Uniform attractors for non-autonomous parabolic equations with delays

Nonlinear Analysis 71 (2009) 2194–2209

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Uniform attractors for non-autonomous parabolic equations with delaysI Jin Li, Jianhua Huang ∗ Department of Mathematics, National University of Defense Technology, Changsha, 410073, PR China

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a b s t r a c t

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Article history: Received 28 May 2008 Accepted 12 January 2009

The present paper is devoted to the asymptotic behavior of non-autonomous parabolic equations with nonlinearity containing general delay and with time-dependent external forces. The existence, uniqueness and regularity of solutions for the equation are obtained. If the time-dependent external forces is a translation compact function in L2loc (R, L2 (Ω )), then, the existences of uniform attractor for non-autonomous dynamics system are provided, and the structures of attractors are also studied, which is the union of all bounded complete trajectory of the equation at a time moment. Finally, the main results are applied to the population dynamics with distributed state-dependent delay and with almost periodic external force, and the uniform attractor and its kernel section are obtained. © 2009 Elsevier Ltd. All rights reserved.

MSC: 35R10 35B40 Keywords: Uniform attractor Translation compact Kernel section Variable delay Distributed delay

1. Introduction In this paper, we consider the following non-autonomous parabolic equation with general delay and external force:

∂ u(t , x) + Au(t , x) + bu(t , x) = F (ut )(x) + g (t , x), x ∈ Ω . ∂t u(τ , x) = u0 (x), u(τ + θ , x) = φ(θ , x), θ ∈ (−r , 0).

(

(1.1)

Here Ω is a bounded domain in Rn0 with smooth boundary, b ≥ 0, and other symbols satisfy the following conditions: 1. A is a densely-defined self-adjoint positive linear operator with domain D(A) ⊂ L2 (Ω ) and with compact resolvent(for example, −∆ with zero Dirichlet data); 2. the nonlinear term, F : L2 (−r , 0; L2 (Ω )) → L2 (Ω ), is a locally Lipschitz continuous for the initial data and there exist k1 , k2 , k3 ≥ 0, such that ∀ξ ∈ L2 (−r , 0; L2 (Ω )), η ∈ L2 (Ω ), one has

|(F (ξ ), η)L2 (Ω ) | ≤ k1 kηk

2 L2 (Ω )

Z

0

+ k2 −r

kξ (θ)k2L2 (Ω ) dθ + k3 ;

3. the external force g is a translation compact function in L2loc (R; L2 (Ω )).

I Supported by the NSF of China (No. 10571175).

∗

Corresponding author. E-mail addresses: [email protected] (J. Li), [email protected] (J. Huang).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.053

(1.2)

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There are three features of the model (1.1). First, it represents a class of parabolic equations arising in mathematical biology. Second, the nonlinearity contains general delay, which is a generalization of distributed state-dependent delay. Third, the system is non-autonomous and the time-dependent external force include a wide variety of functions such as almost periodic function. There are also three main resources contributing to our paper. First, in the article [1] Rezounenko and Wu provide a model of PDEs with distributed state-dependent delay as our prototype, which represents a nonlinear evolution process involving spatial dispersal and time delay that depends on the system’s status in biology. Second, in the articles [2] and [15] Caraballo and Real provide the method of dealing with PDE model with delay in the product phase space L2 (Ω ) × L2 (−r , 0; L2 (Ω )) when consider the long time behavior of the solution. Third, in the book [3] Chepyzhov and Vishik present a general approach to the study of non-autonomous evolution equations and corresponding processes by extending the phase space by using only the hull of the time-dependent coefficients of the equation and then define the uniform attractor. Let us introduce some relevant work in this area. The understanding of asymptotic behavior of dynamical system is one of the most important problems of modern mathematical physics and biology. One way to treat this problem for a system having dissipative properties is to analyze the existence and structure of its global attractor. In the last two decades, autonomous dynamical systems and their attractors have been extensively studied (see, e.g., [4–6]). For autonomous dissipative systems, the global attractor is an invariant compact set which attracts all the trajectories of the system, uniformly on bounded sets. In general, the global attractor has a very complicated geometry which reflects the complexity of the long time behavior of the system (see Chueshov [7], Robinson [4], Teman [6]). Non-autonomous systems are also of great importance and interest as they appear in many applications in the natural science. In the book [8] Haraux considers some special classes of such systems and studies systematically the notion of a uniform attractor paralleling that of a global attractor for autonomous systems. Later on, Chepzhov and Vishik [3] present a general approach that is well suited to study equations arising in mathematical physics. For the pullback attractors, we refer to papers [16,17]. Delays are often considered in the model such as maturation time for population dynamics in mathematical biology and other fields. The theory of ordinary delay differential equations has a rich history and is still one of the actively developing branches of the theory of differential equations. Recently, some efforts have been devoted to the development of theory of PDEs with delay. Such equations are naturally more difficult since they are infinite dimensional both in time and space variables. We refer to the monograph [9] and articles [1,10,2]. Recently the model of PDEs with distributed state-dependent delay has attracted the attention of many researchers. We refer to [1] and references therein. Motivated by the Rezounenko and Wu’s article [1], we study the abstract parabolic equations with general delay and with external force which is translation compact in L2loc (R, L2 (Ω )) in the present paper. The existence, uniqueness and regularity of the solution of the equation is obtained. The existence of the uniform attractor of the non-autonomous system (1.1) is proved and the structure of the uniform attractor is investigated, that is, the union of all the values of the bounded complete trajectory of the equation at a time moment. In application, the corresponding results for Rezounenko and Wu’s model with almost periodic external force as a particular case of our general model are obtained. The rest of this paper is organized as follows. In Section 2 we formulate the model and provide some preliminaries on uniform attractor. Section 3 is devoted to proving the existence, uniqueness and regularity of solutions. The existence of uniform attractor of the non-autonomous delayed parabolic systems are show, and the structure of the uniform attractor are also obtained in Section 4. Finally, the main results are applied to Rezounenko -Wu’s model with almost periodic external force in Section 5. 2. Preliminaries 2.1. Formulation of the model The followings are notations and properties of our model. Since A is a densely-defined self-adjoint positive linear operator with domain D(A) ⊂ L2 (Ω ) and with compact resolvent, A : D(A) → L2 (Ω ) has a discrete spectrum that only contains positive eigenvalues {λk }∞ k=1 satisfying 0 < λ1 ≤ λ2 ≤ . . . , λk → ∞(k → ∞), and the corresponding eigenfunctions {ek }∞ k=1 compose an orthonormal basis of the Hilbert space L2 ( Ω ) . For convenience we use the following notation throughout this paper: H = L2 (Ω ) with norm | · |, and inner product 1

1

1

(·, ·), V = D(A 2 ) with norm k · k, and associated product ((·, ·)), where for u, v ∈ V , ((u, v)) = (A 2 u, A 2 v). Besides, we denote k · kE the norm of a Banach space E, k · kop the operator norm and BE (a, ρ) the closed ball in Banach space {x ∈ E : kx − akE ≤ ρ}. Let α > β . Then the space D(Aα ) is compactly embedded into D(Aβ ) by [7]. In particular, 1

V ⊂ H ≡ H 0 ⊂ V 0 , where the injections are dense and compact and ∀u ∈ V , |u| ≤ λ12 kuk. Given T > τ and u : (τ − r , T ) → H, for each t ∈ (τ , T ) we denote by ut the function defined on (−r , 0) by the relation ut (s) = u(t + s), s ∈ (−r , 0). We also denote L2H = L2 (−r , 0; H ), L2V = L2 (−r , 0; V ), CH = C ([−r , 0]; H ) and CV = C ([−r , 0]; V ).

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In [1] Rezounenko and Wu have already dealt with autonomous PDE model with distributed time-dependent delay. Now we establish suitable hypotheses on the nonlinear term containing the delay with less restriction. Let F : L2H → H satisfy the following assumptions: (I) F satisfy the local Lipschitz continuity for the initial data, i.e., ∀M > 0, ∃LF ,M such that ∀u, v ∈ L2H satisfying (u(0), u) , (v(0), v) ∈ BH ×L2 (0, M ), one has H

  |F (u) − F (v)| ≤ LF ,M |u(0) − v(0)|2 + ku − vk2L2 ; 2

H

(II) ∃k1 , k2 , k3 ≥ 0, such that ∀ξ ∈ L2H , η ∈ H,

|(F (ξ ), η)| ≤ k1 |η|2 + k2

0

Z

|ξ (θ )|2 dθ + k3 .

(2.1)

−r

Note that by the Risez Representation Theorem, we have

|F (ξ )| = kF (ξ )kop = sup (F (ξ ), η) ≤ k1 + k1 |η|=1

Z

0

|ξ (θ )|2 dθ + k3 ,

(2.2)

−r

which implies that F is a bounded operator from L2H to H. Assume that b is a nonnegative constant, g is translation compact in L2 (R, H ), u0 ∈ H and φ ∈ L2H . Observe that g is translation compact implies

|g |2b = kg k2L2 (R,H ) = sup b

t ∈R

t +1

Z

|g (s)|2 ds < ∞. t

Then for each τ ∈ R we consider the problem: d u(t ) + Au(t ) + bu(t ) = F (ut ) + g (t ), dt u(τ ) = u0 , u(τ + θ ) = φ(θ ), θ ∈ (−r , 0).

(

(2.3)

2.2. Preliminaries on uniform attractor We now discuss the theory of uniform attractors, as developed in [3]. In the sequel the abbreviations ‘‘w.r.t.’’ and ‘‘tr.c.’’ are employed as ‘‘with respect to’’ and ‘‘translation compact’’ respectively. Let E be a Banach space and let U (t , τ ) be a process in E. The uniform (w.r.t. τ ∈ R) attractor of the process is then defined. Definition 2.1. A closed set A0 ⊂ E is said to be the uniform (w.r.t. τ ∈ R) attractor of the processes {U (t , τ )} if it is uniformly (w.r.t. τ ∈ R) attracting and it is contained in any closed uniformly (w.r.t. τ ∈ R) attracting set A0 for the processes {U (t , τ )}: A0 ⊂ A0 . Consider a family of processes {Uσ (t , τ )} depending on a parameter σ ∈ Σ . The parameter σ , chosen as the collection of all time-dependent coefficients od the equation, is said to be the symbol of the process {U (t , τ )} and the set Σ is said to be the symbol space. In the sequel we consider symbol space Σ = H (σ0 ), where σ0 (s) is translation compact function in L2loc (R, H ) and H (σ0 ) is the hull of σ0 in L2loc (R, H ), that is, the closure in L2loc (R, H ) defined by:

H (σ0 ) = {σ0 (· + s)|s ∈ R}. We mention that a wide class of functions such as periodic function, quasiperodic function and almost periodic, is tr.c. in appropriate space such as Cb (R) or L2loc (R, H ). By B (E ) we denote the collection of the bounded sets of E. Definition 2.2. A set B0 ∈ E is said to be uniformly (w.r.t. σ ∈ Σ ) absorbing for the family of processes {Uσ (t , τ )|σ ∈ Σ }, if for any τ ∈ R and every B ∈ B (E ) there exists t0 = t0 (τ , B) ≥ τ such that ∪σ ∈Σ Uσ (t , τ )B ⊂ B0 for all t ≥ t0 . A family of processes possessing a compact uniformly absorbing set is called uniformly compact (here we follow the terminology from Haraux [8]). Definition 2.3. A closed set AΣ ⊂ E is said to be a uniform (w.r.t. σ ∈ Σ ) attractor of the family of processes {Uσ (t , τ )|σ ∈ Σ }, if it is uniformly (w.r.t. σ ∈ Σ ) attracting (attracting property) and it is contained in any closed uniformly (w.r.t. σ ∈ Σ ) attracting set A0 of the family of processes {Uσ (t , τ )|σ ∈ Σ }: AΣ ⊂ A0 (minimality property). Let us formulate the theorem on existence of a uniform attractor for a family of processes. Theorem 2.1. If a family of processes {Uσ (t , τ )|σ ∈ Σ } is uniformly (w.r.t. σ ∈ Σ ) compact, then it possesses the uniform (w.r.t. σ ∈ Σ ) attractor AΣ . The set AΣ is compact in E.

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Remark 2.1. Theorem 2.1 holds without any continuity assumption concerning the processes which is essential for the existence of the global attractor for a semigroup. The reason is that the invariance property of the global attractor of a semigroup is replaced by the property of minimality in the definition of the uniform attractor of a family of processes. Definition 2.4. A family of processes {Uσ (t , τ )|σ ∈ Σ } acting in E is said to be E × Σ -continuous, if for all fixed t and τ , t ≥ τ , τ ∈ R the mapping (u, σ ) 7→ Uσ (t , τ )u is continuous from E × Σ to E. The following definition concerns the structure of the uniform attractors. Definition 2.5. A curve u(s), s ∈ R is said to be a complete trajectory of the process {U (t , τ )} if U (t , τ )u(τ ) = u(t ) ∀ t ≥ τ , τ ∈ R.

(2.4)

The kernel K of the process {U (t , τ )} consists of all bounded complete trajectories of the processes {U (t , τ )}:

K = {u(·)|u(·) satisfies (2.4) and kuk∞ < ∞}. The set K (s) = {u(s)|u(·) ∈ K } ⊂ E is said to be the kernel section at a time moment t = s, s ∈ R. We consider two projectors PE : E × Σ → E and PΣ : E × Σ → Σ . Construct the semigroup {S (t )|t ≥ 0} acting in the extended phase space E × Σ that corresponds to the family of processes {Uσ (t , τ )|σ ∈ Σ }: S (t )(u, σ ) = (Uσ (t , 0)u, σ + t ),

t ≥ 0, (u, σ ) ∈ E × Σ .

(2.5)

Let us formulate the main theorem on the attractor of family of processes. Theorem 2.2. Let a family of processes {Uσ (t , τ )|σ ∈ Σ } acting in the space E be uniformly (w.r.t. σ ∈ Σ ) compact and (E × Σ , E )-continuous. Also let Σ = H (σ0 ) be a compact metric space. Then the semigroup {S (t )}, corresponding to the family of processes {Uσ (t , τ )|σ ∈ Σ } and acting on E × Σ (see (2.5)), possesses the compact attractor A, which is strictly invariant with respect to {S (t )}: S (t )A = A for all t ≥ 0. Moreover, (i) PE A = AΣ is the uniform (w.r.t. σ ∈ Σ ) attractor of the family of processes {Uσ (t , τ )|σ ∈ Σ }; (ii) PΣ A = Σ ; (iii) the global attractor satisfies

A=

[

Kσ (0) × {σ };

σ ∈Σ

(iv) the uniform attractor satisfies

AΣ =

[

Kσ (0).

σ ∈Σ

Here Kσ (0) is the section at t = 0 of the kernel Kσ of the process {Uσ (t , τ )} with symbol σ ∈ Σ . Since the uniform (w.r.t. τ ∈ R) absorbing property of the process {U (t , τ )} is equivalent to the uniform (w.r.t. σ ∈ Σ ) absorbing property of the family of processes {Uσ (t , τ )|σ ∈ Σ }, we have the following Theorem concerning the relationship between the uniform (w.r.t. σ ∈ Σ ) attractor for the family of processes {Uσ (t , τ )|σ ∈ Σ } and the uniform (w.r.t. τ ∈ R) attractor for the process {U (t , τ )}. Theorem 2.3. If the family of processes {Uσ (t , τ )|σ ∈ Σ } is (E × Σ , E )-continuous and uniformly (w.r.t. σ ∈ Σ ) compact, then the uniform (w.r.t. σ ∈ Σ ) attractor AΣ of the family of processes {Uσ (t , τ )|σ ∈ Σ } coincides with the uniform (w.r.t. τ ∈ R) attractor A0 of the process {U (t , τ )}:

A0 = AΣ =

[

Kσ (0).

σ ∈Σ

3. Existence and uniqueness of solution Theorem 3.1. Suppose φ ∈ L2H , F satisfy (I)–(II), g ∈ L2loc (R, H ). Then for each τ ∈ R, (a) If u0 ∈ H, then there exists a unique weak solution to (2.3), that is, u ∈ L2 (τ − r , T ; H ) ∩ L2 (τ , T ; V ) ∩ L∞ (τ , T ; H ) ∩ C ([τ , T ]; H ), and the equation is satisfied in the sense of distributions with values in V 0 , i.e., in D0 (τ , +∞; V 0 );

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(b) If u0 ∈ V , then the solution to (2.3) is a strong solution, that is, du

u ∈ L2 (τ , T ; D(A)) ∩ C ([τ , T ]; V ),

∈ L2 (τ , T ; H ),

dt

and the equation is now an equality in L2 (τ , T ; H ). In particular, it holds in H for almost every t ∈ [τ , T ]. Proof. (a) Our proof is based on the Faedo Galerkin approximation and is similar to [1]. We present the proof for the sake of completeness. Let us consider {ek }∞ k=1 , the orthonormal basis of H and all the eigenfunctions of A. The subspace of V spanned by e1 , . . . , em will be denoted Vm . Define the projector Pm : H → Vm as m X (u, ej )ej ,

Pm u =

j=1

and consider the approximation solutions um (t ) =

m X

γ im (t )ej ,

j=1

which satisfy

  um ∈ L2 (−r , T ; Vm ) ∩ C 1 ([0, T ]; Vm ),  d
um (t ), ej + (A1/2 um (t ), A1/2 ej ) + b(um (t ), ej ) = (F (um t ), ej ) + (g (t ), ej ),  dt   m m u (0) = Pm u0 , u (s) = Pm φ(s), s ∈ (−r , 0).

∀j = 1, 2, . . . , m,

(3.1)

Observe that (3.1) is a system of ordinary functional differential equations in the unknown γ m (t ) = (γ m1 (t ), γ m2 (t ), . . . , γ (t )). We can get the existence and uniqueness of the solution by applying the fixed point theorem since F satisfy the local Lipschitz condition. For the detail, we refer the reader to [11]. Suppose the local approximation solution is defined as an interval [−r , t ∗ ]. Now we try to get a priori estimate for the Galerkin approximate solutions. Multiplying in (3.1) by γ im and summing in i, we get mm

d dt

m m |um (t )|2 + kum (t )k2 + b|um (t )|2 = (F (um t ), u (t )) + (g (t ), u (t )).

Using assumption (II), we obtain d dt

|um (t )|2 + kum (t )k2 ≤ k1 |um (t )|2 + k2

0

Z

1

1

2

2

|um (t + θ )|2 dθ + k3 + |g (t )|2 + |um (t )| −r

and d dt

  Z 1 |um (t )|2 + kum (t )k2 ≤ k1 + + k2 2

Rt

τ τ −r

|um (s)|2 ds + k2

t

Z τ

1

|um (s)|2 ds + k3 + |g (t )|2 . 2

Let η(t ) ≡ |u (t )| + τ ku (s)k ds + k3 and note that |ξ | ≤ λ1 kξ k . Then we can rewrite the last estimate as follows m

d dt

2

m

η(t ) ≤ C4 η(t ) + k2

Z

2

τ τ −r

2

−1

2

|um (s)|2 ds + |g (t )|2 .

Multiplying it by e−C4 t , we obtain d dt





2 2 e−C4 t η(t ) ≤ k2 kum · e−C4 t . τ kL2 + |g (t )|




H

Integrating from τ to t and then multiplying by eC4 t , we get

  Z t 2 2 −C 4 s η(t ) ≤ C5 |um (τ )|2 + kum k + k + | g ( s )| e ds · eC 4 t . 3 τ L2 H

τ

2 Notice that |um (τ )| = |Pm u(τ )| ≤ |u(τ )|, kum τ kL2H ≤ kuτ kL2H and g ∈ Lloc (R, H ). So, we have the priori estimate

|u (t )| + m

2

for some C5 > 1.

Z τ

t

  kum (s)k2 ds ≤ C5 |u(τ )|2 + kuτ k2L2 + k˜ 3 · eC4 t − k3 , H

∀t ∈ [τ , t ∗ ],

(3.2)

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Estimate (3.2) gives that, for (u0 , φ) ∈ MH2 , the family of approximate solutions {um (t )}∞ m=1 is uniformly(w.r.t. m ∈ N) bounded in the space L∞ (0, T ; H ) ∩ L2 (0, T ; V ). From (3.2) we also get the continuation of um (t ) on any interval, so (3.2) holds for t ∈ [τ , T ]. (However, the continuation of local solution for functional differential equation is not so natural as the case of ODE. For details see [11].) Notice that d dt

um (t ) = −Aum (t ) − bum (t ) + F (um t ) + g (t ),

d m and F is bounded operator due to the Eq. (2.2). Therefore, dt u (t ) are uniformly bounded in L2 (0, T ; V 0 ). Then, according to the Alaoglu Weak-* Compactness Theorem, there exist a subsequence (still denoted {um }) such that

um * u dum dt

*

in L2 (0, T ; V ), du dt

(3.3)

in L2 (0, T ; V 0 ).

(3.4)

By a standard argument, using the strong convergence um → u in the space L2 (0, T ; H ) which follows from (3.3) and (3.4) and the Doubinskii’s theorem, one can show that any weak-* limit is a solution of problem (2.3) subject to the initial conditions. Now we prove the uniqueness of weak solutions. Let u(t ), v(t ) be two solutions of (2.3). Let w(t ) = u(t ) − v(t ), then w(t ) satisfies d dt

w(t ) + Aw(t ) + bw(t ) = F (ut ) − F (vt ).

Taking inner product with w(t ) and using the local Lipschitz property of F , we obtain 1 d 2 dt

|w(t )|2 + kw(t )k2 + b|w(t )|2 = (F (ut ) − F (vt ), w(t )) ≤ |w(t )| · |F (ut ) − F (vt )| 1 |F (ut ) − F (vt )|2 4λ1  L F ,M  ≤ λ1 |w(t )|2 + |w(t )|2 + kwt k2L2 , H 4λ1

≤ λ1 |w(t )|2 +

hence d dt

|w(t )|2 ≤

L F ,M

0

Z

2λ1

 |w(t + θ )|2 dθ + |w(t )|2 .

r

Integrating from τ to t, we obtain

|w(t )| − |w(τ )| ≤ 2

2

≤ ≤

LF ,M

Z t Z

2λ1 LF ,M

τ

LF ,M 2λ1

|w(s + θ )| dθ ds +

r

 Z

τ

|w(s)| dsdθ + 2

τ −r

r τ −r

t

Z

|w(s)|2 ds + (r + 1)



|w(s)| ds 2

τ



|w(s)| ds 2

τ

r t

t

Z

2

0Z

Z

2λ1

0

t

Z τ

 |w(s)|2 ds .

Gronwall lemma implies

    Z τ rLF ,M (r + 1)LF ,M |w(t )|2 ≤ |w(τ )|2 + |w(s)|2 ds · exp (t − τ ) , 2λ1 τ −r 2λ1

t ∈ [τ , T ],

which gives the uniqueness of solutions. (b) We omit the proof of the strong solution since the idea is essentially the same as previously. The proof of Theorem 3.1 is complete.  Remark 3.1. There are two reasons why we omit some details of the proof for the existence, uniqueness and regularity of PDEs with delay. First, the proof utilizes the standard Faedo-Galerkin method and is similar to [1]. Second, our emphasis is put on the asymptotical behavior of solutions, which is described in the next chapter.

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4. Existence of uniform attractor 4.1. Construction of the associated process Now we will apply the theory of Section 2 to prove the existence of an uniform attractor for our non-autonomous parabolic model with delay. Consider the symbol space Σ = H (g ), where H (g ) is the hull of g in L2loc (R, H ). And the symbol H applied in the sequel will always mean H (g ) unless stated otherwise. H is compact in L2loc (R, H ) since g is tr.c. in L2loc (R, H ). Consider the corresponding family of equations: d u(t ) + Au(t ) + bu(t ) = F (ut ) + g0 (t ), g0 ∈ H dt u(τ ) = u0 , u(τ + θ ) = φ(θ ), θ ∈ (−r , 0).

(

Consider F satisfying (I)–(II) and assume that u0 ∈ H, φ ∈ L2H and τ ∈ R. Then for any g0 ∈ H , Theorem 3.1 ensures that problem (4.1) possesses a unique solution u(· ; τ (u0 , φ), g0 ) which belongs to the space L2 (τ , T ; V ) ∩ C ([τ , T ]; H ) for all T > τ. Consider the product space MH2 = H × L2H where the initial data are taken. This space is a Hilbert space with associated norm

k(u0 , φ)k2M 2 = |u0 |2 + H

Z

0

|φ(s)|2 ds,

∀(u0 , φ) ∈ MH2 .

−r

On the one hand, we can define a process Ug (·, ·) : MH2 → MH2 in the product space as Ug (t , τ )(u0 , φ) = (u(t ; τ , (u0 , φ), g ), ut (·; τ , (u0 , φ), g )) ,

∀(u0 , φ) ∈ MH2 , τ ≤ t ,

and the corresponding family of processes as {Ug0 (·, ·) | g0 ∈ H }. On the other hand, we construct a family of mappings

e Ug0 (·, ·) : MH2 → CH given by

e Ug0 (t , τ )(u0 , φ) = ut (·; τ , (u0 , φ), g0 ),

∀(u0 , φ) ∈ MH2 , t ≥ τ + r

which can help us in the analysis of the long time behavior of our model. Of course, it is sensible to expect that both operators should be related. Let us consider the linear mapping j : φ ∈ CH 7→ j(φ) = (φ(0), φ) ∈ H × CH . This map is obviously continuous from CH into MH2 . Notice that for all (u0 , φ) ∈ MH2 it holds that e Ug0 (u0 , φ) ∈ CH provided that t ≥ τ + r, we then can write U (t , τ )(u0 , φ) = j(e U (t , τ )(u0 , φ)),

∀(u0 , φ) ∈ MH2 , t ≥ τ + r .

The following technical lemma relates the absorption properties for the mapping e Ug0 (·, ·) with those of process Ug0 (·, ·) in a such a way that, proving those for e Ug0 yields similar properties for Ug0 . Lemma 4.1. Assume that the bounded set B in CH is uniformly (w.r.t. g0 ∈ H ) absorbing (w.r.t. B (MH2 )) for the family of operators {e Ug0 (·, ·)|g0 ∈ H }. Then j(B) in H × CH is uniformly (w.r.t. g0 ∈ H ) absorbing (w.r.t. B (MH2 )) for the family of processes {Ug0 (·, ·)|g0 ∈ H }. Proof. Given D ∈ B (MH2 ), there exists T > r such that

e Ug0 (t , τ )(D) ⊂ B,

∀ t − τ ≥ T.

Taking into account that Ug0 (t , τ )(u0 , φ) = j(e Ug0 (t , τ )(u0 , φ)),

∀ t − τ ≥ T,

it follows that Ug0 (t , τ )(u0 , φ) = j(e Ug0 (t , τ )(u0 , φ)) ⊂ j(B),

∀ t − τ ≥ T. 

Remark 4.1. Notice that the word absorbing used here should be interpreted in a generalized sense, since e Ug0 (·, ·) is not a process. The term absorbing applied in the sequel will always mean absorbing w.r.t. B (MH2 ), i.e., the property of absorbing the bounded sets in MH2 unless stated otherwise.

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

2201

4.2. Existence of uniformly (w.r.t. g0 ∈ H ) absorbing set in CH and CV We denote u(·) = u(· ; τ , (u0 , φ), g0 ) when we consider the asymptotical behavior of solutions of problem (2.3) with the external forces g0 ∈ H ⊂ L2loc (R, H ). First, let us make an estimate of solutions. Lemma 4.2. If λ1 + b > k1 + k2 r, then any solution u(t ) of problem (4.1) with arbitrary symbol g0 ∈ H satisfies 2k3

|u(t )|2 ≤ e−c0 (t −τ ) |u0 |2 + 2k2 re−c0 (t −τ −r ) kφk2L2 +

c0

H

1 − e−c0 (t −τ ) +

1




1

2l0 1 − e−c0

|g |2b ,

(4.1)

where l0 , c0 are positive constants only depending on λ1 , k1 , k2 , r , b. Proof. As λ1 + b > k1 + k2 r, we can choose l0 > 0 small enough such that λ1 + b > k1 + k2 r + l0 . Then we take inner product with u(t ) and Eq. (2.3) and obtain 1 d 2 dt

|u(t )|2 + ku(t )k2 + b|u(t )|2 = (F (ut ), u(t )) + (g0 (t ), u(t )) ≤ k1 |u(t )| + k2 2

Z

0

|u(t + θ )|2 dθ + k3 + −r

1 4l0

|g0 (t )|2 + l0 |u(t )|2

(4.2)

and d dt

|u(t )| + 2(λ1 + b − k1 − l0 )|u(t )| ≤ 2k2 2

2

Z

0

|u(t + θ )|2 dθ + 2k3 + −r

Now we can also choose c0 > 0 such that λ1 + b > k1 + k2 rec0 r + l0 + d

c0 . 2

1 2l0

|g0 (t )|2 .

Then

d

(ec0 t |u(t )|2 ) = c0 ec0 t |u(t )|2 + ec0 t |u(t )|2 dt dt   Z c0  c0 t 2 ≤ 2e − λ1 + b − k1 − l0 − |u(t )| + k2 2

0

|u(t + θ )| dθ + k3 + 2

−r

1 4l0

|g0 (t )|

2

Integrating from τ to t, we have e

c0 t

c0 τ

|u(t )| − e 2

Z  c0  t c0 s |u(τ )| ≤ −2 λ1 + b − k1 − l0 − e |u(s)|2 ds 2

2

Z tZ + 2k2

τ

0

e τ

−r

Z tZ

0

c0 s

|u(s + θ )| dθ ds + 2

2k3 c0

τ

(e − e ) + t

Notice that

Z tZ

0

e τ

c0 s

|u(s + θ )| dθ ds ≤ e 2

r

τ

−r

≤e

r

0

Z

ec0 (s+θ) |u(s + θ )|2 dθ ds

−r

Z

−r

 ≤ rer eτ

t

τ −r Z τ

ec0 s |u(s)|2 dsdθ

τ −r

|u(s)|2 ds +

and t

Z τ

e−c0 (t −s) |g0 (s)|2 ds ≤

∞ Z X

≤

e − c0 i

≤

Z

t −i

|g0 (s)|2 ds t −i−1

i=0

≤

e−c0 (t −s) |g0 (s)|2 ds

t −i−1

i=0

∞ X

t −i

1 1 − e − c0 1 1 − e − c0

|g0 |2b |g |2b .

t

Z τ

ec0 s |u(s)|2 ds



1 2l0

t

Z τ

ec0 s |g0 (s)|2 ds.



.

2202

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

Thus, e

c0 t

|u(t )| − e 2

c0 τ

Z  c0  t c0 s r e |u(s)|2 ds |u(τ )| ≤ −2 λ1 + b − k1 − k2 re − l0 − 2 τ Z τ Z t 2k3 c0 t 1 |u(s)|2 ds + (e − ec0 τ ) + ec0 s |g0 (s)|2 ds. + 2k2 rer +τ 2

c0

τ −r

2l0

τ

Therefore,

|u(t )|2 ≤ e−c0 (t −τ ) |u0 |2 + 2k2 re−c0 (t −τ −r ) kφk2L2 +

2k3 c0

H

This completes the proof.

1 − e−c0 (t −τ ) +

1




1

2l0 1 − e−c0

|g |2b .



Theorem 4.1. Assume that (I) and (II) hold and g is tr.c. in L2loc (R, H ). If λ1 + b > k1 + k2 r, then there exists a bounded uniformly (w.r.t. g0 ∈ H ) absorbing set B1 ⊂ CH for the family of mappings {e Ug0 (·, ·)|g0 ∈ H }. Proof. Denote ρH2 =

4k3 c0

+

1 2 l0 1−e−c0

|g |2b . Given D ∈ B (MH2 ), by Lemma 4.2 there exists TH (D) > τ such that

ke Ug0 (t , τ )(u0 , φ)k2CH = max |u(t + s)|2 s∈[−r ,0]

≤ ≤

4k3 c0 4k3 c0

+ +

2 g (1 − e−1 ) 2 g (1 − e−1 )

|g0 |2b |g |2b = ρH2 ,

∀ t > T + r + τ , (u0 , φ) ∈ D, g0 ∈ H .

This means that the closed ball BCH (0, ρH ) = {ξ ∈ CH | kξ kCH ≤ ρH }, which we denote B1 , forms an uniformly (w.r.t. g0 ∈ H ) absorbing set for the mappings {e Ug0 (·, ·)|g0 ∈ H }.  To prove the existence of an uniformly (w.r.t. g0 ∈ H ) absorbing set in CV , we need a bound on the term first. Lemma 4.3. Given D ∈ B (MH2 ), there exist TH (D) and constant IV such that t +1

Z

ku(s)k2 ds ≤ IV ,

∀ t ≥ TH (D) + r + 1.

t

Proof. From Eq. (4.2) we can deduce 1 d 2 dt


1 ku(t )k2 + 1 + (k1 + l0 )λ− ku(t )k2 ≤ k2 1

0

Z

|u(t + θ )|2 dθ + k3 + −r

1 4l0

|g0 (t )|2 .

Integrating from t to t + 1 we obtain 1 2

1 (|u(t + 1)| − |u(t )|) + 1 + (k1 + l0 )λ− 1




t +1

Z

ku(s)k2 ds t

Z

t +1

0

Z

|u(s + θ )|2 dθ ds + k3 +

≤ k2 t

−r

1

t +1

Z

4l0

|g0 (s)|2 ds t

and 1 1 + (k1 + l0 )λ− 1




t +1

Z

ku(s)k2 ds ≤ k2 r t

sup

s∈[t +1,t −τ ]

|u(s)|2 + k3 +

Therefore, there exists TH (D) such that t +1

Z

ku(s)k2 ds ≤ IV ,

∀ t ≥ TH (D) + r + 1,

t

where IV =

1 1 1 + (k1 + l0 )λ− 1

 k2 r +

1 2



ρH2 + k3 +

1 4l0

 |g |2b . 

1 4l0

1

|g |2b + |u(s)|2 . 2

R t +1 t

ku(s)k2 ds

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

2203

Now we proceed in a similar way as we have already done in the previous subsection. Theorem 4.2. Under the assumptions in Theorem 4.1, if k1 < 1, then there exists a bounded uniformly (w.r.t. g0 ∈ H ) absorbing set B2 ⊂ CV for the family of operators {e Ug0 (·, ·)|g0 ∈ H }. Proof. Let D ∈ B (MH2 ). Choose l1 > 0 such that k1 + l1 < 1. Through a ‘‘formal’’ calculation (for details see [4], Lemma 11.2), we can take the inner product with Au(t ) to write 1 d 2 dt

ku(t )k2 + |Au(t )|2 + bku(t )k2 = (F (ut ), Au(t )) + (g0 (t ), Au(t )) Z 0 1 2 |u(t + θ )|2 dθ + k3 + ≤ k1 |Au(t )| + k2 |g0 (t )|2 + l1 |Au(t )|2 . 4l1

−r

Thus d dt

ku(t )k2 + 2 (1 − k1 − l1 ) |Au(t )|2 ≤ 2k2

0

Z

|u(t + θ )|2 dθ + 2k3 + −r

1 2l1

|g0 (t )|2

(4.3)

and d dt

ku(t )k ≤ 2k2 2

0

Z

|u(t + θ )|2 dθ + 2k3 + −r

1 2l1

|g0 (t )|2 .

(4.4)

Theorem 4.1 ensures the existence of TH (D) such that

|u(t )| ≤ ρH ,

for all t > τ + TH (D), g0 ∈ H .

Notice that when t > TH (D) + r + 1 we have t +1

Z

Z

0

|u(s + θ )|2 dθ ds ≤ 2k2 r ρH2 ,

2k2 −r

t t +1

Z

1 2l1

t

|g0 (t )|2 ≤

1 2l1

|g0 |2b ≤

1 2l1

|g |2b ,

and, by Lemma 4.3, t +1

Z

ku(s)k2 ds ≤ IV . t

Now, we can apply the uniform Gronwall lemma (see [6]) and obtain

ku(t )k2 ≤ IV + 2k2 r ρH2 + Denote ρV2 = IV + 2k2 r ρH2 +

1 2l1

1 2l1

|g |2b ,

for all t > τ + TH (D) + r + 1, (u0 , φ) ∈ D, g0 ∈ H .

|g |2b , then

ke Ug0 (t , τ )(u0 , φ)k2CV = max ku(t + θ )k2 θ∈[−r ,0]

≤ IV + 2k2 r ρH2 + ≤ ρV2 ,

1 2l1

|g |2b

for all t > τ + TH (D) + r + 1, g0 ∈ H .

Indeed, the set B2 = BCV (0, ρV ) is uniformly (w.r.t. g0 ∈ H ) absorbing for the family of operators {e Ug0 (·, ·)|g0 ∈ H }.



4.3. Existence and structure of the uniform attractor To prove the existence of the uniform attractor, we have to use some properties of tr.c. functions. First, let us introduce a class of functions that was defined first in [12]. Definition 4.1. A function ξ ∈ L2loc (R; E ) is said to be normal if for any  > 0 there exists η such that t +η

Z

kξ k2E ds < .

sup t ∈R

t

Denote by L2n (R; E ) the set all normal functions in L2loc (R; E ).

2204

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

Remark 4.2. Denote by L2c (R; E ) the class of translation compact functions. It is proved in [12] that L2n (R; E ) is a proper subset of L2c (R; E ), i.e., ξ is tr.c. implies ξ is normal. (For further details see [12].) Theorem 4.3. Assume that the nonlinear term F satisfy the assumptions (I)–(II) and g is tr.c. in L2loc (R, H ). If λ1 + b > k1 + k2 r and k1 < 1, then there exists a uniform attractor AH for the family of processes {Ug0 (·, ·)|g0 ∈ H }. Furthermore, AH ⊂ H × CH . Proof. Let us consider the set B2 . This is a bounded set in CV , which is also uniformly (w.r.t. g0 ∈ H ) absorbing for {e Ug0 (·, ·)|g0 ∈ H }. For each τ ∈ R, construct the set

[

B3 =

Ug0 (τ + r , τ )j(B2 ).

g0 ∈H

Thus, B3 ⊂ B2 ⊂ B1 and B3 is a bounded set in CV which is uniformly absorbing for {e Ug0 (·, ·)|g0 ∈ H }. Let us now prove that B3 is relatively compact in CH . To this aim, we will use the Ascoli–Arzela theorem, in other words, we have to check that B3 is equicontinuous and uniformly bounded. The uniform bound property follows form the definition of B3 immediately. Finally, in order to prove B3 is equicontinuous we proceed with the following estimate:

|e Ug0 (τ + r , τ )(j(φ))(θ1 ) − e Ug0 (τ + r , τ )(j(φ))(θ2 )| = |u(τ + r + θ1 ; τ , (j(φ)), g0 ) − u(τ + r + θ2 ; τ , (j(φ)), g0 )| for θ1 , θ2 ∈ [−r , 0], φ ∈ B2 , g0 ∈ H . Then we obtain(denote u(· ; τ , j(φ), g0 ) by u(·) as usual and assume θ2 > θ1 )

Z τ +r +θ 2 du (s)ds |u(τ + r + θ1 ) − u(τ + r + θ2 )| = τ +r +θ1 dt Z τ +r +θ2 du (s) ds ≤ Z

τ +r +θ1 τ +r +θ2

≤ τ +r +θ1

dt

(|Au(s)| + b|u(s)| + |F (us )| + |g0 (s)|)ds.

Next we will estimate the four terms on the right hand of the equation: (i) g (s) is tr.c. in L2loc (R, H ), then, according to [3] (P. 104, Proposition 3.4.), we have τ +r +θ2

Z

τ +r +θ1

|g0 (s)|ds ≤ sup

t +θ2 −θ1

Z

t ∈R

|g (s)|ds → 0 as θ1 → θ2 for all g0 ∈ H ,

t

and the limit holds since g (s) is normal from the Remark 4.2. (ii) From Eq. (4.3) we deduce

(1 − k1 − l1 )|Au(t )| ≤ k2 2

Z

0

|u(t + θ )|2 dθ + k3 + −r

1 4l1

|g0 (t )|2 .

Since

(1 − k1 − l1 )

Z

τ +r +θ2 τ +r +θ1

|Au(s)|2 ds ≤ k2

Z

0

τ +r +θ2

Z

−r

τ +r +θ1

|u(s + θ )|2 dsdθ + k3 (θ2 − θ1 ) +

≤ k2 r ρH2 (θ2 − θ1 ) + k3 (θ2 − θ1 ) + sup t ∈R

→ 0 as θ2 → θ1 , we have

Z

τ +r +θ2 τ +r +θ1

1

|Au(s)|ds ≤ (θ2 − θ1 ) 2 ·

Z

→ 0 as θ2 → θ1 . (iii) Evidently,

Z

τ +r +θ2 τ +r +θ1

b|u(s)|ds ≤ bρH |θ2 − θ1 |.

τ +r +θ2 τ +r +θ1

|Au(s)|2

 21

t +θ2 −θ1

Z t

1

Z

4l1

|g (s)|ds

τ +r +θ2 τ +r +θ1

|g0 (s)|2 ds

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

2205

(iv) From Eq. (2.2) we deduce

Z

τ +r +θ2

τ +r +θ1

τ +r +θ2

Z

|F (us )|ds ≤



0

Z

|u(s + θ )| dθ + k3 dθ

k1 + k2

τ +r +θ1



2

−r

≤ (k1 + k3 )(θ2 − θ1 ) + k2

0

Z

τ +r +θ2

Z

τ +r +θ1

−r

|u(s + θ )2 |dsdθ

≤ (k1 + k2 r ρH2 + k3 )(θ2 − θ1 ). Comprehensively,

|u(τ + r + θ1 ) − u(τ + r + θ2 )| → 0 as θ2 → θ1 , for all φ ∈ B2 , g0 ∈ H , which implies the needed equicontinuity. Since we prove B3 is relatively compact in CH , B3 , where the closure is taken in CH , is a compact uniformly (w.r.t. g0 ∈ H ) B3 = j(B3 ). Since H × CH ⊂ MH2 and the absorbing set in CH for the family of operators {e Ug0 (·, ·)|g0 ∈ H }. Let us consider e 2 injection is continuous, e B3 is a compact set in MH . Consequently, it is also a compact uniformly (w.r.t. g0 ∈ H ) absorbing set for {Ug0 (·, ·)|g0 ∈ H } in MH2 according to the Lemma 4.1. That ensures the existence of the uniform attractor AH for the family of processes {Ug0 (·, ·)|g0 ∈ H }. Furthermore, AH ⊂ H × CH . The proof of Theorem 4.3 is complete.  Lemma 4.4. The family of processes {Ug0 (·, ·)|g0 ∈ H } is uniformly (w.r.t. g0 ∈ H ) compact and (MH2 × H , MH2 )-continuous. Proof. The uniformly compact follows from the proof of Theorem 4.3 since {Ug0 (·, ·)|g0 ∈ H } processes a compact uniformly absorbing set e B3 . Let us prove the (MH2 × H , MH2 )-continuity. Consider two solutions u and v of (2.3) with symbols g1 and g2 and with initial data (u0 , φ) and (v0 , ψ). The difference w = u − v satisfies the equation d dt

w(t ) + Aw(t ) + bw(t ) = F (ut ) − F (vt ) + g1 (t ) − g2 (t ).

(4.5)

Choosing l1 > 0 such that l1 < λ1 + b and taking the inner product in H of (4.5) and w we obtain 1 d 2 dt

|w(t )|2 + kw(t )k2 + b|w(t )| = (F (ut ) − F (vt ), w(t )) + (g1 (t ) − g2 (t ), w(t )) ≤ |F (ut ) − F (vt )| · |w(t )| + |g1 (t ) − g2 (t )| · |w(t )|  l L F ,M  1 l1 1 ≤ kwt k2L2 + |w(t )|2 + |w(t )|2 + |g1 (t ) − g2 (t )|2 + |w(t )|2 2l1

2

H

2l1

and d dt

|w(t )|2 + 2(λ1 + b − l1 )|w(t )|2 ≤

L F ,M 

 1 kwt k2L2 + |w(t )|2 + |g1 (t ) − g2 (t )|2 .

l1

l1

H

Therefore d dt

|w(t )|2 ≤

L F ,M

Z

l1

0

|w(t + θ )|2 dθ + −r

t

Z τ

 1 |w(s)|2 ds + |g1 (t ) − g2 (t )|2 . l1

Integrating from τ to t we have

|w(t )|2 − |w(τ )|2 ≤

L F ,M

Z t Z

l1

τ

0

|w(s + θ )|2 dθ ds +

Z

|w(s)|2 ds + (r + 1)

Z

t

l1

τ

−r

 Z t 1 |w(s)|2 ds + |g1 (s) − g2 (s)|2 ds, τ

then,

|w(t )|2 ≤ |w(τ )|2 + = |w(τ )|2 +

L F ,M

 Z

t

r

l1

τ −r

rLF ,M l1

kwτ k2L2 ds + H

(r + 1)LF ,M

τ

 Z t 1 |w(s)|2 ds + |g1 (s) − g2 (s)|2 ds

t

Z

l1

t

|w(s)|2 ds +

τ

Note that g1 , g2 ∈ H ⊂ L2loc (R, H ). Therefore ∀ T > τ , we have t

Z τ

|g1 (s) − g2 (s)|2 ds ≤

T

Z τ

|g1 (s) − g2 (s)|2 ds < ∞,

∀ t ∈ [τ , T ].

l1

1 l1

Z τ

τ

t

|g1 (s) − g2 (s)|2 ds.

2

2206

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

Thus



|w(t )| ≤ |w(τ )| + 2

2

rLF ,M l1

kwτ k

2 ds L2H

+

1

T

Z

l1



|g1 (s) − g2 (s)| ds + 2

τ

(r + 1)LF ,M l1

t

Z τ

|w(s)|2 ds,

∀ t ∈ [τ , T ].

Applying Gronwall lemma, we obtain that



|w(t )| ≤ |w(τ )| + 2

2

rLF ,M l1

kwτ k

2 ds L2H

+

1

T

Z

l1

τ



|g1 (s) − g2 (s)| ds · exp 2



(r + 1)LF ,M l1



(t − τ ) .

Finally,

kut − vt k2M 2 = |u(t ) − v(t )|2 + kut − vt k2L2 H

H

≤ |u(t ) − v(t )|2 + r · sup |u(s) − v(s)|2 s∈[t −r ,t ]

 rLF ,M kφ − ψk2L2 ds ≤ (r + 1) |u0 − v0 |2 + l1

+

1

T

Z

l1

|g1 (s) − g2 (s)| ds · exp 2

τ

H





(r + 1)LF ,M l1

 (t − τ ) ,

∀ t ∈ [τ , T ].

Hence, for any fixed t and τ , t > τ , kut − vt kM 2 → 0 if k(u0 , φ) − (v0 , ψ)kM 2 → 0 and kg1 − g2 kL2 (R,H ) → 0. This finishes H H loc the proof.  Thus, all the requirements of Theorem 2.2 are fulfilled for the family of processes {Ug0 (·, ·)|g0 ∈ H } corresponding to problem (2.3). Construct the semigroup {S (t )|t ≥ 0} as (2.5). Finally, we have proved the following: Proposition 4.1. Under the assumptions of Theorem 4.3, the semigroup {S (t )} corresponding to the family of processes {Ug0 (·, ·)|g0 ∈ H } processes the compact attractor A, which is strictly invariant with respect to {S (t )}: S (t )A = A. Moreover, (i) PM 2 A = AH is the uniform (w.r.t. g0 ∈ H ) attractor of the family of processes {Ug0 (·, ·)|g0 ∈ H }; H

(ii) PH A = H ; (iii) the global attractor satisfies

A=

[

Kg0 (0) × {g0 };

g0 ∈H

(iv) the uniform (w.r.t. g0 ∈ H ) attractor AH of the family of processes {Ug0 (·, ·)|g0 ∈ H } coincides with the uniform (w.r.t. τ ∈ R) attractor A0 of the process {U (t , τ )}:

A0 = AH =

[

Kg0 (0),

g0 ∈H

where Kg0 (0) is the section at t = 0 of the kernel Kg0 of the process {Ug0 (t , τ )} with symbol g0 ∈ H . Remark 4.3. Although our model concerns the tr.c. function in L2loc (R, H ), our analysis can be extended to deal with a more general non-autonomous term which is a tr.c. function in corresponding complete metric space such as Cb (R; H ). The technique we have used in the previous subsections can be performed to treat this case in a straightforward way. 5. Application Consider the non-local PDE model for population dynamics with distributed state-dependent delay and almost periodic external force:

∂ u(t , x) + Au(t , x) + d0 u(t , x) = ∂t

Z

0

−r

Z



Ω

b(u(t + θ , y))f (x − y)dy ξ (θ , u(t ), ut )dθ + g1 (t , x)

≡ F (ut )(x) + g1 (t ),

x ∈ Ω.

For the prototype, we refer to Rezounenko and Wu’s model [1]. Follow the assumptions by [1]: (i) b : R → R is locally Lipschitz and there exist constants C1 and C2 so that |b(w)| ≤ C1 |w| + C2 for all w ∈ R, (ii) f : Ω → R is bounded,

(5.1)

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

2207

(iii) ξ : [−r , 0] × H × L2H → R is locally Lipschitz with respect the second and third coordinates, i.e., for any M > 0 there

R0

exists Lξ ,M so that for all θ ∈ [−r , 0] and all (v i , ψ i ) ∈ H × L2 (−r , 0; H ) satisfying kv i k2 + −r kψ i (s)k2 ds ≤ M 2 , i = 1, 2 one has



|ξ (θ , v , ψ ) − ξ (θ , v , ψ )| ≤ Lξ ,M · kv − v k + 1

1

2

2

1

2 2

Z

0

kψ (s) − ψ (s)k ds 1

2

2

 21

,

(5.2)

−r

and there exists Cξ > 0 so that

kξ (· , v, ψ)kL2 (−r ,0) ≤ Cξ for all (v, ψ) ∈ H × L2H .

(5.3)

In addition, we assume (iv) g1 (t ) is an almost periodic function with values in H. Next, we will prove that the nonlinear term F which contains the state-selective delay satisfies our hypotheses (I)–(II). Proposition 5.1. Under the assumptions (i)–(iii) in hypotheses (I) and (II).

[1], the nonlinear term F of the nonlocal PDE model satisfies our

Proof. To avoid confusion we denote k·kL2 by the norm of L2 (Ω ). First we will show F satisfies the local Lipschitz continuity for the initial data. ∀M > 0, u, v ∈ BM 2 (0, M ), we have H

Z

=

kF (ut ) − F (vt )kL2

0

Z

Z

≤

b(u(t + θ , y))f (x − y)dy ξ (θ , u(t ), ut )dθ

Z 0

b(v(t + θ , y))f (x − y)dy ξ (θ , v(t ), vt )dθ

2 −r Ω L

Z  0

b(u(t + θ , y))f (x − y)dy (ξ (θ , u(t ), ut ) − ξ (θ , v(t ), vt )) dθ

Z −

Ω

−r



Ω

−r

Z

+



0

−r

Z Ω

L2

(b(u(t + θ , y)) − b(v(t + θ , y))) f (x − y)dy ξ (θ , v(t ), vt )dθ

2. 

L

Note that

Z



0

b(u(t + θ , y))f (x − y)dy (ξ (θ , u(t ), ut ) − ξ (θ , v(t ), vt )) dθ

2 Ω L Z    12 0

2 2 dθ (C1 |u(t + θ , y)| + C2 ) f (x − y)dy · Lξ ,M ku(t ) − v(t )kL2 + kut − vt kL2

Z

−r

Z

≤



Ω

−r

H

0

1

Z

1



≤ Mf Lξ ,M |Ω | 2

Ω

−r

≤ Mf Lξ ,M |Ω | 2

  12 (C1 |u(t + θ , y)| + C2 ) dydθ · ku(t ) − v(t )k2L2 + kut − vt k2L2

Z Z

H

0

C1 −r



≤ Mf Lξ ,M |Ω | C1 kut kL2

H

   12 ku(t + θ )kL1 dθ + C2 r |Ω | · ku(t ) − v(t )k2L2 + kut − vt k2L2 H

   21 1 + C2 r |Ω | 2 · ku(t ) − v(t )k2L2 + kut − vt k2L2 , H

where |Ω | = Ω 1ds, Mf = max{|f (x)| : x ∈ Ω }. Consider the second term

R

Z



0

(b(u(t + θ , y)) − b(v(t + θ , y))) f (x − y)dy ξ (θ , v(t ), vt )dθ

2 −r Ω L

Z 0 Z

|b(u(t + θ , y)) − b(v(t + θ , y))| · f (x − y)dydθ ≤ Cξ

Z



−r

Ω

≤ Cξ Lb,d Mf |Ω |

L2

1 2

Z

0

−r 1

≤ Cξ Lb,d Mf |Ω | 2

Z

Z Ω

|u(t + θ , y) − v(t + θ , y)|dydθ

0

−r

ku(t + θ ) − v(t + θ )kL1 dθ

L2

2208

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

≤ Cξ Lb,d Mf |Ω |

0

Z

ku(t + θ ) − v(t + θ )kL2 dθ −r

1

≤ Cξ Lb,d Mf |Ω |r 2 kut − vt kL2 . H

Comprehensively,

  |F (u) − F (v)|2 ≤ LF ,M ku(t ) − v(t )k2L2 + ku − vk2L2 , H

n

1 2

where LF ,M = max Cξ Lb,M Mf |Ω |r ,



1

Mf Lξ ,M |Ω | C1 M + C2 r |Ω | 2

o2

. And that implies the local Lipschitz continuity of

nonlinear term F for initial data. The hypothesis (II) follows from (14) in [1]. Namely,

|(F (ut ), v)| ≤ ≤

1 2 1 2



Mf |Ω | kvk2L2 + 2C12 Mf |Ω |Cξ



−r

kvk2L2 + 2C12

≤ k1 kvk2L2 + k2 The proof is complete.

0

Z

Z

Z

 ku(t + θ )k2L2 dθ + 2|Ω |C22 r kξ (· , u(t ), ut )kL2 (−r ,0) 0

−r

ku(t + θ )k2L2 dθ + 2|Ω |C22 r

(14[1] )



0

−r

ku(t + θ )k2L2 dθ + k3 .



Consider Eq. (5.1) with the following initial conditions: u(τ ) = u0 ∈ H ,

u|(τ −r ,τ ) = φ ∈ L2H .

(5.4)

The next result is similar to Theorem 3.1. Proposition 5.2. Problem (5.1), (5.4) has (for each τ ∈ R) a unique weak solution, that is, u ∈ L2 (τ − r , T ; H ) ∩ L2 (τ , T ; V ) ∩ L∞ (τ , T ; H ) ∩ C ([τ , T ]; H ). By the above proposition, for any τ ∈ R we can define a process Ug1 (·, ·) : MH2 → MH2 in the product space as Ug1 (t , τ )(u0 , φ) = (u(t ; τ , (u0 , φ), g1 ), ut (· ; τ , (u0 , φ), g1 )) ,

∀(u0 , φ) ∈ MH2 , τ ≤ t .

It is well-known that by the Bochner-Amerio criterion an almost periodic function g1 (t ) possesses the following characteristic property: the set of all its translations {g1 (·, +s)|s ∈ R} forms a precompact set in Cb (R; H ) (see, for example, Amerio and Prouse [13]). The hull H (g1 ) of the function g1 (t ), defined as the closure in Cb (R; H ) of its translations, is compact in Cb (R; H ). Thus, g1 (t ) is a tr.c. function in Cb (R; H ). Let the symbol space Σ = H (g1 ). Now we can define the corresponding family of processes as {Ug0 (·, ·) | g0 ∈ H (g1 )} and construct the semigroup {S (t )|t ≥ 0} as (2.5). Thus, we can apply our model to obtain the following result. Proposition 5.3. If λ1 > k1 + k2 r and k1 < 1, the semigroup {S (t )} corresponding to the family of processes {Ug0 (·, ·) | g0 ∈ H (g1 )} processes the compact attractor A, which is strictly invariant with respect to {S (t )}: S (t )A = A. Moreover, (i) PM 2 A = AH (g1 ) is the uniform (w.r.t. g0 ∈ H (g1 )) attractor of the family of processes {Ug0 (·, ·) | g0 ∈ H (g1 )}; H

(ii) PH A = H (g1 ); (iii) the global attractor satisfies

A=

[

Kg0 (0) × {g0 };

g0 ∈H (g1 )

(iv) the uniform (w.r.t. g0 ∈ H (g1 )) attractor AH (g1 ) of the family of processes {Ug0 (·, ·)|g0 ∈ H } coincides with the uniform (w.r.t. τ ∈ R) attractor A0 of the process {Ug1 (t , τ )}:

A0 = AH (g1 ) =

[

Kg0 (0),

g0 ∈H (g1 )

where Kg0 (0) is the section at t = 0 of the kernel Kg0 of the process {Ug1 (t , τ )} with symbol g0 ∈ H (g1 ). In particular when the external force g (t , x) ≡ 0, problem (5.1) reduces to Rezounenko and Wu’s model which is a autonomous case. Finally we have the following result.

J. Li, J. Huang / Nonlinear Analysis 71 (2009) 2194–2209

2209

Proposition 5.4. Assume that (i)–(iii) hold and g (t , x) ≡ 0. If, in addition, λ1 > k1 + k2 r and k1 < 1, then the semigroup {S (t )} corresponding to problem (5.1), (5.4) has the global attractor A. Furthermore, the global attractor has the structure:

A = K (0), where the kernel K for (5.1) is the union of all complete bounded solutions (u(t ), ut (·)) of (5.1). Remark 5.1. Concerning the Rezounenko and Wu’s model, there are two differences between our result and their’s. First, our dynamical system is constructed by the weak solution in comparison with that by the mild solution. Thus, our global attractor should be different from their’s. Second, compared to the Rezounenko and Wu’s restriction of parameters (α ≥ 12 and the small enough ρ such that C1 r ≤ ρ ), our parameters region can provide a better understanding (since λ1 , the first eigenvalue of operator A, reflects the averaged spatial effects according to [14]). References [1] A.V. Rezounenko, Jianhong Wu, A non-local PDE model for population dynamics with state-selective delay: Local theory and global attrators, J. Comput. Appl. Math. 190 (2006) 99–113. [2] T. Caraballo, J. Real, Attractors for 2D-Navier–Stokes models with delays, J. Differential Equations (2004) 271–297. [3] V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, AMS, 2002. [4] J.C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, 2001. [5] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., Providence, RI, 1998. [6] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, second ed., Springer, New York, 1997. [7] I.D. Chueshov, Introduction to the theory of infinite-dimensional dissipative systems, Acta (2002). [8] A. Haraux, Systemes Dynamiques Dissipatifs et App;ocations, Masson, Paris, 1991. [9] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. [10] A.V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, J. Math. Anal. Appl. 326 (2007) 1031–1045. [11] A.V. Rezounenko, A Short Introduction to the Theory of Ordinary Delay Differential Equation, in: Lecture Notes, Kharkov Univ. Press, Kharkov, 2004. [12] S.S. Lu, H.Q. Wu, C.K. Zhong, Attractors for non-autonomous 2D Navier–Stokes equations with normal external forces, Discrete Contin. Dyn. Syst. 13 (3) (2005) 701–719. [13] L. Amerio, G. Prouse, Abstract Almost Periodic Functions and Functional Equations, Van Nostrand, New York, 1971. [14] R.S. Cantrell, C. Cosner, Spatial Ecology via Reaction–Diffision Equations, John Wiley & Sons, 2003. [15] T. Caraballo, J. Real, Navier–Stokes equations with delays, Roy. Soc. London Proc. Ser. A Math. Phys. Eng. Sci. 457 (2014) (2001) 2441–2453. [16] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dynam. Differential Equations 9 (2) (1995) 307–341. [17] H.T. Song, C.K. Zhong, Attractors of non-autonomous reaction-diffusion equations in Lp , Nonlinear Anal. 68 (2008) 1890–1897.