4 = l and
(wi.¢j>=0 1f 1%]-
Now suppose < ¢,cp. > =0 for j = l, - - °,m . Then by

31:.

Lemma 2.4, (Ema-111) 1 for i = l,°“,n. But then,

n
since I67) and If n Sud-111) . 111:0. So < .

>
i=1 a

is nondegenerate on 77: .

41

n
Let P(x) = Z <x,wi>« cpi. Then Px em. Also
i=1

n
P(cpi) = .21 <mi'¢i>« cpi = “’1' Since P is linear and
1:

sends each element of a basis of 77: into itself, le is
the identity. So P: W 4 W is a projection onto 772.

Now let y 6 mo. Then since <y,cp> = O for any
m E m. <Ya¢i> = 0 for all i. So Py = 0.

Conversely, suppose Py = 0. Since the cpi form a
basis, they are linearly independent. Using equation

(2.7). (y. $1) = O for each “'1' Since the Vi form a

basis of m <y.z> = 0 for any 2 6 7n. and y 6 7720. Thus

7UP) = mo. Since P is a projection, 7UP) = EDI-P),

which shows that I - P is a projection of W onto 77:0.

0 n k.
Now, 971 = 0 8(4- XI) 1 by Lemma 2.5. Since
i=1

eat“ (x) = a

= 19(4- AI)k eatx, 8(4- u)k is invariant under e

e“(t)(x) for x EDM) and, eatf?(d-ll)k(x)
at

Now let x E W. Then by (2.8)
x: y+z, where y 6777 and z 67710 and
Peatx = Pe«t(y+z) = Pemzy + Peatz

«t at Kt
e y e e

In the same way

42

§6. The Operator 0' and solutions of (1).
Now the connection between the operator 0’ and

solutions of (2.1) will be shown in the special case where

 

A = A and E = Hé({n.
Theorem 2.7. Let X = L t(-°°,O:H1(Q)),
2'ea O
l l
W=XxHO(n). let w EX. and p SHOW).

. . _ . 1
Define P1. W 4 X by P1(w,p) —‘w, and P2.W-+HO(Q)
by P2(w,p) = p .
Let (7 be the operator defined by equation (2.4) , with
A = A and E = Hé(0). Then for any given initial condition

(w,p) E W, the equation

t
(2.9) {1(t) = A(cu(t) —_[ g(t-s)u(s)ds)

1

with u(t) 6 H- (O) has a unique solution given by

(2.10) u(t) = Pzeat(w,p), t 2.0.

Proof: By semigroup theory [13], if (v,q) 6 D(d),

then eat(v,q) E D(d) for all t > 0 and
a%e“<v.q)) = a(e"t(v.q)).
Now let
edt(v.q) = (v(ti).q(t)).

where v(ti) 6 X for each t, and q(t) 6 H3 for each t.

43

By Theorem 2.4,

3—5 v(t,s) = 5% v(t,s) .

Thus
v(t,s) = v(t+s) .
Also since

lim_ v(t,s) = lim_ v(t-bs) = q(t),
s40 s40

it follows that v(t) = q(t), for t >»O. From this,

it follows that

o
q(t) A(c q(t) - j‘ g(-s)v(t.s)ds)

A(C q(t) - f0 g(-s)q(t-+s)ds)

A(c q(t) - ft g(t-s>q<s>ds)

Also, by definition,

q(t) Pzedt (Wop) -

Now take any (w,p) E W. W(d) is dense in ‘W [13],

so there exist xn e‘W such that

lim xn = (w,p).
n4.

44

Then each Pedtxn satisfies

t
xn) = A(c(P2eatxn) - I g(t-s)(P2e

0t <7

d s
E'E(Pze xn) ds).

A is a continuous linear operator from H; to
H_1 and lea%| g keat' for some constants k and a

at

not depending on t [13]. Thus e xn 4 edt(w,p) uniformly

on any interval [0,T], T < m. and

t
at as
A(c(P2e xn) — I g(t-s)(P2e xn)ds)
converges uniformly in the norm of H"1 on h3gr].Therefore
d at d Gk
at”? xn) *5? (P26 X)

by the theorem on uniform convergence of derivatives [15].

So Pzea%(w,p) = u(t) satisfies

t
&(t) = A(c\1un — I g(t-s)u(s)ds)

Plxn 4'w, and szn 41p, so u(t) = w(t) for t < O
and u(O) = p. So u satisfies the initial conditions.

Also, u is unique since if u satisfies (2.9) with

-1

13(t) e H1

the variation of constants formula gives

t s
u(t) = eCAgp —.f AecA(t-s)( II g(s—s’)u(s')ds’)ds.

Integration by parts gives u(t) is a solution of (2.2)

which by Theorem 2.1 is unique.

CHAPTER III

NONHOMOGENEOUS LINEAR PROBLEMS AND PERTURBED
NONLINEAR PROBLEMS

§l. Introduction.

This chapter will first be concerned with nonhomo-

geneous linear equations of the type

t
(3.1) (m) = c Au(t) - f g(t-s)Au(s) ds + f(t).

_m

A solution will be in the same sense as in section 2.6,
that is, u(t) will be in Hém) for each t, {1(t) will
be in H-1(Q).

Once a result has been obtained for (3.1), it will
then be used to Obtain a result on equations of the form

t
(3.2) {1(t) = c Au(t) - I g(t—s)Au(s) ds + f(t,ut,u(t))

These results come easily from the variation of constants

formula for semigroups.

Theorem 3.1. [13]. If A is an infinitesimal generator

on a Banach space E, f: [0,T] 4 E a differentiable

function, and x E D(A), then the equation

f1(t)= Au(t)+ f(t). u(O) = x

45

46

has a unique solution

t
(3.3) u(t) = eAtx + f eA(t-S)f(s)ds.
O

§2. The inhomogeneous linear equation.

Theorem 3.2. Let 0 and Hé(0) be as in section 2,5.
Let a > O,

X = L at(

-a°,0 ; Héfil) and
2,e .

_ l
W — ijHO(O).

Let g:(—w,0] 4 R be differentiable and satisfy

(i) [ [g(s)]zezasds < co.
0

(ii) J [g'(s)]2e2asds < a.

O

For any T > O, x e X, p 6 H3 and continuous

f: [0,T] 4 Hé(fl), the equation

t
c.Au(t) — I g(t—s)Au(s)ds + f(t)

(3.1) h(t)

u(t) = Mt). t < o

u(O) p

has a unique solution on [0,T], with u(t) 6 Hé(0) and
G(t) e H’1(n) for each t 2_o.

47

nggf: Let a' be the infinitesimal generator defined
by equation (2.4) in the case where E = H3(Q) and A = A.
Let v e x, q e H; . (v.q) sum) , and r: [0,T] .. H3)" be
differentiable. Then h: [0,T] 4‘W defined by

h(t) = (O,r(t)) is differentiable so the equation

(3.4) h)(t) a w(t) + h(t)

with initial condition

«u0) (V.q)

has, by the previously stated theorem, the unique solution

t
co(t) = eat(v,q) + I ea(t-S)h(s)ds.
0

Now, as in section 2.6, if w E X , and p EH25 , then let
Pl(W.p) = w P2(W.p) = p.

For each t 2_O, w(t) 6 W. So le(t) E X. This means
P1m(t) is a function on (-w,0]. Let l(t,s) = [le(t)](s),
for s _<_ 0. Then X is defined for s _<_ O and t _>_ 0. Also
let P2m(t) = u(t). Then w(t) = (l(t,o), u(t))-

By equation (3.4) and equation (2.4), since

Sgt-(w(t)) = (( Pig) (to ' ))o (%%) (t) ,

(5-3; x) (t.s) «3%) (t.s)

SO

l(t,s) = l(t+s) .

48

Also by Theorem 3.1, (l(t,°),p(t))‘6D(a) for each t 2_O.

Thus lhn 1(t.8) = u(t). From this l(t) = u(t) and hence

s40

u(t) satisfies t

h(t) = c Amt) —I g(t—8)Au(8)ds +r(t).
setting u(t) = v(t) for t < 0. Also

(1(0) = Pzea‘o’ (v.q) = q.

Now D(d) is dense in W. Also, the differentiable
functions from [0,T] into Hé(fl) are dense in
C[0,T : 1130)] [15]. So let (w,p) e w and let
f(t): [0,T] 4 Hé(fl) be continuous. Choose (vn,qn)<ED(d),

such that lim (v ,q ) 4 (w,p), and let rn be differentiable

11".” n n
and rn 4 f uniformly on [0,T] as n 4 4.. Then
a T ( )
t d t-s
e (vn.qn) + I e (0,vn(s))ds a
o
T
edt(w,p) + I emt"s)(o,f(s))ds
o .

. . _ at
uniformly on [0,T]. Also, if un(t) - Pze (vn,an)

t
un(t) = Pzedtwnan) + I0 ed‘t's’ (o.f(s))ds.

then
t

(Inn) = A((c un(t) - I_¢g(t-s)un(s)ds) + vnm

1

converges uniformly on [0,T] in the norm on H' . Therefore

t
hum 4 3% p2(e"t(w,p) + I e"(t‘s) (O.f(s))ds)
0

49

and thus
a t
(3.5) u(t) = P2(e t(w,p) + I emt’s) (O,f(s))ds
0
satisfies

u(O) = p = Pzeam) (w,p)

t

cAu(t) - I g(t-s) Au(s)ds + f(t).

m

{.(t)

This not only shows a solution of (3.1) exists, it gives a
formula (3.5) for obtaining the solution. Uniqueness comes

from the uniqueness of the homogeneous equation.

§3. The perturbed linear equation.

In this section the equation

(3.2) u(t) = c Au(t) - It g(t-s)Au(s) ds + f(t,ut,u(t))
. _o
u(t) = w(t), t < 0
u(O) = P
will be studied. Again, for each t > O, u(t) E H; and
u(t) 6 H"1 . Using the variation of constants formula in

section 3.2, we have

t
(3.6) u(t) = P2(é7t(w,p) + I edIt-S)(O,f(s,us,u(s))ds).
0

Conversely, if a solution to (3.4) can be found, then by

Theorem 1, (3.2) has the solution u(t).

50

Theorem 3.3. Let 0' be the infinitesimal generator

in Theorem 2.4 for the case where A = A and E = H; .

Let
x = L (-.. O°H1(Q)) and
at l I O I
2,e
W = X XIII-(Q):
O
U be an open set in W , f: R x U 4 Hcl) . Assume that

f satisfies

1. f is continuous in t.
2. There is an L > 0 such that for any t1, for any

x E U ,

1'X2

|f(tl'xl) ' f(tl'xz) 'H. 5- L |"1""2 |w
o

If (Wop) 6 U p then

t
cAu(t) - I g(t-s)Au(s) ds

(3.2) u(t) =
+ f(t,ut.u(t))
u(t) = w(t), t < o
11(0) = P

has a unique solution in [0,t1], for some t1 > 0.

Proof: This is just using the variation of constants

formula (3.4) and Banach's contraction mapping theorem.

K2t

There are K1,K2 such that Ieatl g Kle [13].

51

Choose 6 > O and T > 0 so that

{(S.q) Il(8.q) - (W.p) IW < 6] CU

and

Hem! _I) (w,p) |w $361 for O

I/\

hg'r.

(Since egt(w,p) is continuous in t, this can be done).
K T
6

2 (B+L6) < —

Also let B = max |f(t,w,p)|0‘t‘1,.TK1e 4,

T<l,Iw-wT|<g-, and TeaT<%.

Let S = [q €C[O,T:W]Ilg(t) - (w,p) I _<_ 61 for all
t 6 [0,T], (P,g(t))(s) = Pzg)(s), O < s < t}. Then S is
is a complete metric space since it's a closed subspace of

c[o.T:w].

Define G: S 4 S ‘by G(v(t)) = (gt,.g(t)), where

t
g(t) = P2(eat(w,p) + I ea(t-S)f(s,v(s))ds).
0

It will be shown that G is a strict contraction.on :3. First,

1mm (1:) - (w.p) Iw _<_ Heat-I) (W.p) IW

K T

t
+ | I ed(t"3)f(s,v(s))de| _<_-3+ (TKle 2 )(B+L6) 03-5.

0

by the choice of T. Also

0
|w-P1(G(v) |(t) | _g Iw-wtlw + ( I e"2“‘s P2|Gv(s) |2ds)1b
t

Clearly (P1(Gv)(t))s = P Gv(s), so G: S 4 S. Also, G

52

is a contraction. Let v and q be elements of S. Then

t
IGv-G I g_ sup IP2 I eth—S)(f(s,v(s))-f(s,q(s))ds| 1
‘33 tE[O,T] 0 HO
K T
_<_K1e2L!v-CI| <%;|v-<2|
s

by choice of T, and thus, G is a contraction, and so has
a unique fixed point. This fixed point gives a solution of

(3.6), which by Theorem 3.2 gives a solution of (3.2).

CHAPTER IV

THE SADDLE POINT PROPERTY

§1. The spectrum of the semigroup.
This chapter will be concerned with the saddle point
property for the equation

t
(4.1) u(t) = c Au - I g(t-s)Au (s)ds + f(t,ut,u(t))

with lim f(t,w) = o(|w|).

w40
To prove the saddle point prOperty for (1), the
spectrum of eat must be found. By a general theorem on

semigroups [13], if 1 6 0(0). then e)‘t 6 0(eat). Also,

lt at

if e EPO(e 27”“

) then X + T— E Po(a) for some

integer n. But eat

has continuous spectrum besides 0,
which for a general semigroup does not necessarily arise
from the spectrum of the infinitesimal generator. Hence,

something more is needed to find the spectrum of eat.

When the spectrum of edI' is found, this can be used
to determine the rate of growth of edt restricted to
invariant subspaces.

First, a lemma is needed.

Lemma 441. Let g and A be as in Theorem 2.1 with

the additional restrictions:

53

54

1. g is decreasing

2. c-;(O) >0.

3. There exists a K such that |R(A,l)| < TE—Tc- for

l>-k.

(This means A + k1 generates a contractive semigroup [13] ) .

Assume there is a root 1 of

l = 1:650.) -c) + 33-}:Q

with Re 1 > -a and let

)‘1 = max [Re 1 I). = k(§(l) -c) + 395.1(3)].

Then there is a constant M Z 1 such that, for w 6K and
p 6E the solution u of equation 2.2 with initial condi-

tion (w, p) satisfies

Alt
I“ (t) I _<_ M e lep) IKXE ‘

Proof: We have that u(t) satisfies equation 2.2 so

0 t
u(t) = eCAt(p-% Img(-s)w(s)ds) + 21; Img(t-s)w(s)ds

t O
.. .35: I ecA(t-s)( I g'(s-s ')w(s ’)ds ‘)ds
0 _Q
t t
+ 215- I g(t—s)u(s)ds — HJEQL I eCA(t-S)U(s)ds
O O

55

t s
- %.I eCA(t’S)( I g'(s_s’)u(s’)ds’)ds.
o ‘0

Since A + k‘I generates a contractive semigroup,

‘eCAt I -th

_<_e for t 20 [13]. Also, since g(s) _>_o

and g'(s) _<_ 0, we have, by taking norms, that

o
4‘“ IP -;l;I 9(—8)W(S) ds IE

-0

(4.2) Iu(t) IE _<_ e

O t O
+ I%I g(t-s)w(s)dsIE+-1- I e—kc(t—s) II g'(s-s ')w(s')ds'IEds

C .
O
t t k
+ % I he he + 44 I .- ....) his
0 O
1 t k (t ) ‘S
+ zI e“ c '3 (I) -g'(s-s’> Iu(s’> Ids‘>ds-
0

Using integrationby parts on (4.2) gives

. O
Iu(t) IE 3 e‘k‘Cth -% Img(-S)W(S)ds) IE
0 t O
+ %I'I-mg(t-s)w(s)dsIE + 2]?- IO e-k.c(t-s) II g'(s—s ')w(s ')ds'IE

t S
+ k.< I e‘kc(t‘3)I g(t-s)Iu(s)IEds
o o

t
+ 2 CO) I) e-kc(t-8) Iu(s) [Eds .

Let

56

O
( I (g(-s))2 8-2aeds)a .

G =
and
O 2 -2as fi
G1 = ( Im[g’(_s)] e ds) .
Then
1 O '
IP-g Img(—S)W(s)dsIE _<_ (9+ 1) I(w,p) IKxE.
Also
0 ° 2 2 5
I I g(t-S)W(S)d8| _<_ ( Iwg (t—s)e" asde) IWIK'
Since

0 t
I 92(t-s)e'2asds g I 92(t-s)e-2asds

t
-2at I
e

a.”

g2(t_s)e2a(t-s)ds = e-ZatGZ

O
I g2(t-s)e’2asds)i g e’atG .

Next, consider the term

I

O

t O
e-kc(t-S)I I g'(s-s ‘)w(s ')ds'Ids .

Now as above,

0
II—mg’(s-s ’)w(s ’)<3e’|Hl _<_ (61+ 1) lex e‘39
o

S7

and so
t o
I e4kc(t-S)I I g'(s-s')w(s')ds'Ids

_m

Ite-kc(t—s)e-as(

V\

GI+JJIwads

0
t

= I e-kCte(kC-2a)s(Gla-l)IwIde
O

l I -kct
g, (a-kc) e IGl-I-lIIwIK .

 

From this

0 O
e’kct Ip -%— I.” g(-s)w(s) ds IE +'('1._.'I I a g(t-s)w(s)ds IE

1 t k t O
. .u) .- .( I I
0 _Q

.1:
g .9 e Ct I (W.p) leE

where

1

Q: (G+l) +EG+Ta3c (G1+l)'

i.e. Q is a constant not depending on (w,p). Thus

Akct
Iu(t) IE _<_. QHWIP) ‘KXE e

t s
+ K I eékc(t—s)( I g(s-s')Iu(s')IEds')ds

'o o
2 t 4k
+ q§02 I e C(t‘3)|u(s)|Eds .

O

58

Let v(f) be the solution of

(4.3)

-kCt t -kC (t-S) S o a» I
v(t) = QI(wyp)I e -+K e ( g(s—s )v(s )ds )ds
KxE I6 £0

+ gig—(2)— ‘I‘t e-kc(t—S)v(s)ds
0

then Iu(t)IE g v(t) for t.2 0 by a comparison theorem
for Volterra integral equations [19]. But if v satisfies
(4.3) for t 2;O, setting v(t) = O for t < O and

differentiating gives
t

(4.2) v’(t) = -kcv(t) + Ziggy-v(t) + kI g(t—s)v(s)ds
-a

with v(t) satisfying the initial condition

v(t) = O t < O

v(O)

Q IIWIP) 'KXE.

Hence, by [20], v(t) increases no more rapidly than the
real part of the root of the characteristic equation for
(4.2) with largest real part. to be exact.

It

A
NH 5 Mle Ql (MP) [W

where M1 > 1 is some positive number. Since 1 > 11.

and Iu(t) [E g v(t),

59

xlt lt
Iu(t> IE _<_ Mle 0|(w.p) leE 5 M06 ((w.p) lw.

Now a few definitions are needed. In section 2.5 the set

of normal eigenvalues N 0(V) of an Operator V on a
Banach space E was defined.
The essential spectrum of V, denoted by 06(V)' is

defined as
06(V) = G(V) - NO(V)

and the essential spectral radius of V by
rE(V) = sup {IlII l E O€(V)]

when this exists. If V is a bounded Operator, then G(V)
is bounded, so r€(V) exists.

The Kuratowski measure of noncompactness u(B) for a
bounded subset B of a Banach space E is defined by
u(fi) = inf [rIB can be covered by a finite number of balls
Of radius r} [23].

If V is a bounded linear Operator from E to
E. the measure Of noncompactness, G(V), of V is defined

by
u(V) = inf {r Ia(V(B))'_<_ra (V), for all BCE].

Clearly u(V) g IVI. Also let u1,u2,°'° be the eigenvalue
of the Laplacian on 0, arranged in decreasing order, with
multiple eigenvalues listed once for each multiplicity and

m1.m2.°".mh.--- be the associated eigenvectors.

60

If Bn = span ($1,...,¢h} let V1,n be the

orthogonal projection of Hé(0) onto Bn and let
V2'n = I-V1’n. Now let P1,n: W 4)W be defined by
Pl'n(w,p) = (u,vl'np) where u(s) = Vl,n w(s). Then
P1,nPl,n(w'p) = Pl'n(w,p), so P1,n is a projection.
Let P2,n = 1"Pl,nf These definitions and projec-
tions will be used in the following theorem

Theorem 4.1. Let 9,0, Hc1)(0), X,W and d be as

in Theorem 2.6. Set

3(1) = I e-xsg(s)ds.
0

Assume 9 also satisfies
(1) g is decreasing
(ii) c - 3(0) > 0.

Then the equation c - 341) = O has a largest real root

1 t
l‘ d’ generates a semigroup e“t E(eat) = e 2

l and r
Also, if u > 11, there are only finitely many
elements in 0(a) fl [lIRe 1 >11]. All these points are
elements in the normal spectrum. If P is the projection
defined in Chapter 2, section 5.4, then I(I-P)e«tI sKeut

for some constant K > 0.

Proof: By Theorem 2.4, d’ is an infinitesimal gener-

ator. Let W = P (W) and W = P (W) If

2,n 2,n
(0), then

l,n l,

v 6 Hé(0) and Axle H

OHS

61

V: 2“ <VrcD-> CD-
i=1 1 H1 1
O
and
Av = 2 Hi <V.co-1>Hco1
i=1 HO
Hence
n
Vl'nAv = E ui<v,coi> lmi = avl'nv.

1

H.
II

Ho

Pl,nd(w1p) = (Vl'nw,V1'nA(Cp —Icg(-s)w(s)ds)

47),)“chn —I g(-s) an w(s)ds)

((Vl,n

-a)

a P1,n(w'P)°

 

A130:
n67(w.p) = (I-Pl. n) d (w, p) = 4(1 -P1 ) (w p)
= sz'n (w,p) .
l t
Now suppose r6(e“t) > e)‘t and e 3 = r€(e“t).
11+ 13
Let 12 = -—7f—— Since g(s) is decreasing and positive,
3(1) is decreasing for 1 real and 6(1) I g g(Rel). Let
_ 3.9.12).
1 c
k = 2
c -g(12)

and 2 be a solution of k(c-§(z)) = z - 3%49-1- .

Then

62

k(c-Re§’(z)) = Re(z) - 2 0).

Now
c—Re§(z) _>_c- Ig(z)I_>_c-§(Rez).

Since c-§(Re(z)) > O, and K < O, we have that

 

_ 23(0)
c
kzkgl
c-g(Re z)
x-29.$9l
Since ———-9-—- is increasing on [Mg-Egg], Re z < )2.
c-é‘fm

Suppose x E 3"". Then

2 2
<Ax.x> _<__+ (ln+1)IxI 0_<_+ lel 0'
H1 H1

and hence A + k1 generates a contractive semigroup [13].

Also W2 n is invariant for 4. since
P2,ndx =dP2'nx for x E D(d) . Define

_ 1 _ 1 .
Bn + — (v 6 HOW) I <v,s>Hl - O for all s 6 En] (Bn 1s

0
the orthogonal complement of En).

If v EB; and Av EHCJSUI), then

(onX>H1 = < Z <Xocoi>cpio 2' ui<xomi>mi>

0 n+1 n+1
E 2 I 12 )2
= u.<x,co.> _<_)..L x _<_le .
n+1 1 1 n+1 113(0) Hg)-

SO (A + k1 ) IB 1 generates a contractive semigroup [13].
n

63

Also if x E D(d) n‘w

I

2,n

O
d‘WZ n(W.P) = (W.AIBn1 (Cp-I-mg(-s)w(s)ds)).

1 satisfies the requirements for A in Lemma
-1 t
a -k
IeatI = IP e tI < Q<e It <(2e 2
W 2 -—
2,n
where Q is some constant > 0.

Recall the definition of the measure Of noncompactness

a of an Operator. From inequality (4.4),

a
2,ne

2

u(P t) _<_Qe

Now consider aflw . 0n Bn.’ the Laplacian is a continu-
l,n
ous linear Operator and hence

o
1 = (w,AIanp - I g(-s)w(s)ds))-
In '4”

Thus «IW is the infinitesimal generator of the solution
l,n

semigroup to an ordinary delay differential equation and by

a general theorem [20],

where a is the positive number such that

I g(s)2e2asds < w .
0

By another theorem [22], for any bounded Operators V and

S on a Banach E,

64

n r—--
rE(V) = lim Jaw“)
nam
Also
d(V+ S) SOLW) + u(S).
Therefore
1
r6(e t) = lim (h(enm‘nn _<_ lim (G(Pl en“t)+o.(1>2 emtn
_ n4on n4m ,n ,n
l
1 nt -
< lim (e 2 +0422 ne"““))n
n40° '
l
l t —1 t —-
= 11m e 2 (1 + e 2 d(P2 Mt))n
n4co ,n
Now
1
-l +n ~—
lim (e 2 u(P2 endt)n
n4co ,n
-l t (a—x )t
= e 2 eat _ e 2 < 1'
since 12 > a and hence,
-l tn
lim (e 2 “(P2 en“t)) =o
n4m‘ ,n
and
at "2‘: '7‘2“ hat '31'
r€(e ) < e lim (l4—e G(P2 ))
n4up ,n
1
l t —- l t
=e2 1m (l+0)”=e2
nae
A t - l t
This implies that r€(e“t) g e 2 , contracting r€(e«t) = e 3 .
l t l t l t
Hence, r€(eat) g_e 1 . Since e 1 6 06(4), rc(e“t) = e 1

65

For any u.> 1, eat has only finitely many normal

eigenvalues l ‘with IlI > eut, since
a
N0(e t) nIlIIlI > eut} = N0u(e“t)

is a compact subset of the normal eigenvalues and so has

a
only finitely many points. If e)‘t E Nou(e t), then

 

only finitely many 1 + 2".th can be eigenvalues of a, as
otherwise 1 would have an infinite eigenspace, contradict-

a
ing 1 e N0(e t

). Also. each such point is a normal eigen-
value Of 0’ [13]. Thus. a has only finitely many points Of

spectrum with real part greater than u, all normal eigen—

values.

Now let ll.°",lm be all these eigenvalues, and P
the projection defined in Theorem 2.6. Then Pefit = eatP,
0(Peat) = 0(eat) — ”1"”)..9 [16]. SO tome“) _<_ei"t
< eut. since

rO(Pe“t) = lim n/ Peilnt , and lim (Peat)e_”‘t = O.
n4on t4a
Since lim(PeMt)e'”Lit exists,
tam
I(Peat)e'”tl = e-“tIPeatI

is bounded for t 2.0. Therefore fOr some M.>'O, e-utIPeatI

g M for all t 2 O, that is IPeatI g Meut for t _>_ 0.

Results similar tO this are given in [17] and [18].

66
§2. The Saddle Point property.
In this section, we consider the equation

t
(4.4) u(t) = c Au- I g(t-s)Au(s) ds + h(t,(ut,u(t)))

—Q
where 9 satisfies the hypotheses in Theorem (2.1), and

h: R x W 4 H; is continuous and satisfies:

(4.5) Iu(t.z> —h(t.q)|1_<_on(c)lz-qlW

H

O
for all IzIW , IqIW < C , where a is continuous and
lim u(C) = 0. Let 4' be the infinitesimal generator
C40
associated with the linearized equation (2.1L.and
{ll.-°°.ln} the first n eigenvalues of this Operator

(counting multiplicities), arranged in order Of real part.

Suppose that
(4.6) Reli > u) 1* =max [HO-g(l) =0} 1 = l,'°°,n.

The projection from W onto the subspace generated by
the generalized eigenspaces corresponding to the eigenvalues
{ll.'°°.ln} will be denoted by P. The projection I-P
will be denoted by Q.

We can now state the saddle point property for equation

(4.4):

Thegrem 4.2. Consider the equation (4.4). Assume
that all of the above hypotheses are satisfied. Then there

are constants 6 > O and M 2_1 such that

(1) Let S(u) denote the set of initial values

(W.p) E W such that

67

JL
(a) I(w.p) IW < 2M

(b) If u(t) is the solution Of (4.4) with initial

value (w,p), then Ie-ut u(t)I 1 < 6 for all t 2 0.
Ho
Then S(u) is homeomorphic under the mapping QIS to the

closed ball of radius 3% in Q(W). Also S is tangent

to Q(W) at zero, that is

lim diSt((W.p).QfW))

= 0
[own [40 I ("'P) Iw

and lim e-utIu(t)I 1 = o.
t4¢ H
O
(2) Let U(u) denote the set Of initial values

(w.p) E‘W such that
(c) {(w.p) IW _<_. 5%.;

(d) There is a solution u(t) Of (4.3) defined for
all real numbers t, such that ‘w(t) = u(t) for t < o,

and u(o) = p.
(e) Ie-ut u(t)I < A for all t g;o.

Then U(u) is homeomorphic to the closed ball Of radius 'gk

in P(W), under the mapping PI U is tangent to P(W)

U(u)'

at o and lim Iept u(t)I 4 o.
t4-a

Proof: Let “1 = Pd.42 = Q“, W1 = P(W) and
W2 = Q(W).
First, let (w,p) E S(p). Then for each t by

Theorem 3.1,

68

t
(ut.u(t)) = e“t<w.p) + I e“‘t‘

O

S)(0.1'1(S.(v.ls.u(s)))ds.

Now, (ut,u(t)) = P(ut,u(t)) + Q(ut,u(t)). Using these

projections,
altp t «I(t-s)
P(ut,u(t)) = e (w,p) + I e P(u ,u(s))ds.
s
O
alt
Since Pd’= di is a bounded linear operator, e
exists for all t,. positive and negative. So
«It ~ t 413
e P(ut.u(t)) = P(W.p) + I e P(O,h(s,(uS.U(s))))ds.

0

Now allwl has spectrum [ll,"°,ln]. From this

—41t -l t
I(e Iw )I g.e n for t 2.0. SO
1
.4 t (-l +p)t
. 1 . n -ut _
11m e P(u ,u(t))I 3 11m e sup Ie (u ,u(t))I —O.
This gives P(w,p) = - I e-a,s P(O,h(s,(us,u(s))))ds.

0

Substituting this into the equation for (ut,u(t)) gives

«2t t «2(t-s)
(4.7) (ut.u(t)) = e R + I e Q(O.h(8.(us.u(8))))ds
0
° «I(t-s)
- e P(O.h(8.(us.u(s))))ds
t

where R = Q(w,p).
Now suppose R E'Wé . Then it will be shown (4.7) has
a unique solution u(t) for each R with IRIW.S'§% ,

for p small enough.

69

Let C = {2| 2: [0.00) 4W, 2 continuous,
le-utz(t)lW 4 O as t 4 o}. It will be shown

«2t t «2(t—s)
(TRz)(t) = e R + I e Q(Ooh(Sr(us,u(s))))ds

O

(D “I(t—S)
- f e p(o,h(s.(us.u(s))))ds

t

is a contraction for lR|W small enough.
First, TRz is continuous. e R is continuous in
t, and
t “2(t-s)
I e Q(O,h(s,us,u(s))ds
O
is continuous in t by the same argument as in Theorem
(2.1). The last term is continuous by the usual theorem on
continuous dependence of an integral on a parameter. Now
a
by Theorem 4.1, Ie 2 I _<_ Me-ut for all t _>_ O , where
M is a positive constant 2_1. So
sup Ie-ut TR z(t)| < “:
t20

since

sup e t(TR z(t)) < sup e “te xn+lMIRI

tzo
(t- 8)
+ e “I ln+1 IPIe-usds)a(p)|z|c
0
” —ln (t- s)
93’” (ef Io le‘“sds)a(p) lz Ic
+

.<. IRIM+ “Pita-y #1:] |ch
n

70

So TRz: c 4 c. Also let BC(p) = {z e C||z|C g p}. Then

if z e BC(p). from (4.5),
_hLL. __Efl__ .l
IRRZIC _<_ MIRIW + a (m an-“ + Mm) 1ch _<_ Mm,» 2 p.

if p is chosen so small that K(p)(-h1L-+ ——ELL—) <~£.
)‘n-Ll pl—)‘n+l 2

So choosing IR'W.S'§% ,

TR:BC(p) a BC(p) .

Also TR is a contraction on BC(p). Using (4.5),

— tr t xn+1(t’s) + s
IRRz — Tquc 3 sup e u LI e u(p)IPIe u Iz—qlcds
tzo 0
m —X (t-s)
+ IQII e n a(9)e“(S)IZ-qlcds
t

by choice of p. So T has a unique fixed point z in

R
BC(p). Hence, 2 satisfies
a t d (t-s)
z(t) = e 2R + I e 2 Q(O,h(s,z(s)))ds
0
“ «I(t-s)
- I e P(O,h(s,z(s)))ds
t
or
° —d s
z(t) = ea”) (R—I e 1 P(O,h(s,z(s)))ds)
0

t
+-I e“(t's)(o,h(s,z(s))ds .
O

71

By Theorem 3.1, Pzz(s) gives a solution of (4.3). Also

t
lim le-utz(t)l g eu'gj'tehm'1 MIR]
t-M:
K -ut (u- )t
+ e n+1 '——lQ%T— 0(9) + e kn -—¥E——-a(p) 4 O
xn+1 u u-kna

as t 4 a. Furthermore, if (w,p) 6 s(u), then
Q(wop) " (Wop) = -P(WoP)

, idle
=f e P(0.g(s.us.u(s)))ds.
0

Since. if [RI g-fih. [u(t)Ic S_P by definition 0f S(u).

, S
If e 1 P(0.g(S.uS.u(s)))dsl

1
S m k( Iut:u(t) IC) lut’u(t) [C

,5; u k<2MlPl (w.p))znlpl (w,p)

 

g,

Hence

I (mp) -Q (W. p) lw
lim (w H
[(w,p)l40 I .p 'w

 

s K<2MlPll<w.p) lw) 4 o.

and S(u) is tangent to ‘w2 at O.

FOr the unstable manifold, if (w,p) 6 U(p), then

since u(t) exists for all t‘g 0, if t1 < t,

72

“(t—t1)

(ut,u(t)) = e (u .u(tl))

t1

t a
+ I e (t-S) (0.h(s, (us,u(s))))ds.
t1

Taking projections of this gives

«z(t-tl)
Q(utou(t)) = e Q(utlau(tl))
t «z(t—s)
+ I e Q(O.g(s.(usou(5))))ds.
t1
Since
a (1:4: ) - t
Ie 2 1 Q(ut ,u(t1))| g pe+“te “ MIDI,
l
t "Ant -
and lim peu e MIQI = 0. letting t 4 ~a» gives
t-Om
t «z(t-s)
o<ut.u(t)) = I e o<o.h(s.(us.u(s))>)ds.
Also

0
(w.p) = edflt(ut.u(t)) + I e“(-s)(0.h(s,(us,u(s))))ds.

t

SO

P(w,p) = e P(ut.u<t)) + I e P(o.h(s.(us.u(s))))ds

t
d t
since e exists for all t, this becomes
«11: O «I(t-s)
P(ut.u<t>) = e P(w.p> - I e P(o.h(s.<us.u(s)))).

t

Adding the equations for Q(ut,u(t)) and P(ut,u(t)) gives

73

(4.8) (ut.U(t)) = P(ut.u(t)) + Q(ut.u(t))l
«1t 0 «I(t-s)
= e P(w,p) — I e P(o.h(s.(us.u(s>)>)ds
t
t “z(t—S)
+ I e Q(O).9(S.(us.U(S))‘)ds.

Now computations similar to the ones for (4 4) give the

results for the set U(u).

§3. Illustration of the projections in section 4.1.
Now an example will be given of the projections in
section 1, in the case where Rn is R. This is slightly

simpler than the general case.

Let Q = [0,2W]. Then equation (2.1) is

2 t 2
(53;:- u) (x.t) = c(-—a—2- u) (x.t) - I g(t-s) (~33- u) (x.s)ds
Bx _n as
u(0,t) = u(2v,t) — O
u(x,t) = w(x,t) t.g O.
_ 2 _ . 1
Here n — - n and m — (Sin nx) -—7—.
n n -
V/rr
Any w(x,t) 6 L at(’°°'0 : Hé(0,21r)) can be written
2,e
in the form
w(x,t) = f.(t) sin (jx).
i=1 3
Thus
n
Pl'n(w(x,t)) = jg: fj sin (jx)

74

and

P2 n(w(x,t)) = Z f.(t) sin (j x).
' j=n+l

If u(t,x) is the solution of (2 1) with initial value

w(t,x), then

u(t,x) = Z h.(t) sin (j x)
i=1 3
where hn(t) satisfies
h'(t) = -anh (t) +n2 t (t )h d
n n I. g ‘3 n(S) S.

BI BL IOGRAPHY

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

BIBLIOGRAPHY

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Coleman, Bernard, "Thermodynamics of materials with
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Coleman, Bernard, "On thermodynamics, strain, im-
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Coleman, Bernard and Owen, David K., "On the initial
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Hale, Jack, Functional Differential Equations,
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Hille, F. and Phillips, R.S., Functional Analysis
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Jacobson, N., Basic Algebra, p. 338, W.H. Freeman and
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