On the dynamics of equations with infinite delay - De Gruyter

DOI: 10.2478/s11533-006-0024-7 Research article CEJM 4(4) 2006 635–647

On the dynamics of equations with infinite delay∗ Dalibor Praˇz´ak† Charles University Prague, 186 75 Prague 8, Czech Republic

Received 3 May 2006; accepted 6 June 2006 Abstract: We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.  Keywords: Equations with infinite delay, exponential attractor, fractal dimension MSC (2000): 37L25, 37L30, 34K17

1

Introduction

There are many examples of models-arising in areas such as biology, economics, and materials science-in which the rate of change at time t depends not only on the present state of the system, but also on its history during some time interval [t − τ, t]. Mathematically, these models are characterized by evolutionary equations with delay. The more general case τ = ∞ yields so-called equations with infinite delay. There is extensive literature on the study of various aspects of equations with delay. However, it seems to us that, in most papers, a rather specific form of equation is studied. From the specific form, one can usually obtain a correspondingly specific description of the dynamics at hand. Yet, the general case-delayed equations in abstract form-remains to be subjected to more study. Herein lies the objective of the present paper. We consider systems of ordinary differential equations (ODEs) with infinite delay, where the underlying phase space X is characterized by an exponentially decaying weight. ∗ †

Research supported by the project LC06052 of the Czech Ministry of Education. E-mail: [email protected]ff.cuni.cz

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The main assumption is the Lipschitz continuity of the right-hand side with respect to the norm of X. Note that systems with bounded delay, or equations where the dependence on the past is given by convolution with an exponentially decaying kernel, are special cases of our general formulation. (More on the latter class of models follows.) Our main theorem is the existence of an exponential attractor. We also estimate its fractal dimension in terms of relevant constants. The possible applications of our results are twofold. On the one hand, the technique we use is based on the ”short trajectory” method (cf. Lemma 4.2), which, from what we know, occurs for the first time in [10]. This approach is very elementary and it is not difficult to see that the extension to partial differential equations (PDEs) is possible under suitable conditions. We believe that, in fact, the approach suggests a more general method of constructing exponential attractors, or finite-dimensional attractors-attractors that will cover a wide class of nonlinear dissipative evolutionary PDEs with infinite delay. The idea of constructing attractors in the space of trajectories is not new. Note that, in [1, 2], the existence of ”trajectory attractors” is proved for certain classes of PDEs with infinite delay. With regard to the finite-dimensionality of attractors, we also build on the approach of [11], where problems with bounded delay are studied. On the other hand, we find it no less important to consider the dynamics of ODEs with infinite delay. We observe that a general class of dissipative PDEs (without delay) can, using a suitable projection, be reduced to a system of ODEs with infinite delay, a system that preserves the large time dynamics of the original PDEs [5, 9, 13]. We discuss these problems in the last section. Let us now compare our results with other recent publications and also try to put them into a somewhat broader perspective. In [7], we find interesting results for a system in which dependence on the past is realized by convolution with kernels of the form k (s) =

1 s k0 ,  

(1)

where k0 (·) has exponential decay. The authors show that, for every  ∈ (0, 0 ], the problem has an exponential attractor E . In addition, the authors give rigorous proof of the expectation that, for  → 0+, E approach the attractor of the undelayed system. In our general formulation, we have not been able to recover such a specific result, but have identified an interesting analogy (at least a formal one). The estimate of the dimension of the attractor we obtain depends only on L/γ, where L is the Lipschitz constant for the dependence on the past, and γ measures the rate of the decay of the corresponding weight as s → −∞. From (1), one has L ∼ −1 , γ ∼ −1 , so our estimate remains bounded even if  → 0. Finally, we note that the presence of infinite delay essentially makes the problem hyperbolic in character. This is seen explicitly in the setting where the dependence on the past is given by convolution with a sufficiently well-behaved kernel. One can then rewrite the problem in terms of a new function-a so-called ”summed past history” (see [4])-that satisfies a particular hyperbolic equation.

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While, in our abstract formulation, we have not recovered a certain specificity, we have found a similarity between our approach and [12] or [3], where a nonlinear hyperbolic problem is treated using a similar kind of squeezing property. The paper is organized as follows: In section 2, we describe our system and fix the notation. Section 3 presents preliminary results concerning the theory of exponential attractors. It also gives two covering lemmas that are needed for the explicit dimension estimate. The main result-the existence of an exponential attractor for the system of delayed ODEs and an explicit dimension estimate-is proved in section 4 (Theorem 4.5). In the last section, we briefly discuss an application to systems that arise as suitable projections of dissipative PDEs.

2

Equation

We consider a system of ODEs with infinite delay. Express the system in the form dt p(t) = F (pt ) ,

(2)

where p(t) : R → RM is the unknown function. A natural phase-space for the problem is   X = χ(s) : (−∞, 0] → RM ; χ(s) is continuous , endowed with the norm (γ > 0) χX = sup |χ(s)|eγs .

(3)

s≤0

Throughout the paper, we adopt the convention that, if p(s) is a time-dependent function, then pt is an element of X defined by pt (s) = p(t + s) ,

s ≤ 0.

Our equation is equipped with the initial condition p0 = χ ∈ X .

(4)

We assume that F : X → RM is Lipschitz continuous. That is, |F (χ) − F (χ)| ˜ ≤ Lχ − χ ˜ X

χ, χ˜ ∈ X .

(5)

It is not difficult to adapt standard ODE theory to prove that (2), (4) is well posed and that one has the unique global solution p(t) : R → RM . Denoting St χ = p t , we define the solution operator St : X → X to problem (2). In our abstract formulation, we assume that there exists a more regular set   B = χ ∈ X; |χ(s)| ≤ K1 , Lip χ ≤ K2 ,

(6)

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It is positively invariant and uniformly absorbing, or at least uniformly exponentially attracting for (St , X). This is a natural assumption with regard to applications (cf. the last section). Hence, it will suffice to construct the exponential attractor for the dynamical system (St , B). The dimension estimate we derive will be in terms of M , γ and L, but independent of K1 , K2 . The convention is that c1 , c2 , . . . are generic constants that are independent of M , γ and L, and that change in meaning, depending on context. The reasonable assumption L ≥1 (7) γ simplifies certain expressions.

3

Preliminaries

Let X be a bounded, complete metric space. Typically, X is a closed subset of a Banach space from which it inherits a metric. The fractal dimension of A ⊂ X is defined as dimfX (A) = lim sup →0+

ln NX (A, ) − ln 

where NX (A, ) is the smallest number of closed sets of diameter 2 that cover A. We assume that St : X → X is a semigroup of operators, S0 = I and St+s = St Ss . The set E ⊂ X is called an exponential attractor to (St , X) provided (1) E is compact, (2) St (E) ⊂ E, t ≥ 0, (3) dimfX (E) < ∞, (4) distX (St X , E) ≤ c exp(−βt), t ≥ 0. As is well-known, the concept of an exponential attractor was proposed in [6] in response to specific defects associated with the usual notion of the global (universal) attractor. A primary feature of the exponential attractor is its robustness with respect to various approximations, including singular perturbations (see [8]). Extensive literature focuses on proving the existence of exponential attractors for various dissipative equations. It turns out that, as soon as one can prove that the global attractor has finite fractal dimension, one can almost always construct an exponential attractor. If S : X → X is a continuous mapping, the concept of exponential attractor extends in an obvious way to the (discrete) dynamical system (S n , X ). One usually first constructs the exponential attractor for a certain discrete subgroup of St , generated by S = St∗ with suitably-chosen t∗ . The following lemma gives a useful sufficient condition. Lemma 3.1. Let S : X → X be Lipschitz continuous, and let there exist θ ∈ (0, 1) and a constant K > 0 such that NX (S(F ), θρ) ≤ K (8)

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for any ρ > 0, F ⊂ X with diamX (F ) ≤ 2ρ. Then, the dynamical system (S n , X ) has an exponential attractor E ∗ , and ln K . (9) dimfX (E ∗ ) ≤ − ln θ Proof. Starting with F = X , R = diam(X )/2, one obtains NX (S n (X ), θn R) ≤ K n .

(10)

by induction. However, from this condition-which is, in fact, equivalent to the existence of the exponential attractor-one can employ a standard construction to obtain the conclusion of the lemma, including the explicit estimate (8). See e.g. [11, Theorem 1.1] for details.  Once the exponential attractor is constructed for the discrete subgroup, it is a matter of routine to extend it for the entire dynamics. Lemma 3.2. Let E ∗ be an exponential attractor for (S n , X ), where S = St∗ with some fixed t∗ > 0. Assume that St x is locally Lipschitz continuous w.r. to t and x. Then there exists an exponential attractor E to (St , X) such that dimfX (E) ≤ dimfX (E ∗ ) + 1 .

(11)

Proof. Set E = F(E ∗ × [0, t∗ ]), where F : (x, t) → St x. Given that F is Lipschitz, one has dimfX (E) ≤ dimfX (E ∗ ) + dimfR ([0, t∗ ]) = dimfX (E) + 1. It is straightforward to check the other properties of the exponential attractor. (See [6, Chapter 3].)  Since we want to estimate the dimension of E explicitly, we need to compute some coverings. Lemma 3.3. Let r ≤ R. It follows that BRM (0, R) can be covered by m balls of radius r, where  M 3R m≤ . (12) r Proof. Let Bj , j = 1, . . . m be a maximal system of disjoint balls of radius r/2, centered in BRM (0, R). Clearly, the same balls with radius r cover BRM (0, R). Otherwise, the system is not maximal. Moreover, Bj ⊂ BRM (0, R + r/2) and, by volume comparison,  r M

r M m ≤ R+ 2 2    M M 2R 3R m≤ ≤ , +1 r r using 1 ≤ R/r.





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Lemma 3.4. Let A ≥ 1/5, B > 0. The set   M = χ ∈ C([0, t∗ ]); |χ(t)| ≤ A , Lip χ ≤ B can be covered by K balls of radius 1 in the space C([0, t∗ ]), where ln K ≤ c M (t∗ B + 1) ln(A + 1) .

(13)

Proof. Choose points t0 = 0 < t1 < · · · < tn = t∗ such that ti+1 − ti ≤ δ := 1/(5B). This can be accomplished with n ≤ t∗ /δ + 1 ≤ 5Bt∗ + 1. Furthermore, there exist points xj , j = 1, . . . m in RM such that BRM (xj , 1/5) cover BRM (0, A). According to Lemma 3.3, one has m ≤ (15A)M . Now, consider the set N of functions χ(t) : [0, t∗ ] → RM such that χ(t) takes only the values xj for t = ti and is linear in [ti , ti+1 ]. Clearly, if K = #N , one has ln K = ln mn+1 ≤ M (5t∗ B + 2) ln 15A ≤ c M (t∗ B + 1)(ln A + 1) . The proof will be finished once we show that the balls B(χ, 1), χ ∈ N cover the set M. Let ψ ∈ M be arbitrary. From the preceding discussion, there exists χ ∈ N such that |χ(ti ) − ψ(ti )| ≤ 1/5, i = 0, . . . , n. We claim that |χ(t) − ψ(t)| ≤ 1 for all t ∈ [0, t∗ ]. It suffices to show this for fixed t ∈ [ti , ti+1 ], i. One has |χ(t) − ψ(t)| ≤ |χ(t) − χ(ti )| + |χ(ti ) − ψ(ti )| + |ψ(ti ) − ψ(t)| ≤ |χ(t) − χ(ti )| + 1/5 + δB ≤ |χ(t) − χ(ti )| + 2/5 . Yet, as a consequence of the piecewise linearity of χ, one has |χ(t) − χ(ti )| ≤ |χ(ti+1 ) − χ(ti )|. Also, |χ(ti+1 ) − χ(ti )| ≤ |χ(ti+1 ) − ψ(ti+1 )| + |ψ(ti+1 ) − ψ(ti )| + |ψ(ti ) − χ(ti )| ≤ 1/5 + δB + 1/5 ≤ 3/5 . 

The proof is finished.

4

Exponential attractor

In this section, we establish our main result: we construct an exponential attractor to (2) and give the estimate of its fractal dimension. The dissipation that is encoded in our equation is described in the following lemma. On the surface, it looks more like an assertion about the space X, and not about the equation itself. Lemma 4.1. Let η : (−∞, t] → RM be given. It follows that η t X ≤ e−γt η 0 X + ηC([0,t]) .

(14)

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Proof. By (3), η t X = sup |η(t + s)|eγs s≤0   = max sup |η(t + s)|eγs , sup |η(t + s)|eγs s≤−t

s∈[−t,0]

  ≤ max e−γt η 0 X , ηC([0,t]) ≤ e−γt η 0 X + ηC([0,t]) .

 Note that (14) is a kind of ”squeezing property”. To handle the second term on its right-hand side, we need the continuity of the evolution operator. Without concrete assumptions on its nonlinearity, the only available method is the long-established ”shorttrajectory trick” used in [10]. Lemma 4.2. Set

1 , (15) 2L where L is the Lipschitz constant from (5). Let p and q be solutions to (2), and z = p − q. Then (16) zC([0,]) ≤ 3z 0 X . =

Proof. Subtract the equations for p and q, and multiply by z/|z| to get dt |z(t)| ≤ |F (pt ) − F (q t )| ≤ Lz t X ≤ Lz 0 X + LzC([0,]) ,

∀t ∈ [0, ]

using (14). Pick t0 ∈ [0, ] such that |z(t0 )| = zC([0,]) . Integration over t ∈ [0, t0 ] yields |z(t0 )| ≤ |z(0)| + Lz 0 X + L|z(t0 )| (1 − L)|z(t0 )| ≤ (1 + L)z 0 X as |z(0)| ≤ z 0 X . Given that L = 1/2, the conclusion follows.



Using the equation once more, we estimate the difference of the time derivatives. Thus, as a result of the compact embedding of C 1 ([0, ]) into C([0, ]), we establish a kind of ”smoothing property”. Lemma 4.3. Using the previous notation, one finds dt zC([0,]) ≤ 3L z 0 X . Proof. Note that, in the proof of Lemma 4.1, we established   z t X ≤ max e−γt z 0 X , zC([0,t]) .

(17)

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Taken together with (5) and (16), we have dt zC([0,]) ≤ sup |F (pt ) − F (q t )| t∈[0,]

≤ L sup z t X t∈[0,]

  ≤ L sup max e−γt z 0 X , zC([0,t]) t∈[0,]

≤ 3Lz 0 X .  Now we can establish the ”discrete version” of the exponential attractor. Theorem 4.4. Set S = S . Then the semigroup (S n , B) has exponential attractor E ∗ , and dimfX (E ∗ ) ≤ c M (γ −2 L2 + 1) ln(γ −1 L + 1) . (18) Proof. We will use Lemma 3.1. Let F ⊂ B, diamX (F ) ≤ 2ρ. We cover S(F ) with sets Fj , j = 1, . . . , K that have diamX (Fj ) ≤ 2ρθ, where 1 θ = (1 + e−γ ) , 2

(19)

and K is independent of F , ρ. Clearly, F ⊂ BX (ψ0 , 2ρ), with ψ0 ∈ F arbitrary. By Lemmas 4.2 and 4.3, the set F |[0,] is contained ‡ in ψ0 |[0,] + M, where   M = χ ∈ C([0, ]); |χ(t)| ≤ 6ρ , Lip χ ≤ 6Lρ . We can find ηj ∈ F , j = 1, . . . , K such that the sets   Gj = ψ ∈ F ; ψ − ηj C([0,]) ≤ ρ cover F , where 1 (20)  = (1 − e−γ ) . 2 According to the scaling argument, this covering is equivalent to the problem of Lemma 3.4 with A = 6−1 , B = 6L−1 . Note that, since (7), γ ≤ 1/2 and, consequently,  ≤ 0.2. Thus, A ≥ 1/5, as required in Lemma 3.4. Hence, the number K is estimated by ln K ≤ c M ( L−1 + 1) ln(−1 + 1) .

(21)

Clearly, Fj = S(Gj ) cover S(F ). We claim that diamX (Fj ) ≤ 2ρθ. Let ψ, ψ˜ ∈ Gj be arbitrary. Using (14) with t = , from (19) it follows that (20) ˜ X + ψ − ψ ˜ C([0,]) ˜ X ≤ (θ − )ψ − ψ S(ψ) − S(ψ) ≤ (θ − )2ρ + 2ρ = 2θρ . ‡

To simplify the notation, we implicitly extend the elements of X to t ∈ [0, ] by solving (2).

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There remains the matter of evaluating the estimate for K. Using (7), (15) −1 =

2 γ ≤c . −γ/2L 1−e L

Similarly, − ln θ = − ln[(1 + e−γ/2L )/2] ≥ c

γ . L 

Hence, (9) and (21) yield (18). The main result now follows readily.

Theorem 4.5. Let the assumptions of section 2 hold. Then the dynamical system (St , X) associated with (2) has exponential attractor E. Its fractal dimension is estimated as dimfX (E) ≤ c M L2 γ −2 ln(Lγ −1 + 1) .

(22)

Proof. By a previous theorem, we have an exponential attractor E ∗ to (S n , B), where S ∗ = S . We want to apply Lemma 3.2, thus reducing the problem to the verification of the Lipschitz continuity of St χ w.r. to both χ and t. However, by (14) and (16), we have (t ∈ [0, ]) St χ − St χ ˜ X ≤ e−γt χ − χ ˜ X + χ − χ ˜ C([0,]) ≤ 4χ − χ ˜ X. For continuity in time, we observe St χ − StˆχX = sup |χ(t + s) − χ(tˆ + s)|eγs ≤ K2 |t − tˆ| , s≤0

where χ is an element of B. There remains the matter of evaluating the dimension estimate. Inasmuch as (7), we substitute (18) into (11), yielding (22). 

5

Application to projected PDEs

In the last section, we discuss an interesting application of the previous results. We look at a system of ODEs with infinite delay that arises as an equivalent description of the large time dynamics of an evolutionary PDE. Consider an abstract dissipative PDE in the form dt u + Au + R(u) = f .

(23)

Here, u = u(t) ∈ H is the unknown, where H is a suitable Hilbert space, A is a linear elliptic operator, and R is some lower-order nonlinearity. The dimension of H is infinite, and the relevant question is whether the large time behavior of the solutions is, in some sense, finite-dimensional. A partial answer-one that holds for numerous physically-interesting instances of (23)-is given by the existence of a global attractor A with finite (fractal) dimension (cf. [14]).

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The best kind of attractor is the so-called inertial manifold. Its existence implies that the large time dynamics of a given PDE can be described by a system of ODEs. Unfortunately, the known conditions under which this manifold can be constructed (e.g. large gaps in the spectrum of A) are too restrictive to cover a sufficiently wide class of problems. An important example where the existence of an inertial manifold remains an open problem is given by the 2d Navier-Stokes equations. It turns out, however, that, under general conditions, one can rewrite the large time dynamics of (23) as a system of ODEs with infinite delay. These so-called ”inertial manifolds with delay” were first treated in [5] (see also [9]). Here, following [13], we outline an alternate construction. We omit most technical details. Nonetheless, the key assumption that enables the construction is the squeezing property: there exist θ ∈ (0, 1), t∗ , c1 > 0 and finite-dimensional projector P such that u(t + t∗ ) − v(t + t∗ )H ≤ c1 P (u(t + t∗ ) − v(t + t∗ ))H + θu(t) − v(t)H

(24)

for any u and v solutions to (23) in the global attractor A. Setting p(t) = P u(t), from (23) we obtain dt p + Ap + P R(u) = g .

(25)

We denote g = P f and, henceforth, identify P H with RM . The key step in closing this projected equation in terms of just p is the following ”recovering lemma”. It states that the solutions on the attractor are uniquely determined by the P projections of its past values. Lemma 5.1. Set   A0 = u(s) : (−∞, 0] → H; u(0) ∈ A, u solves (23) ,   T = χ(s) : (−∞, 0] → RM ; ∃u ∈ A0 , χ(s) = P u(s) . Then there exists a mapping E : T → A such that, if u ∈ A0 and χ(s) = P u(s), E(χ) = u(0). Moreover, E is Lipschitz continuous from X (given in (3)) to H as long as γ > 0 is small enough. Proof. Let u(t), v(t) ∈ A0 . In virtue of (24), u(0) − v(0)H ≤ c1 P u(0) − P v(0)H + θu(−t∗ ) − v(−t∗ )H . Applying (24) repeatedly to the last term, one has u(0) − v(0)H ≤ c1

n−1  k=0

θk P u(−kt∗ ) − P v(−kt∗ )H + θn u(−nt∗ ) − v(−nt∗ )H .

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We let n → ∞. Since u and v are bounded, denoting χ = [P u]0 , ψ = [P v]0 enables us to obtain ∞  u(0) − v(0)H ≤ c1 θk |χ(−kt∗ ) − ψ(−kt∗ )| k=0

= c1

∞ 

∗

θk eγkt |χ(−kt∗ ) − ψ(−kt∗ )|e−γkt

k=0

≤ c1 χ − ψX

∞ 

∗

∗

θk−1 eγkt .

k=1

For γ sufficiently small, the last sum converges, thus establishing the desired result.



We see, then, that the elements p(t) of T are the solutions of ˜ t) = g , dt p + Ap + B(p ˜ : T ⊂ X → RM is Lipschitz, one can extend it to a ˜ = P R(E(pt )). Since B where B mapping B : X → RM that has the same Lipschitz constant. Moreover, based on the boundedness of T , we also have B(χ) = 0 if χX ≥ R0 . See [13, Lemma 2.2] for details. Thus, we obtain the projected (reduced) version of (23) dt p + Ap + B(pt ) = g .

(26)

The Lipschitz continuity of B ensures unique solvability-an important specification in proving the one-to-one correspondence of the solutions of (26) on T and the solutions of (23) on A (see [13, Theorem 3.3]). We have already shown that (26) has a global attractor Ar ⊂ X. It is easy to see that T (the P -projected negative trajectories on A ⊂ H) is an invariant subset of Ar . The question was whether the dynamics on Ar are really larger than the dynamics on A. We can now show that the dynamics are, in fact, not substantially larger insofar as (26) has an exponential attractor. In particular, Ar has finite dimension. We first show that there exists a more regular set B ⊂ X, one that attracts the solutions exponentially. Lemma 5.2. There exists K1 , K2 > 0 such that the set   B = χ ∈ X; |χ(s)| ≤ K1 , Lip χ ≤ K2 , is positively invariant and uniformly exponentially attracting for (26). Consequently, for any solution χ, there exist c, t0 > 0, depending only on χ0 X , such that distX (χt , B) ≤ ce−γt ,

∀t ≥ t0 .

Proof. Take K1 such that B(χ) = 0 Ap · p − g · p ≥ |p|2

if χX ≥ K1 ;

(27)

if |p| ≥ K1 .

(28)

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Further, set K2 = |g| + AK1 + sup |B(χ)| . χ∈X

(The supremum is finite because B is Lipschitz and has bounded support.) Note that any solution χ to (26) satisfies |dt χ(t)| ≤ K2 , t > 0. Moreover, the choice of K1 ensures that, if |χ(t)| ≤ K1 for t ≤ 0, then |χ(t)| ≤ K1 for all t. As a result, B is positively invariant. To prove that B is uniformly exponentially attracting, take an arbitrary solution χ. Let t0 ≥ 0 be the smallest number such that |χ(t0 )| ≤ K1 . By (27)-(28), such a t0 exists, depending only on |χ(0)| ≤ χ0 X , and, moreover, |χ(t)| ≤ K1 for ∀t ≥ t0 . Set χ(t), t ≥ t0 χ(t) ˜ = . χ(t0 ), t < t0 Clearly, χ˜t ∈ B for all t ∈ R. We claim that χt − χ˜t X → 0 exponentially, which is the desired conclusion. For t ≥ t0 , we have ˜ + s)|eγs χt − χ˜t X = sup |χ(t + s) − χ(t s≤0

= sup |χ(t + s) − χ(t ˜ + s)|eγs s≤t0 −t   ≤ e−γ(t−t0 ) sup |χ(t + s)| + K1 eγ(t+s−t0 ) s≤t −t  0 t0  −γ(t−t0 ) χ X + K1 . ≤e  Now, we can state the main theorem. Theorem 5.3. Equation (26) has an exponential attractor Er ⊂ X. Proof. This follows immediately from Theorem 4.5. For, writing F (χ) = g − Aχ(0) − B(χ), we see that (26) is a special case of (2). All the assumptions are verified above. 

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