non linear partial differential equations with discrete state dependent delays in a metric space

Contents lists available atScienceDirectNonlinear Analysis journal homepage:www.elsevier.com/locate/na Non-linear partial differential equations with discrete state-dependent delays in a

Contents lists available atScienceDirect

Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Non-linear partial differential equations with discrete state-dependent delays in a metric space

Alexander V Rezounenko

Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody sqr., Kharkov, 61077, Ukraine

a r t i c l e i n f o

Article history:

Received 29 November 2008

Accepted 1 May 2010

MSC:

35R10

35B41

35K57

Keywords:

Partial functional differential equation

State-dependent delay

Well-posedness

Global attractor

a b s t r a c t

We investigate a class of non-linear partial differential equations with discrete state-dependent delays The existence and uniqueness of strong solutions for initial functions from a Banach space are proved To get the well-posed initial value problem we restrict our study to a smaller metric space, construct the dynamical system and prove the existence

of a compact global attractor

© 2010 Elsevier Ltd All rights reserved

1 Introduction

The theory of dynamical systems is a theory which describes qualitative properties of systems, changing in time One

of the oldest branches of this theory is the theory of delay differential equations We refer the reader to some classical monographs on the theory of ordinary delay equations (O.D.E.s) [1–3] A characteristic feature of any type of delay equations

is that they generate infinite dimensional dynamical systems The theory of partial delay equations (P.D.E.s) is essentially less studied since such equations are simultaneously infinite dimensional in both time (as delay equations) and space (as P.D.E.s) variables, which makes the analysis more difficult We refer the reader to some works which are close to the present research [4–7] and to the monograph [8

Recently, much attention was paid to the investigation of a new class of delay equations — equations with a state-dependent delay (SDD) (see e.g [9–16] and also the survey paper [17] for details and references) The study of these equations essentially differs from the ones of equations with constant or time-dependent delays The main difficulty is that non-linearities with SDDs are not Lipschitz continuous on the space of continuous functions — the main space of initial data, where the classical theory of delay equations is developed (see the references above) As a result, the corresponding initial

value problem (IVP) is, in general, not well-posed (in the sense of Hadamard [18,19]) An explicit example of the

non-uniqueness of solutions to an ordinary equation with state-dependent delay (SDD) is given in [20] (see also [17, p.465])

As noticed in [17, p.465] ‘‘typically, the IVP is uniquely solved for initial and other data which satisfy suitable Lipschitz conditions.’’

First attempts to study P.D.E.s with SDDs have been made for delays of different types: for a distributed delay problem

in [21,22] (see also [23]) and for discrete SDDs in [24] (mild solutions, infinite delay) as well as in [22] (weak solutions, finite delay)

The following property of solutions of P.D.E.s (with or without delays) is very important for the study of equations with

a discrete state-dependent delay Considering any type of solution (weak, mild, strong or classical) and having the property

E-mail addresses:rezounenko@univer.kharkov.ua , rezounenko@yahoo.com

0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved.

doi:10.1016/j.na.2010.05.005

u∈C([a,b];X), one cannot, in general, guarantee that the solution is a Lipschitz function u: [a,b] →X This fact brings

about essential difficulties for the extension of the methods developed for O.D.E.s (see the discussion above) That is why

in previous investigations we proposed alternative approaches, i.e approximations of a solution of a P.D.E with a discrete SDD by a sequence of solutions of P.D.E.s with distributed SDDs [21,22], or used an ‘‘ignoring condition’’ for a discrete SDD function [25]

The main goal of the present work is to make a step in extending the approach used for O.D.E.s with SDDs [9,10,17] to the

case of P.D.E.s Our idea is to look for a wider space Y ⊃X such that a solution u: [a,b] →Y is a Lipschitz function (with

respect to the weaker norm of Y ) and to construct a dynamical system on a subset of the space C([a,b];Y) It is interesting

to note that, in contrast to the case for previous investigations, the dynamical system is constructed on a metric space which

is not a linear space

The article is organized as follows Section2contains the formulation of the model and the proof of the existence and uniqueness of strong solutions for initial functions from a Banach space In Section3we construct an evolution operator S t

and study its asymptotic properties Here we restrict the evolution operator to a smaller metric space to get the continuity

of S t, which is not available in the initial Banach space Section4deals with the particular case when the delay time is state independent Here we also compare the results with the state-dependent case See the preprint [26]

2 Formulation of the model and basic properties

Our goal is to present an approach to studying the following partial differential equation with state-dependent discrete delay:

∂

∂t u(t,x) +Au(t,x) +du(t,x) =b([Bu(t− η(u t), ·)]( x)) ≡ (F(u t))( x), x∈Ω, (1)

where A is a densely defined self-adjoint positive linear operator with domain D(A) ⊂L2(Ω)and with compact resolvent,

so A : D(A) → L2(Ω)generates an analytic semigroup,Ω is a smooth bounded domain in R n0, B : L2(Ω) →L2(Ω)is a

bounded operator to be specified later, b:R→R is a locally Lipschitz map and d is a non-negative constant The function

η(·) :C([−r,0];L2(Ω)) → [0,r] ⊂R+represents the state-dependent discrete delay We define C ≡C([−r,0];L2(Ω)),

for short The norms in L2(Ω)and C are denoted byk · kandk · kCrespectively Byh· , ·iwe denote the inner product in

L2(Ω) As usual for delay equations, we denote by u tthe function ofθ ∈ [−r,0]given by the formula u t≡u t(θ) ≡u(t+ θ).

Remark 1 For example, the operator B may be of the following forms (linear examples):

[Bv](x) ≡ Z

or, even simpler,

[Bv](x) ≡ Z

where f :Ω−Ω →R is a smooth function,` ∈C∞

0 (Ω) In the last case the non-linear term in(1)takes the form

(F(u t))( x) ≡b

Z

Ω

u(t− η(u t),y)f(x−y)`(y)dy



We consider Eq.(1)with the following initial conditions:

Main assumptions:

(H.B) We will need the following Lipschitz property of the operator B:

∃L B>0: ∀u, v ∈L2(Ω) ⇒ kBu−Bvk ≤L B· kA−1/ 2(u− v)k. (6) (H.η) The discrete delay functionη :C→ [0,r]satisfies

∃Lη>0: ∀ ϕ, ψ ∈C ⇒ | η(ϕ) − η(ψ)| ≤Lη· max

θ∈[−r, 0 ]

Remark 2 For the term of the form(3), assuming that for all (almost all) x ∈ Ω ⇒ f(· − x)`(·) ∈ D(A1 / 2)and

u∈L2(Ω) ⊂D(A−1/ 2)one gets|hu,f(· −x)`(·)i| ≤ kA−1/ 2uk · kA1/ 2f(· −x)`(·)kwhich implies

Z

Ω

Z

Ω

u(y)f(y−x)`(y)dy

2

dx

!1 / 2

≤ kA−1/ 2uk ·

Z

Ω

kA1/ 2f(· −x)`(·)k2dx

1 / 2

.

Hence, property (H.B) (see(6)) holds with LB≡ RΩkA1/2f(· −x)`(·)k2dx
1/ 2

.

The same arguments hold (with L B≡ RΩkA1/2ef(x, ·)k2dx
1/ 2

) for a more general term of the form(2)

Now we introduce the following:

Definition 1 A vector-function u(t) ∈ C([−r,T];D(A− 1 / 2)) ∩C([0,T];D(A1 / 2)) ∩L2(0,T;D(A))with derivativeu˙ (t) ∈

L∞(0,T;D(A− 1 / 2))is a strong solution of problem(1),(5)on an interval[0,T]if:

• u(θ) = ϕ(θ)forθ ∈ [−r,0];

• for any functionv ∈L2(0,T;L2(Ω))such thatv ∈ ˙ L2(0,T;D(A−1))andv(T) =0, one has

−

Z T

0

hu(t), ˙v(t)idt+

Z T

0

hA1/ 2u(t),A1/ 2v(t)idt= h ϕ(0), v(0)i +

Z T

0

hF(u t), v( t)idt. (8) Let us introduce the following space:

L≡



ϕ ∈C([−r,0];D(A−1/ 2))|sup

s6=t

 kA− 1 / 2(ϕ(s) − ϕ(t))k

|s−t|



< +∞; ϕ(0) ∈D(A1/ 2)



(9) with the natural norm

k ϕkL≡ max

s∈[−r, 0 ]

kA−1/ 2ϕ(s)k +sup

s6=t

 kA− 1 / 2(ϕ(s) − ϕ(t))k

|s−t|



Now we prove the following theorem on the existence and uniqueness of solutions

Theorem 1 Let assumptions(H.B)and(H.η)hold (see(6),(7)) Assume that the function b : R → R is locally Lipschitz and bounded (b(·) ≤M b ).

Then for any initial functionϕ ∈L(the spaceLis defined in(9)) the problem(1),(5)has a unique strong solution on any time interval[0,T] The solution has the property u˙ ∈L2(0,T;L2(Ω)).

Remark 3 Let us notice that we do not assume thatϕ ∈ L2(−r,0;D(A)), but the definition of a strong solution above implies that

Proof of Theorem 1 Let us denote by{e k}∞k=1an orthonormal basis of L2(Ω)such that Ae k= λke k, 0< λ1 < · · · < λk → +∞

Consider Galerkin approximate solutions of order m: u m=u m(t,x) = Pm

k= 1g k,m( t)e k,such that

h˙u m+Au m+du m−F(u m t),e ki =0,

∀k=1, ,m Here g k,m∈C1(0,T;R) ∩L2(−r,T;R), withg˙k,m( t)being absolutely continuous

The system(12)is an (ordinary) differential equation in R mwith a concentrated state-dependent delay for the unknown

vector function U(t) ≡ (g1,m( t), ,g m,m( t))(the corresponding theory is developed in [10,11]; see also the recent review [17])

The key difference between equations with state-dependent and state-independent delays is that equations of the first type are not well-posed in the space of continuous (initial) functions For getting the well-posed initial value problem, the theory [10,11,17] suggests restricting considerations to a smaller space of Lipschitz continuous functions or even to a smaller

subspace of C1([−r,0];R m).

Conditionϕ ∈Limplies that initial data U(·)|[−r, 0 ] ≡P mϕ(·)are Lipschitz continuous as a function from[−r,0]to R m

Here P m is the orthogonal projection onto the subspace span{e1, ,e m} ⊂L2(Ω) Hence we can apply the theory of O.D.E.s with state-dependent delay (see e.g [17]) to get the local existence and uniqueness of solutions of(12)

Now we look for an a priori estimate to prove the continuation of solutions u mof(12)on any time interval[0,T]and then use it for the proof (by the method of compactness; see [27]) of the existence of strong solutions to(1),(5)

We multiply the first equation in(12)byλkg k,m and sum for k=1, ,m to get

1

2

d

dtkA

1 / 2

u m(t)k2+ kAu m(t)k2+d· kA1/ 2

u m(t)k2= hP m F(u m t),Au m(t)i ≤ 1

2kP m F(u m t )k2+1

2kAu

m(t)k2.

Since the function b is bounded (b(·) ≤M b), we havekF(u m

t)k2≤M2|Ω|(here|Ω| ≡ R

Ω1 dx) and, as a result, we conclude

that

d

dtkA

We integrate(13)with respect to t, and use the propertiesϕ(0) ∈D(A1/ 2),u m(0) =P mϕ(0) ∈D(A1/ 2)andkA1/ 2u m(0)k =

kA1 / 2P mϕ(0)k ≤ kA1 / 2ϕ(0)kto get an a priori estimate:

kA1/ 2

u m(t)k2+

Z t

0

kAu m(τ)k2

dτ ≤ kA1/ 2ϕ(0)k2+M b2|Ω| ·T, ∀m, ∀t∈ [0,T] (14) Estimate(14)means that

{u m}∞m=1is a bounded set in L∞(0,T;D(A1/ 2)) ∩L2(0,T;D(A)).

Using this and(12), we get that

{ ˙u m}∞m=1is a bounded set in L∞(0,T;D(A−1/ 2)) ∩L2(0,T;L2(Ω)).

Hence the family{ (u m; ˙u m)}∞

m= 1is a bounded set in

Z1≡ L∞(0,T;D(A1/ 2)) ∩L2(0,T;D(A))
× L∞(0,T;D(A−1/ 2)) ∩L2(0,T;L2(Ω))
(15) Therefore there exist a subsequence{ (u k; ˙u k)}and an element(u; ˙u) ∈Z1such that

The proof that any *-weak limit is a strong solution is standard

Now we prove the uniqueness of strong solutions

Using the propertiesϕ ∈L, the definition of a strong solutionvandv( ˙ t) ∈L∞(0,T;D(A− 1 / 2))(see(16)) we have that for any such solutionvand any T>0 there exists Lv,T>0 such that

kA−1/ 2(v(s1) − v(s2))k ≤Lv,T· |s1−s2| , ∀s1,s2∈ [−r,T] (17)

Consider two strong solutions u andvof(1),(5)(not necessarily with the same initial function)

Assumption (H.B) (see(6)) and the Lipschitz property of b imply

kF(u s1) −F(vs2)k2 =

Z

Ω

b [Bu] (s1,x)
−b [Bv](s2,x)

2

dx

≤ L2b

Z

Ω

| [Bu] (s1,x) − [Bv](s2,x)|2dx

= L2b· k [Bu] (s1, ·) − [Bv](s2, ·)k2≤L2b L2B· kA−1/ 2

u(s1) − v(s2)
k2. (18) Now, for any two strong solutions, we have

F(u t) −F(vt) =b(Bu(t− η(u t))) − b(Bv(t− η(vt ))) ±b(Bv(t− η(u t))).

Using the Lipschitz properties of b,B andη(see(6),(7)), and also(17),(18), one gets

kF(u t) −F(vt)k ≤L b L B



max

s∈[t−r,t]

kA−1/ 2(u(s) − v(s))k + kA−1/ 2(v(t− η(u t)) − v(t− η(vt )))k



≤L b L B kA−1/ 2(u t− vt )kC +Lv,T· | η(u t) − η(vt)|

We write, for short,

Now the standard variation-of-constants formula u(t) =e−At u(0) + Rt

0e−A(t− τ)F(uτ)dτand(19)give

kA−1/ 2(u t− vt)kC ≤ kA−1/ 2(u0− v0)kC +Cv,T·

Z t

0

e−λ 1 (t− τ)kA− 1 / 2(uτ− vτ)kCdτ.

The last estimate (by Gronwall’s lemma) implies

kA−1/ 2(u t− vt)kC ≤ kA−1/ 2(u0− v0)kC ·



1+ Cv,T

Cv,T− λ1

e(Cv, T− λ 1 )t−1



which gives the uniqueness of strong solutions of(1),(5)

The proof ofTheorem 1is complete 

Remark 4 It is very important that the termh1+ Cv, T

Cv, T− λ 1 e(Cv, T− λ 1 )t−1

i

in(21)tends to+∞when Lv,T → +∞, except

for the case Lη =0 (see(20))

Let us get an additional estimate for strong solutions

The standard variation-of-constants formula u(t) = e−At u(0) + Rt

0e−A(t− τ)F(uτ)dτ, (19), (20) and the estimate

kAαe−tAk ≤ α

t

α

e−α(see e.g [28, (1.17), p.84]) give

kA1/ 2(u(t) − v(t))k ≤ e−λ 1tkA1/ 2(u(0) − v(0))k + Z t

0

kA1/ 2e−A(t− τ)k · kF(uτ) −F(vτ)kdτ

≤e−λ 1tkA1/ 2(u(0) − v(0))k +2t1/ 2



1 2

1 / 2

e−1/ 2·Cv,T· kA−1/ 2(u0− v0)kC. (22) Here we usedkA1 / 2e−A(t− τ)k ≤ 

1 / 2

t− τ

1 / 2

e−1 / 2andR0t(t− τ)− 1 / 2dτ =2t1 / 2.

Now estimates(21),(22)give

kA1/ 2(u(t) − v(t))k + kA−1/ 2(u t− vt)kC ≤e−λ 1tkA1/ 2(u(0) − v(0))k +Dv,T· kA−1/ 2(u0− v0)kC. (23) Here we define

Dv,T≡2T1/ 2



1 2

1 / 2

e−1/ 2·Cv,T+



1+ Cv,T

Cv,T− λ1

e(Cv, T− λ 1 )T−1



3 Asymptotic behavior

In this section we study long-time behavior of the strong solutions of the problem(1),(5)

Due toTheorem 1, we define in the standard way the evolution semigroup St :L→L(the spaceLis defined in(9)) by the formula

where u(t)is the unique strong solution of the problem(1),(5)

Remark 5 We emphasize that the evolution semigroup S t :L→Lis not a dynamical system in the standard sense (see e.g [29,30,28]) since St is not a continuous mapping in the topology ofL, i.e the problem(1),(5)is not well-posed in the sense of Hadamard [18,19]

Our first goal is to prove:

Lemma 1 Let all the assumptions of Theorem 1 be satisfied Then for anyα ∈ (1

2,1), there exists a set bounded in the space

C1([−r,0];D(A− 1 / 2)) ∩C([−r,0];D(Aα)),BVα, which absorbs any strong solution of the problem(1),(5)with any initial functionϕ ∈L.

Proof of Lemma 1 UsingkA1 / 2vk2≤ λ− 1

1 · kAvk2, we get from(13)that d

dtkA

1 / 2u m(t)k2+ λ1kA1/ 2u m(t)k2≤M b2|Ω|

We multiply the last estimate by eλ 1tand integrate over[0,t]to obtain

kA1/ 2u m(t)k2≤ kA1/ 2u m(0)k2e−λ 1t+ λ− 1

1 M b2|Ω| ≤ kA1/ 2ϕ(0)k2e−λ 1t+ λ− 1

This and(12)givekA−1/ 2u˙m(t)k2≤2kA1/ 2ϕ(0)k2e−λ 1t+2λ− 1

1 M b2|Ω| +M b2|Ω| The last two estimates imply

kA1/ 2

u m(t)k2+ kA−1/ 2u˙m(t)k2≤3kA1/ 2ϕ(0)k2

e−λ 1t+ (1+3λ− 1

We get an analogous estimate for a strong solution of the problem(1),(5), using the well-known:

Proposition 1 ([ 31 , Theorem 9]) Let X be a Banach space Then any *-weak convergent sequence{ wk }∞n=1 ∈ X∗ *-weak converges to an elementw∞∈X∗andk w∞kX ≤lim infn→∞k wn kX.

Now we consider the space V ≡C1([−r,0];D(A− 1 / 2)) ∩C([−r,0];D(A1 / 2)), fix any positiveε0and obtain that the ball

B0of V

B0≡  v ∈V: k vk2

V ≤R20≡ (1+3λ− 1

1 )M b2|Ω| + ε0

(28)

is absorbing for any strong solution of the problem(1),(5)(see(27))

Now we are in a position to use the arguments presented in [28, Lemma 2.4.1, p.101] and get (for any 12 < α <1) the existence of the absorbing ball

Bα≡  v ∈C([−r,0];D(Aα)) : kvkC([−r, 0 ];D(Aα)) ≤Rα , (29)

where Rα ≡ (α −1/2)α− 1 / 2· hλ− 1 / 2

1 M b√ |Ω| + ε i + α α

1 − α·M b

√

|Ω|with any fixedε > 0 More precisely, the standard

variation-of-constants formula u(t) = e−At u(0) + Rt

0e−A(t− τ)F(uτ)dτ and the estimatekAαe−tAk ≤ α

t

α

e− α (see

e.g [28, (1.17), p.84]) give

kAαu(t+1)k ≤ (α −1/2)α− 1 / 2kA1/ 2

u(t)k + Z t+1

t

t+1− τ

α

kF(uτ)kdτ.

Let us consider any set bounded inL,B Estimateˆ (26)and Proposition 1 givekA1/ 2u(t)k ≤ hλ− 1 / 2

1 M b√ |Ω| + ε ifor all

t ≥t Bˆ(here tˆBdepends onB only) These and the estimateˆ kF(uτ)k ≤M b√ |Ω|imply(29)

The above estimates(28),(29)show that there exists a subset (a ball) of Vα≡C1([−r,0];D(A− 1 / 2)) ∩C([−r,0];D(Aα))

(here 12< α <1),

such that for any strong solution, starting inϕfrom any bounded setBˆ ⊂L, there exists t Bˆ ≥0 such that

The proof ofLemma 1is complete 

We will use the notation

||| ϕ||| ≡sup

s6=t

 kA− 1 / 2(ϕ(s) − ϕ(t))k

|s−t|



forϕ ∈L.

Let us fix R0 > 0 and consider the metric spaceLR0 which is the set

ϕ ∈L: ||| ϕ||| ≤R0

equipped with the metrics (cf.(10))

ρ(ϕ, φ) ≡ max

s∈[−r, 0 ]

One can check that(LR0; ρ)is a complete metric space and any set ϕ ∈L: ||| ϕ||| ≤R1<R0

is closed

We need the following (technical) assumption:

(H.I) There exists R0> bRα(bRαis defined in(30)) such that the set

ϕ ∈L: ||| ϕ||| ≤R0

is positively invariant for the semigroup

S t , i.e.

Remark 6 Discussing the technical assumption (H.I), we notice that even in the case when (H.I) is not satisfied for the

original system,Lemma 1allows one to consider a modified system without modifying the long-term dynamics of S t

(see [32]) More precisely, one chooses [32, p.545] a C∞functionχ : [0, +∞) → [0,1]such that

( χ(s) =1, s∈ [0,1];

χ(s) =0, s∈ [2, +∞);

0≤ χ(s) ≤1, s∈ [1,2]

and sets

eF(ϕ) ≡ χ  ||| ϕ|||

bRα



·F(ϕ).

As a result, the modified system(1)(witheF(ϕ)instead of F(ϕ)) has the same behavior inside of the (absorbing) setBVαand

satisfies (H.I) with R0=2bRα.For more details see section 3.1 in [26]

Our next result is the following:

Theorem 2 Let(H.I)and all the assumptions of Theorem 1 be satisfied Then the evolution semigroup S t :LR0 →LR0(see(25)) possesses a global attractor in the metric space(LR0; ρ).

Proof of Theorem 2 Now we concentrate on the metric space(LR0; ρ)(here R0 > bRα) The reason for this is that the

evolution semigroup S is not continuous on the whole spaceL(seeRemark 5) On the other hand, we notice:

Remark 7 Estimate(23)implies that the evolution semigroup S t is a continuous mapping in the topology of(LR0; ρ), i.e.ρ(S tϕ,S tφ) ≤Dv,T· ρ(ϕ, φ)forϕ, φ ∈LR0, and t ∈ [0,T] Here Dv,Tis defined by(24)(see also(20)) with Lv,T =R0 (cf.(17))

Corollary 4 from [33] implies thatBVαis relatively compact in C([−r,0];D(A− 1 / 2))(see also [33, lemma 1]) This fact and the propertykAαϕ(0)k ≤ bRα,1

2 < α < 1 for allϕ ∈ BVαgive thatBVαis relatively compact in the topology of

(LR0; ρ)

Let us consider the following set:

K≡Cl[BVα]( LR0; ρ),

where Cl[·]( LR0; ρ)is the closure in the topology of(LR0; ρ) The above properties show that K is compact in(LR0; ρ)

We get (see(31)) that for any strong solution, starting inϕfrom any bounded seteB⊂LR0, there exists t

eB≥0 such that

S tϕ ∈BVα⊂K, for all t≥teB.

As a result, we conclude that the evolution semigroup S tis asymptotically compact (and dissipative) in(LR0; ρ) Finally, by the classical theorem on the existence of an attractor (see, for example, [29,30,28]) one gets that(S t; (LR0; ρ))

has a compact global attractor The proof ofTheorem 2is complete 

Remark 8 Discussing the restriction of our study from the linear spaceLto the metric space(LR0; ρ), we notice that it is a natural step even for ordinary differential equations with a discrete state-dependent delay For example, in [9, Proposition 1 and Corollary 1] it is shown that maximal solutions of a scalar delay equation with an SDD constitute a semiflow on the set

{ φ :Lip(φ) ≤k, kφk < w} ⊂C([−r,0] ,R) Here Lip(φ) =supx6=y| φ(x) − φ(y)| · |x−y|− 1.

4 A particular case of a state-independent delay (η =const)

In this particular case, the assumption (H.η) (see(7)) is valid automatically with Lη=0 Following the proof ofTheorem 1, one can see that the assumption sups6=tkA− 1 / 2(ϕ(s) − ϕ(t))k · |s−t|− 1

< +∞is not needed in the caseη =const This implies that for any initial functionϕ ∈H (cf.(9)),

the problem(1),(5)has a strong solution The uniqueness of a strong solution follows from(23)and the fact that Lη = 0

implies Dv,T(defined in(24)) is bounded for anyϕ ∈H (cf.Remark 4and(20)) This fact gives the continuity of St :H→H

(cf.Remark 5) and as a consequence, that the pair(S t;H)is a dynamical system

Following the proofs ofLemma 1andTheorem 2we have the following result

Theorem 3 Assumeη =const Let the assumption(H.B)hold and the function b:R→R be locally Lipschitz and bounded Then for any initial functionϕ ∈H the problem(1),(5)has a unique strong solution on any time interval[0,T] The solution has the property u˙ ∈L2(0,T;L2(Ω)).

Moreover, the pair(S t;H)constitutes a dynamical system which possesses a global attractor The attractor is a bounded set in

C1([−r,0];D(A−1/ 2)) ∩C([−r,0];D(Aα))for anyα ∈ (1

2,1).

Now we can compare two cases (state-dependent and state-independent delays), assuming that

• the assumption (H.B) holds and

• the function b:R→R is locally Lipschitz and bounded.

State-dependent delayη State-independent delay

The existence and uniqueness of solutions ϕ ∈L⊂H and (H.η) ϕ ∈H

The continuity of S tand existence of an attractor S t: (LR0; ρ) → (LR0; ρ) S t :H→H

Remark 9 We notice thatLR0 ⊂L⊂H and the metricρis the natural metric of the space H.

As an application (for both cases of state-dependent and state-independent delays) we can consider the diffusive Nicholson’s blowflies equation (see e.g [34,35]) with state-dependent delays More precisely, we consider Eq.(1)where

−A is the Laplace operator with the Dirichlet boundary conditions,Ω ⊂R n0is a bounded domain with a smooth boundary,

the function f can be, for example, f(s) = √1

4 παe

−s2 / 4 α, as in [36] (seeRemark 2), and the non-linear (birth) function b is

given by b(w) =p· we− w Function b is bounded, so for any delay functionη, satisfying (H.η), the conditions ofTheorems 1 and2are valid (we modify the system according toRemark 6, if necessary) As a result, we conclude that the initial value problem(1),(5)is well-posed in(LR0; ρ)and the dynamical system(S t,LR0; ρ)has a global attractor (Theorem 2)

The author wishes to thank I.D Chueshov for useful discussions of an early version of the manuscript

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