# A Survey of Some Topics Related to Differential Operators

Authors: Denise Huet
Journal: Journal of Operators
 (2013)

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## Abstract

This paper is the result of investigations suggested by recent publications and completes the work of Huet, 2010. The topics, which are dealt with, concern some spaces of functions and properties of solutions of linear and nonlinear, stationary and evolution differential equations, namely, existence, spectral properties, resonances, singular perturbations, boundary layers, and inertial manifolds. They are presented in the alphabetical order. The aim of this document and of  Huet, 2010, is to be a useful reference for (young) researchers in mathematics and applied sciences.


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## Body

## 1. Introduction

The article is divided into several sections entitled: Birman-Schwinger operators; BMO spaces; Bounded variation (functions of); Discrete energy; Dissipative operators; Dynamical systems; Equal- area condition; Inertial manifolds; Mathieu-Hill type equations; Memory (equations with); Nodes, Nodal; Resonances. The development of each entry includes indications on history, definitions, an overview of main results, examples, and applications but is, of course, nonexhaustive. Complements will be found in the references. A prepublication of some entries is presented in Huet [1].


## 2.  Birman-Schwinger Operator

Definition 1.
Consider the Schrödinger operator(1)Hγ=Ho+γV(x),Ho=(-Δ)l,
acting on L2(ℝd), 2l≥d, where x→V(x) is a real-valued continuous function defined on ℝd which is nonnegative and tends to zero, sufficiently fast, as |x|→∞  and  γ is a small negative coupling constant. The operator Ho is self-adjoint and its spectrum is σo=[0,+∞). The Birman-Schwinger operator associated with (1) is the operator
(2)XV(λ)=V1/2Rλ(Ho)V1/2,
where Rλ(Ho) is the resolvent of Ho in (-∞,0). For each λ∈(-∞,0),  XV(λ) is self-adjoint and compact (cf. Arazy and Zelenko [2]).Application. In [2], the authors consider the decomposition XV(λ)=Φ(λ)+T(λ), where Φ(λ) is a finite rank operator and T(λ) an Hilbert-Schmidt operator whose norm is uniformly bounded with respect to λ∈(-δ,0) for some δ>0. An asymptotic expansion of the bottom virtual eigenvalue λo(γ) of Hγ, as γ<0 tends to zero, is deduced from this decomposition: if d is odd, it is of power type, while, when d is even, it involves the log function. Asymptotic estimates are obtained, as γ↑0, for the nonbottom virtual eigenvalues of Hγ, {λk(γ),k∈ℤ+d;0<|k|≤m}, where m=l-((d+1)/2) if d is odd and m=l-(d/2) if d is even. If d is odd, Φ(-t2l) is a meromorphic operator function, and the leading terms of the asymptotic estimates of λk(γ) are of power type. An algorithm, based on the Puiseux-Newton diagram (cf. Baumgärtel [3]), is proposed for an evaluation of the leading coefficients of these estimates. If d is even, two-sided estimates are obtained for eigenvalues with an exponential rate of decay; the rest of the eigenvalues have a power rate of decay. Estimates of Lieb-Thirring type are obtained for groups of eigenvalues which have the same rate of decay, when d is odd or even.


## 3. BMO Spaces

### 3.1. BMO and Related Spaces

Definition 2.
LetsupI be the supremum over all cubes I⊂ℝn with edges parallel to the coordinate axes, ℓ(I) the sidelength of I, and fI the mean value of f over I. The John-Nirenberg’s BMO=BMO(ℝn) space (cf. John and Nirenberg [4]) is the space of locally integrable complex-valued functions f defined on ℝn, such that
(3)∥f∥BMO=(supI(ℓ(I))-n∫I‍|f(x)-fI|2dx)1/2<∞.
The space of functions of bounded mean oscillation, modulo constants, equipped with the above norm, is a Banach space.Definition 3 (real Hardy spaceH1(ℝn)).
A functionf∈H1(ℝn) if and only if
(4)f=∑‍cjαj,
where cj∈ℝ, ∑‍|cj|<∞, each function αj is supported on a ball Bj and has integral zero, and supx∈Bj|αj(x)|≤1/|Bj|. Functions α that satisfy the above properties are called 1-atom (cf. Stein, [5]). With the norm
(5)∥f∥H1=inf∑‍|cj|,
where the inf is taken on all decompositions of f of the form (4), H1(ℝn) is a Banach space. In Fefferman and Stein [6, Theorem 2], it is proved that the dual of H1(ℝn) is BMO. For definitions and properties of Hardy-spaces Hp(ℝn), see [6].
Related spaces are presented in [7].


### 3.2. BMO Nonlinearity (cf. Byun and Wang [8])

Leta(ξ,x) be a nonlinear, real valued, function on ℝn×ℝn. For y∈ℝn and ρ>0, Bρ(y) denotes the open ball of radius ρ centered at y. Set
(6)a-Bρ(y)(ξ)=1|Bρ(y)|∫Bρ(y)‍a(ξ,x)dx
and define the function β(a,Bρ(y)) by
(7)β[a,Bρ(y)](x)=supξ∈ℝn|a(ξ,x)-a-Bρ(y)(ξ)||ξ|+1.Definition 4 ((δ,R)-BMO condition).
The functiona=a(ξ,x) satisfies the (δ,R)-BMO condition if
(8)sup0<ρ≤Rsupy∈ℝn1|Bρ(y)|∫Bρ(y)‍|β[a,Bρ(y)]|2(x)dx≤δ2.Definition 5 (Reifenberg domain).
A bounded open setΩ in ℝn is (δ,R)-Reifenberg flat if, for every xo∈∂Ω and every r∈(0,R], there exists a coordinate system {y1,…,yn} (which can depend on r and xo) so that xo is the origin in this coordinate system and that
(9)Br(0)∩{yn>δr}⊂Br(0)∩Ω⊂Br(0)∩{yn>-δr}
(cf. [9]).Applications. (i) Let Ω be a bounded, open subset of ℝn, f∈Lp(Ω;ℝn), 2≤p<∞, a=a(ξ,x):ℝn×ℝn→ℝn is a vector field measurable in x for almost every ξ and continuous in ξ for each x. In [8], the authors consider the nonlinear boundary value problem
(10)diva(∇u,x)=divf,inΩu=0on∂Ω.
The following conditions are imposed on a(ξ,x):
(11)[a(ξ,x)-a(η,x)]·(ξ-η)≥co|ξ-η|2
for all ξ,η∈ℝn and almost every x∈Ω,
(12)|a(ξ,x)|≤c1(1+|ξ|)
for all ξ∈ℝn and almost x∈ℝn, and
(13)|∇ξa(ξ,x)|≤c2foranyξ∈ℝnandalmosteveryx∈ℝn
for some positive constants co, c1,  and  c2. Then it is proved that there exists δ>0 such that, if a satisfies the (δ,R)-BMO condition and Ω is (δ,R)-Reifenberg flat, the weak solution u∈Ho1(Ω) to (10) belongs to Wo1,p(Ω) with the estimate
(14)∥∇u∥Lp(Ω)≤C(∥f∥Lp(Ω)+1),
where C is independent of u and f.(ii) In [9], S-S Byun extends the previous results to Orlicz spaces. He recalls the following definitions.Definition 6.
A positive functionϕ defined on [0,∞) is called a Young function if it is increasing, convex and satisfies
(15)ϕ(0)=0,ϕ(∞)=limt→∞ϕ(t)=∞,limt→0ϕ(t)t=0,limt→∞ϕ(t)t=∞.Definition 7.
One says that the Young functionϕ∈Δ2∪∇2 if it satisfies the following conditions:
(16)ϕ(2t)≤κϕ(t),∀t≥0,2toϕ(t)≤ϕ(tot),∀t>0,
for some numbers κ,to>1.Definition 8.
Letϕ∈Δ2∪∇2 be a Young function. The Orlicz space   Lϕ(Ω) is the linear space of all measurable functions v:Ω→ℝ satisfying
(17)∫Ω‍ϕ(|v|)dx<∞.
Equipped with the norm
(18)∥v∥Lϕ(Ω)=inf{λ>0:∫Ω‍ϕ(|v|λ)dx≤1},Lϕ(Ω) is a Banach space.The following result is proved in [9]: let ϕ∈Δ2∪∇2 be a Young function. There exist a small δ>0 and a positive constant C such that, if the nonlinearity a satisfies (11), (12), (13), and (8), if Ω satisfies (9), and if |f|2∈Lϕ(Ω), then the unique weak solution u∈Ho1(Ω) to (10) satisfies |∇u|2∈Lϕ(Ω), with the estimate
(19)∫Ω‍ϕ(|∇u|2)dx≤C(1+∫Ω‍ϕ(|f|2)dx).Remark 9.
Ifp>2 and ϕ(t)=tp/2 inequality (19) reduces to only (14).


## 3.1. BMO and Related Spaces

Definition 2.
LetsupI be the supremum over all cubes I⊂ℝn with edges parallel to the coordinate axes, ℓ(I) the sidelength of I, and fI the mean value of f over I. The John-Nirenberg’s BMO=BMO(ℝn) space (cf. John and Nirenberg [4]) is the space of locally integrable complex-valued functions f defined on ℝn, such that
(3)∥f∥BMO=(supI(ℓ(I))-n∫I‍|f(x)-fI|2dx)1/2<∞.
The space of functions of bounded mean oscillation, modulo constants, equipped with the above norm, is a Banach space.Definition 3 (real Hardy spaceH1(ℝn)).
A functionf∈H1(ℝn) if and only if
(4)f=∑‍cjαj,
where cj∈ℝ, ∑‍|cj|<∞, each function αj is supported on a ball Bj and has integral zero, and supx∈Bj|αj(x)|≤1/|Bj|. Functions α that satisfy the above properties are called 1-atom (cf. Stein, [5]). With the norm
(5)∥f∥H1=inf∑‍|cj|,
where the inf is taken on all decompositions of f of the form (4), H1(ℝn) is a Banach space. In Fefferman and Stein [6, Theorem 2], it is proved that the dual of H1(ℝn) is BMO. For definitions and properties of Hardy-spaces Hp(ℝn), see [6].
Related spaces are presented in [7].


## 3.2. BMO Nonlinearity (cf. Byun and Wang [8])

Leta(ξ,x) be a nonlinear, real valued, function on ℝn×ℝn. For y∈ℝn and ρ>0, Bρ(y) denotes the open ball of radius ρ centered at y. Set
(6)a-Bρ(y)(ξ)=1|Bρ(y)|∫Bρ(y)‍a(ξ,x)dx
and define the function β(a,Bρ(y)) by
(7)β[a,Bρ(y)](x)=supξ∈ℝn|a(ξ,x)-a-Bρ(y)(ξ)||ξ|+1.Definition 4 ((δ,R)-BMO condition).
The functiona=a(ξ,x) satisfies the (δ,R)-BMO condition if
(8)sup0<ρ≤Rsupy∈ℝn1|Bρ(y)|∫Bρ(y)‍|β[a,Bρ(y)]|2(x)dx≤δ2.Definition 5 (Reifenberg domain).
A bounded open setΩ in ℝn is (δ,R)-Reifenberg flat if, for every xo∈∂Ω and every r∈(0,R], there exists a coordinate system {y1,…,yn} (which can depend on r and xo) so that xo is the origin in this coordinate system and that
(9)Br(0)∩{yn>δr}⊂Br(0)∩Ω⊂Br(0)∩{yn>-δr}
(cf. [9]).Applications. (i) Let Ω be a bounded, open subset of ℝn, f∈Lp(Ω;ℝn), 2≤p<∞, a=a(ξ,x):ℝn×ℝn→ℝn is a vector field measurable in x for almost every ξ and continuous in ξ for each x. In [8], the authors consider the nonlinear boundary value problem
(10)diva(∇u,x)=divf,inΩu=0on∂Ω.
The following conditions are imposed on a(ξ,x):
(11)[a(ξ,x)-a(η,x)]·(ξ-η)≥co|ξ-η|2
for all ξ,η∈ℝn and almost every x∈Ω,
(12)|a(ξ,x)|≤c1(1+|ξ|)
for all ξ∈ℝn and almost x∈ℝn, and
(13)|∇ξa(ξ,x)|≤c2foranyξ∈ℝnandalmosteveryx∈ℝn
for some positive constants co, c1,  and  c2. Then it is proved that there exists δ>0 such that, if a satisfies the (δ,R)-BMO condition and Ω is (δ,R)-Reifenberg flat, the weak solution u∈Ho1(Ω) to (10) belongs to Wo1,p(Ω) with the estimate
(14)∥∇u∥Lp(Ω)≤C(∥f∥Lp(Ω)+1),
where C is independent of u and f.(ii) In [9], S-S Byun extends the previous results to Orlicz spaces. He recalls the following definitions.Definition 6.
A positive functionϕ defined on [0,∞) is called a Young function if it is increasing, convex and satisfies
(15)ϕ(0)=0,ϕ(∞)=limt→∞ϕ(t)=∞,limt→0ϕ(t)t=0,limt→∞ϕ(t)t=∞.Definition 7.
One says that the Young functionϕ∈Δ2∪∇2 if it satisfies the following conditions:
(16)ϕ(2t)≤κϕ(t),∀t≥0,2toϕ(t)≤ϕ(tot),∀t>0,
for some numbers κ,to>1.Definition 8.
Letϕ∈Δ2∪∇2 be a Young function. The Orlicz space   Lϕ(Ω) is the linear space of all measurable functions v:Ω→ℝ satisfying
(17)∫Ω‍ϕ(|v|)dx<∞.
Equipped with the norm
(18)∥v∥Lϕ(Ω)=inf{λ>0:∫Ω‍ϕ(|v|λ)dx≤1},Lϕ(Ω) is a Banach space.The following result is proved in [9]: let ϕ∈Δ2∪∇2 be a Young function. There exist a small δ>0 and a positive constant C such that, if the nonlinearity a satisfies (11), (12), (13), and (8), if Ω satisfies (9), and if |f|2∈Lϕ(Ω), then the unique weak solution u∈Ho1(Ω) to (10) satisfies |∇u|2∈Lϕ(Ω), with the estimate
(19)∫Ω‍ϕ(|∇u|2)dx≤C(1+∫Ω‍ϕ(|f|2)dx).Remark 9.
Ifp>2 and ϕ(t)=tp/2 inequality (19) reduces to only (14).


## 4. (Functions of) Bounded Variation

The notion of functions of bounded variation is closely related to the notion ofmeasure and the following usual definitions are useful.


### 4.1. Measures

Definition 10.
LetΩ be an open subset of ℝn, and 𝒞c(Ω,ℂ) [resp., 𝒞c(Ω,ℝ)] the space of continuous functions φ:Ω→ℂ [resp., ℝ] with compact support in Ω.(i)
A measureμ on Ω is a linear functional
(20)μ:φ∈𝒞c(Ω,ℂ)⟶μ(φ)(or〈μ,φ〉)∈ℂ
that is continuous; in the following sense, for all K compact in Ω, there exists a constant CK such that
(21)|〈μ,φ〉|≤CK∥φ∥∞
for all φ∈𝒞c(Ω,ℂ) whose support is contained in K. We write μ∈ℳ(Ω). In [10], Schwartz introduces a topology on 𝒞c(Ω,ℂ) and μ is continuous on this topological space.(ii)
μ is said to be bounded on Ω if, in (21), the constant is independent of K. The space of bounded measures on Ω is denoted by ℳ1(Ω).(iii)
The conjugateμ- of μ is given by
(22)〈μ-,φ〉=〈μ,φ-〉¯.(iv)
μ is called real if 〈μ,φ〉∈ℝ, for all φ∈𝒞c(Ω,ℝ).(v)
μ is called positive if 〈μ,φ〉≥0, for all φ∈𝒞c(Ω,ℝ),φ≥0.Definition 11 (absolute value of a measure).
Ifμ is a complex or real measure, its absolute value, denoted by |μ|, is the map
(23)ψ∈𝒞c(Ω,ℝ),ψ≥0⟶〈|μ|,ψ〉=supφ∈𝒞c(Ω,ℂ),|φ|≤ψ{|〈μ,φ〉|}.Definition 12 (total variation of a positive bounded measure).
Letμ be a positive, bounded measure on Ω. Its total variation, denoted by |μ|Ω or ∫Ω‍μ, is
(24)|μ|=supφ∈𝒞c(Ω,ℝ),0≤φ≤1μ(φ).


### 4.2. Space BV(I,ℝ) or BV(I)

LetI=(a,b) be an interval in ℝ. The following alternative definitions are well known. A function f:x∈I→ℝ is of bounded variation:(i)
if it can be expressed in the formϕ-ψ, where ϕ,ψ are nondecreasing bounded functions,(ii)
if the interval(a,b) is divided up by points
(25)a=xo<x1<⋯<xn-1<xn=b,then, there exists a constantC>0, independent of the mode of division, such that
(26)∑ν=0n-1|f(xν+1)-f(xν)|<Cand the upper bound of the sum (26) is called the total variation of f on I and is denoted by V(f,I).For these definitions, see Titchmarsh [11, page 355] and Riesz and Sz-Nagy [12, page 10].We have also the following proposition: letf∈Lloc1(ℝ). Then f defines a distribution T=Tf∈𝒟′(ℝ). In Schwartz [10, page 53] it is proved that, in order for the derivative of T to be a measure, it is necessary and sufficient that f is of bounded variation on every finite interval.


### 4.3. Space BV(Ω,ℝ) or BV(Ω)

LetΩ be an open set in ℝn. The following definitions are equivalent (cf. Brezis [13, page 153]).A functionf∈W11(Ω) is of bounded variation(i)
if all first derivatives off, in the distributional sense, that is, in 𝒟′(Ω), are bounded measures,(ii)
if there exists a constantC such that
(27)|∫Ω‍f∂ϕ∂xidx|≤C∥ϕ∥L∞(Ω)∀ϕ∈Cc∞(Ω),i=1,…,n,(iii)
if there exists a constantC such that
(28)∥τhf-f∥L1(ω)≤C|h|
for all open set ω⊂Ω and all h∈ℝn with |h|<dist(ω,∁Ω). Moreover, in (27) and (28) we can take C=||∇f||L1(Ω). Here τhf(x)=f(x+h), and ∇f is the distributional gradient of f.We have also the following proposition: a functionf∈L1(Ω) is of bounded variation if its first distributional derivatives are bounded measures. Then, the gradient ∇f is a bounded, vector-valued measure whose absolute value|∇f| is the map
(29)ψ∈𝒞c(Ω,ℝ),ψ≥0⟶〈|∇f|,ψ〉=supφ∈𝒞c(Ω,ℂn),∑in|φi|2≤ψ2{|〈∇f,φ〉|},
where 〈∇f,φ〉=∑in‍(∂f/∂xi)φi and |∇f| is a bounded positive measure. The total variation of |∇f| is
(30)∫Ω‍|∇f|=∥∇f∥Ω=supψ∈𝒞c(Ω,ℝ),0≤ψ≤1〈|∇f|,ψ〉
(cf. F. Demengel and G. Demengel [14, page 303]).


### 4.4. Space BV(Ω,ℝn)

This section is already presented in [7]. Let Ω be an open subset of ℝd with a smooth boundary. A function f∈Lloc1(Ω,ℝn) has a bounded variation, that is, f∈BV(Ω,ℝn), if ∇f, in the distributional sense, is a vector-valued Radon measure of finite total mass. Let |f|BV=∫Ω‍|∇f|  be aBV-seminorm. In Dávila [15], the following property of |f|BV is proved: there exists a positive constant K, which depends on d, such that, for every family of nonnegative radial mollifiers ρϵ∈Lloc1((0,∞),ℝ+) satisfying
(31)∫0∞‍ρϵ(r)rd-1dr=1,limϵ→0∫δ∞‍ρϵ(r)rd-1dr=000000000000000000000000000000000000.∀δ>0,
we have
(32)limϵ→0∫Ω2‍|f(x)-f(y)||x-y|ρϵ(|x-y|)dxdy=K|f|BV,0000000000000000000000000000∀f∈BV(Ω,ℝn).


### 4.5. Application

In [16], B. Merlet shows, by means of the above property of |f|BV, that, if u∈BV(Ω,S1), there exists a lifting φ∈BV(Ω,ℝ) of u (i.e., u(x)=eiφ(x), for all x∈Ω) such that |φ|BV≤2|u|BV.


## 4.1. Measures

Definition 10.
LetΩ be an open subset of ℝn, and 𝒞c(Ω,ℂ) [resp., 𝒞c(Ω,ℝ)] the space of continuous functions φ:Ω→ℂ [resp., ℝ] with compact support in Ω.(i)
A measureμ on Ω is a linear functional
(20)μ:φ∈𝒞c(Ω,ℂ)⟶μ(φ)(or〈μ,φ〉)∈ℂ
that is continuous; in the following sense, for all K compact in Ω, there exists a constant CK such that
(21)|〈μ,φ〉|≤CK∥φ∥∞
for all φ∈𝒞c(Ω,ℂ) whose support is contained in K. We write μ∈ℳ(Ω). In [10], Schwartz introduces a topology on 𝒞c(Ω,ℂ) and μ is continuous on this topological space.(ii)
μ is said to be bounded on Ω if, in (21), the constant is independent of K. The space of bounded measures on Ω is denoted by ℳ1(Ω).(iii)
The conjugateμ- of μ is given by
(22)〈μ-,φ〉=〈μ,φ-〉¯.(iv)
μ is called real if 〈μ,φ〉∈ℝ, for all φ∈𝒞c(Ω,ℝ).(v)
μ is called positive if 〈μ,φ〉≥0, for all φ∈𝒞c(Ω,ℝ),φ≥0.Definition 11 (absolute value of a measure).
Ifμ is a complex or real measure, its absolute value, denoted by |μ|, is the map
(23)ψ∈𝒞c(Ω,ℝ),ψ≥0⟶〈|μ|,ψ〉=supφ∈𝒞c(Ω,ℂ),|φ|≤ψ{|〈μ,φ〉|}.Definition 12 (total variation of a positive bounded measure).
Letμ be a positive, bounded measure on Ω. Its total variation, denoted by |μ|Ω or ∫Ω‍μ, is
(24)|μ|=supφ∈𝒞c(Ω,ℝ),0≤φ≤1μ(φ).


## 4.2. Space BV(I,ℝ) or BV(I)

LetI=(a,b) be an interval in ℝ. The following alternative definitions are well known. A function f:x∈I→ℝ is of bounded variation:(i)
if it can be expressed in the formϕ-ψ, where ϕ,ψ are nondecreasing bounded functions,(ii)
if the interval(a,b) is divided up by points
(25)a=xo<x1<⋯<xn-1<xn=b,then, there exists a constantC>0, independent of the mode of division, such that
(26)∑ν=0n-1|f(xν+1)-f(xν)|<Cand the upper bound of the sum (26) is called the total variation of f on I and is denoted by V(f,I).For these definitions, see Titchmarsh [11, page 355] and Riesz and Sz-Nagy [12, page 10].We have also the following proposition: letf∈Lloc1(ℝ). Then f defines a distribution T=Tf∈𝒟′(ℝ). In Schwartz [10, page 53] it is proved that, in order for the derivative of T to be a measure, it is necessary and sufficient that f is of bounded variation on every finite interval.


## 4.3. Space BV(Ω,ℝ) or BV(Ω)

LetΩ be an open set in ℝn. The following definitions are equivalent (cf. Brezis [13, page 153]).A functionf∈W11(Ω) is of bounded variation(i)
if all first derivatives off, in the distributional sense, that is, in 𝒟′(Ω), are bounded measures,(ii)
if there exists a constantC such that
(27)|∫Ω‍f∂ϕ∂xidx|≤C∥ϕ∥L∞(Ω)∀ϕ∈Cc∞(Ω),i=1,…,n,(iii)
if there exists a constantC such that
(28)∥τhf-f∥L1(ω)≤C|h|
for all open set ω⊂Ω and all h∈ℝn with |h|<dist(ω,∁Ω). Moreover, in (27) and (28) we can take C=||∇f||L1(Ω). Here τhf(x)=f(x+h), and ∇f is the distributional gradient of f.We have also the following proposition: a functionf∈L1(Ω) is of bounded variation if its first distributional derivatives are bounded measures. Then, the gradient ∇f is a bounded, vector-valued measure whose absolute value|∇f| is the map
(29)ψ∈𝒞c(Ω,ℝ),ψ≥0⟶〈|∇f|,ψ〉=supφ∈𝒞c(Ω,ℂn),∑in|φi|2≤ψ2{|〈∇f,φ〉|},
where 〈∇f,φ〉=∑in‍(∂f/∂xi)φi and |∇f| is a bounded positive measure. The total variation of |∇f| is
(30)∫Ω‍|∇f|=∥∇f∥Ω=supψ∈𝒞c(Ω,ℝ),0≤ψ≤1〈|∇f|,ψ〉
(cf. F. Demengel and G. Demengel [14, page 303]).


## 4.4. Space BV(Ω,ℝn)

This section is already presented in [7]. Let Ω be an open subset of ℝd with a smooth boundary. A function f∈Lloc1(Ω,ℝn) has a bounded variation, that is, f∈BV(Ω,ℝn), if ∇f, in the distributional sense, is a vector-valued Radon measure of finite total mass. Let |f|BV=∫Ω‍|∇f|  be aBV-seminorm. In Dávila [15], the following property of |f|BV is proved: there exists a positive constant K, which depends on d, such that, for every family of nonnegative radial mollifiers ρϵ∈Lloc1((0,∞),ℝ+) satisfying
(31)∫0∞‍ρϵ(r)rd-1dr=1,limϵ→0∫δ∞‍ρϵ(r)rd-1dr=000000000000000000000000000000000000.∀δ>0,
we have
(32)limϵ→0∫Ω2‍|f(x)-f(y)||x-y|ρϵ(|x-y|)dxdy=K|f|BV,0000000000000000000000000000∀f∈BV(Ω,ℝn).


## 4.5. Application

In [16], B. Merlet shows, by means of the above property of |f|BV, that, if u∈BV(Ω,S1), there exists a lifting φ∈BV(Ω,ℝ) of u (i.e., u(x)=eiφ(x), for all x∈Ω) such that |φ|BV≤2|u|BV.


## 5. Discrete Energy

LetωN=(x1,…,xN), N≥2 be a set of N points on the unit-sphere S2={x∈ℝ3,|x|=1}.


### 5.1. Discrete Energy ofωN

Different discrete energies are associated withωN. Let α∈ℝ, -2<α<2.Definition 13.
The Coulomb [resp., logarithmic] energy associated withωN is
(33)ℋC(ωN)=∑i=1N‍∑j=i+1N‍1|xi-xj|
[resp., ℋL(ωN)=-∑i=1N∑j=i+1Nlog|xi-xj|].Definition 14.
More generally, theα-energy associated with ωN is
(34)E(α,ωN)={∑i=1N‍∑j=i+1N‍|xi-xj|αifα≠0,ℋL(ωN)ifα=0.Remark 15.
We have(35)ℋC(ωN)=E(-1,ωN).Remark 16.
E
(
α
,
ω
N
) is the energy of the N points x1,…,xN, on the surface of the sphere, interacting through a potential V=rα (cf. Rakhmanov et al. [17]).
The article [17] is devoted to extremal energy for ωN:
(36)ℰ(α,N)={infωN∈S2E(α,ωN)ifα≤0,supωN∈S2E(α,ωN)ifα>0.
Bounds for ℰ(α,N), -2<α<2 and explicit formula for N points on S2 that yields good estimates for ℰ(α,N) are obtained. The authors point out important applications of the determination of ℰ(α,N) to geometry, chemistry, physics, and crystallography and give references for the history of related researches.


### 5.2. A More General Discrete Energy-Like Function on the Unit Sphere

In [18], Cheviakov et al. introduce the following definition.Definition 17.
The discrete energy-like function associated withωN is
(37)ℋ(ωN)=ℋC(ωN)+12ℋL(ωN)-12∑i=1N‍∑j=i+1N‍log(2+|xi-xj|).Application. In [18], ℋ(ωN) is related to the mean first passage time (MFPT) v(x) for a Brownian particle in the unit ball Ω in ℝ3 that contains N small locally circular absorbing windows ∂Ωϵj on its boundary ∂Ω=S2. Set ∂Ωa=⋃j=1N∂Ωϵj. The function v(x) is solution to the Dirichlet-Neumann problem:
(38)Δv=-1DonΩ,v=0on∂Ωa,∂v∂n=0on∂Ω∂Ωa,
where D is a diffusivity coefficient. The authors obtain three-term asymptotic expansions for v(x), for the average MFPT v-=(1/|Ω|)∫Ω‍v(x)dx, and for the principal eigenvalue λ of the Laplacian associated with the boundary conditions (38), when the area |∂Ωϵj|~πϵ2aj2, ∂Ωϵj→xj∈S2 as ϵ→0, j=1,…,N and |xi-xj|=𝒪(1), i≠j. For instance, when the windows have common radius ϵ≪1, they obtain
(39)v-=|Ω|4ϵDN[1+ϵπlog(2ϵ)00000000+ϵπ(-9N5+2(N-2)log2+32)00000000+4Nℋ(x1,…,xN)+𝒪(ϵ2logϵ)],λ~1Dv-.
Moreover, v- is minimized and the corresponding λ is maximized at the configuration ωN that minimizes ℋ(ωN). The optimum arrangements {x1,…,xN} that minimize ℋ(ωN) are numerically computed by different methods.


## 5.1. Discrete Energy ofωN

Different discrete energies are associated withωN. Let α∈ℝ, -2<α<2.Definition 13.
The Coulomb [resp., logarithmic] energy associated withωN is
(33)ℋC(ωN)=∑i=1N‍∑j=i+1N‍1|xi-xj|
[resp., ℋL(ωN)=-∑i=1N∑j=i+1Nlog|xi-xj|].Definition 14.
More generally, theα-energy associated with ωN is
(34)E(α,ωN)={∑i=1N‍∑j=i+1N‍|xi-xj|αifα≠0,ℋL(ωN)ifα=0.Remark 15.
We have(35)ℋC(ωN)=E(-1,ωN).Remark 16.
E
(
α
,
ω
N
) is the energy of the N points x1,…,xN, on the surface of the sphere, interacting through a potential V=rα (cf. Rakhmanov et al. [17]).
The article [17] is devoted to extremal energy for ωN:
(36)ℰ(α,N)={infωN∈S2E(α,ωN)ifα≤0,supωN∈S2E(α,ωN)ifα>0.
Bounds for ℰ(α,N), -2<α<2 and explicit formula for N points on S2 that yields good estimates for ℰ(α,N) are obtained. The authors point out important applications of the determination of ℰ(α,N) to geometry, chemistry, physics, and crystallography and give references for the history of related researches.


## 5.2. A More General Discrete Energy-Like Function on the Unit Sphere

In [18], Cheviakov et al. introduce the following definition.Definition 17.
The discrete energy-like function associated withωN is
(37)ℋ(ωN)=ℋC(ωN)+12ℋL(ωN)-12∑i=1N‍∑j=i+1N‍log(2+|xi-xj|).Application. In [18], ℋ(ωN) is related to the mean first passage time (MFPT) v(x) for a Brownian particle in the unit ball Ω in ℝ3 that contains N small locally circular absorbing windows ∂Ωϵj on its boundary ∂Ω=S2. Set ∂Ωa=⋃j=1N∂Ωϵj. The function v(x) is solution to the Dirichlet-Neumann problem:
(38)Δv=-1DonΩ,v=0on∂Ωa,∂v∂n=0on∂Ω∂Ωa,
where D is a diffusivity coefficient. The authors obtain three-term asymptotic expansions for v(x), for the average MFPT v-=(1/|Ω|)∫Ω‍v(x)dx, and for the principal eigenvalue λ of the Laplacian associated with the boundary conditions (38), when the area |∂Ωϵj|~πϵ2aj2, ∂Ωϵj→xj∈S2 as ϵ→0, j=1,…,N and |xi-xj|=𝒪(1), i≠j. For instance, when the windows have common radius ϵ≪1, they obtain
(39)v-=|Ω|4ϵDN[1+ϵπlog(2ϵ)00000000+ϵπ(-9N5+2(N-2)log2+32)00000000+4Nℋ(x1,…,xN)+𝒪(ϵ2logϵ)],λ~1Dv-.
Moreover, v- is minimized and the corresponding λ is maximized at the configuration ωN that minimizes ℋ(ωN). The optimum arrangements {x1,…,xN} that minimize ℋ(ωN) are numerically computed by different methods.


## 6. Dissipative Operators

I restrict myself to definitions (cf. Pazy [19]). Let X be a Banach space whose norm is denoted by ∥·∥, X* its dual, and x*∈X*. The value of x* at x∈X is denoted by 〈x*,x〉=〈x,x*〉=x*(x).Definition 18.
Forx∈X, we define the duality set by
(40)F(x)={x*∈X*;x*(x)=∥x∥2=∥x*∥2}.Definition 19.
A linear operatorA, in X, with domain D(A), is dissipative if, for every x∈D(A), there exists x*∈F(x) such that ℜ〈Ax,x*〉≤0 or, equivalently, if ∥(λI-A)x∥≥λ∥x∥ for all x∈D(A) and λ>0. A dissipative operator A is called m-dissipative if, for all λ>0, the operator λI-A:D(A)→X is surjective.TheLumer-Phillips Theorem states that a densely defined operator A in X is the generator of a Co-semigroup T(t) (i.e., limt→0T(t)x=x) of contractions if and only if it is m-dissipative (cf. [19]). I recall that the domain D(A) of the infinitesimal generator A of T(t) is
(41)D(A)={x∈X:limt→0Tx-xtexists}
and, for any x∈D(A),
(42)Ax=limt→0Tx-xt.


## 7. Dynamical Systems

In this section, I restrict myself to several definitions and examples of dynamical systems and also definitions and some properties of limit and invariant sets. More investigations into this large topic are in preparation.Different definitions of dynamical systems are given in the literature.


### 7.1. Definitions

#### 7.1.1. General Definitions

In (2011), Kloeden and Rasmussen give, in [20], the general definitions:let𝕋=ℝ [resp., ℤ], 𝕋o+=𝕋∩[0,∞), that is, ℝ+ [resp., ℕ] and a metric spaceX.Definition 20.
Adynamical system is a continuous map Φ:𝕋×X→X which satisfies the initial value condition
(43)Φ(0,x)=x∀x∈X
and the group property
(44)Φ(s+t,x)=Φ(s,Φ(t,x))∀s,t∈𝕋,x∈X.When𝕋=ℝ the dynamical system is called continuous and when 𝕋=ℤ the dynamical system is called discrete.Definition 21.
Asemidynamical system is a continuous map Φ:𝕋o+×X→X which satisfies the initial value condition
(45)Φ(0,x)=x∀x∈X
and the semigroup property
(46)Φ(s+t,x)=Φ(s,Φ(t,x))∀s,t∈𝕋o+,x∈X.When𝕋o+=ℝ+ [resp., ℕ] the semidynamical system is called continuous [resp., discrete].


#### 7.1.2. “Classical” Dynamical Systems

LetB be a Banach space. In 1969, Hale [21] gives the following definition.Definition 22.
A classical dynamical system on B is a continuous function u:ℝ+×B→B which satisfies
(47)u(0,φ)=φ,u(t+τ,φ)=u(t,u(τ,φ))
for all t,τ≥0 and φ∈B.Remark 23.
In Definition22, the mapping u is a continuous semidynamical system in terms of Definition 21.


#### 7.1.3. “Abstract” Dynamical System

In 1991, Haraux [22] defines abstract dynamical systems in a complete metric space(E,d) in the following way.Definition 24.
An abstract dynamical system inE is a family {S(t)}t≥0 of mappings on E which satisfy the following properties:(i)
for eacht≥0, S(t):E→E is continuous,(ii)
S(0)=I,(iii)
S(t+s)=S(t)S(s),(iv)
for eachz∈E, t∈[0,∞)→S(t)z∈E is continuous.Remark 25.
In Definition24, the mapping u:(t,z)∈ℝ+×E→u(t,z)=S(t)z∈E satisfies conditions (45) and (46) but it is only separately continuous. Therefore it is not a semidynamical system in terms of Definition 21.


#### 7.1.4. Dafermos Definition

In [23, Definition 4.1], Dafermos gives the following definition in a metric spaceX.Definition 26.
A Dafermos dynamical system inX is a map u:X×ℝ+→X which satisfies the initial condition
(48)u(x,0)=0∀x∈X
the semigroup property
(49)u(x,,s+t)=u(u(x,s),t)∀t,s∈ℝ+,x∈X,
and the continuity condition in x, for all t∈ℝ+.Remark 27.
In the previous definitions the spaceX [resp. B, E] is called the phase space, and the set
(50)γ+(x)=⋃t≥0Φ(t,x)[resp.⋃t≥0u(t,x)]
is called the positive orbit or trajectory through x∈X [resp., x∈B, x∈E].


### 7.2. Examples

#### 7.2.1. Continuous (Semi-)Dynamical Systems

Example 28 (cf. [21, Example 1]).
Letf be a continuous function ℝn→ℝn and ξ∈ℝn. Consider the autonomous differential equation
(51)dxdt=f(x(t)),t≥0,
with the initial condition
(52)x(0)=ξ.
If there exists a unique solution u(t,ξ) of (51) and (52) which depends continuously upon t,ξ, then the mapping
(53)Φ:ℝ+×ℝn⟶ℝndefinedbyΦ(t,ξ)=u(t,ξ)
is a continuous semidynamical system on ℝn (Definitions 21 and 22).Example 29 (cf. [20, Example 1.6]).
LetX be the Banach space of bounded continuous functions μ:[-1,0]→ℝd, under the usual norm ∥·∥∞. Let μ∈X and f a continuous function ℝd→ℝd and suppose that there exists a unique solution u(t,μ) of the autonomous delay differential equation(54)dxdt=f(x(t-1)),t≥0,
with the condition
(55)x(s)=μ(s),s∈[-1,0].
Then
(56)Φ:(t,μ)∈ℝ+×X⟶Φ(t,μ)(s)=u(t+s,μ)∈X,0000000000000000000000000000.t≥0,s∈[0,1],
is a continuous semidynamical system on X.Example 30 (see cf. [22, page 12]).
LetX be a real Banach space, A a m-dissipative operator with dense domain in X (cf. Section 5), T(t) the semigroup of contractions generated by A, and F:X→X a Lipschitzian function on the bounded sets of X. Let u(t,x)∈C([o,τ(x)),X) be the unique maximal solution of
(57)u(t,x)=T(t)x+∫ot‍T(t-s)F(u(s))ds,0000000000000000.t∈[0,τ(x)),x∈X.
Set u(t,x)=S(t)x. Let Y⊂X such that τ(y)=∞ for all y∈Y and ∥S(t)y∥≤M, for all t≥0, y∈Y. For y∈Y, u(t,y)=S(t)y is solution of (57) for x=y. Set Z=⋃y∈Y⋃t≥0{S(t)y}¯ and denote by d the distance induced on Z by the norm in X. Then {S(t)}t≥0 is an abstract dynamical system (Definition 24) on (Z,d).Example 31.
Dafermos dynamical systems are adapted to equations of linear viscoelasticity. See Section10.Example 32 (see cf. [20]).
LetX be a metric space and f:X→X be a continuous function. By iteration, fn+1(x)=f(fn)(x)=fofn(x), for all x∈X, n∈N, is well defined. The mapping
(58)Φ:(n,x)∈ℕ×X⟶Φ(n,x)=fn(x)∈X
is a discrete semidynamical system (Definition 21). Now, suppose f is an homeomorphism; that is, it is continuous and invertible with continuous inverse. Then, the mapping Φ defined by (58) can be extended to ℤ- by
(59)Φ(n,x)=(f-1)-n(x),∀n∈ℤ-,x∈X,
and Φ is a discrete dynamical system.


#### 7.2.2. Discrete-Time Dynamical Systems and Fractals

What is known as adiscrete time dynamical system in Pesin and Climenhaga [24] is related to a map from a set to itself. Let X be any set and f a map f:X→X. By iteration, fn+1(x)=f(fn(x))=(f∘fn)(x) is well defined for every n∈ℕ and we have the semigroup property:
(60)fn+m=fm∘fn=fn∘fm
for any integer m,n≥0.Definition 33.
Letx∈X. The sequence of points fo(x)=x,f(x),f2(x),…, is called the trajectory or orbit of x.


#### 7.2.3.  Examples with Chaotic Behavior

Example 34 (the Cantor set).
Consider, inℝ, the intervals I1=[0,1/3], I2=[2/3,1], and D=I1∪I2 and the map f defined by f(x)=3x  on  I1  and  3x-2  on  I2. The range of f is [0,1] and does not lie in its domain of definition. The domain of definition of f2 is {x;x∈Dandf(x)∈D}, that is, the union of the intervals:
(61)[0,19],[29,13],[23,79],[89,1]
obtained by removing the open middle third of each interval I1 and I2 and so on. The domain on which every iterate fn is defined is exactly the Cantor set (cf. Figure 1).Figure 1
The Cantor set.Example 35 (the Sierpinski triangle (or gasket)).
An equilateral triangle is divided into four smaller triangles, each similar to the first and congruent to each other. Then, the middle triangle is removed, and the iterating procedure is iterated on the remaining three, and so on. The fractalT obtained as the limit of this procedure is the Sierpinki triangle (cf. Figure 2). With this procedure is associated the following algorithm. If x∈T, and only in this case, it is possible to define a function x∈T→f(x)∈ℝ2 such that fn(x)∈T for every n (cf. Schroeder [25] which calls this procedure “Sir Pinski game”).Figure 2
The Sierpinski gasket.The Cantor set and the Sierpinski triangle are examples of fractals (cf. [26, 27]).


### 7.3. Limit and Invariant Sets

Definition 36 (see cf. [20]).
LetΦ:𝕋×X→X be a dynamical system in a metric space X. The ω-limit set [resp., α-limit set] of a point ξ∈X is defined by
(62)ω(ξ)=⋂τ≥0⋃t≥τϕ(t,ξ)¯[resp.,α(ξ)=⋂τ≤0⋃t≤τϕ(t,ξ)¯].
In the case of a semidynamical system (Φ:To+×X→X) the notion of a α-limit set is not defined. In the same way, ω and α-limit sets of a subspace M⊂X are defined by (62) where ξ is replaced by M.Remark 37.
Theω and α-limit sets may be defined as follows:
(63)ω(ξ)={x∈X:thereexistsasequencetn,tn>0,0.tn⟶∞suchthatΦ(tn,ξ)⟶xasn⟶∞},α(ξ)={x∈X:thereexistsasequencetn,tn<0,0.tn⟶-∞suchthatΦ(tn,ξ)⟶xasn⟶∞}.Remark 38.
Similar definitions are valid for classical and abstract dynamical systems.Definition 39.
LetΦ:𝕋o+×X→X be a semidynamical system on the metric spaceX (Definition 21). A subset M⊂X is called invariant under Φ if
(64)Φ(t,M)=M∀t∈𝕋o+.
It is called positively [resp., negatively] invariant if(65)Φ(t,M)⊂M[resp.,M⊂Φ(t,M)],∀t∈𝕋o+.
In the case of discrete-time dynamical system and of Example 32, invariance is equivalent to f(M)=M.Properties of
ω
-Limit and Invariant Sets. Let (X,d) be a metric space, Φ:𝕋o+×X→X a semidynamical system on X, and x∈X. If the trajectory γ+(x) through x is precompact, then ω(x) has the following properties:
(66)Φ(t,ω(x))=ω(x)≠∅,∀t∈𝕋o+,ω(x)isacompactsubsetofX,ω(x)isconnectedif𝕋o+=ℝ+.
If 𝕋o+=ℕ, ω(x) does not need to be connected (cf. [20, Exercise 1.10]).In the case of an abstract dynamical system{(S(t)}t≥0, on a complete metric space(E,d), previous results hold with Φ(t,x)=S(t)x, for x∈E, if γ+(x) is relatively compact in E (see [28, page 122]). But in a complete metric space E a subset A is relatively compact if and only if it is precompact (cf. [29]).


## 7.1. Definitions

### 7.1.1. General Definitions

In (2011), Kloeden and Rasmussen give, in [20], the general definitions:let𝕋=ℝ [resp., ℤ], 𝕋o+=𝕋∩[0,∞), that is, ℝ+ [resp., ℕ] and a metric spaceX.Definition 20.
Adynamical system is a continuous map Φ:𝕋×X→X which satisfies the initial value condition
(43)Φ(0,x)=x∀x∈X
and the group property
(44)Φ(s+t,x)=Φ(s,Φ(t,x))∀s,t∈𝕋,x∈X.When𝕋=ℝ the dynamical system is called continuous and when 𝕋=ℤ the dynamical system is called discrete.Definition 21.
Asemidynamical system is a continuous map Φ:𝕋o+×X→X which satisfies the initial value condition
(45)Φ(0,x)=x∀x∈X
and the semigroup property
(46)Φ(s+t,x)=Φ(s,Φ(t,x))∀s,t∈𝕋o+,x∈X.When𝕋o+=ℝ+ [resp., ℕ] the semidynamical system is called continuous [resp., discrete].


### 7.1.2. “Classical” Dynamical Systems

LetB be a Banach space. In 1969, Hale [21] gives the following definition.Definition 22.
A classical dynamical system on B is a continuous function u:ℝ+×B→B which satisfies
(47)u(0,φ)=φ,u(t+τ,φ)=u(t,u(τ,φ))
for all t,τ≥0 and φ∈B.Remark 23.
In Definition22, the mapping u is a continuous semidynamical system in terms of Definition 21.


### 7.1.3. “Abstract” Dynamical System

In 1991, Haraux [22] defines abstract dynamical systems in a complete metric space(E,d) in the following way.Definition 24.
An abstract dynamical system inE is a family {S(t)}t≥0 of mappings on E which satisfy the following properties:(i)
for eacht≥0, S(t):E→E is continuous,(ii)
S(0)=I,(iii)
S(t+s)=S(t)S(s),(iv)
for eachz∈E, t∈[0,∞)→S(t)z∈E is continuous.Remark 25.
In Definition24, the mapping u:(t,z)∈ℝ+×E→u(t,z)=S(t)z∈E satisfies conditions (45) and (46) but it is only separately continuous. Therefore it is not a semidynamical system in terms of Definition 21.


### 7.1.4. Dafermos Definition

In [23, Definition 4.1], Dafermos gives the following definition in a metric spaceX.Definition 26.
A Dafermos dynamical system inX is a map u:X×ℝ+→X which satisfies the initial condition
(48)u(x,0)=0∀x∈X
the semigroup property
(49)u(x,,s+t)=u(u(x,s),t)∀t,s∈ℝ+,x∈X,
and the continuity condition in x, for all t∈ℝ+.Remark 27.
In the previous definitions the spaceX [resp. B, E] is called the phase space, and the set
(50)γ+(x)=⋃t≥0Φ(t,x)[resp.⋃t≥0u(t,x)]
is called the positive orbit or trajectory through x∈X [resp., x∈B, x∈E].


## 7.1.1. General Definitions

In (2011), Kloeden and Rasmussen give, in [20], the general definitions:let𝕋=ℝ [resp., ℤ], 𝕋o+=𝕋∩[0,∞), that is, ℝ+ [resp., ℕ] and a metric spaceX.Definition 20.
Adynamical system is a continuous map Φ:𝕋×X→X which satisfies the initial value condition
(43)Φ(0,x)=x∀x∈X
and the group property
(44)Φ(s+t,x)=Φ(s,Φ(t,x))∀s,t∈𝕋,x∈X.When𝕋=ℝ the dynamical system is called continuous and when 𝕋=ℤ the dynamical system is called discrete.Definition 21.
Asemidynamical system is a continuous map Φ:𝕋o+×X→X which satisfies the initial value condition
(45)Φ(0,x)=x∀x∈X
and the semigroup property
(46)Φ(s+t,x)=Φ(s,Φ(t,x))∀s,t∈𝕋o+,x∈X.When𝕋o+=ℝ+ [resp., ℕ] the semidynamical system is called continuous [resp., discrete].


## 7.1.2. “Classical” Dynamical Systems

LetB be a Banach space. In 1969, Hale [21] gives the following definition.Definition 22.
A classical dynamical system on B is a continuous function u:ℝ+×B→B which satisfies
(47)u(0,φ)=φ,u(t+τ,φ)=u(t,u(τ,φ))
for all t,τ≥0 and φ∈B.Remark 23.
In Definition22, the mapping u is a continuous semidynamical system in terms of Definition 21.


## 7.1.3. “Abstract” Dynamical System

In 1991, Haraux [22] defines abstract dynamical systems in a complete metric space(E,d) in the following way.Definition 24.
An abstract dynamical system inE is a family {S(t)}t≥0 of mappings on E which satisfy the following properties:(i)
for eacht≥0, S(t):E→E is continuous,(ii)
S(0)=I,(iii)
S(t+s)=S(t)S(s),(iv)
for eachz∈E, t∈[0,∞)→S(t)z∈E is continuous.Remark 25.
In Definition24, the mapping u:(t,z)∈ℝ+×E→u(t,z)=S(t)z∈E satisfies conditions (45) and (46) but it is only separately continuous. Therefore it is not a semidynamical system in terms of Definition 21.


## 7.1.4. Dafermos Definition

In [23, Definition 4.1], Dafermos gives the following definition in a metric spaceX.Definition 26.
A Dafermos dynamical system inX is a map u:X×ℝ+→X which satisfies the initial condition
(48)u(x,0)=0∀x∈X
the semigroup property
(49)u(x,,s+t)=u(u(x,s),t)∀t,s∈ℝ+,x∈X,
and the continuity condition in x, for all t∈ℝ+.Remark 27.
In the previous definitions the spaceX [resp. B, E] is called the phase space, and the set
(50)γ+(x)=⋃t≥0Φ(t,x)[resp.⋃t≥0u(t,x)]
is called the positive orbit or trajectory through x∈X [resp., x∈B, x∈E].


## 7.2. Examples

### 7.2.1. Continuous (Semi-)Dynamical Systems

Example 28 (cf. [21, Example 1]).
Letf be a continuous function ℝn→ℝn and ξ∈ℝn. Consider the autonomous differential equation
(51)dxdt=f(x(t)),t≥0,
with the initial condition
(52)x(0)=ξ.
If there exists a unique solution u(t,ξ) of (51) and (52) which depends continuously upon t,ξ, then the mapping
(53)Φ:ℝ+×ℝn⟶ℝndefinedbyΦ(t,ξ)=u(t,ξ)
is a continuous semidynamical system on ℝn (Definitions 21 and 22).Example 29 (cf. [20, Example 1.6]).
LetX be the Banach space of bounded continuous functions μ:[-1,0]→ℝd, under the usual norm ∥·∥∞. Let μ∈X and f a continuous function ℝd→ℝd and suppose that there exists a unique solution u(t,μ) of the autonomous delay differential equation(54)dxdt=f(x(t-1)),t≥0,
with the condition
(55)x(s)=μ(s),s∈[-1,0].
Then
(56)Φ:(t,μ)∈ℝ+×X⟶Φ(t,μ)(s)=u(t+s,μ)∈X,0000000000000000000000000000.t≥0,s∈[0,1],
is a continuous semidynamical system on X.Example 30 (see cf. [22, page 12]).
LetX be a real Banach space, A a m-dissipative operator with dense domain in X (cf. Section 5), T(t) the semigroup of contractions generated by A, and F:X→X a Lipschitzian function on the bounded sets of X. Let u(t,x)∈C([o,τ(x)),X) be the unique maximal solution of
(57)u(t,x)=T(t)x+∫ot‍T(t-s)F(u(s))ds,0000000000000000.t∈[0,τ(x)),x∈X.
Set u(t,x)=S(t)x. Let Y⊂X such that τ(y)=∞ for all y∈Y and ∥S(t)y∥≤M, for all t≥0, y∈Y. For y∈Y, u(t,y)=S(t)y is solution of (57) for x=y. Set Z=⋃y∈Y⋃t≥0{S(t)y}¯ and denote by d the distance induced on Z by the norm in X. Then {S(t)}t≥0 is an abstract dynamical system (Definition 24) on (Z,d).Example 31.
Dafermos dynamical systems are adapted to equations of linear viscoelasticity. See Section10.Example 32 (see cf. [20]).
LetX be a metric space and f:X→X be a continuous function. By iteration, fn+1(x)=f(fn)(x)=fofn(x), for all x∈X, n∈N, is well defined. The mapping
(58)Φ:(n,x)∈ℕ×X⟶Φ(n,x)=fn(x)∈X
is a discrete semidynamical system (Definition 21). Now, suppose f is an homeomorphism; that is, it is continuous and invertible with continuous inverse. Then, the mapping Φ defined by (58) can be extended to ℤ- by
(59)Φ(n,x)=(f-1)-n(x),∀n∈ℤ-,x∈X,
and Φ is a discrete dynamical system.


### 7.2.2. Discrete-Time Dynamical Systems and Fractals

What is known as adiscrete time dynamical system in Pesin and Climenhaga [24] is related to a map from a set to itself. Let X be any set and f a map f:X→X. By iteration, fn+1(x)=f(fn(x))=(f∘fn)(x) is well defined for every n∈ℕ and we have the semigroup property:
(60)fn+m=fm∘fn=fn∘fm
for any integer m,n≥0.Definition 33.
Letx∈X. The sequence of points fo(x)=x,f(x),f2(x),…, is called the trajectory or orbit of x.


### 7.2.3.  Examples with Chaotic Behavior

Example 34 (the Cantor set).
Consider, inℝ, the intervals I1=[0,1/3], I2=[2/3,1], and D=I1∪I2 and the map f defined by f(x)=3x  on  I1  and  3x-2  on  I2. The range of f is [0,1] and does not lie in its domain of definition. The domain of definition of f2 is {x;x∈Dandf(x)∈D}, that is, the union of the intervals:
(61)[0,19],[29,13],[23,79],[89,1]
obtained by removing the open middle third of each interval I1 and I2 and so on. The domain on which every iterate fn is defined is exactly the Cantor set (cf. Figure 1).Figure 1
The Cantor set.Example 35 (the Sierpinski triangle (or gasket)).
An equilateral triangle is divided into four smaller triangles, each similar to the first and congruent to each other. Then, the middle triangle is removed, and the iterating procedure is iterated on the remaining three, and so on. The fractalT obtained as the limit of this procedure is the Sierpinki triangle (cf. Figure 2). With this procedure is associated the following algorithm. If x∈T, and only in this case, it is possible to define a function x∈T→f(x)∈ℝ2 such that fn(x)∈T for every n (cf. Schroeder [25] which calls this procedure “Sir Pinski game”).Figure 2
The Sierpinski gasket.The Cantor set and the Sierpinski triangle are examples of fractals (cf. [26, 27]).


## 7.2.1. Continuous (Semi-)Dynamical Systems

Example 28 (cf. [21, Example 1]).
Letf be a continuous function ℝn→ℝn and ξ∈ℝn. Consider the autonomous differential equation
(51)dxdt=f(x(t)),t≥0,
with the initial condition
(52)x(0)=ξ.
If there exists a unique solution u(t,ξ) of (51) and (52) which depends continuously upon t,ξ, then the mapping
(53)Φ:ℝ+×ℝn⟶ℝndefinedbyΦ(t,ξ)=u(t,ξ)
is a continuous semidynamical system on ℝn (Definitions 21 and 22).Example 29 (cf. [20, Example 1.6]).
LetX be the Banach space of bounded continuous functions μ:[-1,0]→ℝd, under the usual norm ∥·∥∞. Let μ∈X and f a continuous function ℝd→ℝd and suppose that there exists a unique solution u(t,μ) of the autonomous delay differential equation(54)dxdt=f(x(t-1)),t≥0,
with the condition
(55)x(s)=μ(s),s∈[-1,0].
Then
(56)Φ:(t,μ)∈ℝ+×X⟶Φ(t,μ)(s)=u(t+s,μ)∈X,0000000000000000000000000000.t≥0,s∈[0,1],
is a continuous semidynamical system on X.Example 30 (see cf. [22, page 12]).
LetX be a real Banach space, A a m-dissipative operator with dense domain in X (cf. Section 5), T(t) the semigroup of contractions generated by A, and F:X→X a Lipschitzian function on the bounded sets of X. Let u(t,x)∈C([o,τ(x)),X) be the unique maximal solution of
(57)u(t,x)=T(t)x+∫ot‍T(t-s)F(u(s))ds,0000000000000000.t∈[0,τ(x)),x∈X.
Set u(t,x)=S(t)x. Let Y⊂X such that τ(y)=∞ for all y∈Y and ∥S(t)y∥≤M, for all t≥0, y∈Y. For y∈Y, u(t,y)=S(t)y is solution of (57) for x=y. Set Z=⋃y∈Y⋃t≥0{S(t)y}¯ and denote by d the distance induced on Z by the norm in X. Then {S(t)}t≥0 is an abstract dynamical system (Definition 24) on (Z,d).Example 31.
Dafermos dynamical systems are adapted to equations of linear viscoelasticity. See Section10.Example 32 (see cf. [20]).
LetX be a metric space and f:X→X be a continuous function. By iteration, fn+1(x)=f(fn)(x)=fofn(x), for all x∈X, n∈N, is well defined. The mapping
(58)Φ:(n,x)∈ℕ×X⟶Φ(n,x)=fn(x)∈X
is a discrete semidynamical system (Definition 21). Now, suppose f is an homeomorphism; that is, it is continuous and invertible with continuous inverse. Then, the mapping Φ defined by (58) can be extended to ℤ- by
(59)Φ(n,x)=(f-1)-n(x),∀n∈ℤ-,x∈X,
and Φ is a discrete dynamical system.


## 7.2.2. Discrete-Time Dynamical Systems and Fractals

What is known as adiscrete time dynamical system in Pesin and Climenhaga [24] is related to a map from a set to itself. Let X be any set and f a map f:X→X. By iteration, fn+1(x)=f(fn(x))=(f∘fn)(x) is well defined for every n∈ℕ and we have the semigroup property:
(60)fn+m=fm∘fn=fn∘fm
for any integer m,n≥0.Definition 33.
Letx∈X. The sequence of points fo(x)=x,f(x),f2(x),…, is called the trajectory or orbit of x.


## 7.2.3.  Examples with Chaotic Behavior

Example 34 (the Cantor set).
Consider, inℝ, the intervals I1=[0,1/3], I2=[2/3,1], and D=I1∪I2 and the map f defined by f(x)=3x  on  I1  and  3x-2  on  I2. The range of f is [0,1] and does not lie in its domain of definition. The domain of definition of f2 is {x;x∈Dandf(x)∈D}, that is, the union of the intervals:
(61)[0,19],[29,13],[23,79],[89,1]
obtained by removing the open middle third of each interval I1 and I2 and so on. The domain on which every iterate fn is defined is exactly the Cantor set (cf. Figure 1).Figure 1
The Cantor set.Example 35 (the Sierpinski triangle (or gasket)).
An equilateral triangle is divided into four smaller triangles, each similar to the first and congruent to each other. Then, the middle triangle is removed, and the iterating procedure is iterated on the remaining three, and so on. The fractalT obtained as the limit of this procedure is the Sierpinki triangle (cf. Figure 2). With this procedure is associated the following algorithm. If x∈T, and only in this case, it is possible to define a function x∈T→f(x)∈ℝ2 such that fn(x)∈T for every n (cf. Schroeder [25] which calls this procedure “Sir Pinski game”).Figure 2
The Sierpinski gasket.The Cantor set and the Sierpinski triangle are examples of fractals (cf. [26, 27]).


## 7.3. Limit and Invariant Sets

Definition 36 (see cf. [20]).
LetΦ:𝕋×X→X be a dynamical system in a metric space X. The ω-limit set [resp., α-limit set] of a point ξ∈X is defined by
(62)ω(ξ)=⋂τ≥0⋃t≥τϕ(t,ξ)¯[resp.,α(ξ)=⋂τ≤0⋃t≤τϕ(t,ξ)¯].
In the case of a semidynamical system (Φ:To+×X→X) the notion of a α-limit set is not defined. In the same way, ω and α-limit sets of a subspace M⊂X are defined by (62) where ξ is replaced by M.Remark 37.
Theω and α-limit sets may be defined as follows:
(63)ω(ξ)={x∈X:thereexistsasequencetn,tn>0,0.tn⟶∞suchthatΦ(tn,ξ)⟶xasn⟶∞},α(ξ)={x∈X:thereexistsasequencetn,tn<0,0.tn⟶-∞suchthatΦ(tn,ξ)⟶xasn⟶∞}.Remark 38.
Similar definitions are valid for classical and abstract dynamical systems.Definition 39.
LetΦ:𝕋o+×X→X be a semidynamical system on the metric spaceX (Definition 21). A subset M⊂X is called invariant under Φ if
(64)Φ(t,M)=M∀t∈𝕋o+.
It is called positively [resp., negatively] invariant if(65)Φ(t,M)⊂M[resp.,M⊂Φ(t,M)],∀t∈𝕋o+.
In the case of discrete-time dynamical system and of Example 32, invariance is equivalent to f(M)=M.Properties of
ω
-Limit and Invariant Sets. Let (X,d) be a metric space, Φ:𝕋o+×X→X a semidynamical system on X, and x∈X. If the trajectory γ+(x) through x is precompact, then ω(x) has the following properties:
(66)Φ(t,ω(x))=ω(x)≠∅,∀t∈𝕋o+,ω(x)isacompactsubsetofX,ω(x)isconnectedif𝕋o+=ℝ+.
If 𝕋o+=ℕ, ω(x) does not need to be connected (cf. [20, Exercise 1.10]).In the case of an abstract dynamical system{(S(t)}t≥0, on a complete metric space(E,d), previous results hold with Φ(t,x)=S(t)x, for x∈E, if γ+(x) is relatively compact in E (see [28, page 122]). But in a complete metric space E a subset A is relatively compact if and only if it is precompact (cf. [29]).


## 8. Equal-Area Condition

Equal-area type conditions appear, as sufficient or necessary conditions, in the formation of layers (internal or superficial) in stationary solutions to various singularly perturbed reaction-diffusion systems. In the recent works do Nascimento [30], Crema and do Nascimento [31], and  do Nascimento and de Moura [32], the authors prove the necessity of suitable equal-area condition for the formation of internal or (and) superficial transition layers in this type of problems.Example 40.
A simple particular case of problems studied in [30] is the elliptic boundary value problem
(67)ϵdiv(h(x)∇u)+f(u)=0,x∈Ω,∂u∂n=0on∂Ω,
where Ω is a smooth domain in ℝN, N≥1, f:ℝ→ℝ such that there exist α,β,α>β, with f(α)=f(β)=0.LetΓ⊂Ω be a smooth (N-1)-dimensional compact manifold without boundary. It is proved that, if (67) has a family {uϵ} of solutions which develop an internal transition layer with interface Γ connecting the states α to β, then, necessarily, the simple equal-area condition(68)∫αβ‍f(s)ds=0
is satisfied.Example 41.
In [31], the following stationary system is considered:
(69)ϵdiv(h(x)∇u)+f(x,u,v)=0,x∈Ω,div(k(x)∇v)+g(x,u,v)=0,x∈Ω,∂v∂n=0(orv=0)on∂Ω,
where Ω is a smooth domain in ℝN, N≥1, f∈C1(Ω×ℝ×ℝN), v, k, andg are sufficiently smooth ℝn-valued functions, and k∇v=(k1∇v1,…,kn∇vn).Let𝒰 be an open connected set in Ω, Γ⊂𝒰¯ be an (N-1)-dimensional compact connected orientable manifold whose boundary ∂Γ is such that ∂Γ∩∂Ω is an (N-2)-dimensional submanifold of ∂Ω. The authors give a definition of a family of internal transition layer solutions {(uϵ,vϵ)},0<ϵ<ϵo} to (69) in 𝒰 with interface Γ, depending on two functions α,β∈Co(𝒰), α(x)<β(x) on Γ. They show that, for such a family, there exists uo [resp., vo] such that uϵ→uo [resp., vϵ→vo] on compact sets of 𝒰¯∖Γ [resp., in 𝒰¯]. They also prove that, if a family of internal transition layer solutions to (69) exists, then f(x,uo(x),vo(x))=0 on 𝒰∖Γ and necessarily the equal-area condition(70)∫Γ‍(∫α(x)β(x)‍f(x,s,vo(x))ds)dS=0
is satisfied. Several concrete applications of these results are presented in the paper.Example 42.
In [32], Ω is a bounded domain in ℝN, N≥1, with C2 boundary ∂Ω, S is a C2(N-1)-dimensional surface with a boundary Σ which is assumed to be a C2(N-2)-dimensional compact surface without boundary with Σ=S¯∩∂Ω, and S¯ intersects ∂Ω transversally. The authors define a family of solutions {uϵ}, 0<ϵ<ϵo, to the elliptic boundary value problem:
(71)ϵdiv(a(x)∇vϵ)+f(x,vϵ)=0,x∈Ω,ϵa(x)∂vϵ∂n^=g(x,vϵ),x∈∂Ω,
which develops internal and superficial transition layers, depending on some smooth functions α,β,α(x)<β(x), x∈Ω¯, with interfaces S and Σ, respectively.Herea∈C1(Ω¯), a>0, f:Ω×ℝ→ℝ, and g:∂Ω×ℝ→ℝ are of class C1 and n^ is the exterior normal vector field on ∂Ω. It is proved that the equal-area conditions(72)∫α(x)β(x)‍f(x,t)dt=0∀x∈S,∫α(y)β(y)‍g(y,t)dt=0∀y∈Σ
are necessary for the existence of such solutions.


## 9. Inertial Manifolds

### 9.1. Setup

LetH be a Hilbert space and S(t) the semigroup associated with an evolution equation of the form
(73)ut′+Au+Ru=0
with the initial condition u(0)=uo∈H, where A is a linear operator and R a nonlinear one. When an inertial manifold M exists for problem (73), the restriction of (73) to M reduces to a finite dimensional ordinary differential equation (80), which is an exact copy of the initial system (cf. [33]). The manifold M is usually viewed as the graph of a suitable smooth function Φ:PH→QH, where P is an orthogonal projection on H with finite-dimensional range, and Q=I-P.After some definitions, the following sections will be devoted to the existence of an inertial manifold for (73) and the presentation of results on the existence and the behavior of inertial manifolds for phase-field equations.


### 9.2. Definitions

Let(E,d) be a metric space and S(t) a continuous semigroup on E.Definition 43.
A setM⊂E is an inertial manifold for S(t) if(i)
M is a finite-dimensional Lipschitz manifold in E,(ii)
M is of class C1,(iii)
M is positively invariant under the flow that is, S(t)M⊂M, for all t≥0,(iv)
M is exponentially attracting; that is, there exists a constant co such that, for any uo∈E, there exists a constant c1(uo)>0 such that
(74)distE(S(t)uo,M)≤c1e-cot,∀t≥0,
where distE is the Hausdorff semidistance (see Section 1.8 in Huet [7]); compare Bonfoh et al. [34, page 164].Definition 44 (see Temam [33]).
LetB⊂E and U an open subset of E such that B⊂U; B is said to be absorbing for S(t) in U if, for any bounded set Bo in U, there exists t1(Bo) such that S(t)Bo⊂B, for all t≥t1. In particular, if B is bounded, S(t)B⊂B, for t≥t1(B).Remark 45.
Similar definitions were given in [33] and Luskin and Sell [35], in the case of Hilbert spaces. In [33] the smoothness of M and the continuity of the semigroup S(t) are not parts of the definition of an inertial manifold.


### 9.3. Existence of an Inertial Manifold for Problem (73) (cf. [33])

In a Hilbert spaceH whose norm is denoted by |·|, consider problem (73) where A is an unbounded linear operator in H, strictly positive (i.e., ∃c>0 such that (Au,u)≥c∥u∥H2 for all u in the domain D(A) of A), and self-adjoint. It is assumed that A-1 is compact. Thanks to assumptions on A, it is possible to define its powers As defined on D(As), for all s∈ℝ. When 0<s<1 see also Huet [36], Kato [37], Schechter [38]. For r>0, we denote by Bs,r the ball, in D(As), with center 0 of radius r; that is,
(75)Bs,r={v∈D(As);∥v∥s=|Asv|≤r}.
It is assumed that R satisfies the following properties.H
1. There exists α∈ℝ such that R is Lipschitz on the bounded sets of D(Aα) with values in D(Aα-1/2) and
(76)|Aα-1/2R(u)-Aα-1/2R(v)|≤CK|Aα(u-v)|,∀u,v∈D(Aα)satisfying|Aαu|≤K,|Aαv|≤K.H
2. If uo∈D(Aα) problem (73) has a unique solution u∈C(ℝ+;D(Aα))∩L2((O,T);D(Aα+1/2)), for all T≥0, and the map S(t):uo→u(t) is continuous from D(Aα) into itself, for all t≥0.H
3. The semigroup S(t) possesses an absorbing set Bo⊂D(Aα) such that(1)
S(t)Bo⊂Bo,  for all t≥O,(2)
theω-limit set of Bo(77)ω(Bo)=⋂β≥0⋃S(t)Bot≥β¯
is the maximal attractor for S(t) in D(Aα).(3)
ρ>0 is chosen such that Bo is included in the ball Bα,ρ/2 of D(Aα) (cf. (75)).


#### 9.3.1. The Prepared Equation

The prepared equation is equivalent to the original one fort large. Let θ:ℝ+→[0,1] be a C∞ function such that
(78)θ(s)={1for0≤s≤1,0fors≥2,sups≥0|θ′(s)|≤2.
The aim of the prepared equation is to avoid the difficulties related to the behavior of the nonlinear term R(u) for large values of |Aα(u)|. Let θr(s)=θ(s/r), r>0, and
(79)Rr(u)=θr(|Aα(u)|)R(u)∀u∈D(Aα).
The prepared equation associated with (73) is of the form
(80)ut′+Au+Rρu=0,
where ρ is chosen in H3(3). Let Sρ(t) be the semigroup associated with (80). The following is assumed.H
4. The ball Bα,ρ is absorbing for Sρ(t).Thanks to the assumptions onR, Rρ is a bounded operator and a Lipschitz mapping from D(Aα) into D(Aα-1/2). In particular
(81)|Rρ(u1)-Rρ(u2)|≤Cρ|Aα(u1-u2)|000000000000000000∀u1,u2∈D(Aα),
where Cρ is a constant which depends on ρ.


#### 9.3.2. SpaceℱNblα

LetN∈ℕ, b,l>0 and α defined in H1. Under the assumptions of A, there exists an orthonormal basis wj in H, where wj is the eigenvector of A corresponding to the eigenvalue λj, with 0<λ1≤λ2⋯≤λj→∞, as j→∞. Set
(82)P=PNQ=I-P=I-PN,
where PN is the orthogonal projector, in H, onto the space spanned by w1,…,wN. The projections P and Q commute with Aβ, for all β∈ℝ.Definition 46.
Letℱ be the space defined by
(83)ℱ=ℱNblα={φ:PD(Aα)⟶QD(Aα)},
where φ is a Lipschitz function with supp(φ)⊂{p∈PD(Aα),|Aαp|≤2ρ}, that satisfies
(84)|Aαφ(p)|≤b,|Aαφ(p1)-Aαφ(p2)|≤l|Aα(p1-p2)|
for all p∈PD(Aα), p1,p2∈D(Aα).Remark 47.
(
1
)
P
D
(
A
α
) is finite dimensional.
(
2
)
ℱ is a complete metric space for the distance
(85)d(φ1,φ2)=supp∈PD(Aα)|Aα(φ1(p)-φ2(p))|.


#### 9.3.3. Construction of a Map𝒯:ℱ→ℱ

Letφ∈ℱ, po∈PD(Aα). Thanks to assumptions on φ and Rρ, the problem
(86)dpdt+Ap+PRρ(p+φ(p))=0,p(0)=po,
has a unique solution ppo,φ(t)∈PD(Aα). Then, the problem
(87)dqdt+Aq=-QRρ(ppo,φ(t)+φ(ppo,φ(t)))
has a unique solution qpo,φ(t)∈Cb(ℝ;QD(Aα)). Therefore qpo,φ(0)∈QD(Aα) (cf. Lemma 2.3, page 420 in [33]). The mapping 𝒯 is defined by ψ=𝒯(φ) with
(88)ψ:po∈PD(Aα)⟶qpo,φ(0)∈QD(Aα).
Let σ(t) denote the right-hand side of (87). Then
(89)ψ(po)=-∫-∞0‍etAσ(τ)dτ.


#### 9.3.4. Existence of an Inertial Manifold for (73) or (80)

Under assumptionsH1–H4, if it is possible to find N, b, and  l, such that 𝒯 is a strict contraction of ℱ into itself, that is, ∃L<1 with d(𝒯φ1,𝒯φ2)≤Ld(φ1,φ2), for all φ1,φ2∈ℱ, then, 𝒯 has a fixed point Φ∈ℱ, and the graph of Φ is an inertial manifold for (73) or (80).


### 9.4. Inertial Manifolds for Phase-Field Equations

Example 48 (cf. Bonfoh [39]).
SetΩ=∏i=1n‍(0,Li), Li>0,  and  n≤2 and denote by N the operator -Δ:H2(Ω)∩Ho1(Ω)→L2(Ω). Consider the phase-field equations (Pϵ),
(90)δϕt+Nϕ+g(ϕ)=u,ϵut+ϕt+Nu=0
in Ω, with the boundary conditions
(91)ϕ,u|∂Ω=0
and the initial conditions
(92)ϕ|t=0=ϕo,u|t=0=uo,
where δ>0, ϵ∈(0,ϵo,], and  g∈C2(ℝ) satisfies the conditions
(93)|g′(s)|≤C1(|s|p+1),g′(s)≥-C2,00000C1>0,C2≥0,p>0,∀s∈ℝ.
The limit problem (Po) is given by (90) where ϵ=0:
(94)δϕt+Nϕ+g(ϕ)=u,ϕt+Nu=0,
the boundary conditions (91), and the initial condition ϕ|t=0=ϕo. Set
(95)H1=Ho1(Ω),H2=H2(Ω)∩Ho1(Ω),H3={q∈H3(Ω)∩Ho1(Ω);Δq∈Ho1(Ω)}.
Let ∥·∥ denote the usual norm in L2(Ω) and set ∥q∥r=∥Nr/2q∥, r=1,2,3. The norms ∥·∥r are equivalent to the usual Sobolev Hr(Ω) norms on Hr. Several Banach spaces whose norms depend on ϵ are introduced. In particular
(96)ℋϵo=H1×L2(Ω),withthenorm∥(p,q)∥ℋϵo=(∥p∥12+ϵ∥q∥2)1/2.
Under additional assumptions on the regularity of the data, there exist semigroups Sϵ(t), global attractors 𝒜ϵ, and exponential attractors ℳϵ for problems (Pϵ),  0≤ϵ≤ϵo (cf. Chepyzhov and Vishik [40]). It is proved that 𝒜ϵ and ℳϵ converge as ϵ→0, to some lifting of 𝒜o and ℳo, in ℋ1o (cf. (96)). Moreover, there exists r>0, independent of ϵ, such that
(97)B={u∈H3,∥u∥3≤r},Bϵ={(ϕ,u)∈ℋϵ2,∥(ϕ,u)∥ℋϵ2≤r}
are bounded absorbing sets for S(t) in H3 and Sϵ(t) in ℋϵ2, respectively, where
(98)ℋϵ2=H3×H2withthenorm∥(p,q)∥ℋϵ22=∥p∥32+ϵ∥q∥22
and there exists t1>0 such that S(t)B⊂B, Sϵ(t)Bϵ⊂Bϵ, for t≥t1. In particular
(99)B~=S(t1)B⊂B,B~ϵ=Sϵ(t1)Bϵ⊂Bϵ.
Following the classical method presented in the above section for (73), the authors show the existence of an inertial manifold Mr to problem (Po) and of a family of inertial manifolds Mϵr to problems (Pϵ) such that
(100)Mr⊂B~,Mϵr⊂B~ϵ.
Let (Mr)o defined by
(101)(Mr)o={(ϕ,(I+δN)-1(Nϕ+g(ϕ))),ϕ∈Mr}.
Moreover, it is proved that(1)
for all0<ϵ2<ϵ1 small enough, Mϵ2r is lower and upper semicontinuous at ϵ2, with respect to the metric induced by the ℋϵ10 norm,(2)
Mϵr converges, in a suitable sense, to (Mr)o with respect to the metric induced by the ℋ10 norm, as ϵ goes to 0.


## 9.1. Setup

LetH be a Hilbert space and S(t) the semigroup associated with an evolution equation of the form
(73)ut′+Au+Ru=0
with the initial condition u(0)=uo∈H, where A is a linear operator and R a nonlinear one. When an inertial manifold M exists for problem (73), the restriction of (73) to M reduces to a finite dimensional ordinary differential equation (80), which is an exact copy of the initial system (cf. [33]). The manifold M is usually viewed as the graph of a suitable smooth function Φ:PH→QH, where P is an orthogonal projection on H with finite-dimensional range, and Q=I-P.After some definitions, the following sections will be devoted to the existence of an inertial manifold for (73) and the presentation of results on the existence and the behavior of inertial manifolds for phase-field equations.


## 9.2. Definitions

Let(E,d) be a metric space and S(t) a continuous semigroup on E.Definition 43.
A setM⊂E is an inertial manifold for S(t) if(i)
M is a finite-dimensional Lipschitz manifold in E,(ii)
M is of class C1,(iii)
M is positively invariant under the flow that is, S(t)M⊂M, for all t≥0,(iv)
M is exponentially attracting; that is, there exists a constant co such that, for any uo∈E, there exists a constant c1(uo)>0 such that
(74)distE(S(t)uo,M)≤c1e-cot,∀t≥0,
where distE is the Hausdorff semidistance (see Section 1.8 in Huet [7]); compare Bonfoh et al. [34, page 164].Definition 44 (see Temam [33]).
LetB⊂E and U an open subset of E such that B⊂U; B is said to be absorbing for S(t) in U if, for any bounded set Bo in U, there exists t1(Bo) such that S(t)Bo⊂B, for all t≥t1. In particular, if B is bounded, S(t)B⊂B, for t≥t1(B).Remark 45.
Similar definitions were given in [33] and Luskin and Sell [35], in the case of Hilbert spaces. In [33] the smoothness of M and the continuity of the semigroup S(t) are not parts of the definition of an inertial manifold.


## 9.3. Existence of an Inertial Manifold for Problem (73) (cf. [33])

In a Hilbert spaceH whose norm is denoted by |·|, consider problem (73) where A is an unbounded linear operator in H, strictly positive (i.e., ∃c>0 such that (Au,u)≥c∥u∥H2 for all u in the domain D(A) of A), and self-adjoint. It is assumed that A-1 is compact. Thanks to assumptions on A, it is possible to define its powers As defined on D(As), for all s∈ℝ. When 0<s<1 see also Huet [36], Kato [37], Schechter [38]. For r>0, we denote by Bs,r the ball, in D(As), with center 0 of radius r; that is,
(75)Bs,r={v∈D(As);∥v∥s=|Asv|≤r}.
It is assumed that R satisfies the following properties.H
1. There exists α∈ℝ such that R is Lipschitz on the bounded sets of D(Aα) with values in D(Aα-1/2) and
(76)|Aα-1/2R(u)-Aα-1/2R(v)|≤CK|Aα(u-v)|,∀u,v∈D(Aα)satisfying|Aαu|≤K,|Aαv|≤K.H
2. If uo∈D(Aα) problem (73) has a unique solution u∈C(ℝ+;D(Aα))∩L2((O,T);D(Aα+1/2)), for all T≥0, and the map S(t):uo→u(t) is continuous from D(Aα) into itself, for all t≥0.H
3. The semigroup S(t) possesses an absorbing set Bo⊂D(Aα) such that(1)
S(t)Bo⊂Bo,  for all t≥O,(2)
theω-limit set of Bo(77)ω(Bo)=⋂β≥0⋃S(t)Bot≥β¯
is the maximal attractor for S(t) in D(Aα).(3)
ρ>0 is chosen such that Bo is included in the ball Bα,ρ/2 of D(Aα) (cf. (75)).


### 9.3.1. The Prepared Equation

The prepared equation is equivalent to the original one fort large. Let θ:ℝ+→[0,1] be a C∞ function such that
(78)θ(s)={1for0≤s≤1,0fors≥2,sups≥0|θ′(s)|≤2.
The aim of the prepared equation is to avoid the difficulties related to the behavior of the nonlinear term R(u) for large values of |Aα(u)|. Let θr(s)=θ(s/r), r>0, and
(79)Rr(u)=θr(|Aα(u)|)R(u)∀u∈D(Aα).
The prepared equation associated with (73) is of the form
(80)ut′+Au+Rρu=0,
where ρ is chosen in H3(3). Let Sρ(t) be the semigroup associated with (80). The following is assumed.H
4. The ball Bα,ρ is absorbing for Sρ(t).Thanks to the assumptions onR, Rρ is a bounded operator and a Lipschitz mapping from D(Aα) into D(Aα-1/2). In particular
(81)|Rρ(u1)-Rρ(u2)|≤Cρ|Aα(u1-u2)|000000000000000000∀u1,u2∈D(Aα),
where Cρ is a constant which depends on ρ.


### 9.3.2. SpaceℱNblα

LetN∈ℕ, b,l>0 and α defined in H1. Under the assumptions of A, there exists an orthonormal basis wj in H, where wj is the eigenvector of A corresponding to the eigenvalue λj, with 0<λ1≤λ2⋯≤λj→∞, as j→∞. Set
(82)P=PNQ=I-P=I-PN,
where PN is the orthogonal projector, in H, onto the space spanned by w1,…,wN. The projections P and Q commute with Aβ, for all β∈ℝ.Definition 46.
Letℱ be the space defined by
(83)ℱ=ℱNblα={φ:PD(Aα)⟶QD(Aα)},
where φ is a Lipschitz function with supp(φ)⊂{p∈PD(Aα),|Aαp|≤2ρ}, that satisfies
(84)|Aαφ(p)|≤b,|Aαφ(p1)-Aαφ(p2)|≤l|Aα(p1-p2)|
for all p∈PD(Aα), p1,p2∈D(Aα).Remark 47.
(
1
)
P
D
(
A
α
) is finite dimensional.
(
2
)
ℱ is a complete metric space for the distance
(85)d(φ1,φ2)=supp∈PD(Aα)|Aα(φ1(p)-φ2(p))|.


### 9.3.3. Construction of a Map𝒯:ℱ→ℱ

Letφ∈ℱ, po∈PD(Aα). Thanks to assumptions on φ and Rρ, the problem
(86)dpdt+Ap+PRρ(p+φ(p))=0,p(0)=po,
has a unique solution ppo,φ(t)∈PD(Aα). Then, the problem
(87)dqdt+Aq=-QRρ(ppo,φ(t)+φ(ppo,φ(t)))
has a unique solution qpo,φ(t)∈Cb(ℝ;QD(Aα)). Therefore qpo,φ(0)∈QD(Aα) (cf. Lemma 2.3, page 420 in [33]). The mapping 𝒯 is defined by ψ=𝒯(φ) with
(88)ψ:po∈PD(Aα)⟶qpo,φ(0)∈QD(Aα).
Let σ(t) denote the right-hand side of (87). Then
(89)ψ(po)=-∫-∞0‍etAσ(τ)dτ.


### 9.3.4. Existence of an Inertial Manifold for (73) or (80)

Under assumptionsH1–H4, if it is possible to find N, b, and  l, such that 𝒯 is a strict contraction of ℱ into itself, that is, ∃L<1 with d(𝒯φ1,𝒯φ2)≤Ld(φ1,φ2), for all φ1,φ2∈ℱ, then, 𝒯 has a fixed point Φ∈ℱ, and the graph of Φ is an inertial manifold for (73) or (80).


## 9.3.1. The Prepared Equation

The prepared equation is equivalent to the original one fort large. Let θ:ℝ+→[0,1] be a C∞ function such that
(78)θ(s)={1for0≤s≤1,0fors≥2,sups≥0|θ′(s)|≤2.
The aim of the prepared equation is to avoid the difficulties related to the behavior of the nonlinear term R(u) for large values of |Aα(u)|. Let θr(s)=θ(s/r), r>0, and
(79)Rr(u)=θr(|Aα(u)|)R(u)∀u∈D(Aα).
The prepared equation associated with (73) is of the form
(80)ut′+Au+Rρu=0,
where ρ is chosen in H3(3). Let Sρ(t) be the semigroup associated with (80). The following is assumed.H
4. The ball Bα,ρ is absorbing for Sρ(t).Thanks to the assumptions onR, Rρ is a bounded operator and a Lipschitz mapping from D(Aα) into D(Aα-1/2). In particular
(81)|Rρ(u1)-Rρ(u2)|≤Cρ|Aα(u1-u2)|000000000000000000∀u1,u2∈D(Aα),
where Cρ is a constant which depends on ρ.


## 9.3.2. SpaceℱNblα

LetN∈ℕ, b,l>0 and α defined in H1. Under the assumptions of A, there exists an orthonormal basis wj in H, where wj is the eigenvector of A corresponding to the eigenvalue λj, with 0<λ1≤λ2⋯≤λj→∞, as j→∞. Set
(82)P=PNQ=I-P=I-PN,
where PN is the orthogonal projector, in H, onto the space spanned by w1,…,wN. The projections P and Q commute with Aβ, for all β∈ℝ.Definition 46.
Letℱ be the space defined by
(83)ℱ=ℱNblα={φ:PD(Aα)⟶QD(Aα)},
where φ is a Lipschitz function with supp(φ)⊂{p∈PD(Aα),|Aαp|≤2ρ}, that satisfies
(84)|Aαφ(p)|≤b,|Aαφ(p1)-Aαφ(p2)|≤l|Aα(p1-p2)|
for all p∈PD(Aα), p1,p2∈D(Aα).Remark 47.
(
1
)
P
D
(
A
α
) is finite dimensional.
(
2
)
ℱ is a complete metric space for the distance
(85)d(φ1,φ2)=supp∈PD(Aα)|Aα(φ1(p)-φ2(p))|.


## 9.3.3. Construction of a Map𝒯:ℱ→ℱ

Letφ∈ℱ, po∈PD(Aα). Thanks to assumptions on φ and Rρ, the problem
(86)dpdt+Ap+PRρ(p+φ(p))=0,p(0)=po,
has a unique solution ppo,φ(t)∈PD(Aα). Then, the problem
(87)dqdt+Aq=-QRρ(ppo,φ(t)+φ(ppo,φ(t)))
has a unique solution qpo,φ(t)∈Cb(ℝ;QD(Aα)). Therefore qpo,φ(0)∈QD(Aα) (cf. Lemma 2.3, page 420 in [33]). The mapping 𝒯 is defined by ψ=𝒯(φ) with
(88)ψ:po∈PD(Aα)⟶qpo,φ(0)∈QD(Aα).
Let σ(t) denote the right-hand side of (87). Then
(89)ψ(po)=-∫-∞0‍etAσ(τ)dτ.


## 9.3.4. Existence of an Inertial Manifold for (73) or (80)

Under assumptionsH1–H4, if it is possible to find N, b, and  l, such that 𝒯 is a strict contraction of ℱ into itself, that is, ∃L<1 with d(𝒯φ1,𝒯φ2)≤Ld(φ1,φ2), for all φ1,φ2∈ℱ, then, 𝒯 has a fixed point Φ∈ℱ, and the graph of Φ is an inertial manifold for (73) or (80).


## 9.4. Inertial Manifolds for Phase-Field Equations

Example 48 (cf. Bonfoh [39]).
SetΩ=∏i=1n‍(0,Li), Li>0,  and  n≤2 and denote by N the operator -Δ:H2(Ω)∩Ho1(Ω)→L2(Ω). Consider the phase-field equations (Pϵ),
(90)δϕt+Nϕ+g(ϕ)=u,ϵut+ϕt+Nu=0
in Ω, with the boundary conditions
(91)ϕ,u|∂Ω=0
and the initial conditions
(92)ϕ|t=0=ϕo,u|t=0=uo,
where δ>0, ϵ∈(0,ϵo,], and  g∈C2(ℝ) satisfies the conditions
(93)|g′(s)|≤C1(|s|p+1),g′(s)≥-C2,00000C1>0,C2≥0,p>0,∀s∈ℝ.
The limit problem (Po) is given by (90) where ϵ=0:
(94)δϕt+Nϕ+g(ϕ)=u,ϕt+Nu=0,
the boundary conditions (91), and the initial condition ϕ|t=0=ϕo. Set
(95)H1=Ho1(Ω),H2=H2(Ω)∩Ho1(Ω),H3={q∈H3(Ω)∩Ho1(Ω);Δq∈Ho1(Ω)}.
Let ∥·∥ denote the usual norm in L2(Ω) and set ∥q∥r=∥Nr/2q∥, r=1,2,3. The norms ∥·∥r are equivalent to the usual Sobolev Hr(Ω) norms on Hr. Several Banach spaces whose norms depend on ϵ are introduced. In particular
(96)ℋϵo=H1×L2(Ω),withthenorm∥(p,q)∥ℋϵo=(∥p∥12+ϵ∥q∥2)1/2.
Under additional assumptions on the regularity of the data, there exist semigroups Sϵ(t), global attractors 𝒜ϵ, and exponential attractors ℳϵ for problems (Pϵ),  0≤ϵ≤ϵo (cf. Chepyzhov and Vishik [40]). It is proved that 𝒜ϵ and ℳϵ converge as ϵ→0, to some lifting of 𝒜o and ℳo, in ℋ1o (cf. (96)). Moreover, there exists r>0, independent of ϵ, such that
(97)B={u∈H3,∥u∥3≤r},Bϵ={(ϕ,u)∈ℋϵ2,∥(ϕ,u)∥ℋϵ2≤r}
are bounded absorbing sets for S(t) in H3 and Sϵ(t) in ℋϵ2, respectively, where
(98)ℋϵ2=H3×H2withthenorm∥(p,q)∥ℋϵ22=∥p∥32+ϵ∥q∥22
and there exists t1>0 such that S(t)B⊂B, Sϵ(t)Bϵ⊂Bϵ, for t≥t1. In particular
(99)B~=S(t1)B⊂B,B~ϵ=Sϵ(t1)Bϵ⊂Bϵ.
Following the classical method presented in the above section for (73), the authors show the existence of an inertial manifold Mr to problem (Po) and of a family of inertial manifolds Mϵr to problems (Pϵ) such that
(100)Mr⊂B~,Mϵr⊂B~ϵ.
Let (Mr)o defined by
(101)(Mr)o={(ϕ,(I+δN)-1(Nϕ+g(ϕ))),ϕ∈Mr}.
Moreover, it is proved that(1)
for all0<ϵ2<ϵ1 small enough, Mϵ2r is lower and upper semicontinuous at ϵ2, with respect to the metric induced by the ℋϵ10 norm,(2)
Mϵr converges, in a suitable sense, to (Mr)o with respect to the metric induced by the ℋ10 norm, as ϵ goes to 0.


## 10. Mathieu-Hill Type Equations

### 10.1. Mathieu and Hill Equations

The real Mathieu [resp., Hill] equation has the form(102)y′′+(a+bcos2πx)y=0,
where a and b are constants,
(103)[resp.,y′′+(a+bq(x))y=0],
where q is any smooth periodic function of period 1 with mean 0 (see Coddington and Levinson [41] and Wang and Guo [42]). Physical problems leading to Mathieu or Hill equations often require solutions with periodicity, called oscillatory solutions. Therefore, to find conditions on the data for which the above equations have a fundamental system of periodic solutions is a central problem.


### 10.2. Its Equation

In [43], A. R. Its considers the Schrödinger equation on the positive semiaxis
(104)y′′-[xβp(x1+α)+cx-2]y=0,
where p is a smooth periodic function, with period 1 and mean 0, c is a real number, and the parameters α,β satisfy the relations β-α≥-1 and 2α-β>0 (cf. Figure 3).Figure 3
Theα-β plane.He proves that (104) has oscillatory solutions when β>α-1. If β=α-1, the solutions are oscillatory or not. In all cases, asymptotic formulas for the solutions are stated, as x→∞. His method is based on a transformation which leads to a Hill-type equation and Floquet functions.


### 10.3. Method of Diagonalisation

In [44], Bodine and Lutz investigate (104) by a general method of diagonalization with a L1-regularity for p. Set
(105)t=x1+α,ρ(t)=∫0t‍p(s)1+αds,μ=∫01‍ρ(t)dt,ν=∫01‍(ρ(t))2dt(μ2<ν).
Let
(106)λ1=1/2+c+μ2-ν+1/41+α,λ2=1/2-c+μ2-ν+1/41+α
be the eigenvalues of a suitable matrix which depends on α, c, μ,  and  ν, and Λ the diagonal matrix diag{λ1,λ2}. The matrix Λ is a crucial tool in the diagonalizable method used by the authors. When β=α-1, three cases are studied separately as λ1-λ2 belongs to ℕ, or as it does not belong to ℕ0 or as it is equal to 0. In each case, very sharp asymptotic formulas are obtained for a fundamental system of solutions to (104), as t→∞. In particular, when λ1-λ2∈ℕo, a logarithmic term appears in the formulas. For instance, when λ1-λ2∈ℕ the following formulas are obtained:
(107)y1(x)=xλ1(1+α)[1+∑r=1N-1‍pr1(x1+α)xr(1+α)+𝒪(1xN(1+α))]+η(lnx)y2(x),y2(x)=xλ2(1+α)[1+∑r=1N-1‍pr2(x1+α)xr(1+α)+𝒪(1xN(1+α))],
where pr1,2 are scalar valued, bounded, continuous, periodic functions with period 1 which are recursively calculated and η≠0 is a suitable number. This logarithmic term is missing in Its formulas.


## 10.1. Mathieu and Hill Equations

The real Mathieu [resp., Hill] equation has the form(102)y′′+(a+bcos2πx)y=0,
where a and b are constants,
(103)[resp.,y′′+(a+bq(x))y=0],
where q is any smooth periodic function of period 1 with mean 0 (see Coddington and Levinson [41] and Wang and Guo [42]). Physical problems leading to Mathieu or Hill equations often require solutions with periodicity, called oscillatory solutions. Therefore, to find conditions on the data for which the above equations have a fundamental system of periodic solutions is a central problem.


## 10.2. Its Equation

In [43], A. R. Its considers the Schrödinger equation on the positive semiaxis
(104)y′′-[xβp(x1+α)+cx-2]y=0,
where p is a smooth periodic function, with period 1 and mean 0, c is a real number, and the parameters α,β satisfy the relations β-α≥-1 and 2α-β>0 (cf. Figure 3).Figure 3
Theα-β plane.He proves that (104) has oscillatory solutions when β>α-1. If β=α-1, the solutions are oscillatory or not. In all cases, asymptotic formulas for the solutions are stated, as x→∞. His method is based on a transformation which leads to a Hill-type equation and Floquet functions.


## 10.3. Method of Diagonalisation

In [44], Bodine and Lutz investigate (104) by a general method of diagonalization with a L1-regularity for p. Set
(105)t=x1+α,ρ(t)=∫0t‍p(s)1+αds,μ=∫01‍ρ(t)dt,ν=∫01‍(ρ(t))2dt(μ2<ν).
Let
(106)λ1=1/2+c+μ2-ν+1/41+α,λ2=1/2-c+μ2-ν+1/41+α
be the eigenvalues of a suitable matrix which depends on α, c, μ,  and  ν, and Λ the diagonal matrix diag{λ1,λ2}. The matrix Λ is a crucial tool in the diagonalizable method used by the authors. When β=α-1, three cases are studied separately as λ1-λ2 belongs to ℕ, or as it does not belong to ℕ0 or as it is equal to 0. In each case, very sharp asymptotic formulas are obtained for a fundamental system of solutions to (104), as t→∞. In particular, when λ1-λ2∈ℕo, a logarithmic term appears in the formulas. For instance, when λ1-λ2∈ℕ the following formulas are obtained:
(107)y1(x)=xλ1(1+α)[1+∑r=1N-1‍pr1(x1+α)xr(1+α)+𝒪(1xN(1+α))]+η(lnx)y2(x),y2(x)=xλ2(1+α)[1+∑r=1N-1‍pr2(x1+α)xr(1+α)+𝒪(1xN(1+α))],
where pr1,2 are scalar valued, bounded, continuous, periodic functions with period 1 which are recursively calculated and η≠0 is a suitable number. This logarithmic term is missing in Its formulas.


## 11. (Equations with) Memory

Equations with memory are differential equations which are influenced by the past of one or more variables. In this section, I restrict myself to a few examples. I present the equation with memory studied by Dafermos [45] in the context of linear viscoelasticity and examples of singularly perturbed equations with memory.


### 11.1. Dafermos Results

Definition 49 (cf. Coleman and Noll [46]).
A functionh is called an influence function if(i)
h(s) is defined for 0≤s<∞, h(s)>0,(ii)
there existsr>0 such that lims→∞srh(s)=0, monotonically for large s.The function h(s) is said to be of order r.For example,h(s)=(s+1)-p is an influence function for r<p and h(s)=e-βs; β>0 is an influence function of any order.LetH be a Hilbert space with inner product 〈·〉 and norm ∥·∥. In [45], Dafermos investigates solutions t∈(-∞,+∞)→u(t)∈H, on [0,∞), to the integrodifferential equation related to problems in viscoelasticity:
(108)ddt(ρu˙(t))+Cu(t)+∫-∞t‍G(t-τ)u(τ)dτ=0
with specified “history”
(109)u(τ)=v(τ),τ∈(-∞,0].
Here ρ, C, and G(t) are self-adjoint operators in H, ρ is bounded, and C and G(t) are unbounded with dense domains 𝒟(C) and 𝒟[G(t)], respectively. Moreover, they satisfy the following conditions:
(110)〈ρw,w〉≥ρo∥w∥2,∀w∈H,〈Cw,w〉≥c∥w∥2,∀w∈𝒟(C),〈G(t)w,w〉≤0,∀w∈𝒟[G(t)],t∈[0,∞),
with ρo>0, c>0. Let H2 [resp., H1] be the Hilbert space defined as 𝒟(C) [resp., 𝒟(C1/2)], equipped with the natural norms, and H-1 the dual of H1. The injection H2→H is assumed to be compact. More regularity assumptions on G(t) and G˙(t) imply the existence of a decreasing influence function h∈C0[0,∞) such that
(111)∫0∞‍[∥G(t)∥ℒ(H1,H-1)+∥G˙(t)∥ℒ(H1,H-1)]h-2(t)dt<∞.
In the following, h is a fixed influence function which satisfies (111). Set I=(-∞,0]. For k=0,1,…, in [45], the author introduces the Banach spaces of fading memory type (cf. [46]) 𝒞k [resp., ℬk] of sufficiently smooth functions w:I→H1 [resp., w:I→H2] such that
(112)∥w∥𝒞k=∑j=01‍∑i=0k+1-j‍supI[h(-t)∥w(i)(t)∥j]<∞[resp.,∥w∥ℬk=∑j=02‍∑i=0k+2-j‍supI∥w(i)(t)∥j<∞].
Then, ℬk⊂𝒞k+1⊂𝒞k, and the injection ℬk→𝒞k is compact. Let T>0 and set IT=(-∞,T]. If v∈𝒞k [resp. ℬk], he proves the existence of a unique solution u to (108) on [0,T] and (109) on (-∞,0]. Moreover, si v∈𝒞k [resp., ℬk], u satisfies
(113)∑j=01‍∑i=0k+1-j‍sup[0,T]∥u(i)(t)∥j≤c∥v∥𝒞k[resp.,∑j=02‍∑i=0k+2-j‍sup[0,T]∥u(i)(t)∥j≤c∥v∥ℬk].
The following stability results are obtained. Set
(114)A=C+∫0∞‍G(t)dt.
It is assumed that
(115)〈G.(t)w,w〉≥0∀w∈H1,t∈[0,∞),∫0∞‍∥G(t)∥ℒ(H2;H)dt≡M∞<1,〈Aw,w〉≥ao∥w∥12,ao>0,∀w∈H1.
The new function
(116)ω:(v,ξ)∈𝒞k×[0,∞)⟶ω[(v,ξ)](τ)000=u(ξ+τ)∈𝒞k,τ∈I,
where u is the solution of (108) on [0,∞) and (109) on (-∞,0], satisfying
(117)ω(v,0)=v∀v∈𝒞k,ω(v,ξ+ξ′)=ω[ω(v,ξ),ξ′],∀ξ,ξ′∈[0,∞),v∈𝒞k,
that is, ω looks like a dynamical system on 𝒞k (or ℬk), but it is not necessarily continuous on 𝒞k×[0,∞). Therefore, ω is not a classical dynamical system (cf. Section 6). Nevertheless, it is possible to define the following Lyapunov function Vk(v), for ω, on 𝒞k:
(118)Vk(v)(t)=12〈ρuk+1(t),uk+1(t)〉+12〈Auk(t),uk(t)〉-12∫-∞t‍〈G(t-τ)(uk(t)-uk(τ)),(uk(t)-uk(τ))uk(t)-uk(τ)〉dτ≥0
whose derivative, with respect to t, satisfies V.k(v)(t)≤0. Here u is solution of (108) and (109). Under suitable spectral conditions on G, via an adaptation of Hale’s results [21], the following stability results are obtained for the solution u of (108) on [0,∞) and of (109) on (-∞,0], when v∈𝒞m, m=0,1,…:
(119)u(i)(t)⟶0inH1ast⟶∞,i=0,1,…,m,um+1(t)⟶0inHast⟶∞.


### 11.2. Equations with Memory and Singular Perturbations

Example 50.
In [47], Gatti et al. consider a family of Banach spaces Hϵ and a family of semigroups Sϵ(t):Hϵ→Hϵ, ϵ∈[0,1]. They state sufficient conditions on Hϵ and Sϵ which imply the existence of exponential attractors ℰϵ for Sϵ(t), ϵ∈[0,1] such that
(120)dist-symHϵ(ℰϵ,ℰo)≤Cϵδ,dist-symHϵ2(ℰϵ1,ℰϵ2)≤C(ϵ1-ϵ2)δ,∀ϵ1≥ϵ2>0,
with C≥0 and δ∈(0,1). The results are applied to evolution equations with memory of the form
(121)∂tx(t)+∫0∞‍kϵ(s)Bo(x(t-s))ds+B1(x(t))=O,00000000000000.000000000000000000t∈[0,∞),
with specified history
(122)x(t)=v(t),t∈(-∞,0],
where v is a given function, k:[0,∞)→[0,∞) is a convex summable function with ∫0∞‍k(s)ds=1 and kϵ(s)=(1/ϵ)k(s/ϵ), and B0,B1 are densely defined, possibly nonlinear operators in a suitable Banach space. As ϵ→0, the formal limit to problem (121) and (122) is
(123)∂tx(t)+Bo(x(t))+B1(x(t))=0,x(0)=v(0).Example 51.
This example is already presented in [7, Example 4]. In [48], Conti et al. are interested in the convergence, in an appropriate sense, as ϵ→0, of the solution uϵ(x,t), x∈Ω, t∈ℝ to problem (pϵ):
(124)ut-ωΔu-(1-ω)∫0∞‍kϵ(s)Δu(t-s)ds0000+φ(u)=f,t>0,
with Dirichlet boundary conditions on the boundary of the smooth bounded domain Ω in ℝ3,  ω>0; φ is a suitable nonlinearity, f is a time-independent source term, the memory kernel kϵ converges to the Dirac distribution at the origin, u(x,t) is supposed to be a given datum for t≤0. Let A=-Δ on L2(Ω) with domain 𝒟(A)=Ho1(Ω)∩H2(Ω), Hr=𝒟(Ar/2), r∈ℝ. Following Dafermos [45] and Grasselli and Pata [49], they introduce, under additional assumptions on k,φ,f,(φ(x)=x3-x is allowed), the auxiliary variable ηt(x,s)=∫os‍u(x,t-y)dy, the functions μ(s)=-(1-ω)k′(s), μϵ(s)=(1/ϵ2)μ(s/ϵ), and the Hilbert spaces ℳϵr=Lμϵ2(ℝ+,Hr+1) and ℋϵr=Hr×ℳϵr for ϵ>0, ℋor=Hr. The correct reformulation of pϵ, in the framework of dynamical systems, is (Pϵ): find (uϵ,ηϵ)∈C([0,∞),ℋϵo) solution to
(125)ut+ωAu+∫o∞‍μϵ(s)Aη(s)ds+φ(u)=f,∂tη=-∂sη+u
for t>0, associated with the initial condition (uo,ηo)∈ℋϵo. The existence, for ϵ>0, of a strongly continuous semigroup Sϵ(t) on ℋϵo corresponding to Pϵ and of an exponential attractor ℰϵ for Sϵ(t) is proved. The convergence of ℰϵ is also studied.


## 11.1. Dafermos Results

Definition 49 (cf. Coleman and Noll [46]).
A functionh is called an influence function if(i)
h(s) is defined for 0≤s<∞, h(s)>0,(ii)
there existsr>0 such that lims→∞srh(s)=0, monotonically for large s.The function h(s) is said to be of order r.For example,h(s)=(s+1)-p is an influence function for r<p and h(s)=e-βs; β>0 is an influence function of any order.LetH be a Hilbert space with inner product 〈·〉 and norm ∥·∥. In [45], Dafermos investigates solutions t∈(-∞,+∞)→u(t)∈H, on [0,∞), to the integrodifferential equation related to problems in viscoelasticity:
(108)ddt(ρu˙(t))+Cu(t)+∫-∞t‍G(t-τ)u(τ)dτ=0
with specified “history”
(109)u(τ)=v(τ),τ∈(-∞,0].
Here ρ, C, and G(t) are self-adjoint operators in H, ρ is bounded, and C and G(t) are unbounded with dense domains 𝒟(C) and 𝒟[G(t)], respectively. Moreover, they satisfy the following conditions:
(110)〈ρw,w〉≥ρo∥w∥2,∀w∈H,〈Cw,w〉≥c∥w∥2,∀w∈𝒟(C),〈G(t)w,w〉≤0,∀w∈𝒟[G(t)],t∈[0,∞),
with ρo>0, c>0. Let H2 [resp., H1] be the Hilbert space defined as 𝒟(C) [resp., 𝒟(C1/2)], equipped with the natural norms, and H-1 the dual of H1. The injection H2→H is assumed to be compact. More regularity assumptions on G(t) and G˙(t) imply the existence of a decreasing influence function h∈C0[0,∞) such that
(111)∫0∞‍[∥G(t)∥ℒ(H1,H-1)+∥G˙(t)∥ℒ(H1,H-1)]h-2(t)dt<∞.
In the following, h is a fixed influence function which satisfies (111). Set I=(-∞,0]. For k=0,1,…, in [45], the author introduces the Banach spaces of fading memory type (cf. [46]) 𝒞k [resp., ℬk] of sufficiently smooth functions w:I→H1 [resp., w:I→H2] such that
(112)∥w∥𝒞k=∑j=01‍∑i=0k+1-j‍supI[h(-t)∥w(i)(t)∥j]<∞[resp.,∥w∥ℬk=∑j=02‍∑i=0k+2-j‍supI∥w(i)(t)∥j<∞].
Then, ℬk⊂𝒞k+1⊂𝒞k, and the injection ℬk→𝒞k is compact. Let T>0 and set IT=(-∞,T]. If v∈𝒞k [resp. ℬk], he proves the existence of a unique solution u to (108) on [0,T] and (109) on (-∞,0]. Moreover, si v∈𝒞k [resp., ℬk], u satisfies
(113)∑j=01‍∑i=0k+1-j‍sup[0,T]∥u(i)(t)∥j≤c∥v∥𝒞k[resp.,∑j=02‍∑i=0k+2-j‍sup[0,T]∥u(i)(t)∥j≤c∥v∥ℬk].
The following stability results are obtained. Set
(114)A=C+∫0∞‍G(t)dt.
It is assumed that
(115)〈G.(t)w,w〉≥0∀w∈H1,t∈[0,∞),∫0∞‍∥G(t)∥ℒ(H2;H)dt≡M∞<1,〈Aw,w〉≥ao∥w∥12,ao>0,∀w∈H1.
The new function
(116)ω:(v,ξ)∈𝒞k×[0,∞)⟶ω[(v,ξ)](τ)000=u(ξ+τ)∈𝒞k,τ∈I,
where u is the solution of (108) on [0,∞) and (109) on (-∞,0], satisfying
(117)ω(v,0)=v∀v∈𝒞k,ω(v,ξ+ξ′)=ω[ω(v,ξ),ξ′],∀ξ,ξ′∈[0,∞),v∈𝒞k,
that is, ω looks like a dynamical system on 𝒞k (or ℬk), but it is not necessarily continuous on 𝒞k×[0,∞). Therefore, ω is not a classical dynamical system (cf. Section 6). Nevertheless, it is possible to define the following Lyapunov function Vk(v), for ω, on 𝒞k:
(118)Vk(v)(t)=12〈ρuk+1(t),uk+1(t)〉+12〈Auk(t),uk(t)〉-12∫-∞t‍〈G(t-τ)(uk(t)-uk(τ)),(uk(t)-uk(τ))uk(t)-uk(τ)〉dτ≥0
whose derivative, with respect to t, satisfies V.k(v)(t)≤0. Here u is solution of (108) and (109). Under suitable spectral conditions on G, via an adaptation of Hale’s results [21], the following stability results are obtained for the solution u of (108) on [0,∞) and of (109) on (-∞,0], when v∈𝒞m, m=0,1,…:
(119)u(i)(t)⟶0inH1ast⟶∞,i=0,1,…,m,um+1(t)⟶0inHast⟶∞.


## 11.2. Equations with Memory and Singular Perturbations

Example 50.
In [47], Gatti et al. consider a family of Banach spaces Hϵ and a family of semigroups Sϵ(t):Hϵ→Hϵ, ϵ∈[0,1]. They state sufficient conditions on Hϵ and Sϵ which imply the existence of exponential attractors ℰϵ for Sϵ(t), ϵ∈[0,1] such that
(120)dist-symHϵ(ℰϵ,ℰo)≤Cϵδ,dist-symHϵ2(ℰϵ1,ℰϵ2)≤C(ϵ1-ϵ2)δ,∀ϵ1≥ϵ2>0,
with C≥0 and δ∈(0,1). The results are applied to evolution equations with memory of the form
(121)∂tx(t)+∫0∞‍kϵ(s)Bo(x(t-s))ds+B1(x(t))=O,00000000000000.000000000000000000t∈[0,∞),
with specified history
(122)x(t)=v(t),t∈(-∞,0],
where v is a given function, k:[0,∞)→[0,∞) is a convex summable function with ∫0∞‍k(s)ds=1 and kϵ(s)=(1/ϵ)k(s/ϵ), and B0,B1 are densely defined, possibly nonlinear operators in a suitable Banach space. As ϵ→0, the formal limit to problem (121) and (122) is
(123)∂tx(t)+Bo(x(t))+B1(x(t))=0,x(0)=v(0).Example 51.
This example is already presented in [7, Example 4]. In [48], Conti et al. are interested in the convergence, in an appropriate sense, as ϵ→0, of the solution uϵ(x,t), x∈Ω, t∈ℝ to problem (pϵ):
(124)ut-ωΔu-(1-ω)∫0∞‍kϵ(s)Δu(t-s)ds0000+φ(u)=f,t>0,
with Dirichlet boundary conditions on the boundary of the smooth bounded domain Ω in ℝ3,  ω>0; φ is a suitable nonlinearity, f is a time-independent source term, the memory kernel kϵ converges to the Dirac distribution at the origin, u(x,t) is supposed to be a given datum for t≤0. Let A=-Δ on L2(Ω) with domain 𝒟(A)=Ho1(Ω)∩H2(Ω), Hr=𝒟(Ar/2), r∈ℝ. Following Dafermos [45] and Grasselli and Pata [49], they introduce, under additional assumptions on k,φ,f,(φ(x)=x3-x is allowed), the auxiliary variable ηt(x,s)=∫os‍u(x,t-y)dy, the functions μ(s)=-(1-ω)k′(s), μϵ(s)=(1/ϵ2)μ(s/ϵ), and the Hilbert spaces ℳϵr=Lμϵ2(ℝ+,Hr+1) and ℋϵr=Hr×ℳϵr for ϵ>0, ℋor=Hr. The correct reformulation of pϵ, in the framework of dynamical systems, is (Pϵ): find (uϵ,ηϵ)∈C([0,∞),ℋϵo) solution to
(125)ut+ωAu+∫o∞‍μϵ(s)Aη(s)ds+φ(u)=f,∂tη=-∂sη+u
for t>0, associated with the initial condition (uo,ηo)∈ℋϵo. The existence, for ϵ>0, of a strongly continuous semigroup Sϵ(t) on ℋϵo corresponding to Pϵ and of an exponential attractor ℰϵ for Sϵ(t) is proved. The convergence of ℰϵ is also studied.


## 12. Nodes and Nodal

### 12.1. Nodal Points, Curves, Surfaces, or Nodes

Definition 52.
In Courant and Hilbert [50], they are defined as points [resp., curves, surfaces], on which some eigenfunctions of a differential problem, in a domain G∈ℝN, N=1 [resp., N=2,3], vanish.Example 53.
In the 1970 edition of [50, page 452], a second-order Sturm-Liouville problem is considered. It is proved that the nodes of the nth eigenfunction un divide the domain into no more than n subdomains.


### 12.2. Nodal Sets, Nodal Domains, and Nodal Solutions

Example 54.
In [51], Hoffmann-Ostenhof et al. study the Dirichlet eigenvalue problem
(126)-Δui=λiui,i=1,2,…,u∈Ho1(D),
where D is a bounded domain in ℝn. The nodal setui is defined as
(127)𝒩(ui)={x∈D:ui(x)=0}¯.
The nodal domains of ui are the connected components of D∖𝒩(ui). The authors construct a domain D⊂ℝ2 on which the second eigenvalue has a nodal set disjoint from the boundary, whereas the nodal line conjecture, first mentioned by Payne [52], stated that
(128)𝒩(u2)∩∂D≠∅.Example 55.
In [53], Bartsch and Weth consider the nonlinear elliptic Dirichlet problem:
(129)-ϵ2Δu+au=f(u)inΩ,u∈Ho1(Ω),
where Ω is a bounded domain in ℝN, N≥2, a>0, ϵ is a small positive parameter, and f grows superlinearly and subcritically. They study the number of nodal solutions, that is, sign-changing solutions of (129) and their nodal domains. They show that the number of nodal solutions can be expressed as a Lyusternik-Schnirelman category (cf. Lyusternik and Schnirelman [54]), of a suitable inclusion between two spaces which involve the shape of ∂Ω.


### 12.3. N-Mode Solution

In [55], Urano et al. consider the following ordinary differential problem:
(130)ϵ2u′′(x)+f(x,u(x))=0,x∈]0,1[;u′(0)=u′(1)=0,
where f(x,u)=u(1-u)(u-a(x)); a is a C2[0,1] function such that 0<a(x)<1, a′(0)=a′(1)=0, and the subsets Σ and Λ of (0,1), where a(x)=1/2 and a′(x)=0, respectively, are finite sets and a′(x)≠0 for any x∈Σ. The authors study the asymptotic behavior, as ϵ→0, of N-mode solutions uϵ of (130); that is, solutions such that vϵ=uϵ-a have exactly N zero-points in (0,1), and show that any transition layer [resp., spike] is located in a neighborhood of a point of Σ [resp., Λ].


## 12.1. Nodal Points, Curves, Surfaces, or Nodes

Definition 52.
In Courant and Hilbert [50], they are defined as points [resp., curves, surfaces], on which some eigenfunctions of a differential problem, in a domain G∈ℝN, N=1 [resp., N=2,3], vanish.Example 53.
In the 1970 edition of [50, page 452], a second-order Sturm-Liouville problem is considered. It is proved that the nodes of the nth eigenfunction un divide the domain into no more than n subdomains.


## 12.2. Nodal Sets, Nodal Domains, and Nodal Solutions

Example 54.
In [51], Hoffmann-Ostenhof et al. study the Dirichlet eigenvalue problem
(126)-Δui=λiui,i=1,2,…,u∈Ho1(D),
where D is a bounded domain in ℝn. The nodal setui is defined as
(127)𝒩(ui)={x∈D:ui(x)=0}¯.
The nodal domains of ui are the connected components of D∖𝒩(ui). The authors construct a domain D⊂ℝ2 on which the second eigenvalue has a nodal set disjoint from the boundary, whereas the nodal line conjecture, first mentioned by Payne [52], stated that
(128)𝒩(u2)∩∂D≠∅.Example 55.
In [53], Bartsch and Weth consider the nonlinear elliptic Dirichlet problem:
(129)-ϵ2Δu+au=f(u)inΩ,u∈Ho1(Ω),
where Ω is a bounded domain in ℝN, N≥2, a>0, ϵ is a small positive parameter, and f grows superlinearly and subcritically. They study the number of nodal solutions, that is, sign-changing solutions of (129) and their nodal domains. They show that the number of nodal solutions can be expressed as a Lyusternik-Schnirelman category (cf. Lyusternik and Schnirelman [54]), of a suitable inclusion between two spaces which involve the shape of ∂Ω.


## 12.3. N-Mode Solution

In [55], Urano et al. consider the following ordinary differential problem:
(130)ϵ2u′′(x)+f(x,u(x))=0,x∈]0,1[;u′(0)=u′(1)=0,
where f(x,u)=u(1-u)(u-a(x)); a is a C2[0,1] function such that 0<a(x)<1, a′(0)=a′(1)=0, and the subsets Σ and Λ of (0,1), where a(x)=1/2 and a′(x)=0, respectively, are finite sets and a′(x)≠0 for any x∈Σ. The authors study the asymptotic behavior, as ϵ→0, of N-mode solutions uϵ of (130); that is, solutions such that vϵ=uϵ-a have exactly N zero-points in (0,1), and show that any transition layer [resp., spike] is located in a neighborhood of a point of Σ [resp., Λ].


## 13. Resonances

Only a few aspects of this large topic are presented here. More aspects are in preparation.


### 13.1. The Case of a Vibrating String

#### 13.1.1. Free Motion (cf. Schwartz [56])

Consider a homogeneous vibrating string, with linear densityρ, subjected to a constant tension μ. The position of the string, fixed at the end points x=0, x=L, is given by solutions of the wave equation
(131)1v2∂2u∂t2-∂2u∂x2=0,xin[0,L]
with the boundary conditions
(132)u(t,0)=u(t,L)=0,
where v=μ/ρ has the dimension of a velocity. To solve problem (131) and (132) the initial values of the position and of the velocity of the string are needed; that is,
(133)u(0,x)=uo(x),∂u∂t(0,x)=u1(x).
Solutions u(x,t) of the above initial value problem of the form u(x,t)=U(x)V(t) are
(134)uk(x,t)=sinkπxL(AkcoskπvtL+BksinkπvtL),
where k>0 is an integer and Ak,Bk are constants. The numbers λ=k2π2/L2, such that
(135)U′′(x)+λU(x)=0,U(0)=U(L)=0,
are called the eigenvalues and sin(kπx/L) the eigenfunctions of the problem. The constants Ak,Bk are the Fourier coefficients of suitable extensions of uo and u1 to periodic functions.


#### 13.1.2. The Case of a Forced Motion: Resonance (cf. Courant and Hilbert [50])

We suppose that the vibrating string is under the influence of an external force which has the Fourier expansion:(136)𝒬(x,t)=∑k=1∞‍Qk(t)sinkπxL.
Then, the deflection of the string is solution to
(137)1v2∂2u∂t2-∂2u∂x2=𝒬(x,t),u(0,t)=u(L,t)=0
with the initial conditions (133). We look for a solution of the form
(138)u(x,t)=∑k=1∞‍Nk(t)sinkπxL.
The coefficients Nk(t) must be solutions of the equations
(139)1v2Nk′′(t)+k2π2L2Nk(t)=Qk(t).
The general solution of the homogeneous equation associated with (139) is
(140)akcoskπvtL+bksinkπvtL.
A particular solution of (139) is obtained by the method of variation of constants (cf. Coddington [57]). Finally,
(141)Nk(t)=1(kπ/L)∫0t‍Qk(u)sinkπL(t-u)du+akcoskπvtL+bksinkπvtL,
where the constants ak,bk are determined by the initial conditions (133).Now, suppose𝒬k(t)=αcosωt+βsinωt. Then, if ω≠kπ/L, u(x,t), in (138), is a linear combination of a sinusoidal function of frequency ω and one of frequencies kπ/L. But, if ω=kπ/L, Nk(t) contains terms of the forms tsint and tcost which are unbounded, we say that resonance occurs.


### 13.2. Resonances as Poles of Complex Functions (cf. Zworski [58])

SetX=[0,L] with L=π; for simplicity, PV=-(∂2/∂x2)+V(x), V∈C∞(X), V(x)≥0, and
(142)U(x,t)=(u(t,x)-i∂tu(t,x)),𝒫V=(01PV0).
Consider the system
(143)1i∂tU=𝒫VU,U(t,0)=U(t,π)=0,U(0,x)=(uo(x)-iu1(x)).
The solution to system (143) involves eigenvalues and eigenfunctions of
(144)𝒫VW=λW,W|∂X=0.
Consider the Hilbert space ℋ, closure of Cc∞(X)×Cc∞(X) for the inner product
(145)〈Φ,Ψ〉=〈PVϕ1,ψ1〉+〈ϕ2,ψ2〉,Φ=(ϕ1ϕ2),Ψ=(ψ1ψ2).
Then 𝒫V is an unbounded self-adjoint operator on ℋ and its eigenvalues are real. Let σ(𝒫V) be the spectrum of 𝒫V. It is pointed out, in [58], that
(146)λ∈σ(𝒫V)⟺λisapoleof(𝒫V-z)-1:ℋ⟶ℋ.Remark 56.
WhenV(x)=0, system (143) is equivalent to problem (131)–(133), and the eigenvalues of
(147)𝒫0W=λW,W|∂X=0
are λ=k∈ℤ-{0} which correspond to the eigenvalues k2 of problem (135).More generally, consider the problem with a dissipative terma∂t,a>0:
(148)(-∂2∂t2+a∂∂t-PV)u(t,x)=0,t>0,x∈X,u|∂X=0.
It can be rewritten as
(149)1i∂tU(t,x)=𝒫aU(t,x),u|∂X=0,
where U is defined in (142) and
(150)𝒫a=(01PVia).
The operator 𝒫a is not self-adjoint on ℋ. The eigenvalues and the eigenfunctions of 𝒫a are still defined by
(151)𝒫aW=λW,W|∂X=0,
and the solutions of (149) are given by superpositions of solutions of the form
(152)U(t,x)=exp(-itλ)W(x),
but the eigenvalues λ=α+iβ are not real any more. An elementary calculus shows that λ(λ-ia)=μ2, where μ∈ℝ is an eigenvalue of PV; that is, β=a/2, α2=μ2-(a2/4). Here, again, the eigenvalues of 𝒫a are poles of (𝒫a-z)-1:ℋ→ℋ. Now, since α∉σ(𝒫a), for a given f, the equation
(153)(𝒫a-α)u=f
has a solution. But, if a is small enough, λ=α+(ia/2) is a pole of (𝒫a-z)-1 very close to α, and, for a suitable choice of f, the solution of (153) can be “enormous” (cf. [58]). We say that a resonance occurs.


### 13.3. Quasi-Periodic Schrödinger Operator, the Resonant Case

In their papers [59–61] Fedotov and Klopp study the spectrum of quasi-periodic Schrödinger operators of the form
(154)Hz,ϵ=-d2dx2+V(x-z)+αcos(ϵx),
where α>0, V∈Lloc2(ℝ) is periodic with period 1, z is a real parameter indexing the equations of the family, and ϵ>0 is such that 2π/ϵ is irrational. Let
(155)Ho=-d2dx2+V(x)
be the periodic Schrödinger unperturbed operator. Its spectrum, on L2(ℝ), consists of the union of intervals [E2n-1,E2n], n≥1, such that En→∞, as n→∞. It is assumed that all the spectral gaps (E2n,E2n+1) are open. The spectrum of (154), at an energy E, depends on the relative position of the spectral window ℱ(E)=[E-α,E+α] with respect to the spectrum of Ho. In [59] four cases are distinguished as ℱ(E) lies inside or as it completely covers a spectral band of Ho or it contains exactly one edge or two edges (Figure 4) of a spectral band.Figure 4
Spectral band.In each case, under specific additional assumptions, the authors describe the nature of the spectrum of (154) and state asymptotic formulas, as ϵ→0.In [60], they are interested in the spectrum of (154), in intervals J such that, for all E∈J, ℱ(E) covers the edges of two neighboring spectral bands of Ho and the spectral gap located between them (Figure 4). Let ΓR be the real iso-energy curve associated with (154) and γ0,γπ the connected components of ΓR in a periodicity cell. To each of these loops, one associates a sequence of energies in J, Ean, a={0,π}, and, near each Ean, an exponentially small interval Ian such that the spectrum of (154), in J, is contained in the union of these intervals. The location and the nature of the spectrum of (154) are investigated in the union I0n∪Iπm, in the resonant case, that is, when I0n and Iπm intersect with each other.


## 13.1. The Case of a Vibrating String

### 13.1.1. Free Motion (cf. Schwartz [56])

Consider a homogeneous vibrating string, with linear densityρ, subjected to a constant tension μ. The position of the string, fixed at the end points x=0, x=L, is given by solutions of the wave equation
(131)1v2∂2u∂t2-∂2u∂x2=0,xin[0,L]
with the boundary conditions
(132)u(t,0)=u(t,L)=0,
where v=μ/ρ has the dimension of a velocity. To solve problem (131) and (132) the initial values of the position and of the velocity of the string are needed; that is,
(133)u(0,x)=uo(x),∂u∂t(0,x)=u1(x).
Solutions u(x,t) of the above initial value problem of the form u(x,t)=U(x)V(t) are
(134)uk(x,t)=sinkπxL(AkcoskπvtL+BksinkπvtL),
where k>0 is an integer and Ak,Bk are constants. The numbers λ=k2π2/L2, such that
(135)U′′(x)+λU(x)=0,U(0)=U(L)=0,
are called the eigenvalues and sin(kπx/L) the eigenfunctions of the problem. The constants Ak,Bk are the Fourier coefficients of suitable extensions of uo and u1 to periodic functions.


### 13.1.2. The Case of a Forced Motion: Resonance (cf. Courant and Hilbert [50])

We suppose that the vibrating string is under the influence of an external force which has the Fourier expansion:(136)𝒬(x,t)=∑k=1∞‍Qk(t)sinkπxL.
Then, the deflection of the string is solution to
(137)1v2∂2u∂t2-∂2u∂x2=𝒬(x,t),u(0,t)=u(L,t)=0
with the initial conditions (133). We look for a solution of the form
(138)u(x,t)=∑k=1∞‍Nk(t)sinkπxL.
The coefficients Nk(t) must be solutions of the equations
(139)1v2Nk′′(t)+k2π2L2Nk(t)=Qk(t).
The general solution of the homogeneous equation associated with (139) is
(140)akcoskπvtL+bksinkπvtL.
A particular solution of (139) is obtained by the method of variation of constants (cf. Coddington [57]). Finally,
(141)Nk(t)=1(kπ/L)∫0t‍Qk(u)sinkπL(t-u)du+akcoskπvtL+bksinkπvtL,
where the constants ak,bk are determined by the initial conditions (133).Now, suppose𝒬k(t)=αcosωt+βsinωt. Then, if ω≠kπ/L, u(x,t), in (138), is a linear combination of a sinusoidal function of frequency ω and one of frequencies kπ/L. But, if ω=kπ/L, Nk(t) contains terms of the forms tsint and tcost which are unbounded, we say that resonance occurs.


## 13.1.1. Free Motion (cf. Schwartz [56])

Consider a homogeneous vibrating string, with linear densityρ, subjected to a constant tension μ. The position of the string, fixed at the end points x=0, x=L, is given by solutions of the wave equation
(131)1v2∂2u∂t2-∂2u∂x2=0,xin[0,L]
with the boundary conditions
(132)u(t,0)=u(t,L)=0,
where v=μ/ρ has the dimension of a velocity. To solve problem (131) and (132) the initial values of the position and of the velocity of the string are needed; that is,
(133)u(0,x)=uo(x),∂u∂t(0,x)=u1(x).
Solutions u(x,t) of the above initial value problem of the form u(x,t)=U(x)V(t) are
(134)uk(x,t)=sinkπxL(AkcoskπvtL+BksinkπvtL),
where k>0 is an integer and Ak,Bk are constants. The numbers λ=k2π2/L2, such that
(135)U′′(x)+λU(x)=0,U(0)=U(L)=0,
are called the eigenvalues and sin(kπx/L) the eigenfunctions of the problem. The constants Ak,Bk are the Fourier coefficients of suitable extensions of uo and u1 to periodic functions.


## 13.1.2. The Case of a Forced Motion: Resonance (cf. Courant and Hilbert [50])

We suppose that the vibrating string is under the influence of an external force which has the Fourier expansion:(136)𝒬(x,t)=∑k=1∞‍Qk(t)sinkπxL.
Then, the deflection of the string is solution to
(137)1v2∂2u∂t2-∂2u∂x2=𝒬(x,t),u(0,t)=u(L,t)=0
with the initial conditions (133). We look for a solution of the form
(138)u(x,t)=∑k=1∞‍Nk(t)sinkπxL.
The coefficients Nk(t) must be solutions of the equations
(139)1v2Nk′′(t)+k2π2L2Nk(t)=Qk(t).
The general solution of the homogeneous equation associated with (139) is
(140)akcoskπvtL+bksinkπvtL.
A particular solution of (139) is obtained by the method of variation of constants (cf. Coddington [57]). Finally,
(141)Nk(t)=1(kπ/L)∫0t‍Qk(u)sinkπL(t-u)du+akcoskπvtL+bksinkπvtL,
where the constants ak,bk are determined by the initial conditions (133).Now, suppose𝒬k(t)=αcosωt+βsinωt. Then, if ω≠kπ/L, u(x,t), in (138), is a linear combination of a sinusoidal function of frequency ω and one of frequencies kπ/L. But, if ω=kπ/L, Nk(t) contains terms of the forms tsint and tcost which are unbounded, we say that resonance occurs.


## 13.2. Resonances as Poles of Complex Functions (cf. Zworski [58])

SetX=[0,L] with L=π; for simplicity, PV=-(∂2/∂x2)+V(x), V∈C∞(X), V(x)≥0, and
(142)U(x,t)=(u(t,x)-i∂tu(t,x)),𝒫V=(01PV0).
Consider the system
(143)1i∂tU=𝒫VU,U(t,0)=U(t,π)=0,U(0,x)=(uo(x)-iu1(x)).
The solution to system (143) involves eigenvalues and eigenfunctions of
(144)𝒫VW=λW,W|∂X=0.
Consider the Hilbert space ℋ, closure of Cc∞(X)×Cc∞(X) for the inner product
(145)〈Φ,Ψ〉=〈PVϕ1,ψ1〉+〈ϕ2,ψ2〉,Φ=(ϕ1ϕ2),Ψ=(ψ1ψ2).
Then 𝒫V is an unbounded self-adjoint operator on ℋ and its eigenvalues are real. Let σ(𝒫V) be the spectrum of 𝒫V. It is pointed out, in [58], that
(146)λ∈σ(𝒫V)⟺λisapoleof(𝒫V-z)-1:ℋ⟶ℋ.Remark 56.
WhenV(x)=0, system (143) is equivalent to problem (131)–(133), and the eigenvalues of
(147)𝒫0W=λW,W|∂X=0
are λ=k∈ℤ-{0} which correspond to the eigenvalues k2 of problem (135).More generally, consider the problem with a dissipative terma∂t,a>0:
(148)(-∂2∂t2+a∂∂t-PV)u(t,x)=0,t>0,x∈X,u|∂X=0.
It can be rewritten as
(149)1i∂tU(t,x)=𝒫aU(t,x),u|∂X=0,
where U is defined in (142) and
(150)𝒫a=(01PVia).
The operator 𝒫a is not self-adjoint on ℋ. The eigenvalues and the eigenfunctions of 𝒫a are still defined by
(151)𝒫aW=λW,W|∂X=0,
and the solutions of (149) are given by superpositions of solutions of the form
(152)U(t,x)=exp(-itλ)W(x),
but the eigenvalues λ=α+iβ are not real any more. An elementary calculus shows that λ(λ-ia)=μ2, where μ∈ℝ is an eigenvalue of PV; that is, β=a/2, α2=μ2-(a2/4). Here, again, the eigenvalues of 𝒫a are poles of (𝒫a-z)-1:ℋ→ℋ. Now, since α∉σ(𝒫a), for a given f, the equation
(153)(𝒫a-α)u=f
has a solution. But, if a is small enough, λ=α+(ia/2) is a pole of (𝒫a-z)-1 very close to α, and, for a suitable choice of f, the solution of (153) can be “enormous” (cf. [58]). We say that a resonance occurs.


## 13.3. Quasi-Periodic Schrödinger Operator, the Resonant Case

In their papers [59–61] Fedotov and Klopp study the spectrum of quasi-periodic Schrödinger operators of the form
(154)Hz,ϵ=-d2dx2+V(x-z)+αcos(ϵx),
where α>0, V∈Lloc2(ℝ) is periodic with period 1, z is a real parameter indexing the equations of the family, and ϵ>0 is such that 2π/ϵ is irrational. Let
(155)Ho=-d2dx2+V(x)
be the periodic Schrödinger unperturbed operator. Its spectrum, on L2(ℝ), consists of the union of intervals [E2n-1,E2n], n≥1, such that En→∞, as n→∞. It is assumed that all the spectral gaps (E2n,E2n+1) are open. The spectrum of (154), at an energy E, depends on the relative position of the spectral window ℱ(E)=[E-α,E+α] with respect to the spectrum of Ho. In [59] four cases are distinguished as ℱ(E) lies inside or as it completely covers a spectral band of Ho or it contains exactly one edge or two edges (Figure 4) of a spectral band.Figure 4
Spectral band.In each case, under specific additional assumptions, the authors describe the nature of the spectrum of (154) and state asymptotic formulas, as ϵ→0.In [60], they are interested in the spectrum of (154), in intervals J such that, for all E∈J, ℱ(E) covers the edges of two neighboring spectral bands of Ho and the spectral gap located between them (Figure 4). Let ΓR be the real iso-energy curve associated with (154) and γ0,γπ the connected components of ΓR in a periodicity cell. To each of these loops, one associates a sequence of energies in J, Ean, a={0,π}, and, near each Ean, an exponentially small interval Ian such that the spectrum of (154), in J, is contained in the union of these intervals. The location and the nature of the spectrum of (154) are investigated in the union I0n∪Iπm, in the resonant case, that is, when I0n and Iπm intersect with each other.



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*Source: 562963-2013-12-18.xml*