Đa tạp quán tính đối với một số lớp phương trình tiến hóa tt tiếng anh

MINISTY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
------  ------

BUI XUAN QUANG

INERTIAL MANIFOLDS FOR CERTAIN CLASSES OF
EVOLUTION EQUATIONS
Speciality: Integral and Differential Equations
Code:
9.46.01.03

SUMMARY OF Ph.D. DISSERTATION IN MATHEMATICS

Hanoi – 2020


This dissertation has been completed at the Hanoi National University of Education
Scientific Advisors: Assoc. Prof. Dr. habil. Nguyen Thieu Huy
Dr. Tran Thi Loan

Referee 1:

Assoc. Prof. Dr. Khuat Van Ninh
Hanoi Pedagogical University 2

Referee 2:

Assoc. Prof. Dr. Nguyen Xuan Thao
Hanoi University of Science and Technology

Referee 3:

Assoc. Prof. Dr. Le Van Hien
Hanoi National University of Education

The dissertation will be presented to the examining committee at
the Hanoi National University of Education,
136 Xuan Thuy, Hanoi, Vietnam
on
at

This dissertation is publicly available at HNUE Library Information Centre,
the National Library of Vietnam.


Introduction
1

Motivation

Many phenomena in mechanics, physics, ecology, etc. can be described by partial differential equations. By choosing appropriate function spaces and linear operators, these partial
differential equations can be rewritten into semi-linear evolution equations in an infinitedimensional Banach space whose linear part is the generator of a continuous semigroup and
the nonlinear term satisfies the Lipschitz condition.
The investigation of the asymptotic behavior of solutions to partial differential equations in
large time is one of the central problems of the theory for infinite dimensional dynamic systems.
An important tool for such investigation is the concept of inertial manifolds introduced in
1985 by C. Foias, G.R. Sell & R. Temam (1985) when they studied the asymptotic behavior
of solutions to Navier-Stokes equations. An inertial manifold for an evolution equation is a
(Lipschitz) finite-dimensional manifold which is positively invariant and exponentially attracts
all other solutions of the equation. This fact permits to invoke the reduction principle to study

the asymptotic behavior of the solutions to evolution equations in infinite-dimensional spaces
by comparing with that of the induced equations in spaces of finite-dimension.
Nguyen T.H. (2012) proved the existence of inertial manifolds for the solutions to the
semi-linear parabolic

 du + Au = f (t, u),
t > s,
dt
(1)

u(s) = us ,
when the partial differential operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently large distance between some two successive points of the spectrum,
and the nonlinear forcing term f satisfies the ϕ-Lipschitz conditions.
These lead us to research the topic “Inertial manifolds for certain classes of evolution
equations”.

2
2.1

Overview of the Research Problems
Historical Remarks

1 – The existence of inertial manifolds. The notion of inertial manifolds has been introduced
by C. Foias, G.R. Sell & R. Temam (1985). Inertial manifolds for evolution equations have
been systematically studied in many works, for instance, Chow S.N. & and Lu K. (1988)
considered general equations in Banach spaces with the nonlinear term f bounded and of
class C 1 , but the exponential attractivity towards the manifold was not proved to be uniform
on bounded subsets of the phase space. Mallet-Paret J. & Sell G.R. (1988) introduced the
principle of spatial average to prove the existence of inertial manifolds for reaction-diffusion
equations when the spectral gap condition is not fully satisfied. Also, a more geometric

proof was presented by Constantin P. et al. (1988, 1989) in the Hilbert space case using the
1


concept of spectral barriers in an attempt to refine the spectral gap condition. Demengel E.
& Ghidaglia J.M. (1991) studied the Hilbert space case with A self-adjoint obtaining the first
proof for the case where f is not bounded. Debussche A. & Temam R. (1993) gave another
proof for the case where f is not necessarily bounded, but now in the more general case of
Banach spaces, and f is assumed to be of class C 1 . Other proofs for the nonself-adjoint
case in Hilbert spaces were given by Debussche A. & Temam R. (1991) and Sell G.R. &
You Y. (1992). A nice study of the role of the cone and strong squeezing conditions in the
construction of inertial manifolds in Hilbert spaces was made by Robinson J.C. (1993). Mora
X. (1989) considered damped semilinear wave equations. The notion of inertial manifolds has
been translated and extended to more general classes of differential equations in applications,
for example stochastic inertial manifolds Bensoussan A. & Landoli F. (1995), the existence of
inertial manifolds for non-autonomous evolution equations Koksch N. & Siegmund S. (2011),
or for retarded partial differential equations (1998, 2001).
In of all the above publications, the nonlinear term is assumed to be Lipschitz continuity.
However, for equations arising in complicated reaction- diffusion processes, the Lipschitz coefficients may depend on time. In 2012, Nguyen T.H. considered the parabolic equations (1) and
proved the existence of inertial manifolds when the nonlinear term f (t, u) is ϕ-Lipschitz, i.e.,
i.e., f (t, x)
ϕ(t) 1 + Aβ x and f (t, x) − f (t, y)
ϕ(t) Aβ (x − y) where ϕ belongs
to one of admissible function spaces containing wide classes of function spaces like Lp -spaces,
the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.
2 – Generalizations of inertial manifolds. The notion of inertial manifolds has also been
extended to various concepts such as slow manifolds motivated by phenomena in meteorology,
to inertial manifolds for equations with delay. Moreover, the existence of a new type of inertial
manifolds, namely the admissibly inertial manifolds of E-class, has been proved by Nguyen
T.H. (2013). Such manifolds were consisted of solution trajectories belonging to a Banach

space E, which could be the Lp -space, Lorentz space Lp,q or many other function spaces
occurring in interpolation theory.
3 – Applications of inertial manifolds. Besides the existence of inertial manifolds for specific
partial differential equations, inertial manifolds have found numerous useful applications in
other branches of mathematics. These include the connection of inertial manifold with the
multigrid methods of numerical analysis or an attempt of inertial manifold to describe the
turbulence of fluid mechanics. This dissertation will emphasize the applications of inertial
manifolds in control theory.

2.2

Classes of evolution equations in this dissertation

A. Parabolic equations
du(t)
+ Au(t) = f (t, u(t)).
dt

(2)

B. Partial functional differential equations (with finite delay)
du(t)
+ Au(t) = L(t)ut + g(t, ut ).
dt

(3)

C. Partial neutral functional differential equations
∂
F ut + AF ut = Φ(t, ut ).

∂t
2

(4)


3

Purpose, Objects and Scope of the Dissertation

Purpose. We study the existence of inertial manifolds and the asymptotic behavior of solutions to certain classes of evolution equations in an infinite-dimensional Banach space.
The evolution equations considered with the linear parts is the generator of a semigroup
and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to
admissible function spaces which contain wide classes of function spaces like Lp -spaces,
the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.
Objects. Inertial manifolds and finite-dimensional feedback control for evolution equations
(3), (2) and (4) in admissible spaces.
Scope of the Dissertation. The scope of the dissertation is defined by the following contents
◦ Content 1. Study the existence of inertial manifolds for du(t)
+Au(t) = f (t, u(t)) under
dt
the conditions that the partial differential operator A is positive such that −A is
sectorial with a sufficiently large gap in its spectrum and f is a nonlinear operator
satisfying ϕ-Lipschitz condition.
◦ Content 2. Study the regularity of the inertial manifolds for du(t)
+ Au(t) = f (t, u(t))
dt
and using the theory of inertial manifolds for non-autonomous semi-linear evolution
equations, we construct a feedback controller for a class of control problems for the
one-dimensional reaction-diffusion equations with the Lipschitz coefficient of the

nonlinear term may depends on time and belongs to an admissible space.
◦ Content 3. Study the existence of inertial manifolds for partial functional differential
equation du(t)
+ Au(t) = L(t)ut + g(t, ut ) under the conditions that the partial
dt
differential operator A is positive such that −A is sectorial with a sufficiently large
gap in its spectrum; the operator L(t) is linear, and g is a nonlinear operator
satisfying ϕ-Lipschitz condition for ϕ belonging to an admissible function space.
◦ Content 4. Study the existence of inertial manifolds for partial neutral functional dif∂
ferential equation ∂t
F ut + AF ut = Φ(t, ut ), where the partial differential operator
A is positive definite and self-adjoint with a discrete spectrum having a sufficiently
large gap; the difference operator F : Cβ → X is bounded linear operator, and the
nonlinear delay operator Φ satisfies the ϕ-Lipschitz condition.

4

Research Methods

The dissertation uses the tools of functional analysis, fixed point theorem, semigroup theory
to study the contents. Moreover, we use some special techniques to get our purpose:
• Linear operators: Using semigroup theory, analytic semigroups, the perturbation theory
for strongly continuous semigroups.
• Nonlinear terms: Theory of admissible spaces.
• To prove existence of inertial manifolds : we use fixed point theory and the LyapunovPerron method.

3


5

Dissertation Outline
• Chapter 1. Preliminaries.
• Chapter 2. Inertial manifolds for a class of parabolic equations and applications.
• Chapter 3. Inertial manifolds for a class of partial functional differential equations with
finite delay.
• Chapter 4. Inertial manifolds for a class of partial neutral functional differential equations.

4


Chapter 1
Preliminaries
In this chapter, we present some basic results about linear operators, semigroup theory,
and function spaces.

1.1

Semigroups

In this section we recall the most basic notions of the semigroup theory and their generators.
The main reference is Engel K.J. & Nagel R. (2000) (see also C.T. Anh & T.D. Ke (2016)).

1.2
1.2.1

Linear Operators
Positive Operators with Discrete Spectrum

Assumption A. Let X be a separable Hilbert space and suppose that A is a closed linear

operator on X satisfying the following assumption. We suppose that A is a positive definite,
self-adjoint operator with a discrete spectrum, say
0 < λ1

λ2

· · · λk

···

each with finite multiplicity, and

lim λk = ∞.

k→∞

We assume that {ek }∞
k=1 is the orthonormal basis of X consisting of the corresponding
eigenfunctions of the operator A (i.e., Aek = λk ek ). Let now λN and λN +1 be two successive
and different eigenvalues with λN < λN +1 , let further P be the orthogonal projection onto the
first N eigenvectors of the operator A.

1.2.2

Sectorial Operators and Analytic Semigroups

Definition 1.1. Let X be a Banach space. A closed, linear and densely defined operator
B : X ⊃ D(B) → X is called a sectorial operator of (σ, ω)-type if there exist real numbers
ω ∈ R, σ ∈ 0, π2 and M 1 such that
π

ρ(B) ⊃ Σσ (ω) := z ∈ C : | arg(z − ω)| < σ + , z = ω ,
(1.1)
2
M
R(λ, B)
for all λ ∈ Σσ (ω).
(1.2)
|λ − ω|
To prove the existence of an inertial manifold, we suppose the following assumption.
Assumption B. Let A be a closed linear operator on a Banach space X such that −A is a
sectorial operator of (σ, ω)-type with 0 < σ < π/2 and ω < 0. We suppose that the spectrum
σ(−A) of −A can be decomposed as follows:
σ(−A) = σu (−A) ∪ σc (−A) ⊂ C−
5


with ωu < ωc < ω < 0 where
ωu := sup{Reλ : λ ∈ σu (−A)},

ωc := inf{Reλ : λ ∈ σc (−A)}

(1.3)

and σc (−A) is compact.
Assumption B allows us to choose real numbers κ and µ such that
ωu < κ < µ < ωc < 0.

(1.4)

We recall the Riesz projection (or spectral projection) P corresponding to σc (−A), defined

by
P =

1
2πi

R(λ, −A)dλ,

(1.5)

+

where + is a closed regular curve contained in ρ(−A), surrounding σc (−A) and positively
oriented.
We now recall some properties, called dichotomy estimates, of the analytic semigroup
e−tA t 0 .
Proposition 1.2. Let P be the Riesz projection as mentioned above and choose κ < µ < 0 being
the real numbers as in (1.4). For θ > 0, the following dichotomy estimates hold true:
e−tA P

M1 e−µ|t|

for all

t ∈ R,

(1.6)

Aβ e−tA P

M2 e−µ|t|

for all

t ∈ R,

(1.7)

κt

for all

t

0,

(1.8)

for all

t > 0.

(1.9)

−tA

e

(I − P )

Aβ e−tA (I − P )

Me
N κt
e
tβ

Let A satisfy Assumption A or Assumption Assumption1. Then, we can define the Green
function as follows:
e−(t−τ )A [I − P ] for all t > τ,
G(t, τ ) =
(1.10)
−e−(t−τ )A P
for all t τ.

1.2.3

Auxiliary Results

Theorem 1.3 (Bounded Perturbation Theorem). Let A be the generator of a strongly continuous semigroup etA t 0 on a Banach space X satisfying etA
M eωt for all t 0, ω ∈ R
and some M 1. If B ∈ L(X) then
C := B + A

generates a strongly continuous semigroup (S(t))t
S(t)

D(C) := D(A)

with

M e(ω+M

B )t

0

satisfying
for allt

0.

Theorem 1.4. Let V be a Banach space. Suppose A ∈ L(V ) and G is an open set covering
the spectrum σ(A). Then there exists a δ-neighborhood Uδ (A) of A such that σ(X) ⊂ G for
all X ∈ Uδ (A). Moreover, for any ε > 0 there exists a δ such that Rλ (X) − Rλ (A) < ε for
X ∈ Uδ (A) and λ ∈
/ G.

6


1.3

Admissible Spaces

Denote by B the Borel algebra and by λ the Lebesgue measure on R. The space L1,loc (R)
of real-valued locally integrable functions on R (modulo λ-nullfunctions) becomes a Fr´echet
space for the seminorms pn (f ) = Jn |g(t)|dt, where Jn = [n, n + 1] for each n ∈ Z.
We then define Banach function spaces as follows.
Definition 1.5. A vector space E of real-valued Borel-measurable functions on R (modulo

λ-nullfunctions) is called a Banach function space (over (R, B, λ)) if
(1) E is a Banach lattice with respect to the norm · E , i.e., (E, · E ) is a Banach space, and
if ϕ ∈ E, ψ is a real-valued Borel-measurable function such that |ϕ(·)| |ψ(·)| (λ-a.e.)
then ψ ∈ E and ϕ E
ψ E,
(2) the characteristic functions χA belongs to E for all A ∈ B of finite measure and
sup χ[t,t+1]

E

< ∞,

inf χ[t,t+1]

t∈R

t∈R

E

> 0,

(3) E → L1,loc (R).
We remark that the condition (3) in the above definition means that for each compact
interval J ⊂ R, there exists a number βJ 0 such that
|g(t)|dt

βJ f

for all f ∈ E.

E

J

We now introduce the notion of admissibility in the following definition.
Definition 1.6. The Banach function space E is called admissible if it satisfies
(1) there is a constant M

1 such that for every compact interval [a, b] ⊂ R we have
b

M (b − a)
ϕ
χ[a,b] E

|ϕ(t)|dt
a

E,

(1.11)

(2) for ϕ ∈ E the function
t

Λ1 ϕ(t) =

ϕ(τ )dτ

(1.12)

t−1

belong to E,
(3) the space E is Tτ+ -invariant and Tτ− -invariant where Tτ+ and Tτ− are defined, for τ ∈ R,
by
Tτ+ ϕ(t) := ϕ(t − τ )

for t ∈ R,

(1.13)

Tτ− ϕ(t) := ϕ(t + τ )

for t ∈ R.

(1.14)

Moreover, there are constants N1 and N2 such that
Tτ+

N1

Tτ−

and

7

N2

for all τ ∈ R.


Evolution Equations in Admissible Spaces
Definition 1.7 (ϕ-Lipschitz functions). Let E be an admissible Banach function space on R
and ϕ be a positive function belonging to E. Put Xβ := D(Aβ ) for β ∈ [0, 1) and Cβ :=
C([−h, 0], Xβ ). Then, a function Φ : R × Cβ → X is said to be ϕ-Lipschitz if Φ satisfies
(1) Φ(t, ut )

ϕ(t) 1 + |ut | Cβ for a.e. t ∈ R and for all ut ∈ Cβ ,

(2) Φ(t, ut ) − Φ(t, vt )

ϕ(t)|ut − vt | Cβ for a.e. t ∈ R and for all ut , ut ∈ Cβ .

Let χ(s) be an infinitely differentiable function on [0, ∞) such that χ(s) = 1 for 0 s 1,
χ(s) = 0 for s 2, 0 χ(s) 1 and |χ (s)| 2 for s ∈ [0, ∞). Define the cut-off mapping by
fR (t, u) := χ

Aβ u
R

for all u ∈ D(Aβ ).

f (t, u)

(1.15)

We have
Proposition 1.8. If f (t, x) is locally ϕ-Lipschitz in a ball BR , then fR (t, x) is
Lipschitz.

2R2 +5R+2
ϕ
R

-

Assumption C. Let ϕ be a positive function belonging to E such that
ϕ(τ )

t−1

(t − τ )

R(ϕ, β) := sup
t∈R

2β
1+β

1+β
2β

t

1+β
2

dτ

<∞

trong 0 < β < 1.

Assumption D. Let ϕ be a positive function belonging to E such that
ϕ(τ )

t+θ−1

(t + θ − τ )

R(ϕ, β, h) := sup sup
t∈R θ∈[−h,0]

2β
1+β

1+β
2β

t+θ

1+β
2

Note that, in case β = 0 we do not need this assumptions.

8

dτ

< ∞.

(1.16)


Chapter 2
Inertial Manifolds for a Class of Parabolic Equations
and Applications
Motivated by a two-specie competition model with cross-diffusion in population ecology
with time-dependent environmental capacity, we study the semi-linear parabolic equation of
the form x˙ + Ax = f (t, x) where the linear operator −A generates an analytic semigroup.
Our main task is to prove the existence of inertial manifolds for mild solutions to such an
evolution equation under the conditions that the linear partial differential operator −A has
spectral gap being sufficiently large, and the nonlinear term f satisfies ϕ-Lipschitz condition,
i.e., f (t, x) − f (t, y)
ϕ(t) Aβ (x − y) for ϕ belonging to some admissible space. We then
apply the obtained result to study asymptotic behavior of the above-mentioned competition
model.
Next, we will establish the C 1 -regularity of the inertial manifolds for the above-mentioned
parabolic equation when the nonlinear term is of class C 1 with respect to the state variables
and the linear part is a positive definite, self-adjoint operator with a discrete spectrum (satisfies
Assumption A) or is a sectorial operator with a sufficiently large gap (satisfies Assumption
B). The proof of the regularity will be performed in detail for the case where the linear operator
satisfies Assumption A.
Finally, we present an application of the theory of inertial manifold in the study of a class
of finite-dimensional feedback control problem of a one-dimensional reaction-diffusion system

with distributed observation and control.
The content of this chapter is written based on the papers [3] and [4] in the List of Publications.

2.1

Competition Model with Cross-Diffusion: Sectorial Operators and Setting of the Problem

Consider the parabolic equation

 du(t) + Au(t) = f (t, u(t)),
dt

u(s) = us ,

t > s,

(2.1)

s ∈ R.

where the linear operator A satisfies Assumption A or Assumption B and f : R × Xβ → X is
a nonlinear mapping satisfies Assumption C.
In the case of infinite-dimensional phase spaces, instead of parabolic equation (2.1), we
consider the integral equation
t

u(t) = e−(t−s)A u(s) +

e−(t−ξ)A f (ξ, u(ξ))dξ
s

9

for a.e. t

s.

(2.2)


By a solution of equation (2.2) we mean a strongly measurable function u(t) defined on an
interval J with the values in Xβ that satisfies (2.2) for t, s ∈ J. The solution u to equation
(2.2) is called a mild solution of evolution equation (2.1).
Definition 2.1. An inertial manifold of equation integral (2.2) is a collection of Lipschitz
manifolds M = Mt t∈R in X such that each Mt is the graph of a Lipschitz function
Φt : Pn X → (I − Pn )Xβ , i.e.,
Mt = {x + Φt x : x ∈ Pn X}

for t ∈ R

(2.3)

and the following conditions are satisfied:
(1) The Lipschitz constants of Φt are independent of t, i.e., there exists a constant C
independent of t such that
Aβ (Φt x1 − Φt x2 )

C Aβ (x1 − x2 )

(2.4)

for all t ∈ R and x1 , x2 ∈ Xβ .
(2) There exists γ > 0 such that to each x0 ∈ Mt0 there corresponds one and only one
solution u(t) to (2.2) on (−∞, t0 ] satisfying that u(t0 ) = x0 and
esssup e−γ(t0 −t) Aβ u(t) < ∞.

(2.5)

t t0

(3) The collection Mt t∈R is positively invariant under (2.2), i.e., if a solution x(t), t
to (2.2) satisfies xs ∈ Ms , then we have that x(t) ∈ Mt for t s.

s,

(4) The collection Mt t∈R exponentially attracts all the solutions to (2.2), i.e., for any
solution u(·) of (2.2) and any fixed s ∈ R, there is a positive constant H such that
distXβ (u(t), Mt )

He−γ(t−s)

for t

s,

(2.6)

where γ is the same constant as the one in (2.5), and distXβ denotes the Hausdorff
semi-distance generated by the norm in Xβ .
We now fully state the main results about the existence of an inertial manifold for mild

solutions to the semi-linear evolution equations is as follows.
Theorem 2.2 (see Nguyen T.H. (2012)). Let the operator A satisfying Assumption A and
ϕ belongs to some admissible space E. Let f be ϕ-Lipschitz function such that the function
ϕ satisfying Assumption C. Suppose that there are two successive eigenvalues λn < λn+1 of
linear operator A satisfying
kγ < 1

kγ M 3 N2 λ2β
n Λ1 ϕ ∞
+ kγ < 1,
(1 − kγ )(1 − e−α )

and

(2.7)

where
kγ :=


 M (β β N1 +λβn+1 N1 +λβn N2 )
1−e−α

 M (N1 +N2 )
1−e−α

Λ1 ϕ

Λ1 ϕ

∞

+ M β β R(ϕ, β)

1−β
α(1+β)

1−β
1+β

if 0 < β < 1,
if β = 0.

∞

Then, integral equation (2.2) has an inertial manifold.
10

(2.8)


We will study the Lotka-Volterra competition model of two species with cross-diffusion
which is described by the following system of partial differential equations of parabolic type

∂u
r1


= D1 ∆u + r1 u − u2 − h1 uv,
∂t

K
(2.9)
∂v
r

2 2

= D2 ∆v + r2 v − v − h2 uv.
∂t
K
For brevity, we will write that system in the matrix form
∂ u
D1 ∆
0
=
0
D2 ∆
∂t v

u
r u − rK1 u2 − h1 uv
+ 1
.
v
r2 u − rK2 v 2 − h2 uv

(2.10)

The above system can be rewritten in an operator form as
dx

= Cx + g(x),
dt

(2.11)

where
x :=

U
,
V

C :=

D1 ∆
0
,
0
D2 ∆

and

g(x) =

r1 U − rK1 U 2 − 2 rK1 U u0 − h1 U V − h1 U v0 − h1 u0 V
.
r2 V − rK2 V 2 − 2 rK2 V v0 − h2 U V − h2 U v0 − h2 u0 V

Our aim is apply the Linearization Principle to the system (2.11). We will rewritten the
system as

dx
= (C + Jg (x0 )) x + g(x) − Jg (x0 )x,
(2.12)
dt
u0
and Jg (x0 ) denotes the Jacobian matrix of g(x) at x0 .
v0
The important information that we would like to mention is the linear operator −A with
relevant domain is a sectorial operator.
where x0 =

2.2
2.2.1

Inertial Manifolds for Parabolic Equations with Sectorial
Operators
Lyapunov-Perron Equation

Lemma 2.3. Let the operator A satisfy Assumption B and f : R × Xβ → X be ϕ-Lipschitz
for ϕ satisfying Assumption C. For any fixed t0 ∈ R let x(t), t t0 be a solution to integral
equation such that x(t) ∈ Xβ for t t0 and
esssup e−γ(t0 −t) Aβ x(t) < ∞.
t t0

Then, this solution x(t) satisfies
x(t) = e−(t−t0 )A v1 +

t0

G(t, τ )f (τ, x(τ ))dτ

−∞

where v1 ∈ P X, and G(t, τ ) is the Green’s function.

11

for a.e. t

t0

(2.13)


2.2.2

Existence and Uniqueness of Solutions in Weighted Spaces

We then have the following lemma which describes the existence and uniqueness of certain
solutions belonging to weighted spaces.
Lemma 2.4. Let the operator A satisfy Assumption B and f : R × Xβ → X be ϕ-Lipschitz
with the positive function ϕ satisfied the condition Assumption C. Put

1−β
1+β
1+β
 N N1 +M1 N2
−α 1−β
1−β
if 0 < β < 1,
Λ1 ϕ ∞ + N R(ϕ, β) α(1+β) 1 − e

−α
1−e
k :=
(2.14)
 M N1 +M1 N2
Λ
ϕ
if
β
=
0.
1
∞
−α
1−e
If k < 1, there corresponds to each v ∈ P X one and only one solution x(t) of equation (2.13)
on (−∞, t0 ] satisfying the condition P x(t0 ) = v and
esssup e−γ(t0 −t) Aβ x(t) < ∞.
t t0

2.2.3

The Existence of Inertial Manifolds

The main result of this chapter is stated in the following theorem.
Theorem 2.5. Let the operator A satisfy Assumption B and ϕ belongs to some admissible
space E. Let f be ϕ-Lipschitz satisfying condition Assumption C. If
k<1

and

M kM22 N2
Λ1 ϕ
(1 − k)(1 − e−α )

∞

+k <1

(2.15)

then integral equation has an inertial manifold.

2.3

Applications to the Competition Models with Cross-Diffusion

In this section, we will apply the obtained results to the Lotka-Volterra competition models
with cross-diffusion.

2.4

Regularity of the Inertial Manifolds

Theorem 2.6. If f (t, ·) ∈ C 1 (Xβ , X), then the inertial manifold given in Theorem 2.2 and
Theorem 2.5 is of class C 1 and Φt satisfies the Sacker’s equation
DΦt (y)(−Ay + Pn f (t, y + Φt (y)) + AΦt (y)) = Qn f (t, y + Φt (y))

(2.16)

for all y in the domain of Φt .

2.5
2.5.1

Finite-Dimensional Feedback Control of a Reaction-Diffusion
System via Inertial Manifold Theory
The Open-Loop System

Consider open-loop system of the following nonlinear one-dimensional reaction-diffusion
equation with zero Dirichlet boundary condition and distributed observation and control (as
12


in R. Rosa and R. Temam (1997))

I−1

∂u(t, x)



=
∆u(t,
x)
+
f
(u(t,
x))
+

gi (t)ψi (x), t > 0, 0 < x < π,


∂t

i=1

J−1
J−1
y(t) = (yi (t))j=1 = (u(t, xj ))j=1 ,
t 0,




u(t, 0) = u(t, π) = 0,
t > 0,




u(0, x) = u0 (x),
0 x π,

(2.17)

where u = u(t, x) is the state variable, for x ∈ Ω := (0, π), y is the observation, g = (gi )i is
the control, f is a nonlinear term, and I, J ∈ N. The functions ψi are called the actuators
and are assumed to lye in the Sobolev space H01 (Ω), while the points xj are distinct points in
Ω called the obervation points and assumed to increase with j.

We consider this equation in the phase space X = H01 (Ω) endowed with the norm u =
|Du|, for u ∈ X, where | · | denotes the usual L2 -norm on Ω and Du denotes the derivative
of u. We also denote by ((·, ·)) and (·, ·) the corresponding inner-products in X and L2 (Ω),
respectively.
We consider the linear operator A := −∆ and we have that the linear operator A is
a self-adjoint operator with eigenvalues given by {λn = n2 }n∈N and eigenfunctions {en =
sin(nx)}n∈N . Moreover, we have β = 0 and Xβ = H01 (Ω).
Secondly, let Z1 and Z2 be two finite-dimensional Hilbert spaces. Suppose that Z1 RI−1
and Z2 RJ−1 . We now define two bounded linear operators
I−1

B : Z1 → X

by Bg =

gi ψi (x)

I−1
for g = (gi )i=1
∈ Z1 ,

(2.18)

for u ∈ X.

(2.19)

i=1

C : X → Z2

J−1
by Cu = ((Cu)j )J−1
j=1 = (u(xj ))j=1

We can now write the control problem (2.17) in the Sobolev space X = H01 (Ω),

 du + Au = f (u) + Bg,
dt

y = Cu.

(2.20)

We would like to construct g as a function of y so that the closed-loop system behaves in
a certain desired way. We will expand to the case of non-autonomous control system of the
form

 du + Au = f (t, u) + Bg,
dt
(2.21)

y = Cu.
Here, we consider control g(t, y) depends on both the time t and the observation y.
We further assume that we are given another set of points {˜
xi }Ii=1 , with
0 = x˜0 < . . . < x˜i < x˜i+1 < . . . < x˜I = π,
and that ψi , for i = 1, . . . , I − 1, is given more precisely by

x−˜

xi−1

x ∈ [˜
xi−1 , x˜i ),

 h˜ i ,
−x
ψi (x) = x˜i+1
x ∈ [˜
xi , x˜i+1 ),
˜ i+1 ,
h


0,
otherwise,
13

(2.22)


˜ i = x˜i − x˜i−1 . Set also
where h
hj = xj − xj−1 ,

˜i .
˜ = max h
h

h = max{hj },

j

i

(2.23)

It is not very difficult to show that
B

2.5.2

L(Z1 ,X)

=1

and

C

L(X,Z2 )

= 1.

(2.24)

The Desired Dynamics

Firstly, consider a nonlinear mapping W : R × Pn0 X → Pn0 X satisfies the following conditions
W (t, u) − W (t, v)
DW (t, u) − DW (t, v)

ς1 (t) u − v

L(X)

ς2 (t) u − v

for all u, v ∈ Pn0 X,
ν

for all u, v ∈ Pn0 X,

(2.25)
(2.26)

for some positive valued functions ςi (t), i = 1, 2, belong to an admissible space, and 0 < ν 1.
We now consider the non-autonomous control system

 du + Au = f (t, u) + Bg,
dt
(2.27)

y = Cu
in a separable infinite-dimensional Hilbert space X and finite-dimensional non-autonomous
ordinary differential equation
dz(t)
= W (t, z(t)),
(2.28)
dt
where n0 ∈ N is fixed.

We look forward that the desired dynamics for the system (2.27) will be determined by
system (2.28).

2.5.3

The Input and Output Control Operators

Lemma 2.7. For m and n are two arbitrary natural numbers, we have the following estimates
for the operators B and C

2.5.4

(CPm )−1

L(Z2 ,X)

(Pn B)−1
r

L(Pn X,Z1 )

2
,
1 − 2h2 λm
1
.
˜ 2 λn
1 − 4h

(2.29)

(2.30)

The Finite-Dimensional Feedback Controller

Consider m and n arbitrary such that
m

n > n∗ ,

(2.31)

where n∗ ∈ N such that the Theorem 2.2 is satisfied. This means that the Theorem 2.2 will
be holds for λn∗ and λn∗ +1 . Choose then the xj ’s and the xi ’s such that
√
3
1
˜
h
and h
(2.32)
1/2
1/2
4λn0
2λm
14


so that
1
˜ n

1 − 4hλ

2

and

2
1 − 2hλm

2.

(2.33)

We now define g : R × Z2 → Z1 by
g(t, y) = (Pn B)−1
APn0 (CPm )−1 y + W t, Pn0 (CPm )−1 y − Pn f t, (CPm )−1 y
r

(2.34)

for all y ∈ Z2 and t ∈ R. Thanks to Lemma 2.7, we have g is a globally Lipschitz function
with
Lip(g)
(Pn B)−1
r

L(Pn X,Z1 )

1
˜ n

1 − 4hλ

(CPm )−1

L(Z2 ,X)

APn0

L(X)

+ Lip(W ) + Lip(f )

2
(λn0 + ς1 (t) + ϕ(t))
1 − 2hλm

4 (λn0 + ς1 (t) + ϕ(t)) .
Thus
Lip(g)

2.5.5

ξ(t),

where ξ(t) := 4 (λn0 + ς1 (t) + ϕ(t)) , for all t ∈ R.

(2.35)

Inertial Manifolds for the Closed-Loop System

With g given by (2.34) we can write (2.20) in the closed-loop form
du
+ Au = f (t, u) + Bg(t, Cu).
dt

(2.36)

We shall also consider the following auxiliary parabolic equation
dv
+ Av = Pm f (t, Pm v) + Pm Bg(t, CPm v).
dt

(2.37)

Note that the nonlinear term of both the equations above have Lipschitz coefficient less than
or equal to η(t) := ϕ(t) + ξ(t) for t ∈ R.
We want that, under the suitable conditions, there will be inertial manifolds for parabolic
equations (2.36) and (2.37). Applying Theorem 2.2 for the parabolic equations (2.36) and
t
(2.37), we obtain that, if n∗ is large enough and the norm Λ1 η ∞ = supt∈R t−1 η(τ )dτ is
sufficiently small, then have inertial manifolds, M = Mt t∈R and N = Nt t∈R , respectively
for (2.36) and (2.37).
In more detail, the inertial manifold for the parabolic equation (2.36) is
M = Mt

t∈R

,

where Mt = {p + Φt (p) : p ∈ Pn X},

(2.38)

here Φt : Pn X → Qn X, defined by Φt0 (p) := Qn x(p)(t0 ) where x(p) is the unique solution in
γ,t0 ,θ
L∞
to the equation (2.2) satisfying that Pn x(p)(t0 ) = p. Similarly,
N = Nt

t∈R

,

where Nt = {p + Ψt (p) : p ∈ Pn X}, here Ψt : Pn X → Qn X

is the inertial manifold for the auxiliary parabolic equation (2.37).

15

(2.39)


When the two evolution equations (2.36) and (2.37) have two inertial manifolds, the corresponding inertial forms on Pn X are

and

dp
+ Ap = Pn f (t, p + Φt (p)) + Pn Bg(t, C(p + Φt (p))),
dt

(2.40)

dρ
+ Aρ = Pn f (t, Pm (ρ + Ψt (ρ))) + Pn Bg(t, CPm (ρ + Ψt (ρ))).
dt

(2.41)

Note that
dρ
+ A(Pn − Pn0 )ρ
dt
= −APn0 ρ + Pn f (t, Pm (ρ + Ψt (ρ)))
+Pn B(Pn B)−1
APn0 (CPm )−1 CPm (ρ + Ψt (ρ))
r
+W t, Pn0 (CPm )−1 CPm (ρ + Ψt (ρ)) − Pn f t, (CPm )−1 CPm (ρ + Ψt (ρ)
= −APn0 ρ + Pn f t, Pm (ρ + Ψt (ρ)) +
APn0 Pm (ρ + Ψt (ρ)) + W t, Pn0 Pm (ρ + Ψt (ρ)) − Pn f t, Pm (ρ + Ψt (ρ))
= W (t, Pn0 ρ).
Thus, the inertial form for (2.37) reads
dρ
+ A(Pn − Pn0 )ρ = W (t, Pn0 ρ),
dt

(2.42)

and can be split for ρ = ρ1 + ρ2 , where ρ1 ∈ Pn0 X, ρ2 ∈ (Pn − Pn0 )X as

dρ1



= W (t, ρ1 ),
dt

 dρ2 + A(Pn − Pn )ρ2 = 0.
0
dt

(2.43)

The system (2.43) above is now decoupled with
2 (t−s)

ρ2 (t) = e−(t−s)A(Pn −Pn0 ) ρ2 (s) = O e−(n0 +1)

,

as t → ∞.

Hence, the long-time dynamics of the inertial form and, hence, of the auxiliary equation (2.37)
1 (t)
is given by the system dρdt
= W (t, ρ1 ).
Concerning the inertial form (2.40), we can write it as
dp
+ A(Pn − Pn0 )p = W (t, Pn0 p) + ε(t, p),
dt

(2.44)

where ε(t, p) is regarded as an error term given by
ε(t, p) = Pn f (t, p + Φt (p)) + Pn Bg(t, C(p + Φt (p)))
−Pn f (t, p + Pm Ψt (p)) − Pn Bg(t, C(p + Pm Ψt (p))).
By using dichotomy estimates and admissibility of function spaces we can obtain
ε(t, p)

η(t)
1/2

λm

for all p ∈ Pn X,

(c1 + c2 p )
16

(2.45)


Dε(t, p)

L(Pn X)

c3

η(t)

1/2
λm

c4

+

for all p ∈ Pn X,

ν/2

λm

(2.46)

where the ci ’s are constant such that
ci = ci (n0 , n, Λ1 ϕ

∞,

c4 = c4 (n0 , n, Λ1 ϕ2

∞,

Λ1 ς1

∞) ,

Λ1 ς2

for i = 1, 2, 3,

∞ , ν)

Thus for each t ∈ R, we have ε(t, p) → 0 and Dε(t, p) L(Pn X) → 0 as m → ∞.
We will summarize the above events in the following main results:
Theorem 2.8. Consider the open-loop system (2.17). Let non-autonomous ordinary differential
equation (2.28) be given with n0 ∈ N and W satisfying (2.25) and (2.26). Suppose that n∗ is
the natural number that the conditions in the Theorem 2.2 satisfied with λn∗ and λn∗ +1 , and
conditions (2.31) and (2.32) hold.
If a feedback law g = g(t, y) is given by (2.34), then the closed-loop equation (2.36) has an
inertial manifold whose inertial form (2.44) is C 1 -close to (2.42), which has essentially the
same dynamics as (2.28), in a weighted metric for the vector fields as estimated in (2.45) and
(2.46).
Similar to the work R. Rosa and R. Temam (1997), we state the following result about
structural stability of the dynamical systems.
Theorem 2.9. Assume the hypotheses in Theorem 2.8 hold and the nonlinear funtion W satisfies condition, for some r0 > 0,
−α z

((W (t, z), z))
and that the flow induced by

dz
dt

for all z

r0 , and for some α > 0,

= W (t, z) for z restricted to the ball
Brn00 := {z ∈ Pn0 X : z

r0 }

is structurally stable.
If feedback law g = g(t, y) is given by (2.34) with m chosen large enough, then the longtime dynamics of the inertial form (2.44) of the closed-loop equation (2.36) is contained in
the ball Brn0 = {p ∈ Pn X : p
r0 } and the corresponding flow restricted to this ball Brn0
is topologically equivalent to the flow given by (2.42), so that the dynamics of the closed-loop
= W (t, z).
system is essentially that of dz
dt

17


Chapter 3
Inertial Manifolds for a Class of
Partial Functional Differential Equations
with Finite Delay
In this chapter, we prove the existence of inertial manifolds for the partial functional
differential equations of the form du(t)
+ Au(t) = F (t)ut + g(t, ut ) under the conditions that the
dt
partial differential operator A is positive such that −A is sectorial with a sufficient large enough
gap in its spectrum, t → F (t) is an operator-valued function, and g is a nonlinear operator
satisfying ϕ-Lipschitz condition, i.e., g(t, ψ)
ϕ(t) 1 + |ψ| Cβ and g(t, ψ) − g(t, φ)
ϕ(t)|ψ − φ| Cβ for Cβ := C([−h, 0], D(Aβ ). Here, F (·) and ϕ belong to an admissible
function space. Our main methods are based on Lyapunov-Perron equations combining with
analytic semigroups and admissibility of function spaces.
The content of this chapter is written based on the paper [1] in the List of Publications.

3.1

Setting of the Problem

We study partial functional differential equation (PFDE) of the form

 du(t) + Au(t) = F (t)u + g(t, u ), t > s,
t
t
dt

us = φ ∈ Cβ , s ∈ R = (−∞, +∞),

(3.1)

where Cβ := C([−h, 0], Xβ ), with Xβ := D(Aβ ), 0 β < 1, being the domain of the fractional
power of the positive operator A having the property that −A generates an analytic semigroup
{e−tA }t 0 on X, F (t) : Cβ → X is bounded linear operator for each t ∈ R, g : R× Cβ → X is a
nonlinear operator, and ut is the history function defined as ut (θ) := u(t+θ) for all θ ∈ [−h, 0].
We consider the integral equation
t
−(t−s)A

u(t) = e

e−(t−ξ)A F (ξ, u(ξ))dξ

u(s) +

for a.e. t

s.

(3.2)

s

In this section we will prove the existence of the inertial manifolds for partial functional
differential equations with finite delay. Then, we state of the main result of this chapter.
We suppose that A satisfies Assumption B and consider the spectral projection P defined
as in (1.5), we define the projector Pˆ in Cβ by
Pˆ φ = (Pˆ φ)(θ) := e−θA P φ(0),

(3.3)

where θ ∈ [−h, 0] and φ = φ(θ) is an element of Cβ .
Motivated by the definition of inertial manifolds in the case without delay, we then make
precisely the notion of inertial manifolds in the following definition.
18


Definition 3.1. The inertial manifold of equation (3.1) is a collection of surfaces M = {Mt }t∈R
in Cβ of the form
Mt = pˆ(θ) + Γt (ˆ
p(θ)) : pˆ(θ) ∈ Pˆ Cβ

⊂ Cβ

for all t ∈ R

(3.4)

where Γt (·) is a mapping from P X into (I − Pˆ ) Cβ , possessing the following properties:
(1) For every t ∈ R, Mt is the graph of a Lipschitz function with the Lipschitz constants
of Γt (·) are independent of t, i.e., there exists a constant C independent of t such that
|Γt (p1 ) − Γt (p2 )| Cβ

C p1 − p2

β.

(2) There exists γ > 0 such that to each u0 ∈ Mt0 there corresponds one and only one
solution u(t) to (3.2) on (−∞, t0 ] satisfying that u(t0 ) = u0 and
sup e−γ(t0 −t) Aβ u(t) < +∞.

(3.5)

t t0

(3) {Mt }t∈R is invariant under equation (3.2), i.e., if u(t), t ∈ R is a solution to equation
(3.2) satisfying conditions that us ∈ Ms and supt s ut Cβ < +∞ for some s ∈ R then
we have ut ∈ Mt for all t ∈ R.
(4) {Mt }t∈R exponentially attracts all the solutions to (3.2), i.e., for any solution u(·) of
(3.2) and any fixed s ∈ R, there exists a solution ut ∈ Mt and a positive constant H
such that
(3.6)
|ut − ut | Cβ He−γ(t−s) for t s
where γ is the same constant as the one in (3.5).

3.2

Lyapunov-Perron Equation

We can now construct the form of the solutions of equation (3.2) which are rescaledly
bounded on the half-line (−∞, t0 ] in the following lemma.
Lemma 3.2. Let the operator A satisfy Assumption B and g : R × Cβ → X be ϕ-Lipschitz
for ϕ satisfy Assumption D. For fixed t0 ∈ R, let u(t) be a solution of integral equation (3.2)
such that u(t) ∈ Xβ for all t ∈ (−∞, t0 ], and
sup

eγ(t−t0 ) u(t)

β

< +∞.

t∈(−∞,t0 ]

Then, for t ∈ (−∞, t0 ], the solution u(t) can be rewritten in the form
u(t) = e−(t−t0 )A p +

t0

G(t, τ )[F (τ )uτ + g(τ, uτ )]dτ
−∞

where p ∈ P X and G(t, τ ) is Green’s function.

19

for all t ∈ (−∞, t0 ]

(3.7)


3.3

Existence and Uniqueness of Solutions in Weighted Spaces

For t0 ∈ R we introduce the space
L−
γ,t0 :=

v ∈ C (−∞, t0 ], D(Aβ ) :

sup

eγ(t−t0 ) v(t)

β

< +∞ .

t∈(−∞,t0 ]

and put
|v|−
γ :=

eγ(t−t0 ) v(t) β .

sup
t∈(−∞,t0 ]

−
Then the pair (L−
γ,t0 , | · |γ ) is Banach space.
We have the following lemma which describes the existence and uniqueness of certain
solutions belonging to weighted spaces.

Lemma 3.3. Let the operator A satisfy Assumption B and nonlinear term g : R × Cβ → X
satisfy Assumption D. If < 1 then for any fixed t0 ∈ R and any p ∈ P X, there exists
a unique function v(p) ∈ L−
γ,t0 satisfying the integral equation (3.7) for all t ∈ (−∞, t0 ] with
P v(p)(t0 ) = p. Moreover,
|v(p)|−
γ < +∞,

(3.8)
M2
1−

|v(p) − v(q)|−
γ

3.4

p−q

β.

(3.9)

The Existence of Inertial Manifolds

The main result of this chapter is stated as follows.
Theorem 3.4. Let the operator A satisfy Assumption B and nonlinear term g be ϕ-Lipschitz
satisfy Assumption D. If
<1
here

and

M M22 N2 eγh
(1 − )(1 − e−α )

Λ1 ϕ

∞

+ Λ1 F (·)

:= eγh k and

N N1 +M1 N2

[ Λ1 ϕ ∞ + Λ1 F (·) ∞ ]



 1−e−α
k :=
+N [R(ϕ, β, h) + R( F (·) , β, h)]



 M N1 +M1 N2 [ Λ ϕ + Λ F (·)
]
1
∞
1
1−e−α
∞

1−β
α(1+β)

1−β
1+β

∞

+ < 1,

eαh

(3.10)

if 0 < β < 1,
if β = 0

then equation (3.2) has an inertial manifold.

3.5

Applications to the Modified Hutchinson Equation
with Diffusion

As an example we consider the modified Hutchinson

∂ 2w
∂w



−
d
= −ψ(t)w(t − 1, x)(1 + w),
 ∂t
∂x2
w(t, 0) = w(t, π) = 0,




w(θ, x) = φ(θ, x),
where d > 0.
20

equation with diffusion

t

s,

t

s,

x ∈ (0, π),

θ ∈ [−1, 0], x ∈ (0, π),

(3.11)


Chapter 4
Inertial Manifolds for a Class of
Partial Neutral Functional Differential Equations
In this chapter, we prove the existence of an inertial manifold for the partial neutral func∂
F ut + AF ut = Φ(t, ut ), where the partial differential
tional differential equations of the form ∂t
operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently
large gap; the difference operator F : Cβ → X is bounded linear operator, and the nonlinear delay operator Φ satisfies the ϕ-Lipschitz condition, i.e., Φ(t, φ)
ϕ(t)(1 + |φ| Cβ ) and
Φ(t, φ) − Φ(t, ψ)
ϕ(t)|φ − ψ| Cβ , where ϕ belongs to an admissible function space defined on R. Our main method is based on Lyapunov-Perron’s equations combined with the
admissibility of function spaces and the technique of choosing F -induced trajectories.
The content of this chapter is written based on the paper [2] in the List of Publications.

4.1

Setting of the Problem

In this section we will prove the existence of the inertial manifolds for partial functional
differential equations with finite delay. Then, we state of the main result of this chapter.
Consider the partial neutral functional differential equation

 ∂ F u + AF u = Φ(t, u ), t > s,
t
t
t
∂t
(4.1)

us = φ, s ∈ R.
In the case of infinite-dimensional phase spaces, instead of (4.1), we consider the integral
equation
t

F ut = e

−(t−s)A

e−(t−ξ)A Φ(ξ, uξ )dξ

F us +

for all λ-a.e. t

s.

(4.2)

s

We suppose that A satisfies Assumption A and consider the orthogonal projection P (defined
as in Assumption A), we define the projector Pˆ in Cβ by
Pˆ φ = (Pˆ φ)(θ) := e−θA P φ(0),

(4.3)

where θ ∈ [−h, 0] and φ = φ(θ) is an element of Cβ .
Definition 4.1. The inertial manifold of equation (4.1) is a collection of surfaces M = {Mt }t∈R
in Cβ of the form
Mt = pˆ(θ) + Γt (ˆ
p(θ)) : pˆ(θ) ∈ Pˆ Cβ

⊂ Cβ

for all t ∈ R

where Γt (·) is a mapping from P X into (I − Pˆ ) Cβ , possessing the following properties:
21

(4.4)


1. For every t ∈ R, Mt is the graph of a Lipschitz function with the Lipschitz constants of
Γt (·) are independent of t, i.e., there exists a constant C independent of t such that
|Γt (p1 ) − Γt (p2 )| Cβ

C p1 − p2

β.

2. There exists γ > 0 such that to each u0 ∈ Mt0 there corresponds one and only one
solution u(t) to (4.2) on (−∞, t0 ] satisfying that u(t0 ) = u0 and
sup e−γ(t0 −t) Aβ u(t) < ∞.

(4.5)

t t0

3. {Mt }t∈R is positively F -invariant under equation (4.2), i.e., if u(t), t ∈ R is a solution
to equation (4.2) satisfying conditions that ut0 ∈ Mt0 and supt t0 ut Cβ < ∞ for some
t0 ∈ R, then we have ut ∈ Mt for all t ∈ R, where the function ut is defined as in Lemma
4.3 with t0 being replaced by t, i.e.,
ut (θ) := F ut−θ

for all θ ∈ [−h, 0] and t ∈ R.

4. {Mt }t∈R exponentially attracts all the solutions to (4.2), i.e., for any solution u(·) of
(4.2) and any fixed s ∈ R, there exists a solution ut ∈ Mt and a positive constant H
such that
|ut − ut | Cβ He−γ(t−s) for t s
(4.6)
where γ is the same constant as the one in (4.5).

4.2

Lyapunov-Perron Equation

We can now construct the form of the solutions of equation (4.2) which are rescaledly
bounded on the half-line (−∞, t0 ] in the following lemma.
Lemma 4.2. Let the operator A satisfy Assumption A and Φ : R × Cβ → X be ϕ-Lipschitz
for ϕ satisfy Assumption D. For fixed t0 ∈ R, let u(t) be a solution of integral equation (4.2)
such that u(t) ∈ Xβ for all t ∈ (−∞, t0 ], and
eγ(t−t0 ) u(t)

sup
t∈(−∞,t0 ]

where γ =

λN +1 +λN
.
2

β

<∞

Then, for t ∈ (−∞, t0 ], the solution u(t) can be rewritten in the form

F ut = e−(t−t0 )A p +

t0

G(t, τ )Φ(τ, uτ )dτ

for all t ∈ (−∞, t0 ],

(4.7)

−∞

where p ∈ P X and G(t, τ ) is Green’s function defined by as in (1.10).

4.3

Existence and Uniqueness of Solutions in Weighted Spaces

For each t0 ∈ R we introduce the Banach space
Q−
γ,t0 :=

v ∈ C (−∞, t0 ], D(Aβ ) :

sup
t∈(−∞,t0 ]

22

eγ(t−t0 ) v(t)

β

<∞ .

endowed with the norm
|v|−
γ :=

eγ(t−t0 ) v(t) β .

sup
t∈(−∞,t0 ]

We have the following lemma which describes the existence and uniqueness of certain
solutions belonging to weighted spaces.
Lemma 4.3. Let the operator A satisfy Assumption A and nonlinear term Φ : R × Cβ → X
satisfy Assumption D. If
:= eγh k < 1
(4.8)
where

β
β
N1 +λβ

N N2 )
 M (β N1 +λN +1
Λ1 ϕ
1−e−α
k :=

 M (N1 +N2 ) Λ ϕ
1
∞

1−e−α

β
∞ + M β R(ϕ, β, h)

1−β
α(1+β)

1−β
1+β

eαh

if 0 < β < 1,
if β = 0

then for any fixed t0 ∈ R and any p ∈ P X, there exists a unique function v(p) ∈ Q−
γ,t0
satisfying the integral equation (4.7) for all t ∈ (−∞, t0 ] with P v(p)(t0 ) = p. Moreover,
|v(p)|−
γ < ∞,
|v(p) − v(q)|−
γ

4.4

(4.9)
M λβN
Aβ (p − q) .
1−

(4.10)

The Existence of Inertial Manifolds

The main result of this chapter is stated as follows.
Theorem 4.4. Let the operator A satisfy Assumption A and nonlinear term Φ be ϕ-Lipschitz
satisfy Assumption D. If
<1

and

1
1− Ψ

2γh
M 3 N2 λ2β
N e
Λ1 ϕ
(1 − Ψ − )(1 − e−α )

∞

+

<1

(4.11)

where


β
β
N1 +λβ

N N2 )
 M (β N1 +λN +1
Λ1 ϕ
1−e−α
k :=

 M (N1 +N2 ) Λ ϕ
1
∞
1−e−α

β
∞ + M β R(ϕ, β, h)

1−β
α(1+β)

1−β
1+β

eαh

if 0 < β < 1,
if β = 0

:= eγh k
then equation (4.2) has an inertial manifold.

4.5

An Illustrative Example

Consider the following neutral partial functional differential equation

∂w(x, t)
∂w(x, t − 1)
∂ 2 w(x, t)
∂ 2 w(x, t − 1)


−k
=
−k


∂t
∂t
∂x2
∂x2



0

ln(1 + |w(x, t + θ)|)dθ, x ∈ [0, π], t s,

+ bte−α|t|
−1





w(0, t) = w(π, t) = 0,
t s,



ws (x, θ) = w(x, s + θ) = ψ(x, θ),
x ∈ [0, π], θ ∈ [−1, 0].
where k is real constant with |k| < 1, the given function ψ is continuous.
23