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

By choosing appropriate function spaces and linear operators, these partialdifferential equations can be rewritten into semi-linear evolution equations in an infinite-dimensional Banach

MINISTY OF EDUCATION AND TRAININGHANOI NATIONAL UNIVERSITY OF EDUCATION

- 

-BUI XUAN QUANG

INERTIAL MANIFOLDS FOR CERTAIN CLASSES OF

This dissertation has been completed at the Hanoi National University of EducationScientific Advisors: Assoc Prof Dr habil Nguyen Thieu Huy

Dr Tran Thi Loan

Referee 1: Assoc Prof Dr Khuat Van Ninh

Hanoi Pedagogical University 2Referee 2: Assoc Prof Dr Nguyen Xuan Thao

Hanoi University of Science and TechnologyReferee 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

This dissertation is publicly available at HNUE Library Information Centre,

the National Library of Vietnam

1 Motivation

Many phenomena in mechanics, physics, ecology, etc can be described by partial ential equations By choosing appropriate function spaces and linear operators, these partialdifferential equations can be rewritten into semi-linear evolution equations in an infinite-dimensional Banach space whose linear part is the generator of a continuous semigroup andthe nonlinear term satisfies the Lipschitz condition

differ-The investigation of the asymptotic behavior of solutions to partial differential equations inlarge 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 attractsall other solutions of the equation This fact permits to invoke the reduction principle to studythe 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 thesemi-linear parabolic

(1)

when the partial differential operator A is positive definite and self-adjoint with a discrete trum having a sufficiently large distance between some two successive points of the spectrum,and the nonlinear forcing term f satisfies the ϕ-Lipschitz conditions

spec-These lead us to research the topic “Inertial manifolds for certain classes of evolutionequations”

2 Overview of the Research Problems

2.1 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 havebeen 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 ofclass C1, 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 theprinciple of spatial average to prove the existence of inertial manifolds for reaction-diffusionequations when the spectral gap condition is not fully satisfied Also, a more geometricproof was presented by Constantin P et al (1988, 1989) in the Hilbert space case using the

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 firstproof for the case where f is not bounded Debussche A & Temam R (1993) gave anotherproof for the case where f is not necessarily bounded, but now in the more general case ofBanach spaces, and f is assumed to be of class C1 Other proofs for the nonself-adjointcase 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 theconstruction 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 hasbeen translated and extended to more general classes of differential equations in applications,for example stochastic inertial manifolds Bensoussan A & Landoli F (1995), the existence ofinertial 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 coef-ficients may depend on time In 2012, Nguyen T.H considered the parabolic equations (1) andproved the existence of inertial manifolds when the nonlinear term f (t, u) is ϕ-Lipschitz, i.e.,i.e., kf (t, x)k 6 ϕ(t) 1 + Aβx
and kf (t, x) − f (t, y)k 6 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 beenextended 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 inertialmanifolds, namely the admissibly inertial manifolds of E -class, has been proved by NguyenT.H (2013) Such manifolds were consisted of solution trajectories belonging to a Banachspace E , which could be the Lp-space, Lorentz space Lp,q or many other function spacesoccurring in interpolation theory

3 – Applications of inertial manifolds Besides the existence of inertial manifolds for specificpartial differential equations, inertial manifolds have found numerous useful applications inother branches of mathematics These include the connection of inertial manifold with themultigrid methods of numerical analysis or an attempt of inertial manifold to describe theturbulence of fluid mechanics This dissertation will emphasize the applications of inertialmanifolds in control theory

2.2 Classes of evolution equations in this dissertation

3 Purpose, Objects and Scope of the Dissertation

Purpose We study the existence of inertial manifolds and the asymptotic behavior of tions 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 semigroupand the Lipschitz coefficient of the nonlinear term may depend on time and belongs toadmissible function spaces which contain wide classes of function spaces like Lp-spaces,the Lorentz spaces Lp,qand 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

solu-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)dt +Au(t) = f (t, u(t)) underthe conditions that the partial differential operator A is positive such that −A issectorial with a sufficiently large gap in its spectrum and f is a nonlinear operatorsatisfying ϕ-Lipschitz condition

◦ Content 2 Study the regularity of the inertial manifolds for du(t)dt + Au(t) = f (t, u(t))and using the theory of inertial manifolds for non-autonomous semi-linear evolutionequations, we construct a feedback controller for a class of control problems for theone-dimensional reaction-diffusion equations with the Lipschitz coefficient of thenonlinear term may depends on time and belongs to an admissible space

◦ Content 3 Study the existence of inertial manifolds for partial functional differentialequation du(t)dt + Au(t) = L(t)ut + g(t, ut) under the conditions that the partialdifferential operator A is positive such that −A is sectorial with a sufficiently largegap in its spectrum; the operator L(t) is linear, and g is a nonlinear operatorsatisfying ϕ-Lipschitz condition for ϕ belonging to an admissible function space

◦ Content 4 Study the existence of inertial manifolds for partial neutral functional ferential equation ∂t∂F ut+ AF ut = Φ(t, ut), where the partial differential operator

dif-A is positive definite and self-adjoint with a discrete spectrum having a sufficientlylarge gap; the difference operator F : Cβ → X is bounded linear operator, and thenonlinear 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 theoryfor strongly continuous semigroups

• Nonlinear terms: Theory of admissible spaces

• To prove existence of inertial manifolds : we use fixed point theory and the Perron method

Lyapunov-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 withfinite delay

• Chapter 4 Inertial manifolds for a class of partial neutral functional differential tions

equa-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 Linear Operators

1.2.1 Positive Operators with Discrete Spectrum

Assumption A Let X be a separable Hilbert space and suppose that A is a closed linearoperator on X satisfying the following assumption We suppose that A is a positive definite,self-adjoint operator with a discrete spectrum, say

0 < λ1 6 λ2 6 · · · λk6 · · · each with finite multiplicity, and lim

k→∞λk = ∞

We assume that {ek}∞

k=1 is the orthonormal basis of X consisting of the correspondingeigenfunctions of the operator A (i.e., Aek = λkek) Let now λN and λN +1 be two successiveand different eigenvalues with λN < λN +1, let further P be the orthogonal projection onto thefirst 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

σ(−A) = σu(−A) ∪ σc(−A) ⊂ C−

e−tAP 6 M1e−µ|t| for all t ∈ R, (1.6)

Aβe−tAP 6 M2e−µ|t| for all t ∈ R, (1.7)

e−tA(I − P ) 6 Meκt for all t > 0, (1.8)

Aβe−tA(I − P ) 6 N

tβeκt for all t > 0 (1.9)Let A satisfy Assumption A or Assumption Assumption1 Then, we can define the Greenfunction as follows:

kS(t)k 6 Me(ω+M kBk)t

for allt > 0

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

X ∈ Uδ(A) and λ /∈ G

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´echetspace for the seminorms pn(f ) =R

J n|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 (moduloλ-nullfunctions) is called a Banach function space (over (R, B, λ)) if

(1) E is a Banach lattice with respect to the norm k·kE, i.e., (E, k·kE) is a Banach space, and

if ϕ ∈ E, ψ is a real-valued Borel-measurable function such that |ϕ(·)| 6 |ψ(·)| (λ-a.e.)then ψ ∈ E and kϕkE 6 kψkE,

(2) the characteristic functions χA belongs to E for all A ∈ B of finite measure and

supt∈R

kχ[t,t+1]kE < ∞, inf

t∈Rkχ[t,t+1]kE > 0,(3) E ,→ L1,loc(R)

We remark that the condition (3) in the above definition means that for each compactinterval J ⊂ R, there exists a number βJ > 0 such that

ZJ

|g(t)|dt 6 βJkf kE for all f ∈ E

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

Z b a

|ϕ(t)|dt 6 M (b − a)kχ

[a,b]kE kϕkE, (1.11)(2) for ϕ ∈ E the function

Λ1ϕ(t) =

Z t t−1

belong to E,

(3) the space E is T+

τ -invariant and Tτ−-invariant where T+

τ 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

kT+

τ k 6 N1 and kTτ−k 6 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 Rand 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) kΦ(t, ut)k 6 ϕ(t) 1 + |ut|Cβ
for a.e t ∈ R and for all ut ∈ Cβ,

(2) kΦ(t, ut) − Φ(t, vt)k 6 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 6 s 6 1,χ(s) = 0 for s > 2, 0 6 χ(s) 6 1 and |χ0(s)| 6 2 for s ∈ [0, ∞) Define the cut-off mapping by

fR(t, u) := χ A

βuR

Z t+θ t+θ−1

Next, we will establish the C1-regularity of the inertial manifolds for the above-mentionedparabolic equation when the nonlinear term is of class C1 with respect to the state variablesand the linear part is a positive definite, self-adjoint operator with a discrete spectrum (satisfiesAssumption A) or is a sectorial operator with a sufficiently large gap (satisfies AssumptionB) The proof of the regularity will be performed in detail for the case where the linear operatorsatisfies 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 systemwith distributed observation and control

The content of this chapter is written based on the papers [3] and [4] in the List of cations

Publi-2.1 Competition Model with Cross-Diffusion: Sectorial

Oper-ators and Setting of the Problem

Consider the parabolic equation

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), weconsider the integral equation

u(t) = e−(t−s)Au(s) +

tZs

e−(t−ξ)Af (ξ, u(ξ))dξ for a.e t > s (2.2)

By a solution of equation (2.2) we mean a strongly measurable function u(t) defined on aninterval 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 Lipschitzmanifolds M = Mt

t∈R in X such that each Mt is the graph of a Lipschitz function

Φt: PnX → (I − Pn)Xβ, i.e.,

Mt= {x + Φtx : x ∈ PnX} 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 Cindependent of t such that

Aβ(Φtx1− Φtx2) 6 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 onesolution u(t) to (2.2) on (−∞, t0] satisfying that u(t0) = x0 and

esssupt6t 0

e−γ(t0 −t)

(3) The collection Mt

t∈R is positively invariant under (2.2), i.e., if a solution x(t), t > s,

to (2.2) satisfies xs ∈ Ms, then we have that x(t) ∈ Mt for t > s

(4) The collection Mt

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

distXβ(u(t), Mt) 6 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 Hausdorffsemi-distance generated by the norm in Xβ

We now fully state the main results about the existence of an inertial manifold for mildsolutions 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 oflinear operator A satisfying

kγ < 1 and kγM

3N2λ2βn kΛ1ϕk∞(1 − kγ)(1 − e−α) + kγ < 1, (2.7)where

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



=D1∆ 0

0 D2∆

 uv

+r1u − r1

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

system as

dx

dt = (C + Jg(x0)) x + g(x) − Jg(x0)x, (2.12)where x0 =u0

v0

and Jg(x0) denotes the Jacobian matrix of g(x) at x0.The important information that we would like to mention is the linear operator −A with

relevant domain is a sectorial operator

2.2 Inertial Manifolds for Parabolic Equations with Sectorial

Operators

2.2.1 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 6 t0 be a solution to integral

equation such that x(t) ∈ Xβ for t 6 t0 and

esssupt6t 0