Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space

Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of \textit{the solution manifold} proposed for ordinary equations in [H.-O. Walther, The solution manifold and $C\sp 1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, {195}(1), (2003) 46--65]. The existence of a compact global attractor is proven.


Introduction
The partial differential equations (PDEs) with delays have attracted a lot of attention during the last decades as many processes of the real world (like an automatically controlled furnace, bi-directional associative memory (BAM) neural networks, reaction-diffusion processes) can be described by such kind of equations. Studying these equations is based on the well-developed approaches to the ordinary differential equations (ODEs) with delays [11,7,1] and PDEs without delays [8,9,15,14]. Under certain assumptions both types of equations describe a kind of dynamical systems that are infinite-dimensional, see [2,30,6] and references therein; see also [31,4,5,3] and to the monograph [37] that are very close to this work. 1 AMS Subject Classification: 35R10, 35B41, 35K57 In many evolution systems arising in applications the presented delays are frequently state-dependent (SDDs). The theory of such equations, especially the ODEs, is rapidly developping and many deep results have been obtained up to now (see e.g. [32,33,34,16,18,35] and also the survey paper [12] for details and references). The underlying main mathematical difficulty of the theory of PDEs with SDDs lies in the fact that the functions describing state-dependent delays are not Lipschitz continuous on the space of continuous functions -the main space, on which the classical theory of equations with delays is developed. This implies that the corresponding initial value problem (IVP) is not generally well-posed in the sense of J. Hadamard [8,9].
The partial differential equations with state-dependent delays were first studied in [21] (the case of distributed delays, weak solutions), [13] (mild solutions, infinite discrete delay), and [22] (weak solutions, finite discrete and distributed delays). An alternative approach to the PDEs with discrete SDDs is proposed in [24]. This paper is a continuation of the work [25] and its goal is to study the approach used for ODEs with SDDs [32,33,12] in the case of PDEs. The main idea lies in finding a wider space Y ⊃ X such that a solution u : [a, b] → Y be a Lipschitz function (with respect to a weaker norm of Y ), and constructing a dynamical system on a subset of the space C([a, b]; Y ). It should be emphasized that the dynamical system is constructed on a metric space that is nonlinear. More precisely, the existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct the aforementioned dynamical system. This subset is an analogue of the solution manifold proposed in [33], see also [12]. We use the same class of non-local in space variables nonlinear PDEs as in [25].
The paper is organized as follows. The section 2 is devoted to the formulation of the model. The proof of the existence and uniqueness of (strong) solutions for initial functions from a Banach space forms a main part of the section 3. In the section 4, an evolution operator S t is constructed and its asymptotic properties in different functional spaces are investigated. The dissipativeness is obtained in a Banach space, while the existence of a global attractor is proven on a smaller metric space (the solution manifold). The choice of this smaller space is different from that proposed in [25]. 2 The model with discrete state-dependent delay and preliminaries Consider the following non-local partial differential equation with a discrete state- where A is a densely-defined self-adjoint positive linear operator with domain D(A) ⊂ L 2 (Ω) and compact resolvent, which means that A : D(A) → L 2 (Ω) generates an analytic semigroup, Ω ⊂ Ê n 0 is a smooth bounded domain, B : L 2 (Ω) → L 2 (Ω) denotes a bounded operator that will be defined later, b : Ê → Ê stands for a locally Lipschitz map, d ∈ Ê, d ≥ 0, and the function η : ). Norms defined on L 2 (Ω) and C are denoted by || · || and || · || C , respectively, and ·, · stands for the inner product in L 2 (Ω). As usually, Remark 1. The operator B may for example be of the following forms (linear operators) or even simpler where f : Ω → Ê is a smooth function and ℓ ∈ C ∞ 0 (Ω). In the last case the nonlinear term in (1) is of the form 2 Consider the equation (1) with the initial condition and let Let further be a norm defined on the space H and D(A α ) denote the domain of the operator A α . In the sequal the following assumptions will play an important role.

The existence and uniqueness of solutions
As in [25] we need the following is a (strong) solution to the problem defined by (1) and (5) Now we prove the following theorem on the existence and uniqueness of solutions.  (1) and (5) has a unique solution on any time interval [0, T ] such thatu ∈ L 2 (0, T ; L 2 (Ω)).

Remark 3.
Notice that ϕ does not assume ϕ ∈ L 2 ([−r, 0]; D(A)). However, the definition of a strong solution implies that 2 Proof of Theorem 1. We follow the proof of Theorem 1 in [25]. Notice that the assumption (H1.η) is slightly more general than the assumption (H.η) in [25]. This implies some changes in the proof of the uniqueness of solutions.
The system (17) is a system of (ordinary) differential equations in Ê 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 [33,34] and also a recent review [12]).
The key difference between equations with state-dependent and state-independent (concentrated) delays is that the first type of equations is not well-posed in the space of continuous (initial) functions. To get a well-posed initial value problem, it is better [33,34,12] to use a smaller space of Lipschitz continuous functions or even a smaller The condition ϕ ∈ L implies that the function U(·)| [−r,0] ≡ P m ϕ(·), which defines initial data, is Lipschitz continuous as a function from [−r, 0] to Ê m . Here P m is the orthogonal projection onto the subspace span {e 1 , . . . , e m } ⊂ L 2 (Ω). Hence, we can apply the theory of ODEs with discrete state-dependent delay (see e.g. [12]) to get the local existence and uniqueness of solutions to (17).
Next, we will get an a priory estimate to prove the continuation of solutions u m to (17) on any time interval [0, T ] and then use it for the proof (by the method of compactness, see [15]) of the existence of strong solutions to (1) and (5). To that end, multiply the first equation in (17) by λ k g k,m and sum for k = 1, . . . , m to get Integrating (18) with respect to t and using the relationships The above relationship (19) means that Using this fact and (17), it follows that Hence, the family {(u m ;u m )} ∞ m=1 is a bounded set in Therefore, there exist a subsequence {(u k ;u k )} and an element (u;u) ∈ Z 1 such that The proof that any *-weak limit is a strong solution is standard. To prove the property  (21)), it follows that for any such a solution v and any T > 0, there exists L v,T > 0, such that In the light of (11) give as e −1/2 and similarly, The last estimate and (23) give (just the case when q = 1 is shown for the purpose of clarity) It follows, from Lemma 1, that where L F 1 ,v,T is defined in the same way as L F 1 in (13), just with ℓ = L v,T instead of L ϕ -see (13) and (22).
It should be emphasized how the Lipschitz constant L v,T ≡ |||v||| [−r,T ] of a strong solution v is taken into account in (26) (see (22) and (11)). Let Then the relationships (24) and (25) lead to the following estimate Then Moreover, if α is non-decreasing, then It follows, from the above lemma and equality t 0 (t − τ ) −1/2 dτ = 2t 1/2 , that which implies, ∀t ∈ [0, T ], that where see (26) for the definition of L F 1 ,v,T ≡ L F 1 L v,T . This proves the uniqueness of the solution to (1) and (5), and completes the proof of theorem 1.

Asymptotic properties of solutions
This section is devoted to studies of the asymptotic behavior of solutions in different functional spaces. We define first (in a standard way) the evolution semigroup S t : L → L (the space L is defined in (9)) by the formula where u(t) is a unique solution to the problem (1) and (5) (see definition 1).
The estimate (27) means the continuity of the evolution operator S t in the norm of the space H (see (6)), i.e.
The aim now is to get a more precise estimate, e.g. the continuity of S t in the norm of the space L (see (9), (10)). Consider the definition of the Galerkin approximate solution (see (17)). It gives and Lemma 1 implies An analogous estimate for a solution to the problem (1) and (5), can be obtained from (21) and the following Proposition 2. [38, Theorem 9] Let X be a Banach space. Then any *-weak convergent sequence {w k } ∞ n=1 ∈ X * *-weak converges to an element w ∞ ∈ X * and w ∞ X ≤ lim inf n→∞ w n X .

From that and
which finally means that for any T ≥ 0 there exists a constant C T > 0 such that ∀t ∈ [0, T ] it gives The last inequality means the continuity of the evolution operator S t in the norm of the space L (see (29) and compare with (30)).

Remark 4.
It should be noted that the evolution operator and, more generally, the timeshift is not a (strongly) continuous mapping in the norm of the space L (see (9)). This can be illustrated by the following simple (scalar) example.
Obviously, when t = 0 one considers h → 0 + , while for t = T , the case h → 0 − should be investigated.
To prove the claim, we must show that (34) does not hold, i.e.
∃v ∈ Lip ([−r, T ; Ê) and ∃t 0 ∈ [0, T ] for which lim Thus, consider the case t 0 = 0, h → 0 + and the function It can be seen Hence, ||v t 0 +h − v t 0 || Lip = ||v t 0 +h || Lip = h + 1 and finally lim (h + 1) = 1 = 0, which means that (34) does not hold. In the space L, we would proceed analogously. 2 Remark 5. In the same way as in the previous remark one can show that the time-shift is not a (strongly) continuous mapping in the topology of L ∞ (−r, 0). One could consider the and t 0 = 0 to show that lim By the way, v = d dt v, where, as usually, the time-derivative is understood in the sense of distributions. 2 The above remarks show that despite of the existence and uniqueness of solutions in the space L and even strong continuity of the evolution operator S t in the norm of L (see (33)), the pair (S t ; L) does not form a dynamical system since S t is not strongly continuous as a mapping of time variable.
The methods, developed for ordinary delay equations in [33] suggest to restrict our considerations to a smaller subset of the space of Lipschitz functions. In this paper we follow this suggestion and consider the evolution operator S t on the following subset of L Here the equalityφ(0) + Aϕ(0) + dϕ(0) = F 1 (ϕ) is understood as an equality in D(A −1/2 ).

Remark 6.
The set X is an analogue of the solution manifold introduced in [33] for the case of ODEs with state-dependent delays. 2 To show that the set X is invariant under the evolution operator S t , we first have to establish an additional smoothness property of the solutions of problem (1), (5).
Consider the following auxiliary linear system without delay In the same way as in (17), the Galerkin approximate solution v m = v m (t, x) = m k=1 g k,m (t)e k of order m to (39) can be defined such that where g k,m ∈ C 1 (0, T ; Ê) ∩ L 2 (−r, T ; Ê) andġ k,m (t) is absolutely continuous.
The difference between approximate solutions u m and v m lies in that v m are solutions just to linear system (40). So, for any two approximate solutions v n and v m (solutions to (40) of different orders n and m), one has g k,n (t) ≡ g k,m (t), which is denoted by g k (t).
Thus, there exists a unique solution v(t) (v ≡ lim n→∞ v n ) to the linear system (39), which On the other hand, the nonlinear delay system (1), (5) with the initial function ϕ has also a unique solution. From the construction of p(t) (see (38)), it follows that u(t) ≡ v(t) for all t ∈ [0, T ], which gives (37) and completes the proof of Lemma 3.
Lemma 3 particularly shows that the set X, defined by (36), is invariant under the evolution operator S t (see (29)). This fact allows to define an evolution operator (denoted again by S t ) S t : X → X in the same way as in (29). Now, if the natural norm on X is taken into account, then Theorem 1, Lemma 3, and Proposition 1 give the continuity of S t with respect to t in the norm of X. Hence, (S t ; X) defines a dynamical system. Now we will pay attention to the long-time asymptotic behavior of the constructed evolution semigroup S t : X → X.
Theorem 2. Using the above notation and under the assumptions of Theorem 1, Proof of Theorem 2. It will be shown first that (S t , X) is a dissipative dynamical system. To that end, the below proposition is needed. Second, to apply the classical theorem on the existence of a global attractor (see, for example [2,30,6]), we show that (S t , X) is asymptotically compact. Consider therefore any solution u(t) to the problem (1) and (5) with ϕ ∈ BV α as an initial function. We will show that for any δ > r > 0 and any T > δ the set U ≡ {u t = S t ϕ | ϕ ∈ BV α , t ∈ [δ, T ]} is relatively compact in X.
As an application we can consider the diffusive Nicholson blowflies equation (see e.g. [29]) with state-dependent delays, i.e. the equation (1) where −A is the Laplace operator with the Dirichlet boundary conditions, Ω ⊂ Ê n 0 is a bounded domain with a smooth boundary, the nonlinear (birth) function b is given by b(w) = p · we −w . The function b is bounded, so for any delay function η satisfying (H1.η), the conditions of Theorem 1 and Theorem 2 are satisfied. As a result, we conclude that the initial value problem (1) and (5) is well-posed in X and the dynamical system (S t , X) has a global attractor (Theorem 2).