Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. We show that the model considered generates an evolution operator semigroup $S_t$ on a space $CL$ of Lipschitz type functions over delay time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous for $t$ large enough. Our main result shows that the semigroup $S_t$ possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.


Introduction
Differential equations with different types of delay attract much attention during last decades.
Including delay terms in differential equations is a natural step of taking into account that the majority of real-world problems depends on the pre-history of the evolution. Delay terms in an equation reflect a well-understood phenomenon that evolution of a state of a system depends not only on this state but rather on the states during some previous interval of time (memory of the system). This leads to infinite-dimensional dynamics even in the case of ordinary differential equations. The general theory of delay differential equations was initially developed for the simplest case of constant delays. We cite just classical monographs [2,13,18] on ordinary differential equations (ODEs) and milestone articles [16,37] on partial differential equations (PDEs) with constant delays. On the other hand it is clear that the constancy of the delay is just an extra assumption made to simplify the study, but it is not really well-motivated by real-world models.
To describe a process more naturally a new class of state-dependent delay models was introduced and intensively studied during last decades. We mention works on ODEs [14,20,22,38] and on PDEs [12,29,30,32,33] with state-dependent delays.
The simplest case of a state-dependent delay is a delay explicitly given by a real-valued function η : R → R + which depends on the value x(t) at the reference time t but not on previous values of the solution {x(τ ), τ ≤ t}. This leads to terms of the form f (x(t− η(x(t))) in the model considered.
Even in this case the non-uniqueness could appear (see the scalar ODE example constructed by R.Driver [14] in 1963 for initial data from the space of continuous functions on the delay interval).
The standard way for general models to avoid non-uniqueness in the case of infinite-dimensional dynamics is to consider smoother (narrower) classes of solutions. However in this case the existence problem may become critical. The main task is to find a good balance between these two issues.
In this paper we deal with a certain abstract parabolic problem with the state dependent delay term of a rather general structure. Our considerations are motivated by several biological models, see the discussion and the references in [3], [17] and [33]. Our main goal in this paper is to find appropriate phase spaces in which we can establish the well-posedness of our model and study its long time (qualitative) dynamics.
Our first result (Theorem 3.3) states well-posedness of the problem and allows us to define an evolution semigroup S t of closed mappings on a certain Banach space of functions on the delay time interval with values in an appropriate Hilbert space. In some sense this result extends the well-posedness statements in [30,32,33] to more general delay terms. The main result of the paper (Theorem 4.2) states the existence of a global finite-dimensional attractor, the object which is responsible for long-time dynamics. We also show that the model possesses an exponential fractal global attractor (see the definition in the Appendix).
Although for some parabolic problems with state-dependent delay the existence of compact global attractors was established earlier in [30,33], to the best of our knowledge, results on finitedimensional behavior for parabolic state-dependent delay problems were not known before. The main difficulty is related to the fact the corresponding delay term is not Lipschitz on the natural energy balance space. We also mention that our Theorem 4.2 can be applied in the situation considered in [33] and gives the finite-dimensionality of the global attractor constructed in that paper.
We note that the evolution operators S t we construct are not continuous mapping on the phase space for t small enough. Therefore to prove the existence of a compact global attractor we use the extension of the standard theory suggested in [28]. As for dimension issues we apply the idea of the method of quasi-stability estimates developed earlier in [6,7,8,9] for the second order in time evolution models which generate continuous evolution semigroups. This is possible in our case due to the continuity of evolution operator for large times. We note that in the delay case the quasi-stability method was applied earlier in [10,11,12] for second order models, see also [5, Chapter 6].

Model description
We deal with well-posedness and long-time dynamics of abstract evolution equations with delay of the formu in some Hilbert space H. Here the dot over an element means time derivative, A is linear and F , G are nonlinear operators, h ∈ H. The term F (u t ) represents (nonlinear) delay effect in the dynamics.
As usually for delay equations, the history segment (the state) is denoted by We define the spaces H α which are D(A α ) for α ≥ 0 (the domain of A α ) and the completions of H with respect to the norm A α · when α < 0 (see, e.g., [25]). Here and below, · is the norm of H, and ·, · is the corresponding scalar product. For r > 0, we denote for short We also write C = C 0 .
globally Lipschitz for α = 0 and α = −1/2, i.e., there exists L F > 0 such that and where L G : R + → R + is a nondecreasing function. We also assume that G is a potential mapping, the latter means that there exists a (Frechét differentiable) functional Π(u) : Moreover, we assume that (a) there exist positive constants c 1 and c 2 such that and (b) there exist δ > 0 and m ≥ 0 such that G : Our main motivating example of a system with discrete state-dependent delay is the following one: -the diffusive model of Hematopoiesis -blood cell production, the Lasota-Wazewska-Czyzewska model in hematology) with state-dependent delay we refer to [17] and [33] and to the references therein. We note that several special cases of the model in (6) were studied in [30,31,32,33]).
For instance it was assumed in [33] that g(s) ≡ 0, b(s) is a bounded function, and B is an integral compact linear operator. This leads to nonlocal (in space) models. Our assumptions covers the non-compact case. We can take b(s) = s and B = Id, for instance. We also note that if we equip (6) with the Dirichlet boundary condition, then the dissipativity property in (5) holds provided g ∈ C 1 (R), g(0) = 0 and the derivative g ′ (s) is bounded from below. This follows by the standard integration by parts. Thus population dynamics models with nonlinear sink/source feedback terms can be included in consideration. For this kind of biological models, but with state-independent delay, we refer to [39].
We equip the equation (1) with the initial condition and for initial data ϕ consider the space where and We equip the space CL with the natural norm We note that the delay term F (ϕ) ≡ F 0 (ϕ(−η(ϕ))) in (1) is well-defined for every ϕ ∈ C and possesses the property (see (2) with c 1 = ||F (0)|| and c 2 = L F . However it is not Lipschitz on the space C. One can only show that the delay term F satisfies the inequality for every ϕ ∈ CL and ψ ∈ C. Using the terminology of [26] we can call this mapping F "almost Lipschitz" from C into H −1/2 , see also discussion in [20].

Remark 2.2
We can also include in (1) a delay term M (u t ) which is defined by a globally Lipschitz function from C(−r, 0; H 1/2 ) into H. We will not pursue this generalization because our main goal is state-dependent delay models.

Well-posedness
In this section we prove the existence and uniqueness theorem and study properties of solutions.
Then we use these results to construct the corresponding evolution semigroup and describe its dynamical properties.
We introduce the following definition.
is said to be a (strong) solution to the problem defined by (1) and (7) Then it follows from (12) and also from (4) and (10) that This allows us to conclude from (12) and (13) thaṫ Moreover, the relation in (13) implies that u(t) satisfies (1) for almost all t ∈ [0, T ] as an equality in H. We also note that relations (12) and (14) yield for every strong solution u with initial data ϕ from the space CL defined in (8).
Our first result is the following theorem on the existence and uniqueness of solutions.
and satisfies the estimate for all t ∈ [0, T ] and ||A 1/2 ϕ(0)|| 2 + |ϕ| 2 C ≤ R 2 . Moreover, for every two strong solutions u 1 and u 2 with initial data ϕ 1 and ϕ 2 from CL we have that for all ϕ i such that |ϕ i | CL ≤ R.
Proof. To prove the existence we use the standard compactness method [24] based on Galerkin approximations with respect to the eigen-basis {e k } of the operator A (see Assumption 2.1 (A)).
We define a Galerkin approximate solution of order m by the formula where the functions g k,m are defined on [−r, T ], absolutely continuous on [0, T ] and such that the The equation in (19) is a system of (ordinary) differential equations in R m with a concentrated (discrete) state-dependent delay for the unknown vector function U (t) ≡ (g 1,m (t), . . . , g m,m (t)) (for the corresponding theory see [38] and also the survey [20]).
The condition ϕ ∈ CL implies that the function U (·)| [−r,0] ≡ P m ϕ(·), which defines initial data, is Lipschitz continuous as a function from [−r, 0] to R m . Here P m is the orthogonal projection onto the subspace Span {e 1 , . . . , e m }. Hence, we can apply the theory of ODEs with discrete state-dependent delay (see e.g. [20]) to get the local existence of solutions to (19).
Next, we derive an a priori estimate which allows us to extend solutions u m to (19) on an arbitrary time interval [0, T ]. We also use it for the compactness of the set of approximate solutions.
We multiply the first equation in (19) by λ k g k,m and sum for k = 1, . . . , m to get Due to (10) and (5) this implies that Integrating the last inequality we can easily see that the function Therefore Gronwall's lemma gives us the a priori estimate Now we establish additional a priori bounds. Using (20), (4) and (10) from the first equation in (19) we obtain that Thus by (20) we obtain the estimate for all t ∈ [0, T ] and ||A 1/2 ϕ(0)|| 2 + |ϕ| 2 C ≤ R 2 . It also follows from (19) that Thus Hence, there exist a subsequence {(u k ;u k )} and an element (u;u) By the Aubin-Dubinski theorem [34,Corollary 4] we also have Now the proof that any *-weak limit u(t) is a solution is standard. To make the limit transition in the nonlinear terms F and G we use relation (11) and Assumption 2.1(Gb).
The property u(t) ∈ C([0, T ]; H 1/2 )) follows from the well-known continuous embedding (see also [25 The continuity ofu in H −1/2 follows from equation (1) and from continuity of u in H 1/2 . Thus the existence of strong solutions is proved. It is easy to see from (21) and (22) that the strong solution constructed satisfies (17).
Now we use this fact to prove the uniqueness.
Let u 1 and u 2 be two solutions (at this point we do not assume that they have the same initial data). Then the difference is a strong solution to the linear parabolic type (non-delay) equatioṅ (4) and (11) using (15) we also have that Thus using the standard multiplier z in (23) we obtain that This implies uniqueness of strong solutions.
As a by-product the uniqueness yields that any strong solution satisfies (17). Therefore we can apply (24) with ̺ > R + C T (R) to obtain (18).
Thus the proof of Theorem 3.3 is complete.
Theorem 3.3 allows us to define an evolution semigroup S t on the space CL (see (8)) by the formula where u(t) is the unique solution to the problem (1) and (7). We note that (18) implies that S t is almost locally Lipschiz on C, i.e., However, it seems that a similar bound is not true in the space CL. We can only guarantee that ϕ → S t ϕ is a continuous mapping on CL for all t > r. Moreover, the following assertion shows that the mapping ϕ → S t ϕ is even 1 2 -Hölder on CL with respect to ϕ when t > r.
for all t ∈ [r, T ] and for all initial data ϕ i such that |ϕ i | CL ≤ R. This implies that for every t > r the evolution semigroup S t is 1 2 -Hölder continuous in the norm of CL. In the case when t ∈ (0, r] we can guarantee the closeness of the evolution operator S t only. This means 1 (see, e.g., [28]) that the properties ϕ n → ϕ and S t ϕ n → ψ in the norm of CL as n → ∞ imply that S t ϕ = ψ.
Proof. Multiplying (23) by Az and using (17) and (4) we obtain that From (17), (2) and (3) we also have that for every t ≥ r. Therefore Integrating over interval [τ, t] with τ ≥ r and using (18) we obtain that Now we integrate (28) with respect to τ over [r, t], change the order of integration, and use (18) to get 1 We refer to the Appendix for a discussion of closed evolutions. Here we only mention that any continuous mapping is closed and a mapping can be closed but not continuous, see examples in [28] and also in [5, Sect.1.1].
Using the expression forż from (23) and also the bounds in (18) and (27) we have that This implies (26).
The 1 2 -Hölder continuity of the evolution semigroup S t in the norm of CL follows from (26). The closedness of S t for t ∈ (0, r] easily follows from (18). (27) we can obtain a 1 2 -Hölder continuity relation like (26) for all t ≥ 0 if we assume in addition that one of initial data ϕ i possesses the propertyφ i ∈ L 2 (−r, 0; H).

Remark 3.5 As it follows from
In this case the argument above leads to the relation for all t ∈ [0, T ] and for all initial data ϕ i such that |ϕ i | CL + |φ i | L 2 (−r,0;H) ≤ R. Moreover, one can also see that the set is forward invariant with respect to S t . Thus ϕ → S t ϕ is a 1 2 -Hölder continuous mapping for each t ≥ 0 on the Banach space CL 0 endowed with the norm |ϕ| CL 0 = |ϕ| CL + |φ| L 2 (−r,0;H) . Hence the dynamical (in the classical sense, see, e.g., [1,4,36]) system (CL 0 , S t ) arises. However we prefer to avoid propertyφ ∈ L 2 (−r, 0; H) in the description of the phase space. The point is that our goal is long-time dynamics and it is well-known (see, e.g., [1,4,36]) that the existence of limiting objects requires some compactness properties. Unfortunately we cannot guarantee these properties in the space CL 0 without serious restrictions concerning the delay term. This is why we prefer to use the observation made in [28] concerning closed evolutions. Remark 3.6 A similar problem as above we have with time continuity of evolution operator S t .
It is clear from (12) and (16) that t → S t ϕ is continuous for every ϕ ∈ CL when t > r. To guarantee the continuity t → S t ϕ for all t ≥ 0 we need make further restriction 2 on initial data. The main restriction is a compatibility condition at time t = 0. To describe this condition we introduce the following (complete) metric space Here the compatibility conditionφ(0) + Aϕ(0) + F (ϕ) + G(ϕ(0)) = 0 is understood as an equality in H −1/2 . The distance in X is given by the relation One can see that X is a closed subset in the Banach space CL and the topology generated by the metric dist X coincides with the induced topology of CL (see 9).
In the following assertion we collect several dynamical properties of the evolution semigroup S t which are direct consequences of Theorem 3.3 and Proposition 3.4 and Remark 3.6.
Proposition 3.7 Under the conditions of Theorem 3.3 problem (1) generates an evolution semigroup S t of closed mappings on CL such that (a) S t CL ⊂ X for every t ≥ r and the set S t B is bounded in X for each t ≥ r when B is bounded in the space CL; (b) the set X is forward invariant: S t X ⊂ X; (c) the mapping ϕ → S t ϕ is a 1 2 -Hölder continuous on CL (and hence on X) for all t > r; (d) the trajectories t → S t ϕ are continuous for t > r and ϕ ∈ CL. If ϕ ∈ X, then these trajectories are continuous for all t ≥ 0.

Long time dynamics
This section is central for the whole paper. Here we study long-time dynamics of the delay model generated by (1) and (7). The main result stated below in Theorem 4.2 deals with finite-dimensional global and exponential attractors. We refer to the Appendix for the corresponding definitions and the auxiliary facts which we use in our argument.
Moreover, we assume that (b) for every η > 0 there exists C η > 0 such that In the case of parabolic models like (6) examples of functions g(u) such that the corresponding Nemytskii operator satisfies Assumptions 2.1(G) and 4.1 can be found in [1] and [36]. The simplest one is g(u) = u 3 + a 1 u 2 + a 2 u with arbitrary a 1 , a 2 ∈ R in the case when Ω is a 3D domain.
Our main result is the following assertion.
(B) There exists a fractal exponential attractor A exp .
We devote the remaining subsections to the proof of Theorem 4.2.

Existence of a global attractor
To prove the existence of a global attractor it is sufficient to show that the evolution operator possesses a compact absorbing set. In this case we can apply the standard existence result in the form given in [28] for closed semigroups (see the Appendix for more details).
We start with the existence of a bounded absorbing set. (1) and (7) with ϕ ∈ CL.

Proposition 4.3 (Bounded dissipativity) Assume that u(t) solves
Then one can find ℓ 0 > 0 such that for every delay time r such that m F r < ℓ 0 the following property holds: there exists R * such that for every bounded set B in CL there is t B such that for all t ≥ t B and for all initial data ϕ ∈ B. This yields that the evolution semigroup S t is dissipative on both CL and X provided m F r < ℓ 0 .
Proof. We use the Lyapunov method to get the result. For this we consider the following functional defined on strong solutions u(t) for t ≥ r. The positive parameter µ will be chosen later. We note that the main idea behind inclusion of an additional delay term in V is to find a compensator for the delay term in (1). This idea was already applied in [12] for second oder in time models with state-dependent term, see also [8, p.480] and [10] for the case of a flow-plate interaction model which contains a linear constant delay term with the critical spatial regularity. The corresponding compensator is model-dependent.
One can see from (33) that there is 0 < c 0 < 1 and c, c 1 > 0 such that We consider the time derivative of V along a solution. One can easily check that The last terms are due to (1): By the definition of m F in (36) for any number M F greater than m F we can find C(M F ) such that and thus In a similar way we also have that The relations in (34) and (35) with small enough η > 0 (and η ∈ [0, 1)) yield for some a i > 0 with a 0 independent of M F . Thus it follows from the relations above that for some a i . Thus using the right inequality in (41) we arrive at the relation Therefore taking µ = 1/4 and fixing γ ≤ a 0 c −1 1 we obtain that provided γr + 4a 2 M 2 F r 2 ≤ 1. Thus under the condition 4a 2 m 2 F r 2 < 1 we can choose γ ∈ (0, a 0 c −1 1 ] and M F > m F such that (43) holds. In particular we have that when m F r < ℓ 0 . Using (41) and (17) we can conclude that | V (r)| ≤ C B for all initial data from a bounded set B in CL. Hence (see (1)) there exists R such that for every initial data from a bounded set B in CL we have that Moreover, it follows from (43) that To get this one should multiply (43) by e γt , integrate over [t, t + 1] and multiply by e −γt . Then ultimate boundedness of V (t) (see (44)) and the relation 1 ≤ e γ(τ −t) for τ ≥ t give the last estimate.
These relations imply (40) and allow us to complete the proof of Proposition 4.3.

Remark 4.4
If the mapping F 0 has sublinear growth in H, i.e., there exists β < 1 such that then the linear growth parameter m F given by (36) is zero. Thus in this case we have no restrictions concerning r in the statement of Proposition 4.3. In particular, this is true in the case of bounded mappings F 0 . Moreover, in the latter case the argument can be simplified substantially (we can use a Lyapunov type function without delay terms). For more details we refer to [31,33].
We use Proposition 4.3 to obtain the following assertion which means that the evolution semigroup S t is (ultimately) compact.
Proposition 4.5 (Compact dissipativity) As in Proposition 4.3 we assume that m F r < ℓ 0 .
Using the mild form of the problem and also the bound in (10) one can also show that for every δ > 0, where u(t) is a solution possessing property (40).

Dimension and exponential attractor
The proof of finite-dimensionality is based on the notion of quasi-stability which says that the semigroup is asymptotically contracted up to a homogeneous compact additive term. For the convenience we remind the corresponding abstract result in the Appendix.
We can assume that there exists a forward invariant closed absorbing set D 0 which belongs to D R α,β for an appropriate choice of the parameters (see Proposition 4.5). We also note that the restriction of S t on D 0 is continuous in both t and initial data in the topology induced by CL (see (9)). Thus a dynamical system (S t , D 0 ) in the classical (see [1,4,19,36]) sense arises. Therefore we can apply the quasi-stability method developed earlier in [5,6,7,8,9] for continuous evolution models.
Proposition 4.6 (Quasi-stability) Let Assumptions 2.1 and 4.1 be in force. Assume that (38) and (39) are valid. Then Proof. Using the mild form presentation for u i (t) and (39) we have that As in (27) we also have that for every t ≥ 0. Therefore Using (1), (4) and (11) we also have that This completes the proof of Proposition 4.6.
In order to prove the finite dimensionality of the attractor A we apply Theorem A.6 on the attractor with an appropriate choice of operators and spaces. Indeed, let T > 0 be chosen such that q ≡ C R e −λ1T < 1 where C R is the constant from (47). We define the Lipschitz mapping by the rule Kϕ = u(t), t ∈ [0, T ], with u be the unique solution of (1) and (7) with initial function ) by the Arzelà-Ascoli theorem (see, e.g., [34]).
If we take To prove the existence of a fractal exponential attractor we first use (48) on the set D 0 and then apply Theorem A.7 to show that there exists a finite-dimensional set A θ ⊂ D 0 such that (50) holds. Then as in the standard construction (see, e.g., [15] or [27]) we suppose Since V = S T it is easy to see that A exp is exponentially attracting, see (49) in the Appendix.
Since D 0 is included in the set D R α,β given by (37), we have that t → S t ϕ is α-Hölder on D 0 and Therefore in the standard way (see, e.g., [15] or [27]) we can conclude that A exp has finite fractal dimension in Y .
This completes the proof of Theorem 4.2.
Definition A.1 (Closed semigroup) Let X be a complete metric space. A closed semigroup on X is a one-parameter family of (nonlinear) operators S t : X → X (t ∈ R + ) (or t ∈ N) satisfying the conditions (S.1) S 0 = Id X -identical operotor; (S.2) S t+τ = S t S τ for all t, τ ∈ R + ; (S.3) for every t ∈ R + the relations x n → x and S t x n → y imply that S t x = y.
Assumptions (S.1) and (S.2) are the semigroup properties, while (S.3) says that S t is a closed (nonlinear) map. We note the operator closeness is a well-known concept in the theory of linear (unbounded) operators. To our best knowledge in the context of evolution operators this notion was appeared in [1] as a (weak) closeness of an evolution (strongly continuous) semigroup (see also [4]).
The following notions are standard in the theory of infinite-dimensional evolution semigroups and dynamical systems (see, e.g., [1,4,19,23,36]). We recall ( [1,36]) that a set K ⊂ X is called attracting for S(t) if, for any bounded set B ⊂ X , The following assertion is a reformulation of Corollary 6 [28] which also takes into account the statement of [28, Theorem 2]).
Theorem A.4 (Existence of a global attractor) Assume that S t : X → X is a closed semigroup possessing a compact connected absorbing set K abs ⊂ X . Then there exists a compact global attractor A for S t . This attractor is a connected set and A = ω(K abs ) = t∈R τ ≥t S τ K abs .
One of the desired qualitative properties of an attractor is its finite-dimensionality. We remind the following definition. (ii) There exist a Lipschitz mapping K from M into some Banach space Z and a compact seminorm for any v 1 , v 2 ∈ M , where 0 < γ < 1 is a constant. Then M is a compact set in Y of a finite fractal dimension and where L K > 0 is the Lipschitz constant for K: and m Z (R) is the maximal number of elements z i in the ball {z ∈ Z : ||z i || Z ≤ R} possessing the property n Z (z i − z j ) > 1 when i = j.
We recall (see [15]) that a compact set A exp ⊂ CL is said to be fractal exponential attractor for S t iff A exp is a positively invariant set whose fractal dimension is finite and for every bounded set D there exist positive constants t D , C D and γ D such that For details concerning fractal exponential attractors in the case of continuous semigroups. we refer to [15] and also to the recent survey [27]. We only mention that (i) a global attractor can be non-exponential and (ii) an exponential attractor is not unique and contains the global attractor.
The dimension theorem discussed above pertains to negatively or strictly invariant sets M (M ⊆ V (M )). To prove the existence of exponential attractors we need an analog of Theorem A.6 for positively invariant sets. More precisely we need the following assertion which was established in [5] and is a version of the result proved in [7] for metric spaces.
for some constant r > 0. Moreover, where we use the same notations as in Theorem A.6.