Pullback attractors of reaction-diffusion inclusions with space-dependent delay

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.


1.
Introduction. As a qualitative motivation from biology, consider inhomogeneous cell tissue under the influence of a chemical substance. Whenever the concentration u(t, x) of that substance at a position x exceeds a given threshold, the consequences for the cell there are significant, maybe even fatal (like apoptosis). Hence it is not just the present concentration u(t, x) which influences the cells, but also its history -for a while, at least. Neighboring cells have a recovering effect on the tissue at position x, and the duration of such a process depends on the cell type, i.e., which position x in the tissue we consider. From the mathematical point of view, the evolution depends on ess sup

PETER E. KLOEDEN AND THOMAS LORENZ
with a given non-negative function Θ(·) of space. Strictly speaking, the threshold c should also come into play, i.e., the special interest is on the question whether the threshold c ≥ 0 has been exceeded in recent history or not. We will express the relevant non-local term by means of a cut-off function ess sup Considering any bounded open domain Ω ⊂ R d and time period T ≥ 0, this nonlocal term is well-defined for any Lebesgue measurable function u : (−∞, T ]×Ω −→ R, but it leads to mathematically challenging questions since it clearly differs from standard forms of "delay" or "memory". Motivated by this model problem with the biological background, the focus of interest in this paper is the existence of a pullback attractor of the following nonautonomous reaction-diffusion inclusion on a bounded open domain Ω ⊂ R d with smooth boundary ∂ t u − ∆u ∈ G t, x; u(t, x), ess sup a.e. in (t 0 , T ) ×Ω u = u 0 a.e. in (−∞, t 0 ] ×Ω In particular, we prefer a differential inclusion to its single-valued counterpart, i.e., a differential equation, in support of the possibility that the concept of relaxation is applied later whenever required in modeling, e.g., when terms are switched on or off. Obviously, it is a parabolic differential inclusion subject to homogeneous Dirichlet boundary conditions. The "reaction term" on the right-hand side, however, does not depend just on the current state u(t) ∈ L 2 (Ω) (in the pointwise or functional way) as usual. It is rather the "history" of the desired solution u : (−∞, T ] −→ L 2 (Ω) which comes into play. Hence, this problem can be regarded as a non-autonomous differential inclusion with delay (as, e.g., in [7,8,30]) or with memory (as, e.g., in [1,29]) or as a retarded functional differential inclusion (similarly to, e.g., [22,40]). (From our point of view, the distinction between "delay" and "memory" is not handled uniformly in the literature about differential equations and inclusionsparticularly if initial conditions are not specified.) The remarkable aspect of problem (1) is the influence of the "history" depending on space, i.e., the duration of the time interval in the past depends on the spatial position x ∈ Ω and is assumed to be strictly positive. In more detail, the set-valued reaction term G in inclusion (1) uses the representation Θ(x) := θ + Θ(x) with a fixed constant θ > 0 (indicating a positive lower bound) and a given nonnegative function Θ : Ω −→ R.
The existence of strong solutions to problem (1) can be concluded directly from our earlier results in [37], essentially due to the positive lower bound θ. (The details are explained in section 3 below). In this article, we focus on sufficient conditions for a pullback attractor in an appropriate sense of convergence.
This topic is analytically challenging. Indeed, standard tools for establishing the existence of attractors attractors start with the reformulation in terms of dynamical systems. (In our case of a parabolic differential inclusion, we consider a multivalued non-autonomous dynamical system as described in section 5 below.) In the framework of dynamical systems, the essential challenge is then to specify a norm or, more generally, a metric on the state space which serves two different purposes simultaneously: First, the dynamical system is appropriately (semi-) continuous w.r.t. initial states. Second, an additional compactness property enables us to draw conclusions about accumulation points whenever time is tending to ∞ or −∞ respectively. Pullback asymptotic compactness belongs to the most popular forms in this regard (see, e.g., [8,9,11,38]).
In the special case of parabolic differential inclusion (1), we can specify a metric on the state space and adapt arguments from [8] which guarantee the existence of a pullback attractor. This result is formulated in Theorem 4.2 below, but it is based on a technical assumption about the set-valued mapping G : R × Ω × R × R × L 2 (Ω) ; R in the reaction term, which plays the role of a monotonicity condition w.r.t. the retarded argument: x; y, z 1 , w) ⊂ G(t, x; y, z 2 , w).
(see hypothesis (G7) in section 2 for its complete formulation). From our current point of view, there is no transparent way how to apply the "standard method" for pullback attractors of dynamical systems in metric spaces to parabolic inclusion problem (1) if we dispense with this special inclusion assumption (2). This is essentially due to the fact that the solution u : (−∞, T ] −→ L 2 (Ω) enters the reaction term at time t ∈ (−∞, T ] in form of Apparently, the latter function is bounded on the bounded domain Ω ⊂ R d and so, it also belongs to L 2 (Ω). But it is not obvious at all that its L 2 (Ω) norm depends continuously on one of the popular norms of u(· + T ) ∈ C γ ⊂ C 0 (−∞, 0], L 2 (Ω) (i.e., which are well established in the literature like the weighted supremum norm · γ in (8) below, see, e.g., [8,30]). Even under additional assumptions for Θ ∈ L 1 (Ω, R + ), there can always be some positions x ∈ Ω in which the delay reaches infinitely long in the past. Hence on the one hand, the standard approaches to norms like exponentially decaying weights are to fail when verifying an appropriate form of (semi-) continuity of the dynamical system. On the other hand, we do not see how to draw conclusions about pullback asymptotic compactness if the metric of functions considers delay which is just integrably bounded in space.
Due to this observation, we consider problem (1) without monotonicity assumption (2) as a touchstone for a new approach to proving pullback attractors. This metric concept is the main contribution of our article to the general theory of nonautonomous dynamical systems.
Indeed, Zhong et al. have published interesting proofs for the existence of pullback attractors by supplying domain and value space of dynamical systems with different topologies. In [60], for example, so-called norm-to-weak continuous semigroups on a Banach space are used for investigating nonlinear parabolic differential equations. This idea has then been extended to non-autonomous dynamical systems or processes (see, e.g., [41,52,56,55]). The joint key idea is assuming the condition on the dynamical system that for at each time instant, a norm converging sequence in the domain state space is always mapped to a (just) weakly convergent sequence of values in the same state space. This modification of topologies facilitates verifying the general conditions in concrete applications like reaction-diffusion equations.
We have borrowed this idea of different topologies in domain and value space, but (re-) interpret it in a purely metric setting. In particular, we do not use the norm or weak convergence in a Banach space in the standard meaning of functional analysis based on linear forms.
Indeed, the well-known consequence of Hahn-Banach theorem : Y −→ R linear and continuous, Lin(Y,R) ≤ 1 . for any Banach space Y, · Y [6, Corollary 1.4] reveals a relationship between weak and norm convergence which we use as a starting point in a purely metric setting. Each continuous linear functional : Y −→ R induces a "distance function" quantifying how "far" two vectors are away from each other: This is a pseudo-metric on Y , i.e., it is reflexive, symmetric and satisfies the triangle inequality (see, e.g., [31]). But in general, it is not positive definite and so, it is not a metric. The kind of a "distance function", however, leads to the metric induced by the norm if we take the pointwise supremum for all continuous linear functionals : Y −→ R with Lin(Y,R) ≤ 1. Shortly speaking, these linear functionals : Y −→ R with Lin(Y,R) ≤ 1 can be regarded as a parameter for the family of "distance functions". Such a connection via pointwise supremum is now considered in a purely metric setting: Consider a nonempty set X and a family (d X,j ) j∈J of functions X × X −→ [0, ∞) whose parameter j always belongs to a nonempty set J . Each d X,j (j ∈ J ) represents a form of quantifying distances between two elements of X, but we do not assume that d X,j satisfies all standard conditions on a metric. (This is what we just call a "distance function" in the following.) For subsequent conclusions about pullback attractors, however, the pointwise supremum of this family and so, they can clearly differ from the standard norms in L ∞ [−j, 0]; L 2 (Ω) , L ∞ (−∞, 0]; L 2 (Ω) respectively. Our choice of the second term is motivated by the special structure of the fourth argument of G in parabolic inclusion (1). The joint first term, i.e., the weighted supremum metric for all s ≤ 0, induces a topology on C γ ⊂ C 0 (−∞, 0], L 2 (Ω) which is finer than the compact-open topology, but there is no obvious relationship with the convergence in L 2 Ω; L ∞ ([−j, 0]) , L 2 Ω; L ∞ ((−∞, 0]) ) respectively. Preferring such an alternative family (d X,j ) j∈J , however, implies that some standard tools from linear functional analysis require a counterpart in the new metric setting.
The Lemma of Mazur has proved to be a very useful implication from weak to norm convergence [57, § V.1 Theorem 2]. Indeed, for any weakly converging sequence in a Banach space, there exists another sequence converging strongly to the same limit and, additionally, such a strongly converging sequence can be represented in terms of the original sequence by means of convex combinations. In other words, we can obtain strong convergence to the same limit if we are willing to consider a possibly different sequence constructed by means of convex combinations. This gist can be extended to the purely metric setting of a nonempty set X and the family (d X,j ) j∈J : We start with a sequence converging to an element x ∈ X pointwise w.r.t. the parameter j ∈ J , but for conclusions about continuity later on, we would like to represent the same element x ∈ X as the limit of a (possibly different) sequence converging uniformly w.r.t. the parameter j ∈ J . In the purely metric setting, however, we should feel free to rely on construction approaches different from convex combinations, which is established in vector spaces only. In connection with parabolic inclusion (1), for example, we will use a form of concatenation in the past in which the given limit function in C γ is involved explicitly (see Lemma 8.26 below for more details).
Strictly speaking, our focus of interest is on pullback attractors and so, we do not need a metric counterpart of Mazur's Lemma for any sequence converging pointwise w.r.t. the parameter, but just for those sequences that occur in the construction of pullback ω-limit points, i.e., they are related to a sequence of initial time instants tending to −∞. This is an additional feature which can prove to be very useful for constructing alternative sequences whenever standard results from functional analysis cannot be applied to the respective convergence or if standard constructions such as convex combinations are no longer available in a purely metric setting.
Thus, we suggest a counterpart criterion for sequences tending to pullback ω-limit points as a definition in the purely metric setting of (d X,j ) j∈J and its supremum d X on a nonempty set X (see Definition 6.7 below). This condition, however, has to be verified for each investigated metric example individually. Indeed, even if X is a Banach space and if d X is induced by its norm, then we cannot just apply the Lemma of Mazur (in its standard form) unless the family (d X,j ) j∈J metrizes the weak topology on X. But this will not be the case in our example related to parabolic problem (1) (the key steps of verifying the metric counterpart are presented in Lemma 8.26 below, which is essentially based on concatenating solutions appropriately).
In this article, we develop an existence theory for pullback attractors in a purely metric setting using essentially the assumption that the underlying multi-valued non-autonomous dynamical system (MNDS) is sequentially continuous in (possibly) different senses of convergence in the domain and the value space respectively. This concept is then used for proving sufficient conditions on the parabolic differential inclusion (1) for pullback attractors. Our main results about parabolic problem (1), i.e., Theorems 4.1 and 4.2 below, cover the following examples: ψ(y) u(t, y)dy· [1,2] with some ψ ∈ C 0 c (Ω), Λ ∈ L 2 (R).
These examples indicate the basic difference between the two theorems: If the set-valued mapping G in the reaction term satisfies the inclusion principle (2) in addition, then we can even draw conclusions about the pullback attractor for examples with infinite delay (i.e., θ = ∞ and so, Θ ≥ 0 arbitrary). Whenever not supposing such a monotonicity property of G, we need a form of decaying influence of the past or "vanishing" memory -for technical reasons in connection with asymptotic compactness. This is represented by the non-negative Lebesgue integrable function Θ of space.
The article has the following structure. In section 2, we list all assumptions used for differential inclusion (1) later. This concerns the set-valued mapping G of the reaction term in particular. Section 3 summarizes the aspects concerning the existence of (both strong and mild) solutions to (1) and, we briefly sketch how they result from our earlier arguments in [37]. These solutions induce a multi-valued non-autonomous dynamical system (MNDS) on a function space and, here we aim at its pullback attractors. Section 4 provides two selections of assumptions from section 2 each of which is sufficient for the existence of a pullback attractor. In particular, Theorems 4.1 and 4.2 specify the sense of convergence to which their attraction refers. Section 5 suggests how to reformulate the parabolic differential problem (1) as an abstract evolution inclusion in a metric vector space. Its metric proves to be the pointwise supremum of a family of "distance functions". This metric setting is the starting point for investigating pullback attractors in section 6. There we extend the concept of so-called norm-to-weak processes by Zhong et al. beyond Banach spaces.
For the sake of transparency, all proofs required in this article are collected in the last sections. We have aimed at a self-contained and detailed way of presentation. Section 7 provides all the proofs of results about attractors formulated in section 6. Finally section 8 consists of all the remaining proofs of statements in sections 3 -5. In other words, we there verify the properties step by step required for applying our general metric theory (from section 6) to the original parabolic differential problem (1).
This class of parabolic inclusion problem with memory clearly differs from what was investigated by Chepyzhov and collaborators, for example (see, e.g., [14,15,16] and references therein).
3. Existence of solutions to the reaction-diffusion inclusions with delay. The existence of solutions has already been investigated in [37] for the special case of constant finite delay θ.
Let Ω ⊂ R d be a bounded open set with smooth boundary. In addition to T, θ > 0 and c ≥ 0 given, suppose that the set-valued map G : R×Ω×R×R×L 2 (Ω) ; R satisfies the hypotheses (G1) -(G5).
Then, for every initial state u 0 ∈ C 0 [−θ, 0], L 2 (Ω) , there exists a continuous curve u : [−θ, T ] → L 2 (Ω) such that u induces a strong solution to the following inclusion problem Indeed, this theorem results from essentially the same proof as [37,Corollary 3.3]. The only two differences are that first, the essential supremum there is taken over t − T ≤ s ≤ t (with the same time extent T as the interval of existence [0, T ]) and second, the initial state function u 0 there is constant w.r.t. time. These two modifications, however, do not have any significant influence on the proof in [ (4) is characterized equivalently in terms of integral solutions (in the sense of Bénilan [5]) and mild solutions u : [−θ, T ] → L 2 (Ω) of the single-valued parabolic problem for a selector f ∈ L 2 0, T ; for Lebesgue-almost all t ∈ [0, T ] and x ∈ Ω. Hence we have the additional representation where S(t) t≥0 denotes the heat semigroup on L 2 (Ω) with homogeneous Dirichlet boundary conditions. Parabolic differential equations are known to have some smoothing effect on their (weak) solutions and so, better results about the regularity in space are known (see, e.g., [4,39,42] for the general theory and [27], [37, § 4] for some more details). This aspect, however, is not really relevant now for specifying pullback attractors. Next we extend the existence result from inclusion problem (4) to the case (1) with infinite delay by means of a simple piecewise argument: Let Ω ⊂ R d be a bounded open set with smooth boundary, T ≥ t 0 , θ > 0, c ≥ 0 and Θ ∈ L 1 (Ω, R + ). Suppose that the set-valued map G : R × Ω × R × R × L 2 (Ω) ; R satisfies the hypotheses (G1) -(G5).
Proof. It results directly from Theorem 3.1 in a piecewise way. Without loss of generality we assume t 0 = 0. Indeed, we first focus on the wanted solution just in 0, θ 2 and then, the local result can be iterated until we have found a solution up to final time T after finitely many steps. This preliminary restriction to 0, θ 2 has the advantage that every u : −∞, θ 2 → L 2 (Ω) with u (−∞,0] = u 0 satisfies for each t ∈ 0, θ 2 and x ∈ Ω ess sup Hence we just have to apply the preceding existence statement to the auxiliary map G : R × Ω × R × R × L 2 (Ω) ; R defined by G(t, x; y, z, v) := G t, x; y, max ess sup which obviously also satisfies conditions (G1) -(G5).

4.
Main results. This paper focuses on the existence of pullback attractors of the non-autonomous dynamical system induced by parabolic differential inclusion (1). For notational consistency with the literature a set-valued mapping A : R ; X for some space X will often be considered as a family A = {A(t) : t ∈ R} of subsets of X with the subsets of X denoted by A(t) instead of A(t). Then there exists a set-valued mapping A : R ; C 0 (−∞, 0], L 2 (Ω) , t → A(t) with following features: 1. For every t ∈ R, the set A(t) ⊂ C 0 (−∞, 0], L 2 (Ω) is non-empty and compact with respect to locally uniform convergence in (−∞, 0]. For every function ψ ∈ A(t), the limit of e γ s ψ(s) for s −→ −∞ exists in L 2 (Ω).

2.
A is negatively invariant, i.e., fixing τ ≤ T arbitrarily, every ψ ∈ A(T ) which is a strong solution to the parabolic differential inclusion with space-dependent delay and

4.
A is pullback attracting in the following sense: ∞ and the limit of e γ s ψ k (s) ∈ L 2 (Ω) for s → −∞ exists for every k ∈ N. Each ψ k , k ∈ N, initializes a strong solution u k : (−∞, T ]×Ω → R of parabolic differential inclusion (7) in A similar existence result can be shown for the problems with infinite delay (instead of space-dependent delay), i.e., depending on ess sup −∞ < s ≤ t [u(s, x)] c in the fourth argument on the right-hand side. In particular, hypothesis (G7) specifies a monotonicity condition on the set-valued mapping G with respect to its fourth argument, in which the essential supremum term usually occurs. From the conceptual point of view, we consider the reaction-diffusion inclusion (1) with memory in the so-called history space setting, i.e., a past history variable is introduced as additional component [16] and so, the underlying state space consists of (some) continuous functions (−∞, 0] −→ L 2 (Ω). This notion was proposed by Dafermos [19].
The first aspect of convergence in condition (4.) is usually called locally uniform convergence and, it is exactly the same topology as in many publications about socalled trajectory attractors (see, e.g., [14,17,18,26,34,58] and references therein). Chepyzhov and Miranville suggested trajectory attractors for hyperbolic and parabolic differential equations with memory [15,16]. Their underlying state space and the semigroup are similar to our considerations, but their criterion of attraction is based on convergence in forward time direction (see [16,Definition 0.1 (3)], in particular).
Our main results, however, focus on pullback attraction, i.e., convergence w.r.t. Hausdorff semi-distance for the initial time tending to −∞ while the end time is fixed. To the best of our knowledge, the only statements combining trajectory attractors with pullback convergence so far were published by Zhao and Zhou [59]. They consider trajectories in forward time direction and so, their underlying state space consists of continuous functions on [0, ∞) −→ Y (with a Banach space Y ).
In a word, main Theorems 4.1 and 4.2 specify sufficient conditions for a "pullback trajectory attractor" combining the modified gist of Chepyzhov and Miranville [16] with pullback attraction -but in a new way quite different from [59].
Our proofs are based on the concept of multi-valued non-autonomous dynamical systems in metric spaces (see § 5.2 below for details). It is worth mentioning here that the general theory of their pullback attractors provides some more information about the convergence. Indeed, the semi-distance between sets used in criterion (17) below implies, for example, that the convergence in statement (4.) is uniform with respect to all strong solutions u k : (−∞, T ] × Ω → R initialized by the same ψ k .

5.
A way to a multi-valued non-autonomous dynamical system.

5.1.
Reformulation as an abstract evolution inclusion in a metric vector space. The inclusion system (1) can be reformulated as an abstract evolution equation on the function space where γ > 0 is a constant specified in a moment. C γ supplied with the weighted supremum norm is a separable Banach space [8] since the transformation argument on [30, page 15] can be extended to function values in a separable Banach space such as L 2 (Ω) here.
We reformulate system (1) as the abstract evolution inclusion where A is defined in terms of the minus Laplace operator on Ω with homogeneous Dirichlet boundary condition, u t := u(· + t) : (−∞, 0] → L 2 (Ω) and the set-valued map G : R × C γ ; L 2 (Ω) is defined by for almost every x ∈ Ω The dependence of G(t, φ) on the essential supremum is rather difficult to handle by means of the weighted supremum norm · γ on C γ . It seems to be more recommendable to consider the metric d c,γ : .
(11) In regard to pullback attractors, however, it has proved to be useful to model some "effect of vanishing influence of delay" or "vanishing memory" in the sense that the past becomes less and less relevant. Indeed, the first term in d c,γ (u, v) takes this aspect into consideration in form of the weighting factor e γ s (with s ≤ 0 and fixed γ > 0). For implementing it also in the second term with essential supremum, we introduce additionally the following family of metrics d c,γ,τ : Levi's theorem about monotone convergence provides directly a connection between these metrics: The representation in (12) via supremum is analogous to a standard result about norms in any real Banach space Y , which is a consequence of Hahn-Banach theorem: : Y −→ R linear and continuous, Lin(Y,R) ≤ 1 . (13) Indeed, one distance can be represented as supremum of a family of some other distance functions (each of which need not satisfy all typical conditions on metrics) and so, the related convergence is uniform w.r.t. the underlying parameter of the family. For avoiding misunderstandings with terms established in functional analysis, the convergence in C γ , d c,γ is called "uniform w.r.t. the parameter" here. If the convergence w.r.t. d c,γ,τ holds for each τ > 0 separately, we call it "pointwise in parameter" abbreviated as "p.i.p.". (In [43, § 0.3.4 step (F)], the terms "strong" and "weak convergence" are used instead, but it is worth pointing out that this interpretation should be understood in the purely metric context, i.e., the linear structure of C γ is not relevant in this generalizing interpretation and so, the "weak convergence" a.k.a. "convergence pointwise in parameter τ " here is not based on some dual space as usual in functional analysis.) From the topological point of view, the condition on a sequence in C γ to converge w.r.t. each metric d c,γ,τ , τ > 0, can be expressed by means of just a single metric on C γ , e.g., (14) in which we have slightly modified the Fréchet product metric [21,25]. It is worth into consideration -and not the standard norm of L ∞ [−τ, 0]; L 2 (Ω) . At first glance, this similarity does not imply that d c,γ,p.i.p. metrizes the compact-open topology for continuous functions (−∞, 0] −→ L 2 (Ω), · L 2 (Ω) .
The proof of this lemma is postponed to § 8.1. Finally, we specify solutions to evolution inclusion (9) in C γ : Fix initial state φ ∈ C γ and initial time t 0 ∈ R and, choose final time T > t 0 arbitrarily. Under appropriate growth conditions on G, each solution u : (−∞, T ] −→ L 2 (Ω) to the parabolic differential inclusion The equivalent mild characterization (6) leads to the following representation of the solution y t (·) : with a selector f : By definition, φ ∈ C γ implies that the limit of e γ s φ(s) ∈ L 2 (Ω) for s −→ −∞ exists and so,

Multi-valued system generated by the evolution inclusion. Define
and consider any metric space (X, d X ). A set-valued map U : R 2 ≥ × X ; X is called a multi-valued non-autonomous dynamical system (MNDS) if it satisfies the following conditions: In particular, U induces a set-valued mapping R 2 ≥ × P(X) −→ P(X) (with P(X) denoting the power set of X) according to This notion is already underlying the right-hand side of the inclusion in condition (3.).
A multi-valued non-autonomous dynamical system U : R 2 ≥ × X ; X is said to be strict if even equality holds in condition (3.), i.e., Further details about this setting are presented in [8,9,38,45], for example. Now the solutions to evolution inclusion (9) prove to induce a multi-valued nonautonomous dynamical system on C γ if γ > 0 is fixed sufficiently large (see § 8.2 for the details of the proof): with Θ ≥ 0 and, let γ > 0 be a constant larger than the smallest eigenvalue λ 1 > 0 of the negative Laplacian operator with homogeneous Dirichlet boundary conditions in Ω. For 6. The general theory of pullback attractors in spaces with (possibly) two metrics.
6.1. Pullback attractors of MNDS in a (standard) metric space (X, d X ): Definition and general existence via pullback asymptotic compactness. Now we specify the concept of pullback attractors with respect to a universe of sets and establish a condition sufficient for their existence. These considerations are continuing the lines of [9,10], [38,Ch.9], for example. Existence Theorem 6.3 below slightly extends [8, Theorem 3.3] because we assume closed graph of U (t, t 0 , ·) : C γ ; C γ instead of upper semi-continuity for each tuple (t, t 0 ) ∈ R 2 ≥ . For the general setting consider any metric space (X, d X ) again and let U : R 2 ≥ × X ; X be a multi-valued non-autonomous dynamical system. A set-valued mapping D : R ; X (equivalently denoted by D = {D(t) : t ∈ R}) is said to be negatively, strictly, or positively invariant (resp.) for the MNDS U if B ∈ D is said to be pullback D-absorbing if for every D ∈ D and every t ∈ R, there exists some T : Combining pullback absorption and some form of compactness is to lay the basis for verifying pullback attraction. For this purpose we need an additional feature of the families D ∈ D for which the limit is required. Following a suggestion by Schmalfuß [50], a subset D ⊂ D of multi-valued mappings R ; X is called a universe if it satisfies the following inclusion for every family D = {D(t) : t ∈ R} ∈ D in addition: Every further mapping D ∈ D with D (t) ⊂ D(t) for each t ∈ R also belongs to D.
A is said to be a strict global pullback D-attractor if the invariance property in (iii) is strict.
It is worth mentioning that the concept of pullback attractor does not imply any general equivalence to forward attractors or to so-called trajectory attractors, both of which have already been extended to non-autonomous dynamical systems (see, e.g., [14,16,17,26,38,58]). As an obvious formal difference, pullback attractors consist of subsets of state space X corresponding to a set-valued mapping of time whereas (uniform) trajectory attractors are defined as a set of (usually continuous) functions [0, ∞) −→ X specified by means of the translation semigroup. There are examples showing that pullback and forward attractors do not have to coincide (see, e.g., [36,38]) and, the relation between global forward and trajectory attractors (in the sense of Chepyzhov and Vishik) is investigated in [34], for example.
Following the standard approach to pullback attractors as presented in [8,38], for example, the main tool to prove the existence of an attractor is the concept of pullback ω-limit set. For a multi-valued mapping D : R ; X, we define the pullback ω-limit set as the t-dependent set Λ( D, t, d X ) ⊂ X given by where the closure is considered w.r.t. d X . Obviously this set is closed in (X, d X ), but it may be empty. It can be proved that y ∈ Λ( D, t, d X ) if and only if there exist sequences t n +∞ in R and y n ∈ U (t, t − t n , D (t − t n )) ⊂ X with lim n→+∞ y n = y.
The next lemma extends [8, Lemma 3.2] to closed graphs of U (t, τ, ·) and thus, it is a generalization of [9, Theorem 6 and Lemma 8] to the case of a general universe D (instead of just the bounded sets of X). It will be extended to a more general metric setting in Lemma 6.8 below and so, the proof is not presented in detail here.
≥ × X ; X be a multi-valued non-autonomous dynamical system in the metric space (X, d X ). Assume for each (t, τ ) ∈ R 2 ≥ that U (t, τ, ·) : X ; X has closed graph. Let B : R ; X be a multi-valued mapping such that U is pullback asymptotically compact w.r.t. B, i.e., for any real sequence t n → +∞, every sequence y n ∈ U t, t − t n , B(t − t n ) has a converging subsequence.
Then, for every t ∈ R, the pullback ω-limit set Λ B, t, d X ⊂ X is non-empty, compact, and We can now specify a condition sufficient for the existence of pullback attractors. It results from the preceding lemma for exactly the same reasons as [8,Theorem 3.3] as we show for a more general situation in section 7 below. Theorem 6.3. In addition to the hypotheses in Lemma 6.2, suppose that B ∈ D is pullback D-absorbing and has closed values. Then, the set-valued mapping A : R ; X, t → A(t) given by is a pullback D-attractor. Moreover, A is the unique element from D with these properties. In addition, if U is a strict MNDS, then A is strictly invariant.

6.2.
Pullback flattening for MNDS in metric spaces (not necessarily Banach spaces). Obviously, some form of compactness has to be guaranteed for covering the asymptotic features of the dynamical systems independently of the respective initial time instants. In Lemma 6.2, we used the concept of pullback asymptotic compactness, which represents a form of sequential compactness: For any sequence t n → +∞ in R, every sequence y n ∈ U t, t − t n , B(t − t n ) has a converging subsequence (see, for example, [8,9,10,38]).
An alternative criterion is based on the Kuratowski measure of non-compactness. Zhong et al. suggested the related "condition (C)" for attractors of autonomous dynamical systems in Banach spaces (e.g., [44,53,60]) and then extended it to pullback attractors of non-autonomous dynamical systems (e.g., [41,51,52,54,55,56]). It has the basic notion in common with the flattening property, which was introduced for proving the existence of random attractors in [35] and which is presented in [11, § 2.4] as well as [38,Ch. 12] for the general setting in Banach spaces. Here we adapt [11,Definition 2.24] to the multi-valued case in metric spaces, which do not have to be Banach spaces: ≥ × X ; X a multi-valued non-autonomous dynamical system and D a universe in X.
U is said to be D-pullback flattening w.r.t. d X if for any t ∈ R fixed arbitrarily, every B = {B(s) : s ∈ R} ∈ D and ε > 0, there exist T 0 = T 0 (t, ε, B) > 0 and a nonempty subset M ε ⊂ X such that (i) M ε is relatively compact, For the sake of transparency, we formulate explicitly that whenever X is a Banach space, the established definition (as, e.g., in [11,38,41,54,55,56,60]) is just a special case. This implication is a rather obvious consequence of the compactness theorem of Heine-Borel and so, its proof is skipped. Lemma 6.5. Let X be a real Banach space, U : R 2 ≥ × X ; X a MNDS and D a universe in X.
Then U is D-pullback flattening (in the sense of Definition 6.4) if it satisfies the following condition: For any t ∈ R, every B = {B(s) : s ∈ R} ∈ D and each ε > 0, there exist some T 0 = T 0 (t, ε, B) > 0, a finite-dimensional subspace X ε ⊂ X and a bounded projector P ε : Lemma 6.6. Let U be a MNDS in a metric space (X, d X ) and D be a universe.
Whenever U is D-pullback flattening, then U is also pullback asymptotically compact w.r.t. any B ∈ D (in the sense of Lemma 6.2).
The lemma adapts [11,Theorem 2.25] to our metric setting. If X is a uniformly convex Banach space, then the opposite inclusion can be verified by arguments similar to those used for [11,Theorem 2.25] or [38,Theorem 12.12] (the latter for skew-product flows). However, we are not going to use this opposite inclusion here.
Motivated by equation (13), which is well established in Banach spaces, we regard a sequence (x n ) n∈N in X as converging to some x ∈ X "uniformly w.r.t. the parameter If the index set J is at most countable, we assume J ⊂ N without loss of generality and, the p.i.p. convergence can be expressed in terms of a single distance function d X,p.i.p. : X × X → [0, 1], e.g., It is worth mentioning here that this interpretation does not require d X,j or d X,p.i.p. to satisfy all three typical conditions on metrics. Their essential purpose is just to quantify distances between any two points of X. If each d X,j , j ∈ J , is a metric on X, however, then so is the proposed d X,p.i.p. (the triangle inequality results from the same arguments as in [57, page 27]).
In this setting, we want to specify sufficient conditions for the existence of pullback attractors similar to Theorem 6.3. The main idea is to select the metrics d X , d X,p.i.p. in a way which is more convenient to verify in the respective context. In regard to suitable (asymptotic) compactness, for example, the "p.i.p." metric d X,p.i.p. induces a weaker condition on the considered system than the "strong" metric d X , which is often used in Banach spaces (like the examples in [8,41]). With respect to (semi-) continuity or closed graph, however, publications using norm-to-weak continuity in Banach spaces (e.g., [41,52,56,55,60]) inspire us to supply domain and value space with different metrics, namely (X, d X ) ; (X, d X,p.i.p. ). To the best of our knowledge, this aspect is new in the metric setting.
The basic idea of using diverse metrics simultaneously leads to the question how to specify pullback ω-limit sets. Definition 6.7. Let (X, d X ) be a metric space and d X,j : X × X −→ [0, ∞), j ∈ J ⊂ N, a family of distance functions with d X = sup j∈J d X,j and, define d X,p.i.p. by (22).
A multi-valued non-autonomous dynamical system U : R 2 ≥ × X ; X is said to have the pullback ω-Mazur property with respect to a mapping B : R ; X and d X , d X,p.i.p. if the related pullback ω-limit sets satisfy the following condition for every tuple (t, τ ) ∈ R 2 ≥ : By definition (19), every point y ∈ Λ B, τ, d X,p.i.p. is related to sequences s n → +∞, Obviously d X,j ≤ d X implies for each j ∈ J that the closure w.r.t. d X on the right-hand side of (19) is contained in its counterpart w.r.t. d X,j and so, Λ B, t, d X ⊂ Λ B, t, d X,p.i.p. is satisfied at every time t ∈ R. Condition (i) specifying the pullback ω-Mazur property ensures the equality of these pullback ω-limit sets, i.e., each element of Λ B, t, d X,p.i.p. can be characterized as a limit w.r.t. d X : Condition (ii) provides a connection between the respective "flows" of the related approximating sequences. It considers the Hausdorff semi-distance between sets w.r.t. d X,p.i.p. and so, it can be interpreted as some form of sequential upper semicontinuity. Here it is worth mentioning that this condition (ii) does not concern the set-valued mapping U (t, τ, ·) : X ; X in general, but merely those sequences which are related to pullback ω-limit points and, they may depend on t, τ, y, (y n ) n∈N . This is an essential difference from other standard assumptions like norm-to-weak upper semi-continuity.
The designation "ω-Mazur property" is motivated by two aspects: First we focus essentially on the asymptotic features of sequences as the underlying initial time tends to −∞. This is indicated in the letter ω. Secondly, in any normed linear spaces, the well-known Lemma of Mazur implies an alternative characterization of weakly converging sequences in terms of strongly converging convex combinations (see, e.g., [57, Ch.V.1 Theorem 2 on page 120]). It is this type of relationship between weak and strong convergence which we extend beyond vector spaces here. The next step is to extend Lemma 6.2 to this nonlinear setting with (possibly) several metrics. Obviously Lemma 6.2, which goes essentially back to [8], is the special case with d X = d X,j for all indices j ∈ J since d X and d X,p.i.p. are then topologically equivalent and the pullback ω-Mazur property is trivial (just setting y n := y n , n ∈ N). This adapted lemma, however, leads to a further theorem about the existence of pullback attractors: Lemma 6.8. Consider the metric space (X, d X ), the family of metrics (d X,j ) j∈J and d X,p.i.p. as in Definition 6.7. Let U : R 2 ≥ × X ; X be a multi-valued nonautonomous dynamical system. Suppose B : R ; X to be a multi-valued mapping satisfying (1.) U is pullback asymptotically compact w.r.t. B and d X,p.i.p. , i.e., for any real sequence t n → +∞, every sequence y n ∈ U t, t − t n , B(t − t n ) has a subsequence converging w.r.t. d X,p.i.p. .
Then, for every t ∈ R, the pullback ω-limit set Λ B, t, d X, Assume in addition that .
Theorem 6.9. In addition to all the assumptions of Lemma 6.8, let D be a universe in X. Suppose B ∈ D to be pullback D-absorbing and to have closed values w.r.t. d X .
Then, the set-valued mapping A : As an immediate consequence of Lemma 6.6, assumption (1.) about asymptotic pullback compactness is satisfied whenever U is D-pullback flattening and so, we obtain: Corollary 6.10. Consider the metric space (X, d X ), the family of metrics (d X,j ) j∈J and d X,p.i.p. as in Definition 6.7. Suppose D to be a universe in X and B : R ; X to belong to D. Let U : R 2 ≥ × X ; X be a multi-valued non-autonomous dynamical system satisfying hypotheses (2.), (3.) in Lemma 6.8. Furthermore assume U to be D-pullback flattening w.r.t. d X,p.i.p. (in the sense of Definition 6.4).
Then the conclusions of Theorem 6.9 hold, i.e., A : R ; X, t → Λ B, t, d X,p.i.p. is a pullback D-attractor w.r.t. d X,p.i.p. . Furthermore, A is the unique element from D with these properties. In addition, if U is a strict MNDS, then A is strictly invariant.
7. Proofs about general existence of pullback attractors (stated in § 6). Proof of Lemma 6.6. Let U be a MNDS on a metric space (X, d X ) and D be a universe such that U is D-pullback flattening (in the sense of Definition 6.4). Fix t ∈ R and B = {B(s) : s ∈ R} in D arbitrarily. For each sequence (t n ) n∈N in [0, ∞) with t n → ∞, it remains to show that any sequence (y n ) n∈N with y n ∈ U t, t − t n , B(t − t n ) has a convergent subsequence. Due to [57, Ch. 0.2 Theorem on page 13], it is sufficient to verify a subsequence (y n k ) k∈N such that the set {y n k | k ∈ N} ⊂ X is totally bounded, i.e., for every ρ > 0, the set {y n k | k ∈ N} which can be covered by finitely many balls of radius ρ w.r.t. d X .
For each index k ∈ N, there exist some T k > 0 and a nonempty compact subset We can select a monotone sequence of indices n k ∞ with t n k ≥ max{T 1 , . . . , T k } for every k ∈ N. Consider the subsequence (y n k ) k∈N .
Fixing ρ > 0 arbitrarily, there is some K = K(ρ) ∈ N with 1 K < ρ 2 . The compact subset M K ⊂ X can be covered by finitely many open balls of radius ρ 2 (w.r.t. d X ). Thus the union of open balls with the same respective centers and radius ρ covers y n k k ≥ K ⊂ s≤s0 U (t, s, B(s)) with s 0 := t − max{T 1 , . . . , T K }. Obviously, the set {y n k | k ∈ N} can also be covered by finitely many open balls of radius ρ.
Proof of Lemma 6.8. We follow essentially the arguments of [8, Lemma 3.2]. Consider any sequence y n ∈ U t, t − t n , B(t − t n ) with t n → +∞. As U is pullback asymptotically compact w.r.t. B and d X,p.i.p. , there exists a subsequence converging w.r.t. d X,p.i.p. and, its limit y belongs to Λ B, t, d X,p.i.p. , i.e., Λ B, t, d X,p.i.p. is non-empty.
We now prove that Λ B, t, d X,p.i.p. is compact in (X, d X,p.i.p. ). For any sequence (y n ) n∈N in Λ B, t, d X,p.i.p. , there exist sequences t n → +∞ and z n ∈ U t, t − t n , B(t − t n ) with d X,p.i.p. (y n , z n ) < 1 n for each n ∈ N. Now the pullback asymptotic compactness of U again implies the existence of a subsequence z n k k∈N whose limit z w.r.t. d X,p.i.p. is also contained in the closure Λ B, t, d X,p.i.p. . Then, The claimed limit in (23) is now proved by contradiction. If (23) does not hold, then there exist ε > 0 and sequences t n → +∞, y n ∈ U t, t − t n , B(t − t n ) such that dist X,p.i.p. y n , Λ B, t, d X,p.i.p.

Def.
= inf d X,p.i.p. (y n , ξ) ξ ∈ Λ B, t, d X,p.i.p. is larger than ε. As U is pullback asymptotically compact w.r.t. B and d X,p.i.p. , there exists a subsequence (y n k ) k∈N whose limit w.r.t. d X,p.i.p. is also contained in the closed set Λ B, t, d X,p.i.p. , but this contradicts the preceding lower distance bound ε.
Fix (t, τ ) ∈ R 2 ≥ and y ∈ Λ B, t, d X,p.i.p. arbitrarily. Then, by definition (19), there exist sequences t n → +∞, y n ∈ U t, t − (t n − τ ), x n , x n ∈ B t − (t n − τ ) with d X,p.i.p. (y n , y) −→ 0. For all t n ≥ t, the standard process property implies U t, t − t n + τ, x n ⊂ U t, τ, U (τ, t − t n + τ, x n ) , and then y n ∈ U t, τ, z n with some z n ∈ U τ, t − t n + τ, x n . As before, the pullback asymptotic compactness w.r.t. B and d X,p.i.p. ensures a subsequence (again denoted by) (z n ) n∈N which converges w.r.t. d X,p.i.p. . Its limit z is contained in Λ B, τ, d X,p.i.p. .
Assuming the pullback ω-Mazur property w.r.t. B, d X and d X,p.i.p. , this element z is related to (possibly different) sequences t k → +∞, Considering appropriate subsequences (with the same notation) instead, we obtain index sequences n j ∞, k j ∞ such that Finally the graph of U (t, τ, ·) : X, d X ; X, d X,p.i.p. is supposed to be closed and so, the respective limits for j → ∞ reveal Proof of Theorem 6.9. As in the proof of [8, Theorem 3.3], we first verify pullback D-attraction, i.e., for every D ∈ D, Indeed, thanks to the limit (23) in Lemma 6.8, for every ε > 0 and t ∈ R, there exists T 1 (t, ε) > 0 such that for τ ≥ T 1 (t, ε), B ∈ D is assumed to be pullback D-absorbing and so, every D ∈ D has some Hence we have for all τ > 0 sufficiently large Condition (iii) in Definition 6.1, i.e., negative invariance of pullback D-attractors, follows from inclusion (24). Pullback D-absorption of B ∈ D implies some

Hence we have the inclusion A(t)
Def.
This observation leads to the conclusion that A is unique. Indeed, suppose that A ∈ D is another pullback D-attractor w.r.t. d X,p.i.p. , then as we have that A (t) ⊂ A(t) for each t ∈ R. Exchanging A and A , it follows that A = A .
Finally, assume that the multi-valued non-autonomous dynamical system U is strict (in addition). It remains to show U (t, r, A(r)) ⊂ A(t) for any real t ≥ r. Indeed, we have for every τ ≥ 0 U t, r, A(r) ⊂ U t, r, U (r, r − τ, A(r − τ )) = U t, r − τ, A(r − τ ) .
Since ε > 0 was fixed arbitrarily, we conclude from the set A(t) ⊂ X being closed w.r.t. d X,p.i.p. that U (t, r, A(r)) ⊂ A(t), as required.
The limit for j −→ ∞ reveals v(s) − e γ s u(s) L 2 (Ω) ≤ 2 ε, for each s ≤ σ, i.e., v is also the limit of e γ s u(s) for s −→ −∞. Hence, u ∈ C γ . Since the index K depends only on ε, we have u k − u γ −→ 0. It is worth mentioning that this unique limit function u : (−∞, 0] −→ L 2 (Ω) does not depend on the parameter τ and so, limits of Cauchy sequences never depend on τ . In particular, the conclusions about d c,γ,τ with any τ > 0 also hold for the joint metric d c,γ,p.i.p. as well as the "strong" metric d c,γ .
It remains to prove for τ > 0 fixed initially that (u k ) k∈N converges to u w.r.t.  (9). According to Corollary 3.2, every state φ ∈ C γ at any time t 0 ∈ R initializes at least one strong solution to parabolic differential inclusion (1) (with space-dependent delay). Hence, the set-valued mapping U specified in Proposition 5.3 satisfies U (t, t 0 , φ) = ∅.
In regard to conclusions about closed values of U , we require the Nemytskii operator adapted to the arguments on the right-hand side of evolution problem (9) -similarly to [ .) (iii) For every ω ∈ Ω, the set-valued map F(ω, ·) : E 1 ; E 2 is Hausdorff upper semi-continuous. (iv) There exist α ∈ L q ( Ω) and a constant c > 0 such that F(ω, z) E2 ≤ a(ω) + c · z p q E1 holds for µ-almost all ω ∈ Ω and every z ∈ E 1 . Then, the set-valued map F : L p ( Ω, E 1 ) ; L q ( Ω, E 2 ) defined by is Hausdorff upper semi-continuous, i.e., for every u 0 ∈ L p ( Ω, E 1 ) and ε > 0, there exists a radius ρ > 0 such that every u ∈ L p ( Ω, This general statement has already laid the basis for the following conclusions in [37]: Consider the set-valued mapping G : R × L 2 (Ω) 3 ; Then, the set-valued mapping G has the following properties:  (iv) y j (·) j∈N converges to some y(·) weakly in L 1 (0, T ; Y ).

8.3.
Closed graphs of U (t, t 0 , ·) with respect to various metrics on C γ . In this and the following subsections, U : R 2 ≥ × C γ ; C γ always denotes the strict multi-valued non-autonomous dynamical system induced by the parabolic differential inclusion (15) as specified in the mild representation (16). The coefficient mapping G : R × Ω × R × R × L 2 (Ω) ; R is supposed to fulfill hypotheses (G1) -(G5). Fix θ > 0 and Θ ∈ L 1 (Ω) with Θ ≥ 0. Let γ > 0 be larger than the smallest eigenvalue λ 1 > 0 of the negative Laplacian operator with homogeneous Dirichlet boundary conditions in Ω -as in Proposition 5.3. The metrics d c,γ , d c,γ,τ and d c,γ,p.i.p. on C γ are defined in § 5.1.
From now on, our essential goal is to verify in several steps that U satisfies the assumptions of Theorem 6.9. This implies the main Theorem 4.1. Finally, the main Theorem 4.2 (about globally infinite delay) can be concluded from (standard) Theorem 6.3 by means of the additional assumption (G7) and its implied modifications, of which the most significant one is formulated in Lemma 8.9 below.
Proof. We use essentially the same arguments as in the proof of Lemma 8.7, but now take a converging sequence for the argument of U (t, t 0 , ·) into additional consideration. Choose any sequences (φ (k) ) k∈N , (u (k) ) k∈N in C γ with u (j) ∈ U (t, t 0 , φ (j) ) for each index j ∈ N and for a.e. s ∈ [t 0 , t] and

Hypothesis (G5) implies
for Lebesgue-almost every s ∈ [t 0 , t] and so, (f k ) k∈N is bounded in L ∞ t 0 , t; L 2 (Ω) due to Due to Alaoglu's theorem, there exists a sequence of indices k j ∞ such that (f kj ) j∈N converges to some f ∈ L 2 t 0 , t; L 2 (Ω) weakly in L 2 t 0 , t; L 2 (Ω) . We conclude from Lemmas 8.5 and 8.6 that f (s) ∈ G s, u( · + s − t)| (−∞,0] holds for Lebesgue-almost every s ∈ [t 0 , t]. In regard to the claim u ∈ U (t, t 0 , φ), it remains to verify The statement for s < −(t − t 0 ) is rather obvious since the convergence of φ (k) k∈N w.r.t. d c,γ implies the convergence in · γ and hence pointwise convergence in time.
Its proof is based on essentially the same arguments. We just replace preceding Lemma 8.5 by the following auxiliary result. Indeed, the additional assumption (G7) about the inclusion property of G (w.r.t. its fourth argument) now provides the essential tool for extending this upper semi-continuity to the set-valued map G on the right-hand side of evolution inclusion (9). It helps us to overcome the obstacle that C γ → L 2 (Ω), u → ess sup −∞ < s ≤ t [u(s, ·)] c is not continuous w.r.t. d c,γ,p.i.p. : Then for each t 0 ∈ R, the mapping G(t 0 , ·) : (C γ , d c,γ,p.i.p. ) ; L 2 (Ω) specified in (10) is Hausdorff upper semi-continuous with nonempty convex closed values.
has the following properties: Then B is pullback D-absorbing with respect to U .
The radius function ρ : R −→ R here is chosen as the unique solution to the initial value problem In statement (2.), we consider universes which may grow even exponentially (in backward time direction), but their exponential growth rate is bounded from above by the given parameters of U .
(2.) For q ∈ 0, λ 1 − β fixed arbitrarily, choose any family D = {D(t) : t ∈ R} of nonempty closed bounded sets in C γ with C D := sup e q t D(t) γ t < 0 < ∞. Then we conclude from Lemma 8.11 that for any (t, τ ) ∈ R 2 ≥ and φ ∈ D(τ ), Whenever we keep t ∈ R fixed and consider τ → −∞, we obtain due to As a consequence, B is pullback absorbing w.r.t. U . 8. 6. Standard results about a priori estimates for solutions to parabolic differential equations.
An a priori estimate of the time derivative for weak solutions to parabolic equations will be used. The following bounds are special cases of inequalities (6.3), (6.6), respectively, in [39, Ch. III § 6] and, the existence is stated in [39, Ch. III Theorem 6.1]: Lemma 8.15. Consider the parabolic problem with h ∈ L 2 ([0, T ] × Ω) and Lebesgue measurable coefficients a ij : [0, T ] × Ω → R satisfying the condition of uniform parabolicity and Then, for every u 0 ∈ W 1,2 0 (Ω), there exists a unique weak solution u : [0, T ]×Ω → R in W 1,2 ([0, T ] × Ω) and, it satisfies the estimates for every t ∈ (0, T ] Uniqueness implies that the several types of solutions coincide as mentioned in [37, § 2.1]. Hence, for integral solutions to the non-homogeneous heat equation, we obtain: Let Ω ⊂ R d be a bounded open set. Consider an integral solution u : [0, T ] → L 2 (Ω) to non-autonomous problem (29) with f ∈ L 2 0, T ; L 2 (Ω) ⊂ L 1 0, T ; L 2 (Ω) and u 0 ∈ W 1,2 0 (Ω). Then, u is both a strong and a weak solution. In particular, u ∈ W 1,2 ([0, T ] × Ω) and, it satisfies the following a priori estimate for every t ∈ [0, T ] estimates due to assumption (G3 ). Assumption (G3 ) about Lipschitz continuity of G w.r.t. the third and fourth argument implies condition (G2) about Hausdorff upper semi-continuity directly. Now we formulate some further conclusions from (G3 ) which will be used for proving main Theorem 4.1 later.
For proving Lemma 8.17, we use several standard tools in set-valued analysis and evolution equations for imitating conclusions which are usually formulated as Filippov's theorem about differential inclusions (e.g., [ Let Ω be an arbitrary domain in R d and let A be an elliptic differential operator on L 2 (Ω) subject to homogeneous Dirichlet boundary conditions. Then, the solutions of ∂ t u = A u form a strongly continuous semigroup S(t) t≥0 of bounded linear operators on L 2 (Ω). For every t ≥ 0, the intersection L 1 (Ω) ∩ L ∞ (Ω) is invariant w.r.t. S(t), i.e., The restriction S(t) L 1 (Ω)∩L ∞ (Ω) t≥0 can be extended to a unique positive nonexpansive semigroup on L p (Ω) for every p ∈ [1, ∞], i.e., If the domain Ω ⊂ R d is bounded and A the Laplace operator, then the heat semigroup S(t) t≥0 with homogeneous Dirichlet boundary conditions satisfies for every p ∈ [1, ∞], t ≥ 0 and v ∈ L p (Ω) and for every t > 0, v ∈ L p (Ω), 1 ≤ p < q ≤ ∞ with 1 Proof of Lemma 8.17. Fix (T, t 0 ) ∈ R 2 ≥ and φ ∈ C γ with φ(0) ∈ W 1,2 0 (Ω) arbitrarily and let u : (−∞, T ] −→ L 2 (Ω) be any mild solution to parabolic differential inclusion (15), i.e., there exists some f ∈ L 2 t 0 , T ; L 2 (Ω) with for Lebesgue-almost all t ∈ [t 0 , T ] and x ∈ Ω such that we have the representation In particular, u is also a weak solution to a non-homogeneous heat equation with homogeneous Dirichlet condition. Hence in combination with linear growth assumption (G5) and Gronwall inequality, Corollary 8.16 implies u ∈ W 1,2 ([t 0 , T ] × Ω). Due to additional assumption (G3 ), Proposition 8.19 about measurable/Lipschitz parametrizations leads to a constant c ≥ 0 (independent of G) and a function g : x; y, z, v; η) η ∈ [−1, 1] for any t ∈ [t 0 , T ], x ∈ Ω, y, z ∈ R and v ∈ L 2 (Ω), (2.) for a.e. t ∈ [t 0 , T ] and any η ∈ [−1, 1], the function g(t, · ; · , · ; η) : holds for Lebesgue-almost all t ∈ [t 0 , T ] and x ∈ Ω. Now Corollary 3.2 is applied to a single-valued mapping on the right-hand side of (4) and so, for each ψ ∈ C γ , we obtain the existence of a (both strong and mild) solution v : (−∞, T ] → L 2 (Ω) of Similarly to u, we conclude v ∈ W 1,2 ([t 0 , T ] × Ω) from Corollary 8.16 and linear growth assumption (G5). The difference w := u − v : [t 0 , T ] → R also belongs to  [37,Lemma 4.4]). In particular, we obtain for Lebesgue-almost every x ∈ Ω and every t ∈ [t 0 , T ] Hence ess sup Thus, the function x → ess sup t0 ≤ s ≤ t |w(s, x)| belongs to L 2 (Ω) for each t ∈ [t 0 , T ] and, it satisfies In regard to an explicit upper L 2 (Ω) bound of this essential supremum, Lemma 8.15 will lead to an a priori estimate of ∂ s w 2 L 2 ([t0,t]×Ω) in a moment since w = u − v solves an initial-boundary value problem of the non-homogeneous heat equation. In more details, Lipschitz hypothesis (G3 ) and its consequences for g imply for Due to the general inequality α 2 j for all α 1 , . . . , α 4 ∈ R, the integration w.r.t. x ∈ Ω reveals with some constant C = C(Ω, c) > 0 x) and t → v(t, x) are absolutely continuous in [t 0 , T ] for Lebesgue-almost every x ∈ Ω. In particular, this implies for every t ∈ [t 0 , T ], i.e., with a modified constant (again denoted by) C = C(Ω, c) . Now Lemma 8.15 provides an a priori estimate of ∂ s w 2 ds.
Due to equation ( satisfies the following implicit inequality with a constant C = C(Ω, c) The Gronwall inequality for monotone increasing (and not necessarily continuous) functions (see, e.g., [43,Proposition A.1]) bridges the gap to an explicit estimate and leads to the following upper bound of δ(t) for each t ∈ [t 0 , T ] with adapted constants (always denoted by) C = C(Ω, c, T − t 0 ) ≥ 1. As a direct consequence of inequality (35), the proof of estimate (31) in Lemma 8.17 is completed.
Proof. Fix t ∈ R arbitrarily. Consider any sequences t n → +∞ and (φ n ) n∈N , (y n ) n∈N in C γ with y n ∈ U t, t − t n , φ n and φ n ∈ B(t − t n ) for each n ∈ N, i.e., in particular, φ n γ ≤ ρ(t − t n ) and for a.e. τ . Corollary 8.13 (1.) states the positive invariance of B w.r.t. U and so, holds for every s ∈ [t − t n , t].
As a consequence of Cantor's diagonal method, it suffices to show for every ε > 0 and infinite index set J ⊂ N that there exist an infinite subset J ε ⊂ J and some ζ ∈ C γ with y j − ζ γ < ε for all j ∈ J ε . Obviously, it holds for every n ∈ N sup s < −tn e γ s · φ n (s + t n ) L 2 (Ω) ≤ sup s < −tn e γ s · φ n γ e −γ (s+tn) Furthermore, we obtain for every n ∈ N and s ∈ [−t n , 0] e γ s · S(t n + s) φ n (0) L 2 (Ω) e γ s−λ1 (tn+s) In addition, the following estimate is satisfied for all n ∈ N and s ∈ [−t n , 0] t+s t−tn Due to the assumption γ > λ 1 (G6) > β > 0 and the a priori estimate we can choose N ε ∈ N, T ε > 1 and δ ε ∈ ]0, 1[ successively such that for all n ≥ N ε and −t n < −T ε − 2, This implies sup s ≤ −Tε e γ s · y n (s) L 2 (Ω) < ε 4 for every n ≥ N ε .
Last, but not least, the sequence of functions is also relatively compact in C 0 [−T ε , 0], L 2 (Ω) . Indeed, we consider the underlying functions of two real variables is bounded and so, the union of the image sets is relatively compact since S(δ ε ) : L 2 (Ω) −→ L 2 (Ω) is a compact linear operator. Moreover, the same conclusions from (27) and y n γ ≤ ρ(t) as on our way to inequality (39) reveal that and so, the family of integral functions It is worth mentioning here that this Lipschitz constant is uniform w.r.t. both n ≥ N ε and σ ∈ [−T ε , 0]. Next, we verify equicontinuity of the family F n (·, s)| [s,0] n ≥ N ε , s ∈ [−T ε , 0] because then the triangle inequality leads to the equicontinuity of (F n ) n≥Nε on their lays the basis for the same arguments as before: The auxiliary set results from a bounded set in L 2 (Ω) being mapped by the compact linear operator S(δ ε ) and so, it is relatively compact in L 2 (Ω). Due to [24, Lemma I.5.2] (again), the function is uniformly continuous. This implies equicontinuity of all the functions for n ≥ N ε , s ∈ [−T ε , 0] by means of the simple estimate for any σ 1 , σ 2 ∈ [s, 0] Together with the Λ ε -Lipschitz continuity of all the functions F n (σ, ·) [−Tε,σ] , n ≥ N ε , σ ∈ [−T ε , 0], we conclude from the triangle inequality (similarly to the proof of [37, Lemma 2.21]) that are equicontinuous (as functions of two variables though). The relative compactness of the image set in L 2 (Ω) has already been verified and so, the theorem of Arzelà-Ascoli guarantees that (F n ) n≥Nε is relatively compact in C 0 [−T ε , 0] 2 ∩R 2 ≥ , L 2 (Ω) .
Thus, the respective restrictions to the "diagonal", i.e., are also relatively compact in C 0 [−T ε , 0], L 2 (Ω) . Finally we again consider the complete representation and conclude from these two aspects of relative compactness in C 0 [−T ε , 0], L 2 (Ω) that for every infinite index set J ⊂ N, there is a sequence n k ∞ in J with the following three features: Due to the underlying Cauchy property w.r.t. the supremum norm on [−T ε , 0], there is an index κ = κ(ε, J) ∈ N sufficiently large such that for all k ≥ κ, Together with the preceding estimate (40) about s ≤ −T ε , we obtain for all indices k ≥ κ y n k − y nκ γ = sup s ≤ 0 e γ s y n k (s) − y nκ (s) L 2 (Ω) < ε .
Let Ω ⊂ R d be bounded open with smooth boundary, and suppose hypotheses (G1) -(G5). Fix real t 0 < t, 0 < τ < t − t 0 , and consider a sequence Then every sequence [y t (·; t 0 , φ k )] c k∈N induced by respective solutions to parabolic differential inclusion (15) is relatively compact with respect to the norm of The basic idea for proving Lemma 8.23 is to find a candidate for a converging subsequence by means of standard compactness theorems for L 2 ([−τ, 0] × Ω) first. Then we will use former results from [37] to conclude that this subsequence converges to the same limit function even w.r.t. the norm of L 2 Ω; L ∞ ([−τ, 0]) . In more detail, we will apply the following direct consequence of [37,Lemmas 4.6,4.7] which is stated here without proof: Then, the sequence of functions x → ess sup −τ ≤ s ≤ 0 v k (s, x), k ∈ N, defined on Ω converges to x → ess sup −τ ≤ s ≤ 0 v(s, x) in L 2 (Ω).
Furthermore consider the set-valued mapping B = {B(t) : t ∈ R} of "balls" B(t) := φ ∈ C γ φ γ ≤ ρ(t) specified in Corollary 8.13. Then U is pullback asymptotically compact w.r.t. B and d c,γ,p.i.p. due to recent Lemma 8.25. Hence, all the assumptions of Lemma 6.2 are satisfied -as required for Theorem 6.3.
Finally for q ∈ [0, λ 1 − β) fixed arbitrarily, we consider the universe D consisting of all families D = {D(t) : t ∈ R} of nonempty closed bounded sets in C γ satisfying sup e q t D(t) γ t < 0 < ∞.
Then a final step of relabeling the sequence ( y k ) k∈N (if required) leads to all the claimed features.
Next, aiming at property (i ) by means of estimate (33) in Corollary 8.18, we start with the distance ρ k := ϕ k − ϕ k γ + ess sup
The additional advantage of this construction is y k (s) = y(s) for every s ≤ −σ k Def.
8.13. The proof of main Theorem 4.1. Briefly speaking, it is an immediate consequence of Theorem 6.9, which specifies sufficient conditions for the existence of pullback attractors of multi-valued non-autonomous dynamical systems in terms of two metrics.
For each (t, τ ) ∈ R 2 ≥ the set-valued mapping U (t, τ, ·) : C γ , d c,γ ; C γ , d c,γ,p.i.p. has a closed graph according to Lemma 8.8. Now consider the set-valued mapping B = {B(t) : t ∈ R} of "balls" B(t) := φ ∈ C γ φ γ ≤ ρ(t) specified in Corollary 8.13. Then U is pullback asymptotically compact w.r.t. B and d c,γ,p.i.p. due to Lemma 8.25. The recently proved Lemma 8.26 states that U has the pullback ω-Mazur property w.r.t. B, d X , d X,p.i.p. (in the sense of Definition 6.7). Thus, all the assumptions of Lemma 6.8 are fulfilled -as required for Theorem 6.9.
Finally for q ∈ [0, λ 1 − β) fixed arbitrarily, we (again) consider the universe D of all families D = {D(t) : t ∈ R} of nonempty closed bounded sets in C γ satisfying sup e q t D(t) γ t < 0 < ∞. Due to Corollary 8.13 (2.), B is pullback D-absorbing with respect to U .
Hence, Theorem 6.9 ensures that U has a pullback D-attractor with respect to d c,γ,p.i.p. , which is even strictly invariant -as claimed. 2