A general class of evolutionary equations
Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of retarded and neutral type. Specific illustrations are given using trans...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509373243064320 |
|---|---|
| author | Hale, J. K. Хале, Й. К. |
| author_facet | Hale, J. K. Хале, Й. К. |
| author_sort | Hale, J. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:50:43Z |
| description | Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of retarded and neutral type. Specific illustrations are given using transmission lines nearest-neighbor coupled at the boundary and the theory of heat transfer in solids. Some spectral theory for linearization of the equations is also discussed. |
| first_indexed | 2026-03-24T02:40:04Z |
| format | Article |
| fulltext |
UDC 517.9
J. K. Hale (USA)
A GENERAL CLASS OF EVOLUTIONARY EQUATIONS
ZAHAL\NYJ KLAS EVOLGCIJNYX RIVNQN\
Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined
which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of
retarded and neutral type. Specific illustrations are given using transmission lines nearest neighbor coupled at
the boundary and the theory of heat transfer in solids. Some spectral theory for linearization of the equations
also is discussed.
Za dopomohog spostereΩuvanyx velyçyn ta zminno] stanu dynamiçnoho procesu vyznaçeno zahal\ne
evolgcijne rivnqnnq, wo uzahal\ng[ klasyçni zvyçajni dyferencial\ni rivnqnnq, dyferencial\ni
rivnqnnq z çastynnymy poxidnymy ta spadkovi systemy iz zapiznennqm i systemy nejtral\noho typu.
Navedeno specyfiçni ilgstraci] z vykorystannqm linij transmisi] iz zçeplennqm ,,najblyΩçyx susidiv”
na meΩi ta teori] teploperenosu u tverdyx tilax. Rozhlqnuto takoΩ pevnu spektral\nu teorig dlq
linearyzaci] rivnqn\.
1. Introduction. Motivated by the fact that a dynamical system may evolve through an
observable quantity rather than the state of the system, a general class of evolutionary
equations is defined. This class includes standard ordinary and partial differential equa-
tions as well as functional differential equations of retarded and neutral type. In this way,
the theory serves as a unification of these classical problems.
Included in this general formulation is a general theory for the evolution of temper-
ature in a solid material. In the general case, temperature is transmitted as waves with a
finite speed of propagation. Special cases include a theory of delayed diffusion.
We describe also in some detail a lattice on a circle where each point on the lattice
is a transmission line for current and voltage whose dynamics is governed by a linear
hyperbolic equation on [0, 1] with dynamic boundary conditons given by the circuitry on
the line. The systems are coupled to their nearest neighbor at the end point 1 through
resistors. A limiting process letting the distance between the lattice points approach zero
leads to an interesting set of partial differential equations on [0, 1] × S1 with a hyper-
bolic equation on [0, 1] and a parabolic equation on S1. These equations have not been
analyzed in detail. However, we can show that the voltage at 1 satisfies a partial neutral
functional differential equation. We analyze some properties of these equations including
synchronization and the behavior of solutins near periodic orbits.
There are other applications which involve partial differential equations on lattices for
which the dynamics on each lattice point is governed by a partial differential equation on
a bounded domain Ω. These systems could be coupled to neighbors through interaction
on some subset of the boundary ∂Ω of Ω. If certain limiting processes are justified, one
can obtain a partial differential equation on Ω together with another partial differential
equation on another domain Ω1 (determined by the nature of the lattice). We do not dis-
cuss this in the text, but mention it only to suggest that there are very interesting problems
associated with such equations.
When the abstract evolutionary equation is linear, one arrives at an interesting spectral
problem. We give some special results for the evolution of temprature and the partial
neutral functional differential equation mentioned above.
In the discussion below of the dynamical system generated by an evolutionary equa-
tion, we will often enquire about the possibility of the dynamical system being a condi-
tional α-contraction. Dynamical systems which are conditional α-contractions play an
important role in the development of a qualitative theory; for example, the existence of a
c© J. K. HALE, 2007
268 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 269
compact global attractor, behavior near equilibria or periodic orbits, etc. (see, for exam-
ple, [1]).
For the benefit of the reader not familiar with this concept, we recall the definition. If
X is a Banach space, the Kuratowski measure of noncompactness α(B) of a bounded set
B ⊂ X is defined as
α(B) = inf
{
d : B a finite cover of diameter < d
}
.
A bounded set B has compact closure if and only if α(B) = 0.
If S : X → X is a continuous map, then S is said to be a conditional α-contraction
if there is a k ∈ [0, 1) such that, for any bounded set B for which SB is bounded,
α(SB) ≤ kα(B). The map is conditionally compact if it is a conditional α-contraction
with k = 0.We remark that a linear bounded map on X is a conditional α-contraction if
and only if the radius of the essential spectrum is < 1.
A dynamical system T (t) : X → X, t ≥ 0, is a conditional α-contraction if there is
a t0 such that T (t0) is a conditional α-contraction.
This paper originated from a lecture at a conference at the University of São Paulo,
São Carlos, Brazil in February, 2006 celebrating the life and work of Dan Henry who died
on May 4, 2002 at the young age of 57.
2. A class of evolutionary equations. In this section, we describe a general class
of evolutionary equations and show by examples how it unifies the discussion of many
different types of equations that have been considered in the literature.
Let Y be a Banach space which we refer to as the observable space and let X be a
Banach space which is called the state space. If D : D(D) ⊂ X → Y and F : D(F ) ⊂
⊂ X → Y are given functions, we define an abstract evolutionary equation for a function
u(t) ∈ X as
∂t(Du(t)) = F (u(t)). (2.1)
In this abstract form, one cannot hope to have much of a general theory and the class
must be restricted. We consider a special case of which we refer to as quasilinear: that is,
the equation
∂tDu(t) = Lu(t) +G(u(t)), (2.2)
where D : D(D) ⊂ X → Y, L : D(L) ⊂ X → Y, are linear operators and G : X → X
(the domain of G is X).
The first problem is to define and obtain the existence of a solution of (2.2). To make
sure that the linear equation,
∂tDu(t) = Lu(t)), (2.3)
has a solution, we make the following hypothesis:
H1) Equation (2.3) defines a C0-semigroup onX. Denote this semigroup by eAD,Lt,
where AD,L is the infinitesimal generator.
To obtain an evolutionary equation on X, we make the following hypotheses:
H2) There is a bounded linear operator M : Y → X such that (2.2) is equivalent to
the equation on X:
∂tu = AD,Lu+MG(u). (2.4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
270 J. K. HALE
Definition 2.1. A mild solution of (2.4) with initial value u0 ∈ X at t = 0 is a
solution of the integral equation (of course, assuming that it makes sense)
u(t) = eAD,Ltu0 +
t∫
0
eAD,L(t−s)MG(u(s)) ds. (2.5)
Assuming that solutions of (2.5) exist for each u0 ∈ X and t ≥ 0 and the solution
u(t, u0) depends continuously on (t, u0), we let TD,L,G(t), t ≥ 0, be the dynamical
system on X defined by TD,L,G(t)u0 = u(t, u0).
One of the first objectives in the theory of dynamical systems is understand basic
properties of TD,L,G(t). For example, if it is asymptotically smooth or, more particularly,
a conditional α-contraction, then we can make use of many known results from dynam-
ical systems concerning, for example, the existence of compact global attractors, local
behavior near hyperbolic equilibria and periodic orbits, etc. We will mention later some
of the problems involved in obtaining such a characterization of TD,L,G(t). Before doing
this, we give several illustrations of classical equations that fall into this general class of
equations.
3. Examples.
Example 3.1. For X = Y, D = I, and D(F ) = X, equation (2.1) is the standard
type of evolutionary equation, which includes ordinary differential equation (ODE) in
X = R
n as well as the case where X has infinite dimension provided that F is a smooth
function.
Example 3.2. IfX = Y, D = I, A is the infinitesimal generator of aC0-semigroup
onX andG : X → X, this is a quasilinear equation which includes many types of partial
differential equations.
Example 3.3 ( RFDE). Fix r ≥ 0 and let X = C = C
(
[−r, 0],Rn
)
and Y = R
n.
For any observable continuous function z : [−r, α) → R
n, let zt ∈ C be the state of z
defined by zt(θ) = z(t + θ), θ ∈ [−r, 0]. For any continuous function F : C → R
n, we
define a retarded functional differential equation (RFDE) by the relation
∂tz(t) = F (zt). (3.1)
This is a special case of (2.2) with with Dϕ = ϕ(0) for ϕ ∈ C.
This type of equation has been discussed extensively (see, for example, [2, 3] and the
references therein).
For any ϕ ∈ C, there is a solution z(t, ϕ) of (3.1) through ϕ at t = 0 defined on a
maximal interval [−r, α), α > 0. If the solution exists globally in time, then the mapping
T (t) : C → C, t ≥ 0, defined by
T (t)ϕ = zt(·, ϕ), t ≥ 0, (3.2)
is a dynamical system on the state space C.
For t ≥ r, the solution z(t) of (3.1) is continuously differentiable. If F : C → R
n
takes bounded sets to bounded sets, the the Arzela – Ascoli theorem implies that T (t) is a
conditionally compact operator for t ≥ r.
Let C0 =
{
ϕ ∈ C : ϕ(0) = 0
}
. The equation ∂tz(t) = 0 defines a C0-semigroup
S(t) on C0. For any ψ ∈ C, ψ − ψ(0) ∈ C0 and S(t)
[
ψ − ψ(0)
]
= 0 for t ≥ r and, for
any β > 0, there is a constant K such that
∣∣S(t)
[
ψ − ψ(0)
]∣∣ ≤ Ke−βt|ψ| for t ≥ 0 and
all ψ ∈ C. Since the solution of (3.1) with initial data ϕ ∈ C is given by
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 271
z(t) = ϕ(0) +
t∫
0
F (zs) ds, t ≥ 0,
z(t) = ϕ(t), t ∈ [−r, 0],
it is easy to see that
T (t) = S(t)
[
ϕ− ϕ(0)
]
+ U(t)ϕ, (3.3)
where U(t) is conditionally compact for t ≥ 0. Therefore, T (t) is a conditional α-
contraction for all t > 0. From this fact, it has been possible to obtain a qualitative theory
for RFDE which is similar to the theory of ODE.
Is it possible to write (3.1) as an evolutionary equation in C? We consider (3.1) as
a perturbation of the linear equation ∂tz(t) = 0 on C. We have noted that this equation
defines a C0-semigroup S(t) on C0. The infinitesimal generator A of S(t) is easily seen
to satisfy the following:
D(A) =
{
ϕ ∈ C1
(
[−r, 0],Rn
)
: ϕ̇(0) = 0
}
,
Aϕ = ϕ̇, ϕ ∈ D(A).
(3.4)
This makes it apparent that it is not possible to work in the space C and obtain an abstract
evolutionary equation for the state variable zt of (3.1) since each ϕ in the domain of A
must satisfy the boundary condition ϕ̇(0) = 0.
As motivation for further discussion, observe that a solution z(t) of (3.1) is continu-
ously differentiable on the interval [−r,∞) if and only if zt ∈ D(A) for all t ≥ 0. If this
is the case, then, formally,
∂tzt = Azt +X0F (zt), (3.5)
where
X0(θ) =
{
0 if θ ∈ [−r, 0),
I if θ = 0,
(3.6)
I is the n× n identity matrix.
Even though this if formal, it was used extensively in the early development of the
qualitative theory of (3.1) by considering a natural integral equation from (3.5) which is
valid for each θ ∈ [−r, 0] (see [2 – 4] for further discussion and references).
To obtain an abstract evolutionary equation for the state variable of (3.1), we need
to consider a larger space which permits the rows of the matrix function X0 to be in the
space. We choose the space to be
X̃ ≡ R
n × C
and extend the definition of the infinitesimal generator A to an operator à defined by
D(Ã) =
{
ϕ ∈ C1([−r, 0],Rn)
}
,
Ãϕ = ϕ̇−X0ϕ̇(0).
(3.7)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
272 J. K. HALE
With this notation, we can now show that the RFDE on X̃ is equivalent to an abstract
evolutionary
∂tzt = Ãzt +X0F (zt). (3.8)
This is a special case of (2.4) withM equal to X0.
If we replace (3.1) by
∂tz(t) = Lzt +G(xt), (3.9)
where L : C → R
n is a bounded linear operator, then the semigroup defined by the linear
equation
∂tz(t) = Lzt
has the infinitesimal generator A the same as above except that it is required that ϕ̇(0) =
= Lϕ if ϕ ∈ D(A). If we make the same transformation as in (3.7) replacing −X0ϕ̇(0)
by X0
[
Lϕ− ϕ̇(0)
]
, then we obtain the same evolutionary equation with the new Ã.
The transformation (3.7) was first used by Chow and Mallet – Paret (1977) in the study
of the integral averaging and Hopf bifurcation. Generalizations of normal form theory
also follow from (3.8) (see [3] and the references therein). For more information about
RFDE, see these same references.
Abstract evolutionary equations have been associated to (3.1) using deeper concepts
in functional analysis (see [6]).
Example 3.4 (NFDE). With the notation as in Example 3.3, a quasilinear neutral
functional differential equation (NFDE) is defined as
∂tDzt = F (zt), (3.10)
where D : C → R
n is a continuous linear operator which is nonatomic at zero or, equiv-
alently,
Dϕ = ϕ(0) −
0∫
−r
[
dη(θ)
]
ϕ(θ), (3.11)
where η is an n× n matrix function of bounded variation which is nonatomic at zero.
It is not too difficult to show that there is a solution of (3.10) with initial function
ϕ ∈ C at t = 0, defined on a maximal interval [−r, αϕ), αϕ > 0 (see, for example, [2]).
If the solution through ϕ is defined for all t ≥ 0 and, if we define T (t) as in (3.2),
then T is a dynamical system on the state space C.
Let TD(t) be the semigroup defined by the linear functional equation
Dwt = 0, wt ∈ CD = {ϕ ∈ C : Dϕ = 0}. (3.12)
It is known (see, for example, [2]) that the semigroup T (t) defined by (3.10) can be
represented as
T (t) = TD(t)Ψ + U(t), t ≥ 0, (3.13)
where U(t) is conditionally compact for t ≥ 0 and
Ψ = I − ΦD, Φ = (ϕ1, . . . , ϕn), ϕj ∈ C, 1 ≤ j ≤ n, DΦ = I. (3.14)
It follows that T (t) is a conditional α-contraction if
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 273
r
(
σess(TD(1))
)
< 1. (3.15)
If Dϕ = ϕ(0) +
∑N
j=1
βkϕ(−rk), then condition (3.15) is equivalent to saying that the
zero solution of (3.12) is exponentially stable.
Using the same ideas as in the case of RFDE, one can obtain an abstract evolutionary
equation of the form (3.8) in X̃ for the solutions of (3.10). The matrix function X0 is
the same as before and the corresponding operator à has the same domain with Ãϕ =
= ϕ̇−X0Dϕ̇.
In contrast to RFDE, some solutions of (3.10) may not become differentiable for any
t > 0 if Dϕ �= ϕ(0). On the other hand, the development of a qualitative theory for
NFDE satisfying (3.15) which is similar to the one for ODE relies primarily on the fact
that TD(t) is a conditional α-contraction. Many results are known but the theory is not
nearly as complete as for RFDE (see, for example, [2, 3]).
Example 3.5 (More general quasilinear RFDE). With the notation as above, if Y is
the observable Banach space and X = C
(
[−r, 0], Y
)
and A : D(A) ⊂ Y → Y and
f : Y → Y is continuous, then then one can define a quasilinear RFDE on X as
∂tz(t) = Az(t) + f(zt). (3.16)
If A is the generator of a C0-semigroup eAt, then a mild solution of (3.16) with initial
value f at t = 0 is a solution of the equation
z(t) = eAtϕ(0) +
t∫
0
eA(t−s)f(zs) ds, t ≥ 0,
z0 = ϕ ∈ X.
Assuming that each solution z(t, ϕ) exists for all t ≥ −r and is continuous in (t, ϕ),
then TA,f (t) ≡ zt(·, ϕ) : X → X is a dynamical system onX. If TA,f (t) is a conditional
α-contraction, there is the possibility of developing a qualitative theory similar to ODE. It
is known that TA,f (t) is a conditional α-contraction for the situation in which A = ∆BC
on a bounded domain Ω ⊂ R
N with boundary conditions BC and Y is an appropriate
space of functions on Ω. On can find a detailed discussion in [7].
One can also obtain a conditional α-contraction in a linearly damped hyperbolic equa-
tion on a bounded domain Ω provided that f satisfies some growth conditions.
Example 3.6 (more general quasilinear NFDE). Let Y be an observable Banach spa-
ce, X = C
(
[−r, 0], Y
)
. If Dj : C
(
[−r, 0], Y
)
→ Y, j = 1, 2, are bounded linear oper-
ators, A : D(A) ⊂ Y → Y is a linear operator and f : Y → Y is continuous, then a
quasilinear NFDE on X is
∂tD1zt = AD2zt + f(zt). (3.17)
The simplest case to consider is when
Djϕ = ϕ(0) −
0∫
−r
[
dηj(θ)
]
ϕ(θ), j = 1, 2, (3.18)
where each ηj is nonatomic at zero. Such an assumption on the ηj makes easier the
verification of the existence of a solution of the initial value problem.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
274 J. K. HALE
We have mentioned in Section 2 a procedure for obtaining the existence of solutions.
Each situation will require spectial consideration of the linear equation
∂tD1zt = AD2zt, (3.19)
and conditions on D1 and D2 which imply that the solutions generate a C0-semigroup
SD1,D2(t) on X.
Furthermore, to obtain a qualitative theory, the spectrum of TD1,D2(1) will play an
important role. We will discuss some aspects of this problem later, but now we now
discuss in some detail two examples illustrating the importance of considering equations
as general as (3.18). The first one invovles transmission lines on a circle with resistive
nearest neighbor coupling and leads to (3.18) with D1 = D2. The second example deals
with a theory of heat conduction in a solid where D1 is in general different from D2.
4. Transmission lines on a scalar domain with boundary coupling. 4.1. Loss-
less transmission line and a NFDE. The current i and voltage v in this system can be
described by the telegraph equation
L∂ti = −∂xv, C∂tv = −∂xi, 0 < x < 1, t > 0, (4.1)
with the boundary conditions expressing the circuitry at the end points of the line given
by
E − v(0, t) −Ri(0, t) = 0, C1∂tv(1, t) = i(1, t) − g(v(1, t)). (4.2)
It has been known for a long time that the undamped wave equation in one space
dimension with nonlinear boundary conditions can be reduced to a NFDE of the type
considered above (see, for example, [8 – 12]). We give two ways of doing this since
the manner of reduction is not unique and distinct equations are obtained. However, the
qualitative dynamics of the two types of equations are the same.
Define the constants
s = (LC)−1/2, z =
(
L
C
)1/2
, K =
z −R
z +R
, α =
2E
z +R
.
The general solution of the partial differential equation (PDE) is given by
v(x, t) = ϕ(x− st) + ψ(x+ st),
i(x, t) =
1
z
[
ϕ(x− st) − ψ(x+ st)
]
or
2ϕ(x− st) = v(x, t) + zi(x, t),
2ψ(x+ st) = v(x, t) − zi(x, t).
This implies that
2ϕ(−st) = v
(
1, t+
1
s
)
+ zi
(
1, t+
1
s
)
,
2ψ(−st) = v
(
1, t− 1
s
)
+ zi
(
1, t− 1
s
)
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 275
Using these expressions in the general solution and using the first boundary condition at
t− 1/s, we obtain
i(1, t) −Ki
(
1, t− 2
s
)
= α− 1
z
v(1, t) − K
z
v
(
1, t− 2
s
)
.
Inserting the second boundary condition and letting u(t) = v(1, t),we obtain the equation
u̇(t) −Ku̇
(
t− 2
s
)
= f
(
u(t), u
(
t− 2
s
))
,
where, if δ = 2/s,
C1f
(
u(t), u(t− δ)
)
= α− 1
z
u(t) − K
z
u(t− δ) − g
(
u(t)
)
+ g
(
u(t− δ)
)
.
If generalized solutions are considered in the original equation (4.1), then the function
u would not have a derivative and we can only expect that the difference u(t)−Ku(t−δ)
is differentiable and we obtain the NFDE
∂t
[
u(t) −Ku(t− δ)
]
= f
(
u(t), u(t− δ)
)
. (4.3)
If we let Dϕ = ϕ(0) −Kϕ(−δ), then we remark that this equation can be written in the
equivalent form
∂tDut = α− 2
z
u(t) +
1
z
Dut −D(g ◦ u)t,
where (g ◦ u)(t) = g
(
u(t)
)
.
The equation Dut = 0 in the space CD =
{
ϕ ∈ C
(
[−δ, 0],R
)
: Dϕ = 0
}
defines a
C0-semigroup. We say that D is exponentially stable if the zero solution of this equation
is exponentially stabe. The operator D is exponentially stable if and only if K < 1. This
will be the case if there is nonzero resistance in the line.
Let us give another NFDE which will describe qualitatively the dynamics of (4.1) in
the case where K < 1; that is, D is exponentially stable. Let p be the unique constant
solution of the equationDpt = Ez/(z+R); that is, p = zE/2R.Using the first boundary
condition at t− 1/s and the general solution of (4.1), we obtain
ϕ(1 − st) = − z
z +R
E −Kψ(st− 1).
If w(t) = ψ(1 + st) − p, then evaluation in the general solution gives
v(1, t) = w(t) −Kw(t− δ), i(1, t) =
1
z
w(t) − K
z
w(t− δ) + q,
where zq = −(1+K)p+(z/(z+R))E.Using the second boundary condition, we obtain
the equation
∂tDwt = q − (δ +K)w(t) +
(
δ
2
+ 1
)
Dwt − g(w(t)). (4.4)
In (4.4), the nonlinear function g appears only with the argument w(t), whereas in (4.3)
it also occurs with the argument u(t− δ). This can sometimes be useful in trying to make
estimates on the magnitude of solutions (see, for example, [12]).
IfK < 1, then the operatorD is exponentially stable and the semigroup generated by
either (4.3) or (4.4) is a conditional α-contraction.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
276 J. K. HALE
4.2. Ring of lossless transmission lines with resistive coupling. Following Wu and
Xia [13], we consider a ring of N mutually coupled lossless transmission lines intercon-
nected by a common resistor R0 at the right end of the line. For 1 ≤ k ≤ N, the system
of PDE is
L∂tik = −∂xvk, C∂tvk = −∂xik, 0 < x < 1, t > 0, (4.5)
with the boundary conditions
E − vk(0, t) −Rik(0, t) = 0,
C1∂tvk(1, t) = ik(1, t) − g(vk(1, t)) −
1
R0
(vk+1 − 2vk + vk−1)(1, t).
(4.6)
If we make the above reduction to NFDE, we have the system
∂tDuk,t = f
(
uk(t), uk(t− δ)
)
+
1
R0C1
D(uk+1,t − 2ukt + uk−1,t), (4.7)
where Dukt = uk(t) −Kuk(t− δ) and k = 1, 2, . . . , N.
Wu and Xia [13] discussed the existence of periodic solutions for (4.6) with K < 1;
that is,D is exponentially stable. IfD is exponentially stable, then the semigroup defined
on C
(
[−δ, 0],RN
)
is a conditional α-contraction.
4.3. Transmission lines on a circle. The terms on the right-hand side of (4.6) sug-
gest an approximation to the Laplacian operator on S1. Following Hale [14], we suppose
that h is the spacing between the transmission lines and that there is a constant d such that
(R0)−1 = dC1/h
2. If we let s represent distance on S1 and take the limit as h→ 0, then
we obtain the following interesting partial differential equation for i(x, s, t), v(x, s, t),
L∂ti(x, s, t) = −∂xv(x, s, t),
C∂tv(x, s, t) = −∂xi(x, s, t), 0 < x < 1, s ∈ S1, t > 0,
(4.8)
with the boundary conditions
E − v(0, s, t) −Ri(0, s, t) = 0,
C1∂tv(1, s, t) = i(1, s, t) − g(v(1, s, t)) − d∂2
sv(1, s, t), s ∈ S1.
(4.9)
As for the discrete version, we remark that the original dynamics on the line could contain
terms not involving the derivatives.
4.4. A partial NFDE. If we make the above reduction to a NFDE, we obtain the
following partial NFDE for u(t, s),
∂tDut(·, s) = d∂2
sDut(·, s) + f(u(t, s), u(t− δ, s) (4.10)
with s ∈ S1.
Hale [15] discussed the existence and uniqueness of solutions of equations more gen-
eral than (4.10) in the space X = C
(
[−δ, 0], H1(S1)
)
; namely,
∂tDut = d∂2
sDut +H(ut) (4.11)
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A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 277
with s ∈ S1, u = u(s, t) and ut(θ, s) = u(t+ θ, s), s ∈ S1. The function H : X → R
n
is Lipschitzian andD : X → R
n is a continuous linear functional which is atomic at zero
which without loss of generality implies that we can assume
Dϕ = ϕ(0) −
0∫
−δ
dη(θ)ϕ(θ) (4.12)
where η is an n× n matrix of bounded variation with no atom at zero.
A mild solution of (4.11) is defined to be a solution of the equation
Dut = edAstDϕ+
t∫
0
edAs(t−τ)H(uτ )dτ, (4.13)
where As = d∂2
s on its domain in S1.
Let TD,H(t), t ≥ 0, be the dynamical system on X generated by the solutions of
(4.11). Also, let TD,0(t), t ≥ 0, be the dynamical system generated by the equation
∂tDut = d∂2
sDut. (4.14)
If
r(σess(TD,0(1))) < 1, (4.15)
then one can show that TD,H(t) is a conditional α-contraction. We will show lat that this
if true if the solution of the functional equation Dwt = 0 on CD =
{
ϕ ∈ C
(
[−δ, 0],R
)
,
Dϕ = 0
}
is exponentially stable. This implies that the contribution from the partial
derivatives in x is a compact perturbation of the functional equation.
4.5. Spectral properties of TD,0(t). IfAD is the infinitesimal generator of TD,0(t),
then it is not difficult to show that
D(AD) =
{
ϕ ∈ X : ϕ ∈ C1, Dϕ̇ = d∂2
sDϕ
}
,
ADϕ = ϕ̇.
(4.16)
The operator AD has compact resolvent. We use eigenfunction expansions to determine
the spectrum σ(AD).
If (µk, ek), k = 1, 2, . . . , is a complete set of eigenpairs of −d∂2
s on S1, then µk → ∞
as k → ∞. LetX(k) be the span of the eigenfunction ek inX. If σk denotes the spectrum
of AD
∣∣X(k), then
σ(AD) = ∪k≥1σk. (4.17)
A point λ ∈ σk if and only if there is a nonzero function ϕ ∈ X(k) such that ADϕ = λϕ.
For any nonzero ϕ ∈ X(k), there is a nonzero w ∈ C([−δ, 0],R) such that ϕ = wek.
From (4.16), we see that ϕ̇ = λϕ, and Dϕ̇ = −µkDϕ, Therefore, ẇ = λw, Dẇ =
= −µkDw. As a consequence, we see that
σk =
{
λ ∈ C : λDeλ· = −µkDeλ·
}
,
σ(AD) = ∪k≥1σk.
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278 J. K. HALE
That part of the spectrum of Ad corresponding to a fixed k coincides with the eigen-
values of the NFDE
∂tDwt = µkDwt. (4.18)
The spectrum of TD,0(1)|X(k) can be shown to satisfy
σ
(
TD,0(1)
∣∣X(k)
)
= Cl eσk .
We have noted earlier that the essential spectrum of TD,0(1)
∣∣X(k) associated with (4.18)
coincides with the essential spectrum of SD(1) where SD(t), t ≥ 0, is the semigroup on
C
(
[−δ, 0],R
)
generated by the functional equation Dwt = 0 on the space CD. We also
note that λ = −µk is an element of σk. Since µk → ∞ as k → ∞, this implies that
r
(
σess(TD,0(1))
)
= r
(
σess(SD(1))
)
.
Therefore, TD,0(t) is a conditional α-contraction if the zero solution of the functional
equation Dwt = 0 in CD is exponentially stable. As a consequence, TD,H(t) is a condi-
tional α-contraction if the same property holds.
We summarize this in the following statement.
Proposition 4.1. The semigroup TD,H(t) on X defined by (4.11) is a conditional
α-contraction if and only if the zero solution of Dwt = 0 on CD is exponentially stable.
4.6. Synchronization for the partial NFDE. In Proposition 4.1, we have observed
that TD,H(t) is a conditional α-contraction if D is exponentially stable. From [1], this
implies that there is the compact global attractor AD,H if the dynamical system TD,H(t)
is point dissipative and, for any bounded set B ⊂ X, there is a t0 = t0(B) such that
γ+(TD,H(t0)B) is bounded. We recall that AD,H is the compact global attractor if it is
compact, TD,H(t)AD,H = AD,H for all t, and for any bounded set B in X,
lim
t→∞
distX(TD,H(t)B,AD,H) = 0.
In this section, w discuss some properties of the compact global attractor as a function
of the diffusion coefficient d in (4.11). Therefore, we fix D, H and denote the dynamical
system by Td−1 and the corresponding compact global attractor by Ad−1 . We make the
following hypotheses:
H3) The NFDE
d
dt
Dzt = H(zt) has the compact global attractor A0 in
C
(
[−δ, 0],R
)
.
H4) There is a d1 > 0 such that the family of compact sets {Ad−1 , d ≥ d1} ∪ A0 is
bounded in X.
We say that the system (4.11) is synchronized if each element of Ad−1 is independent
of the spatial variable x ∈ S1; that is, Ad−1 = A0.
The following result is proved in [14].
Theorem 4.1. If hypotheses H3) and H4) are satisfied, then there is a d2 ≥ d1 such
that, for each d ≥ d2, system (4.11) is synchronized.
We outline the proof. Let X = X0 ⊕ X1, where X0 consists of functions which
are independent of the spatial variable and X1 consists of all functions in X which are
orthogonal to the constant functions. If we let wt = w1
t +w2
t , w
j
t ∈ Xj , then one obtains
the equations
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A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 279
∂
∂t
D(q)w1
t = H(w1
t ) + π−1
π∫
0
[
H
(
w1
t + w2
t (·, x)
)
−H(w1
t )
]
dx,
∂
∂t
D(q)w2
t = d
∂2
∂s2
D(q)w2
t+
+H(w1
t + w2
t ) − π−1
π∫
0
[
H
(
w1
t + w2
t (·, x)
)]
dx ≡
≡ d ∂
2
∂s2
D(q)w2
t + F (w1
t , w
2
t ).
We first observe that solutions on the attractors must satisfy some special properties.
It is not difficult to see from (4.13) that, for every d, any solution on Ad−1 must satisfy
Dw2
t =
t∫
−∞
edAs(t−τ)F (w1
τ , w
2
τ )dτ (4.19)
for all t ∈ R.
Since F (ϕ0, 0) = 0 for each ϕ0 ∈ X, there is a constant k0 such that∣∣F (ϕ0, ϕ1)
∣∣
Y
≤ k0|ϕ1| + k1 ∀ϕ0 + ϕ1 ∈ {Ad−1 , d ≥ d1} ∪ A0. (4.20)
From (4.19), (4.20), the fact that
‖edAst‖L(X1,X1) ≤ e−dt
for t ≥ 0 and wt is bounded for t ∈ R, one easily shows that w2
t = 0 for all t ∈ R
provided that d is sufficiently large.
4.7. Synchronization in the hyperbolic PDE with parabolic PDE boundary con-
ditions. Theorem 4.1 allows one to obtain a type of synchronization for system (4.8),
(4.9). Suppose that system (4.8), (4.9) has the compact global attractor Ãd−1 . we say that
(4.8), (4.9) is synchronized if each element of the compact global attractor is independent
of y ∈ S1. Using the relationships between the solutions of (4.8), (4.9) and (4.11), one
can prove the following result.
Theorem 4.2. Under the hypotheses of Theorem 4.1, the solutions of (4.8), (4.9)
are synchronizeed for d ≥ d2.
It would be interesting to give a proof of Theorem 4.2 directly on the equations (4.8),
(4.9) without using the partial NFDE. The method employed probably could be used to
discuss other PDE with interactions through the boundary.
It also is possible to consider equations (4.11) for a nonlinearity H(s, ut) which de-
pends upon s. In this case, one can show that distX(Ad−1 ,A0) → 0 as d → ∞, where
A0 is the attractor for the “averaged” NFDE
∂
∂t
Dyt = H̄(yt), yt ∈ C
(
[−δ, 0],R
)
,
H̄(ϕ) =
1
π
π∫
0
H(s, ϕ) ds.
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280 J. K. HALE
The interested reader can obtain much more information about synchronization of
ODE and PDE by consulting the references in [16].
4.8. Behavior near and perturbation of periodic orbits. Consider an abstract evo-
lutionary equation, suppose that u0 is a T0-periodic solution and let Γ0 =
{
u0(s) : s ∈
∈ [0, T0)
}
. It is desirable to define a large class of such equations for which we can obatin
a theory similar to the ODE case describing the behavior of the solutions near Γ0 and the
effects of autonomous as well as nonautonomous perturbations. For the ODE case, a first
step in the development of the theory is to define a rotating coordinate system around the
periodic orbit using the “angle” s (the parameter describing the orbit Γ0) and an element
in a transversal to Γ0 at u0(s). One can then apply the theory of invariant manifolds of
Bogoliubov and Mitropolsky [17] to treat both the autonomous and nonautonomous case.
For parabolic PDE, Henry [18] has shown that one can obtain the same type of results
by using similar methods. Of course, there are many more technical obstacles that must
be overcome. His methods should be applicable to many other types of PDE. For RFDE,
Stokes [19] has given partial results in the same spirit. His results are not as general as for
the ODE because the differential equation for the angle coordinate involved delays in the
angle. For the partial NFDE that are similar to the ones arising in transmission lines, Hale
[15] discussed some elementary properties for a hyperbolically stable periodic orbit un-
der autonomous perturbations without using coordinate systems. For NFDE on R
n, Hale
and Weedermann [20], have given a coordinate system around a periodic orbit for which
the derivative of the angle variable does not involve the delays. In this way, the spirit of
Bogoliubov and Mitropolsky [17] can be followed with modifications of the techniques
of Henry [18]. The construction of this coordinate system will be described later and it
should be applicable to the partial NFDE above as well as more general ones.
If the perturbations are autonomous, why not use the standard Poincaré transversal
map and study the neighborhood of a fixed point of the map? This is a very common
approach in ODE in R
n. We recall definition of the Poincaré map π. One chooses a
transversal Σ at u0(0) so that, for any v ∈ Σ, there is a t0(v) > 0 such that the solution
u(t), u(0) = v, satisfies u(t0(v)) ∈ Σ. One then defines πv = u(t0(v)). Using the
differentiability properties of π on the transversal, one can discuss the local behavior near
the fixed point u0(0). For general evolutionary equations (or even NFDE in R
n), the map
π is not differentiabe and other methods must be employed. In many important situations,
one can prove that u0 is continuously differentiable and this is sufficient to obtain the
rotating coordinate system and then consider the method of integral manifolds mentioned
above.
For convenience, let C = C
(
[−δ, 0],Rn
)
. Consider now a NFDE (3.10) on R
n with
D exponentially stable. Suppose that p(t) is a T0-periodic solution and let Γ =
{
pt, t ∈
∈ [0, T0)
}
. It is known that p(t) is a C1-function (actually as smooth in t as F ) and,
therefore, Γ is a C1-manifold (see [21] for a proof as well as references to previous work
on smoothness of Γ).
The linear variational equation about p(t) is given by
d
dt
Dyt = L(t)yt, (4.21)
where L(t) = F ′(pt) : C → R
n is a bounded linear operator which is continuous in t and
T0-periodic.
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A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 281
Let T (t, s)ϕ = yt(·, s, ϕ), where y(t, s, ϕ) is the solution of (4.21) with initial value
ϕ at t = s. Define U(s) = T (t + s, s), s ∈ R. The point spectrum σP (U(s)) is in-
dependent of s. An element µ �= 0, µ ∈ σP (U(0)), is called a Floquet multiplier of
(4.21). Since ṗ is a T0-periodic solution of (4.21), µ = 1 is a Floquet multiplier. The
orbit Γ is nondegenerate if 1 is a simple Floquet multiplier. The orbit Γ is hyperbolic if
(σP (U(0)) \ {1})∩S1 = ∅, where S1 is the unit circle in the complex plane with center
zero and radius 1.
Even though it is not necessary, we assume for simplicity that Γ is nondegenerate.
There is a closed subspace Q(s), s ∈ R, T0-periodic, such that for every t ≥ s,
1) C = [ṗ0] ⊕Q(s),
2) T (t, s)Q(s) ⊂ Q(s),
3) σ((U(s)|Q(s)) = σ(U(s)) \ {1},
4) Q(s) is homeomorphic to Q(t).
One can give an explicit representation of the decomposition (1) using the formal
adjoint equation of (4.21) and the classical bilinear form 〈·, ·〉 associated with (4.21) and
the adjoint equation. In fact, there is T0-periodic solution q of the adjoint equation such
that 〈qs, ṗs〉 = 1 and Q(s) is the set of ϕ ∈ C for which 〈qs, ϕ〉 = 0.
One can easily show that there is a neighborhood V of Γ such that, for any ϕ ∈ V,
there is a unique s = s(ϕ) and a unique ψ = ψ(ϕ) ∈ Q(s) such that ϕ = p(s) + ψ. One
can now use this representation to change coordinates for solutions u(t) of (3.10) in V
as ut = p(s(t)) + zt with zt ∈ Q(s(t)). This is essentially the same as the one used by
Stokes [19]. As remarked earlier, the differential equation for s(t) involves delays in s.
Hale and Weedermann [20] avoid this difficulty in the following way.
Let M ⊂ C be a linear closed subspace of C of codimension 1 (one could choose
M = Q(0), for example). From the decomposition of C by 1) above, for any s ∈ [0, T0),
there is a bounded linear isomorphism Ls :M → Q(s) such that, for any w ∈M,
w = 〈qs, w〉ṗs + Lsw.
It is not difficult to observe that the following result is valid.
Proposition 4.2 (A local coordinate system around Γ). Suppose that the codimen-
sion 1 subspace M of C and the operator Ls are defined as above. Then there exist
a neighborhood V of Γ such that, for any ϕ ∈ V, there exists a unique pair (s, w) ∈
∈ [0, T0) ×Mε, Mε =
{
w ∈M : |w| < ε
}
, such that
ϕ = ps + Lsw. (4.22)
Suppose now that u(t) is a solution of (4.21) with initial value in the neighborhood V
of Proposition 4.2. As long as ut remains in Γ, relation (4.22) implies that
ut = ps(t) + Ls(t)w(t).
It is possible to obtain the differential equation for s(t) and w(t) and see that these equa-
tions depend only upon s(t) and no delays in s(t). The introduction of the subspace M
transferred all of the dependence upon the past history to the function w(t) ∈M (see [20]
for details and applications).
This same type of transformation should be applicable to many other types of equa-
tions including the partial NFDE above.
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282 J. K. HALE
5. Heat conduction in a solid. The remarks given below about the physical derivation
of the linear theory of heat conduction in a solid is taken from the review of Joseph and
Preziosi [22, 23] which also contains extensive references on the history of the problem.
As we will see, the equations are a special case of our general equations in Section 2. The
objective is to obtain a physical model for rigid heat conductors that can propagate waves.
The methods come from continuum mechanics and thermodynamics.
We begin with a discussion of the simplest situation. Denote by θ the temperature,
q the heat flux, τ the relaxation time and k = k1 + k2 the thermal conductivity, where
k1 is the effective thermal conductivity and k2 is the elastic conductivity. Let Ω ⊂ R
N
be a bounded domain. The objective is to determine an evolutionary equation which will
serve to determine the evolution of temperature θ(t, x), x ∈ Ω and θ satisfies specified
conditions on the boundary ∂Ω of Ω.
If e is the internal energy, then it is assumed that
∂te = −div q. (5.1)
For a solid, it is reasonable to suppose that small changes in e are proportional to small
changes in temperature; that is, there is a constant γ > 0 such that
∂te = γ∂tθ. (5.2)
From (5.1) and (5.2),
γ∂tθ = −div q. (5.3)
The equation governing the evolution of the temperature is obtained by specifying the
manner in which the heat flux depends upon the temperature.
The simplest situation is Fourier’s law:
q = −k∇θ, (5.4)
where k > 0 is a constant. Relations (5.3) and (5.4) yield the classical heat equation
∂tθ =
k
γ
∆θ. (5.5)
This equation has its drawbacks due to infinite speed of propagation.
Cattaneo’s law specifies that
τ∂tq + q = −k∇θ (5.6)
for the relaxation constant τ.
From (5.3) and (5.6), we obtain the linearly damped wave equation
τγ∂2
t θ + γ∂tθ − k∆θ = 0. (5.7)
For given boundary conditions, this equation will generate a C0-semigroup Sτ (t), t ≥ 0,
on a Banach space X and the radius, r(σess(Tτ (1))), of the essential spectrum of Tτ (1)
is less than one. Also, there is a finite speed of propagation of temperature.
One can arrive at (5.7) in the following way by specifying that the heat flux is deter-
mined with a delay time τ (delayed diffusion):
q(t+ τ, x) = −k∇θ(t, x). (5.8)
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A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 283
Relations (5.8) and (5.3) imply that
∂tθ(t+ τ, x) = −k∆θ(t, x). (5.9)
Formally, one obtains (5.7) from (5.9) by relacing θ(t + τ, x) by θ(t, x) + τθ(t, x).
This formal procedure has been used by several authors. Unfortunately, the behavior of
the solutions of (5.7) and (5.9) are completely different. To see this, we write (5.9) in the
equivalent form of an equation with the delay in the diffusion term:
γ∂tθ(t, x) = −k∆θ(t− τ, x). (5.10)
Let us analyze the behavior of some of the eigenvalues of (5.10). If (µk, ek) is a
complete set of eigenpairs of −∆ with the boundary conditions, then we may order the
µk so that µk → ∞ as k → ∞. For any fixed k, there is a solution eλtεk of (4.10) if and
only if λ = (k/γ)e−λτµk or, equivalenty, if λ = µkζ, then
ζ =
k
γ
e−ζµkτ . (5.11)
Fix τ > 0. There is a k0 such that µk > 0 for all k ≥ k0. For each such k, let ζk =
= ζ∗(µkτ) be the unique real solution of (5.11). Then ζk > 0 and ζk → 0 as k → ∞. As
a consequence, there are infinitely many positive eigenvalues of (5.10) which accumulate
at zero and, in particular, r(σess(Tτ (t))) = 1 for all t ≥ 0. The behavior of solutions is in
stark contrast to the damped hyperbolic equation (5.7).
These remarks indicate that neither of the above models are appropriate for heat con-
duction if it is required that there is a finite speed of propagation.
The difficulty arose in the specification of the manner in which the heat flux q depends
upon θ. Let us assume that q depends upon the past history through an expression of the
form
q(t, x) =
0∫
−∞
[
dη(s)
]
θ(t+ s, x), (5.12)
where η is a function of bounded variation. Relations (5.3) and (5.12) imply that
∂tθ(t, x) =
k
γ
∆
0∫
−∞
[
dη(s)
]
θ(t+ s, x). (5.13)
The internal energy should also depend upon the history of the temperature. If we
assume that
e(t, x) = γ
0∫
−∞
[
dµ(s)
]
θ(t+ s, x), (5.14)
where µ is of bounded variation, then the equation for the conduction of heat is given by
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284 J. K. HALE
∂t
0∫
−∞
dµ(s)θ(t+ s, x) =
k
γ
∆
0∫
−∞
dη(s)θ(t+ s, x), (5.15)
which is of the form (3.19).
From our formal discussion of a delayed heat equation, conditions must be imposed
on η and µ in order for this equation to define a dynamical system for which it is possible
to have the radius of the essential spectrum be < 1. To attain this goal we assume that
each of these functions have an atom at zero. In this case, (5.12) and (5.14) can be written
as
e(x, t) = b+ γθ(x, t) +
0∫
−∞
[
dE(s)
]
θ(x, t+ s),
q(x, t) = −k2θ(x, t) −
0∫
−∞
[
dQ(s)
]
∇θ(x, t+ s),
(5.16)
and (5.15) as
γ∂tθ(x, t) +
0∫
∞
[
dE(s)
]
θ(x, t+ s) = k1∆θ(x, t) + ∆
0∫
∞
[
dQ(s)
]
∇θ(x, t+ s), (5.17)
where E and Q are of bounded variation with no atom at 0.
This equation was introduced by Nunziato [24]. For k1 = 0, Gurtin and Pipkin [25]
introduced the equation as a model. Integrals of this type for E = 0 are used in the
Boltzman theory of linear viscoelasticity to express the present value of stress in terms of
past values of strain.
If there is a δ > 0 such that the functions E(s) and Q(s) are constant for s ≤ −δ,
then (5.17) involves only finite delays. If this is not the case, then all of the past history is
required to determine a solution of (5.17). At the present time, there is no general theory
available for the case of infinite delay. For RFDE of retarded type in R
n with infinite
delay, there is an extensive theory in a Banach space X satisfying certain properties and
also conditions on the space X which will ensure that the corresponding semigroup is a
conditional α-contraction (see [26]). Equation (5.17) will exhibit finite speed of propaga-
tion for most kernels E, Q. The proof of this fact requires the discussion of some spectral
theory in an appropriate Banach space for which (4.8) defines a semigroup. We briefly
discuss this problem in the next section for the case of finite delays.
6. Spectrum of linear equations. In this section, we consider the equation
∂tD1zt = AD2zt (6.1)
on an observable space Y and the state space X = C
(
[−r, 0], Y
)
, where A : D(A) ⊂
⊂ Y → Y is the generator of a C0-semigroup on Y and D1, D2 are bounded linear
operators from X to Y.
We also assume that (6.1) generates a C0-semigroup TD1,D2(t), t ≥ 0, on the state
space X and denote by AD1,D2 the infinitesimal generator.
If ϕ ∈ D(AD1,D2), then
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A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 285
AD1,D2ϕ = lim
t→0+
1
t
[
TD1,D2(t)ϕ− ϕ
]
.
If −r ≤ θ < 0, then
lim
t→0+
1
t
[
TD1,D2(t)(θ)ϕ− ϕ(θ)
]
=
= lim
t→0+
1
t
[
ϕ(t+ θ) − ϕ(θ)
]
= ϕ̇(θ) = (AD1,D2ϕ)(θ).
Since ϕ ∈ D(AD1,D2) ⊂ X, it follows that
D(AD1,D2) ⊂
{
ϕ ∈ X : ϕ ∈ C1
}
.
If ϕ ∈ D(AD1,D2), then
∂tTD1,D2(t)ϕ = AD1,D2TD1,D2(t)ϕ
for all t ≥ 0. Therefore,
∂tD1TD1,D2(t)ϕ = D1∂tTD1,D2(t)ϕ = D1AD1,D2(t)TD1,D2(t)ϕ.
Also,
∂tTD1,D2(t)ϕ = AD2TD1,D2(t)ϕ.
Letting t→ 0+, we see that D1ϕ̇ = AD2ϕ and we have shown that
DAD1,D2 =
{
ϕ ∈ X : ϕ ∈ C1, D1ϕ̇ = AD2ϕ
}
,
AD1,D2ϕ = ϕ̇.
The operator AD1,D2 has compact resolvent and the spectrum consists of only point
spectrum which is given by the set
σ(AD1,D2) =
{
λ ∈ C : ∃ϕ ∈ X, ϕ �= 0: λD1ϕ = AD2ϕ
}
,
where A is the operator in (6.1).
We make the following hypothesis:
σ(TD1,D2(1)) = Cl eσ(AD1,D2 ) plus possibly {0}.
From hypothesis (bfH), it is very important to determine the spectrum of AD1,D2 .
We now discuss in some detail a special case of (6.1). However, it will be clear that much
of the analysis is valid in a more general context.
Assume that Ω ⊂ R
N is a bounded domain with smooth boundary and A = ∆, the
Laplacian, with elements in the domain of ∆ satisfying homogeneous Neumann bound-
ary conditions. Let Y = H1(Ω) be the observable space and let X = C
(
[−r, 0], Y
)
.
Consider the equation
∂tD1zt = ∆D2zt (6.2)
We assume that D1 considered as a map from C
(
[−r],R
)
is atomic at zero.
Assume that (6.2) defines a C0-semigroup on X. Let (−µk, ϕk), k = 1, 2, . . . , be a
complete set of eigenpairs of ∆, µ1 < µ2 ≤ µ3 ≤ . . . , µk ≤ . . . , µk → ∞ as k → ∞.
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286 J. K. HALE
Let X(k) = [ϕk], the span of ϕk and let A(k)
D1,D2
= AD1,D2 |X(k). Each subspace
X(k) is invariant under the solutions of (6.2). If zt ∈ X(k), then there is a yt ∈ C([−r, 0],
R) such that zt = ytϕk for all t ≥ 0 and y(t) satisfies the NFDE
∂tD1yt = −µkD2yt. (6.3)
The spectrum σk of the generator AD1,D2 of the semigroup defined by this equation is
given by
σk =
{
λ ∈ C : λD1e
λ· = −µkD2e
λ·}.
It also is clear that
σ(AD1,D2) = ∪k≥1 σk.
If T (k)
D1,D2
(t) = TD1,D2(t)X
(k) and SD1(t) is the C0-semigroup on C0 ≡
{
ϕ ∈
∈ C([−r, 0],R) : ϕ(0) = 0
}
generated by the functional equation D1wt = 0, then we
know that
r
(
σess
(
T
(k)
D1,D2
(1)
))
= r
(
σess(SD1(1)
)
∀k ≥ 1.
In Section 4, we have discussed the case in which D1 = D2 and observed that
r(σess(T
(k)
D1,D1
(1)) = {0} ∪ r(σess(SD1(1)).
In the general case, there are several possibilities for the behavior of the spectrum. It
is instructive to consider some simple examples.
Let us consider the case where D1ϕ = ϕ(0); that is, the equation
∂tz(t) = ∆D2zt. (6.4)
The spectrum of the generator on X(k) is given by the solutions of the equation
λ
µk
= −D2e
λ·, k = 1, 2, . . . . (6.5)
Suppose further that
D2ϕ = ϕ(0) + βϕ(−r), ϕ ∈ C
(
[−r, 0],R
)
. (6.6)
The behavior of the spectrum of the generator of (6.5) is well known (see, for example,
[2]). Using this information, we can make the following remarks. For D2 as in (6.6), if
|β| > 1, there is a k0 such that, for k ≥ k0, there are elements of σk with real parts > 0.
This implies that there is no way to obtain exponential decay of the “delayed diffusion
equation” (6.4), (6.6).
If D2 satisfies (6.6) and if we let ζ = λ/µk in (6.5), then
ζ = −1 − βe−µkrζ . (6.7)
This is the characteristic equation for the RFDE
∂tz(t) = −z(t) − βz(t− µkr). (6.8)
Thus, in order to obtain the exponential stability of all solutions of (6.4), (6.6), we must
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A GENERAL CLASS OF EVOLUTIONARY EQUATIONS 287
have a δ > 0 such that, for every k, all solutions of (6.7) satisfy Re ζ ≤ −δ. To have this
property, it is necessary and sufficient that |β| < 1 (see, for example, [2]). This is the
same as saying that the solutions of the RFDE
∂tz(t) = −z(t) − βz(t− γ) (6.9)
is exponentially stable independent of the delay γ. More precisely, there must be posi-
tive constants K,α such that for any γ ∈ [o,∞), each solution of (6.9) satisfies |zt| ≤
≤ Ke−αt|z0|.
In the same way, if D2 is a general difference operator,
D2ϕ = −ϕ(0) −
M∑
j=1
βjϕ(−rj), (6.10)
then it is not difficult to see that each solution of the equation (6.4), (6.10) approaches
zero exponentially if and only if the zero solution of the RFDE
∂tz(t) = −ϕ(0) −
M∑
j=1
βjz(t− αrj), (6.11)
is exponentially stable independent of α.
It can be shown that a sufficient condition for this to be true is that
∑M
j=1
|βj | <
< 1; that is, the solutions of (6.11) is exponentially stable independently of the delays
r1, r2, . . . , rk.
Consider now equation (6.3) with
D1ϕ = ϕ(0) − αϕ(−1), D2ϕ = ϕ(0) + βϕ(−1). (6.12)
In this case, for every k, we must solve the equation
λ+ µ
λα− µkβ
= e−λ
and determine conditions on α, β so that each solution of (6.3), (6.12) approaches zero
exponentially. If 0 < |α| < 1, then there is a constant R such that |Reλ| < R for all k.
Therefore, there is finite propagation speed. Furthermore, if µ1 > 0 and |β| < 1, then the
zero solution of (6.3), (6.12) is exponentially stable uniformly in k.
These examples suggest the following conjecture.
Conjecture. Suppose that each Dj is a difference operator with delays 0 < r1 <
< r2 < . . . < rn. The radius of the essential spectrum of TD1,D2(1) is < 1 if the
zero solution of each of the equations D1wt = 0 and D2wt = 0 is exponentially stable
independently of the delays.
The results mentioned above depend very strongly on the fact that z in (6.3) is a scalar.
The situation for z ∈ R
n is much more complicated. In the vector situation, there are
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
288 J. K. HALE
known results, there are some results on the exponential stability of difference equations
independent of delays (see [2]). These results apply to some special equations.
The spectral properties of these equations certainly needs to be discussed in more
detail.
1. Hale J. K. Asymptotic behavior of dissipative systems. – Providence: Amer. Math. Soc., 1988.
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4. Hale J. K. Theory of functional differential equations // Ibid. – 1977. – 3.
5. Chow S.-N., Mallet-Paret J. Integral averaging and bifurcation // J. Different. Equat. – 1977. – 26. –
P. 112 – 159.
6. Diekmann O., van Gils S. A., Verduyn-Lunel S. M., Walther H.-O. Delay equations: functional-, complex-
and nonlinear analysis // Appl. Math. Sci. – 1995. – 110.
7. Wu J. Theory and applications of partial functional differential equations // Ibid. – 1996. – 119.
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– 1960. – 50 (92). – P. 423 – 442.
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49. – P. 1281 – 1291.
10. Brayton R. K. Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral
type // Quart. Appl. Math. – 1966. – 24. – P. 215 – 244.
11. Cooke K. L., Krumme D. W. Differential-difference equations and nonlinear initial-boundary value prob-
lems for linear hyperbolic partial differential equations // J. Math. Anal. and Appl. – 1968. – 24. – P. 372 –
387.
12. Lopes O. Asymptotic fixed point theorems and forced oscillations in neutral equations: Ph. D. Thesis. –
Providence RI, 1973.
13. Wu J., Xia H. Self-sustained oscillations in a ring array of lossless transmission lines // J. Different. Equat.
– 1996. – 124. – P. 247 – 278.
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put. Sci. Eng.). – 2004. – 38. – P. 225 – 232.
15. Hale J. K. Coupled oscillators on a circle // Resenhas IME-USP. – 1994. – 1. – P. 441 – 457.
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and Breach, 1961.
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19. Stokes A. Local coordinates around a limit cycle of functional differential equations applications // J.
Different. Equat. – 1977. – 24. – P. 153 – 177.
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2004. – 197. – P. 219 – 246.
21. Hale J. K., Scheurle J. Smoothness of bounded solutions of nonlinear evolution equations // Ibid. – 1985.
– 56. – P. 142 – 163.
22. Joseph D. D., Preziosi L. Heat waves // Rev. Modern Phys. – 1989. – 61. – P. 41 – 83.
23. Joseph D. D., Preziosi L. Addendum to heat waves // Ibid. – 1990. – 62. – P. 375 – 391.
24. Nunziato. Quart. Appl. Math. – 1971. – 29. – P. 187.
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1991. – 1473.
Received 30.11.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
|
| id | umjimathkievua-article-3307 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:04Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/92/a1cf832d57301469a7ea0f8c14f0e192.pdf |
| spelling | umjimathkievua-article-33072020-03-18T19:50:43Z A general class of evolutionary equations Загальний клас еволюційних рівнянь Hale, J. K. Хале, Й. К. Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of retarded and neutral type. Specific illustrations are given using transmission lines nearest-neighbor coupled at the boundary and the theory of heat transfer in solids. Some spectral theory for linearization of the equations is also discussed. 3a допомогою спостережуваних величин та змінної стану динамічного процесу визначено загальне еволюційне рівняння, що узагальнює класичні звичайні диференціальні рівняння, диференціальні рівняння з частинними похідними та спадкові системи із запізненням і системи нейтрального типу. Наведено специфічні ілюстрації з використанням ліній трансмісії із зчепленням „найближчих сусідів" на межі та теорії теплопереносу у твердих тілах. Розглянуто також певну спектральну теорію для лінеаризації рівнянь. Institute of Mathematics, NAS of Ukraine 2007-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3307 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 2 (2007); 268–288 Український математичний журнал; Том 59 № 2 (2007); 268–288 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3307/3359 https://umj.imath.kiev.ua/index.php/umj/article/view/3307/3360 Copyright (c) 2007 Hale J. K. |
| spellingShingle | Hale, J. K. Хале, Й. К. A general class of evolutionary equations |
| title | A general class of evolutionary equations |
| title_alt | Загальний клас еволюційних рівнянь |
| title_full | A general class of evolutionary equations |
| title_fullStr | A general class of evolutionary equations |
| title_full_unstemmed | A general class of evolutionary equations |
| title_short | A general class of evolutionary equations |
| title_sort | general class of evolutionary equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3307 |
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