# Global Attractor for Partial Functional Differential Equations with State-Dependent Delay

Authors: Zhimin He; Bo Du
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This work aims to investigate the existence of global attractors for a class of partial functional differential equations with state-dependent delay. Using the classic theory about global attractors in infinite dimensional dynamical systems, we obtain some sufficient conditions for guaranteeing the existence of a global attractor.


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## Body

## 1. Introduction

Partial functional differential equations with state-dependent delay appear frequently in applications as models of various phenomena, such as biological, chemical, and physical systems, which are characterized by both spatial and temporal variables. For this reason, the study of this kind of equation has received much attention in recent years. For more details, see for instance [1–5] and the references therein. However, it is worth pointing out that all of the papers mentioned above are mainly devoted to the existence of solutions or mild solutions. The literature related to global attractors is limited.It is known that the global attractor is a very useful tool, which is valid for more general situations than those for stability to study the asymptotical behavior. In the present paper, we are devoted to investigating the existence of a global attractor for a type of partial functional differential equations with state-dependent delay as follows:(1)u′(t)=Au(t)+F(t,uρ(t,ut)),t≥0,u0=ϕ∈C,
where C:=C([-r,0],E),r>0, is the space of continuous functions from [-r,0] to the Banach space E, equipped with the uniform norm ∥ϕ∥=sup-r≤θ≤0|ϕ(θ)|, and   A is a linear operator on a Banach space E satisfying the following well-known Hille-Yosida condition:(H1)
there existM0≥1 and ω∈ℝ such that
(2)∥(λI-A)-n∥≤M0(λ-ω)nforλ∈ℛ(A),λ>ω,whereℛ(A) is the resolvent set of A.Consider thatρ:[0,+∞)×C→[0,+∞) satisfies the following properties.(H2)
Letℒ(ρ)={ρ(s,ϕ):(s,ϕ)∈[0,+∞)×C,ρ(s,ϕ)≥0}. There exists a continuous and bounded function Jϕ:ℒ(ρ)→[0,+∞) such that
(3)∥φρ(t,xt)∥≤Jϕ(t)∥ϕt∥,∀t∈ℒ(ρ).Consider thatF:[0,+∞)×C→E satisfies the following properties.(H3)
(i) For everyϕ∈C, the function t→F(t,ϕ) is strongly measurable.(ii) For each t∈[0,+∞],    F(t,·):C→E is continuous.(iii) There exist a positive constant c and a bounded function m:[0,+∞)→[0,+∞) such that
(4)|F(t,ϕ)|≤c+m(t)∥ϕ∥.For everyt≥0, the history function xt∈C is defined by
(5)xt(θ)=x(t+θ),forθ∈[-r,0].
In the present paper, we will obtain some sufficient conditions for guaranteeing the existence of a global attractor to (1) with A being a Hille-Yosida operator but not necessarily densely defined.


## 2. Preliminaries

We recall some definitions and results from the integrated semigroup.Definition 1 (see [6]).
LetT>0. A function x:[-r,T]→E is said to be an integral solution of (1) if (i)
∫0t‍x(s)ds∈D(A) for t≥0,(ii)
x(t)=ϕ(0)+A(∫0t‍x(s)ds)+∫0t‍F(s,xρ(s,xs))ds,(iii)
x0=ϕ.Remark 2.
Clearly, ifx is an integral solution of (1), then xt(0)=x(t)∈D(A)¯ for t∈[0,T]. So ϕ(0)∈D(A)¯, which is a necessary condition for the existence of an integral solution.Let us introduce the partA0 of the operator A in D(A)¯ which is defined by
(6)D(A0)={x∈D(A):Ax∈D(A)¯},A0x=Axforx∈D(A0).Lemma 3 (see [7]).
A
0 generates a strongly continuous semigroup (T0(t))t≥0 on D(A)¯.Based on the previous abstract results, we give some concrete results for (1); see [6].Definition 4.
LetT>0. For any given ϕ∈C with ϕ(0)∈D(A)¯, the function x(·):=x(·,ϕ):[-r,T]→E is said to be an integral solution of (1) with initial function ϕ at  t=0 if
(7)x(t)={T0(t)ϕ(0)+limλ→+∞∫0tT0(t-s)×λ(λI-A)-1F(s,xρ(s,xs))ds,0≤t≤T,ϕ(t),-r≤t≤0.Lemma 5 (see [6]).
Under the assumptions (H1)–(H3), if ϕ∈C with ϕ(0)∈D(A)¯, then (1) posseses a unique global integral solution x(·,ϕ):[-r,+∞)→E with initial function ϕ at t=0, which can be expressed by (7).According to Remark2, denote Σ0={ϕ∈C:ϕ(0)∈D(A)¯}. Then from Lemma 5, for each ϕ∈Σ0, we define the following operator on Σ0 by
(8)U(t)ϕ=xt(·,ϕ),t≥0,
where xt(·,ϕ) is a unique global integral solution of (1) in Lemma 5. Clearly, (U(t))t≥0 is a strongly continuous semigroup on Σ0.Definition 6 (see [8]).
An invariant set𝒜 is said to be a global attractor if 𝒜 is a maximal compact invariant set which attracts each bounded set B⊂X.Definition 7 (see [8]).
A semigroupU(t):X→X,t≥0, is said to be point dissipative if there is a bounded set B⊆X that attracts each point of E under U(t).Lemma 8 (see [9]).
If(i)
there is at0≥0 such that U(t) is compact for t>t0,(ii)
U(t) is point dissipative in X,
then there exists a nonempty global attractor𝒜 in X.


## 3. The Global Attractor

In this section, we will obtain the existence of a global attractor to (7) by using Lemma 8. For the convenience of the proof, we give some assumptions and lemmas.(H4)
ForC0-semigroup T0(t),t≥0, there exists positive constant α such that
(9)∥T0(t)∥≤e-αtfort≥0.(H5)
T0(t) is compact for t>0.Lemma 9 (see [10]).
If(10)x(t)≤h(t)+∫t0tk(s)x(s)ds,t∈[t0,T),
where all the functions involved are continuous on [t0,T), T≤+∞, and k(t)≥0, then x(t) satisfies
(11)x(t)≤h(t)+∫t0th(s)k(s)e∫st‍k(u)duds,t∈[t0,T).Lemma 10.
Assume that assumptions (H1)–(H4) hold. Then, for each ϕ∈Σ0, if α≠maJaeγr, there exists a constant γ>α such that the integral solution x(·,ϕ) of (1) satisfies the following inequality:
(12)∥xt∥≤ceγrα-maJaeγr+eγr(∥ϕ∥-cα-maJaeγr)e(maJaeγr-α)t,t≥0,
where ma=maxs∈[0,∞)m(s) and Ja=maxs∈ℒ(ρ)Jϕ(s).Proof.
By(H3), for each ϕ∈Σ0, we have
(13)|F(t,ϕ)|≤c+m(t)∥ϕ∥.
In the following proof, for simplicity, takeM0=1,n=1 in (H1); then
(14)∥(λI-A)-1∥≤1λ-ωforλ>ω.
Instead of considering the norm ∥xt∥ directly, we firstly estimate ∥eγ·xt∥ for some constant γ>α.
Case  1. For 0≤t≤r,  by (7), we have
(15)sup-r≤θ≤0|eγθxt(θ)|=max{sup-r≤θ≤-t|eγθϕ(t+θ)|,sup-t≤θ≤0|eγθxt(θ)|}≤max{e-γt∥ϕ∥,sup-t≤θ≤0eγθe-α(t+θ)|ϕ(0)|max++sup-t≤θ≤0eγθlimλ→+∞∫0t+θ‍e-α(t+θ-s)∥λ(λI-A)-1∥max×××[c+m(s)Jϕ(s)∥xs∥]dse-γt∥ϕ∥,sup-t≤θ≤0eγθe-α(t+θ)|ϕ(0)|}≤max{sup-t≤θ≤0maJae-α(t+θ)eγθe-γt∥ϕ∥,e-αt|ϕ(0)|≤max++sup-t≤θ≤0ce-α(t+θ)eγθlimλ→+∞∫0t+θ‍λλ-ωeαsds≤max++sup-t≤θ≤0maJae-α(t+θ)eγθ×××limλ→+∞∫0t+θ‍λλ-ωeαs∥xs∥ds}≤e-αt∥ϕ∥+ce-αt∫0t‍eαsds+maJae-αt∫0t‍eαs∥xs∥ds=e-αt∥ϕ∥+cα(1-e-αt)+maJae-αt∫0t‍eαs∥xs∥ds.
Case  2. For t≥r, we have
(16)sup-r≤θ≤0|eγθxt(θ)|≤sup-r≤θ≤0eγθe-α(t+θ)|ϕ(0)|+sup-r≤θ≤0eγθlimλ→+∞∫0t+θ‍e-α(t+θ-s)∥λ(λI-A)-1∥limλ→+∞×[c+maJa∥xs∥]ds≤e-αt∥ϕ∥+cα(1-e-αt)+maJae-αt∫0t‍eαs∥xs∥ds.
Therefore, from (15) and (16), for t≥0, we get
(17)sup-r≤θ≤0|eγθxt(θ)|≤e-αt∥ϕ∥+cα(1-e-αt)+maJae-αt∫0t‍eαs∥xs∥ds.
On the other hand, we have
(18)sup-r≤θ≤0|eγθxt(θ)|=sup-r≤θ≤0eγθ|xt(θ)|≥sup-r≤θ≤0e-γr|xt(θ)|=e-γr∥xt∥,
which combines with (17) and yields
(19)e-γr∥xt∥≤e-αt∥ϕ∥+cα(1-e-αt)+maJae-αt∫0t‍eαs∥xs∥ds.
So we get
(20)eαt∥xt∥≤eγr[∥ϕ∥+cα(eαt-1)]+maJaeγr∫0t‍eαs∥xs∥ds.
Using Lemma 9, we have
(21)eαt∥xt∥≤eγr∥ϕ∥+cαeγr(eαt-1)+maJaeγr∫0t‍[eγr∥ϕ∥+cαeγr(eαs-1)]+maJaeγr∫0t‍×e∫st‍maJaeγrduds≤ceγreαtα-maJaeγr+eγr(∥ϕ∥-cα-maJaeγr)emaJaeγrt.
Thus, (12) holds.Lemma 11.
Assume that the conditions of Lemma10 are satisfied; furthermore, α>maJaeγr, where γ is the constant defined by Lemma 10. Then (U(t))t≥0 is point dissipative.Proof.
From Lemma10, we find that for each ϕ∈Σ0, since α>maJaeγr, there exits a t0:=t0(ϕ)>0 such that for t>t0,
(22)∥xt∥≤ceγrα-maJaeγr+1(independentofϕ).
Therefore,
(23)BX0(0,ceγrα-maJaeγr+1)∩X0
attracts each point of X0, where BX0(0,(ceγr/(α-maJaeγr))+1) denotes the open ball in Σ0 with center 0 and radius (ceγr/(α-maJaeγr))+1.Now, we show the compactness of the operatorU(t).Lemma 12.
Assume that assumptions (H1)–(H5) hold. Then, U(t) is compact for t>r.Proof.
Lett>r and let {ϕn} be any bounded sequence of Σ0. We will use the Ascoli-Arzelà theorem to show that {U(t)ϕn:n∈ℕ} is precompact in Σ0 by two steps.
Step  1.  Show that for any θ∈[-r,0], the set
(24)Z(θ)={((U(t)ϕn)(θ):n∈ℕ)}
is precompact. For t>r and θ∈[-r,0], by (8), we have
(25)(U(t)ϕn)(θ)=T0(t+θ)ϕn(0)+limλ→+∞∫0t+θ‍T0(t+θ-s)×λ(-A)-1×λ(λI-A)-1F(xρ(s,xs)n)ds,
where xn(·) is the integer solution of (1) with initial function ϕn. From (H4) and the boundedness of {ϕn}, we know that {T0(t+θ)ϕn(0):n∈ℕ} are precompact. Now, considering the second term in (25), for sufficiently small ε>0, we have
(26)limλ→+∞∫0t+θ‍T0(t+θ-s)λ(λI-A)-1F(xρ(s,xs)n)ds=T0(ε)limλ→+∞∫0t+θ-ε‍T0(t+θ-s-ε)λ(λI-A)-1T0(ε)limλ→+∞×F(xρ(s,xs)n)ds+limλ→+∞∫t+θ-εt+θ‍T0(t+θ-s)λ(λI-A)-1F(xρ(s,xs)n)ds.
Note that from Lemma 10, we have
(27)supn∈ℕ∥xsn∥<∞,s∈[0,t].
By (H2) and (H3), we get
(28)|F(xρ(s,xs)n)|≤c+maJa∥xsn∥.
Therefore, there exist some constants M1,M2>0 such that
(29)|limλ→+∞∫0t+θ-ε‍T0(t+θ-s-ε)λ(λI-A)-1F(xρ(s,xs)n)ds|≤M1,|limλ→+∞∫t+θ-εt+θ‍T0(t+θ-s)λ(λI-A)-1F(xρ(s,xs)n)ds|≤M2ε
which yields
(30)T0(ε){limλ→+∞∫0t+θ-ε‍T0(t+θ-s-ε)λ×()-1×λ(λI-A)-1F(xρ(s,xs)n)ds:n∈ℕlimλ→+∞∫0t+θ-ε‍}⊂Γε,
where Γε is a compact set. Thus, Z(θ) is precompact.
Step  2.   Show the equicontinuity of {U(t)ϕn:n∈ℕ}. Let  -r≤θ1<θ2≤0;  we have
(31)(U(t)ϕn)(θ2)-(U(t)ϕn)(θ1)=[T0(t+θ2)-T0(t+θ1)]ϕn(0)+limλ→+∞∫0t+θ2‍T0(t+θ2-s)λ(λI-A)-1F(xρ(s,xs)n)ds-limλ→+∞∫0t+θ1‍T0(t+θ1-s)λ(λI-A)-1F(xρ(s,xs)n)ds=T0(t+θ1)[T0(θ2-θ1)-I]ϕn(0)+limλ→+∞∫t+θ1t+θ2‍T0(t+θ2-s)λ(λI-A)-1F(xρ(s,xs)n)ds+limλ→+∞∫0t+θ1‍[T0(t+θ2-s)-T0(t+θ1-s)]+limλ→+∞∫0t+θ1‍×λ(λI-A)-1F(xρ(s,xs)n)ds,
which leads to
(32)|(U(t)ϕn)(θ2)-(U(t)ϕn)(θ1)|≤∥T0(t+θ1)[T0(θ2-θ1)-I]∥+limλ→+∞∫t+θ1t+θ2‍|F(xρ(s,xs)n)T0(t+θ2-s)λ(λI-A)-1×limλ→+∞∫t+θ1t+θ2‍×F(xρ(s,xs)n)|ds+∥T0(θ2-θ1)-I∥×limλ→+∞∫0t+θ1|F(xρ(s,xs)n)T0(t+θ1-s)λ(λI-A)-1×F(xρ(s,xs)n)|ds.
Since the mapping t→T0(t) is norm continuous for t>0, for some δ∈(0,t-r), put
(33)T0(t+θ1)[T0(θ2-θ1)-I]=T0(t+θ1-δ)[T0(θ2-θ1+δ)-T0(δ)].
Then
(34)∥T0(θ2-θ1+δ)-T0(δ)∥⟶0asθ2⟶θ1.
Thus
(35)∥T0(t+θ1)[T0(θ2-θ1)-I]∥⟶0asθ2⟶θ1.
By the boundedness of |T0(t+θ2-s)λ(λI-A)-1F(xρ(s,xs)n)|, then
(36)limλ→+∞∫t+θ1t+θ2‍|(xρ(s,xs)n)T0(t+θ2-s)λ(λI-A)-1limλ→+∞∫t+θ1t+θ2‍×F(xρ(s,xs)n)|ds⟶0asθ2⟶θ1.
Obviously, limλ→+∞∫0t+θ1‍|T0(t+θ1-s)λ(λI-A)-1F(xρ(s,xs)n)|ds belongs to a compact subset of E; we have
(37)∥T0(θ2-θ1)-I∥×limλ→+∞∫0t+θ1‍|(xρ(s,xs)n)T0(t+θ1-s)λ(λI-A)-1∥×-I∥limλ→+∞∫0t+θ1‍×F(xρ(s,xs)n)|ds⟶0∥-I∥limλ→+∞×ds0⟶0∫0t+θ1‍asθ2⟶θ1.
Hence {U(t)ϕn:n∈ℕ} is equicontinuity.Here, we state our main theorem of this paper, which is an immediate consequence of Lemmas8, 11, and 12.Theorem 13.
Assume that assumptions (H1)–(H5) hold. If α>maJaeγr, then (1) has a nonempty global attractor 𝒜.As applications, we give the following example.LetE=C([0,π],ℝ) and C=C([-r,0],E).(38)∂∂tu(t,x)=∂2∂x2u(t,x)-μu(t,x)+f(u(t-r,x)),0≤x≤π,t≥0,u(t,0)=u(t,π)=0,t≥0,u(θ,x)=ϕ(θ)x,0≤x≤π,-r≤θ≤0,
where μ>0 is a constant, u(t,·)∈E,   and  f:C→E satisfies (H3). Define the operator A by
(39)D(A)={q∈C2([0,π],ℝ):q(0)=q(π)=0},A(q)=q′′
and F:C→E with
(40)F(φ)(x)=f(φ(-r)(x)),x∈[0,π].
Then A satisfies the Hille-Yosida condition in E and
(41)∥(λI-A)-1∥≤1λforλ>0.
Moreover, the part A0 of A in D(A)¯ is the infinitesimal generator of a strongly continuous semigroup (T0(t))t≥0 on E such that
(42)∥T0(t)∥≤e-tfort≥0.
According to Theorem 13, if there exist γ>μ, ma and Ja such that maJaeγr<μ, then (38) has a global attractor.



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