# Long-Time Behaviour of Solutions for Autonomous Evolution Hemivariational Inequality with Multidimensional “Reaction-Displacement” Law

Authors: Pavlo O. Kasyanov; Luisa Toscano; Nina V. Zadoianchuk
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider autonomous evolution inclusions and hemivariational inequalities with nonsmooth dependence between determinative parameters of a problem. The dynamics of all weak solutions defined on the positive semiaxis of time is studied. We prove the existence of trajectory and global attractors and investigate their structure. New properties of complete trajectories are justified. We study classes of mathematical models for geophysical processes and fields containing the multidimensional “reaction-displacement” law as one of possible application. The pointwise behavior of such problem solutions on attractor is described.


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## Body

## 1. Introduction

Let a viscoelastic body occupies a bounded domainΩ⊂ℝd,d=2,3 in applications, and it is acted upon by volume forces and surface tractions (this section is based on results of [1] and references therein). The boundary Γ of Ω is supposed to be Lipschitz continuous, and it is partitioned into two disjoint measurable parts ΓD and ΓN such that means (ΓD)>0. We consider the process of evolution of the mechanical state on the interval (0,+∞). The body is clamped on ΓD and thus the displacement vanishes there. The forces field of density f0 act in Ω, the surface tractions of density g0 are applied on ΓN. We denote by u=(u1,…,ud) the displacement vector, by σ=(σij) the stress tensor and by ɛ(u)=(ɛij(u)) the linearized (small) strain tensor (ɛij(u)=(1/2)(∂jui+∂iuj)), where i,j=1,…,d.The mechanical problem consists in finding the displacement fieldu:Ω×(0,+∞)→ℝd such that
(1.1)u′′(t)-div⁡σ(t)=f0inΩ×(0,+∞),(1.2)σ(t)=Cɛ(u′(t))+Eɛ(u(t))inΩ×(0,+∞),(1.3)u(t)=0on⁡ΓD×(0,+∞),(1.4)σn(t)=g0on⁡ΓN×(0,+∞),(1.5)u(0)=u0,u′(0)=u1inΩ,
where 𝒞 and ℰ are given linear constitutive functions and n being the outward unit normal vector to Γ.In the above model, the dynamic equation (1.1) is considered with the viscoelastic constitutive relationship of the Kelvin-Voigt type (1.2) while (1.3) and (1.4) represent the displacement and traction boundary conditions, respectively. The functions u0 and u1 are the initial displacement and the initial velocity, respectively. In order to formulate the skin effects, we suppose (following [2, 3]) that the body forces of density f0 consists of two parts: f1 which is prescribed external loading and f2 which is the reaction of constrains introducing the skin effects, that is, f0=f1+f2. Here, f2 is a possibly multivalued function of the displacement u. We consider the reaction-displacement law of the form:
(1.6)-f2(x,t)∈∂j(x,u(x,t))inΩ×(0,+∞),
where j:Ω×ℝd→ℝ is locally Lipschitz function in u, and ∂j represents the Clarke subdifferential with respect to u. Let 𝒴d be the space of second-order symmetric tensors on ℝd.We consider the following problem: examine the long-time (ast→+∞) behavior of all (weak, generalized) solutions for (1.1)–(1.5) and (1.6).We remark that existence solutions theorems for evolution equations and inclusions where considered in [1–40] (see works and references therein).In [1] for finite time interval, it was presented the hemivariational formulation of problems similar to (1.1)–(1.6) and an existence theorem for evolution inclusions with pseudomonotone operators. We give now variational formulation of the above problem. To this aim let H=L2(Ω,ℝd), H1=H1(Ω,ℝd), ℋ=L2(Ω,𝒴d), and V be the closed subspace of H1 defined by
(1.7)V={v∈H1:v=0on⁡ΓD}.OnV we consider the inner product and the corresponding norm given by
(1.8)(u,v)V=〈ɛ(u),ɛ(v)〉H,‖v‖V=‖ɛ(v)‖H,foru,v∈V.From the Korn inequality∥v∥H1≤C1∥ɛ(v)∥ℋ for v∈V with C1>0, it follows that ∥·∥H1 and ∥·∥V are the equivalent norms on V. Identifying H with its dual, we have an evolution triple V⊂H⊂V* (see, e.g., [8]) with dense and compact embeddings. We denote by 〈·,·〉V the duality of V and its dual V*, by ∥·∥V* the norm in V*. We have 〈u,v〉V=(u,v)H for all u∈H and v∈V.We admit the following hypotheses.H(𝒞)
the linear symmetric viscosity operator𝒞:Ω×𝒴d→𝒴d satisfies the Carathéodory condition (i.e., 𝒞(·,ɛ) is measurable on Ω for all ɛ∈𝒴d, and 𝒞(x,·) is continuous on 𝒴d for a.e. x∈Ω) and
(1.9)C(x,ɛ):ɛ≥C2‖ɛ‖Yd2∀ɛ∈Yd,a.e.x∈ΩwithC2>0.H(ℰ)
the elasticity operatorℰ:Ω×𝒴d→𝒴d is of the form ℰ(x,ɛ)=𝔼(x)ɛ (Hooke's law) with a symmetric elasticity tensor 𝔼∈L∞(Ω), that is, 𝔼=(gijkl),i,j,k,l=1,…,d with gijkl=gjikl=glkij∈L∞(Ω). Moreover,
(1.10)E(x,ɛ):ɛ≥C3‖ɛ‖Yd2∀ɛ∈Yd,a.e.x∈ΩwithC3>0.H(j)
j:Ω×ℝd→ℝ is a function such that
(i)
j(·,ξ) is measurable for all ξ∈ℝd and j(·,0)∈L1(Ω);(ii)
j(x,·) is locally Lipschitz and regular [5] for all x∈Ω;(iii)
∥η∥≤C4(1+∥ξ∥) for all η∈∂j(x,ξ),x∈Ω with C4>0;(iv)
j0(x,ξ;-ξ)≤C5(1+∥ξ∥) for all ξ∈ℝd,x∈Ω, with C5≥0, where j0(x,ξ;η) is the directional derivative of j(x,·) at the point ξ∈ℝd in the direction η∈ℝd.H(f)
f1∈V*,g0∈L2(ΓN;ℝd),u0∈V and u1∈H.Next we need the spaces𝒱=L2(τ,T;V),ℋ̂=L2(τ,T;H), and 𝒲={w∈𝒱:w′∈𝒱*}, where the time derivative involved in the definition of 𝒲 is understood in the sense of vector-valued distributions, -∞<τ<T<+∞. Endowed with the norm ∥v∥𝒲=∥v∥𝒱+∥v′∥𝒱*, the space 𝒲 becomes a separable reflexive Banach space. We also have 𝒲⊂𝒱⊂ℋ̂⊂𝒱*. The duality for the pair (𝒱,𝒱*) is denoted by 〈z,w〉𝒱=∫τT〈z(s),w(s)〉Vds. It is well known (cf. [8]) that the embedding 𝒲⊂C([τ,T];H) and {w∈𝒱:w′∈𝒲}⊂C([τ,T];V) are continuous. Next we define g∈V* by
(1.11)〈g,v〉V=〈f1,v〉V+〈g0,v〉L2(ΓN;Rd)forv∈V.
Taking into account the condition (1.6), we obtain the following variational formulation of our problem:
(1.12)〈u′′(t),v〉V+(σ(t),ɛ(v))H+∫Ωj0(x,u(t);v)dx≥〈g,v〉V∀v∈V,a.e.t∈(0,+∞),σ(t)=C(ɛ(u′(t)))+E(ɛ(u(t)))fora.e.t∈(0,+∞),u(0)=u0,u′(0)=u1.
We define the operators A:V→V* and B:V→V* by
(1.13)〈A(u),v〉V=(C(x,ɛ(u)),ɛ(v))Hforu,v∈V,〈Bu,v〉V=(E(x,ɛ(u)),ɛ(v))Hforu,v∈V.
Obviously, the bilinear forms (1.13) are symmetric, continuous and coercive.Let us introduce the functionalJ:L2(Ω;ℝd)→ℝ defined by
(1.14)J(v)=∫Ωj(x,v(x))dxforv∈L2(Ω;Rd).
From [5, Chapter II] under Assumptions H(j), the functional J defined by (1.14) satisfiesH(J)
J:L2(Ω;ℝd)→ℝ is a functional such that:
(i)
J(·) is well defined, locally Lipschitz (in fact, Lipschitz on bounded subsets of L2(Ω;ℝd)) and regular;(ii)
ζ∈∂J(v) implies ∥ζ∥L2(Ω;ℝd)≤C6(1+∥v∥L2(Ω;ℝd)) for v∈L2(Ω;ℝd) with C6>0;(iii)
J0(v;-v)≤C7(1+∥v∥L2(Ω;ℝd)) for v∈L2(Ω;ℝd) with C7≥0, where J0(u;v) denotes the directional derivative of J(·) at a point u∈L2(Ω;ℝd) in the direction v∈L2(Ω;ℝd).We can now formulate the second-order evolution inclusions associated with the variational form of our problem.(1.15)Findu∈C([0,+∞);V)withu′∈C([0,+∞);H)∩L2loc(0,+∞;V),u′′∈L2loc(0,+∞;V*)suchthatu′′(t)+Au′(t)+Bu(t)+∂J(u(t))∋ga.e.t∈(0,+∞),u(0)=u0,u′(0)=u1.
In this paper, we study a general autonomous evolution inclusion which includes (1.15).


## 2. Setting of the Problem

LetV and H be real separable Hilbert spaces such that V⊂H with compact and dense embedding. Let V* be the dual space of V. We identify H with H* (dual space of H). For the linear operators A:V→V*, B:V→V*, and locally Lipschitz functional J:H→ℝ we consider a problem of investigation of dynamics for all weak solutions defined for t≥0 of nonlinear second order autonomous differential-operator inclusion:
(2.1)u′′(t)+Au′(t)+Bu(t)+∂J(u(t))∋0¯.
We need the following hypothesesH(A)̲A:V→V* is a linear symmetric such that there exists cA>0:〈Av,v〉V≥cA∥v∥V2 for all v∈V;H(B)̲B:V→V* is linear, symmetric, and there exists cB>0:〈Bv,v〉V≥cB∥v∥V2 for all v∈V (we remark that operators A and B are continuous on V [8, Chapter III]);H(J)̲J:H→R is a function such that
(i)
J(·) is locally Lipschitz and regular [5, Chapter II], that is,
(a)
for anyx,v∈H, the usual one-sided directional derivative J′(x;v)=limt↘0(J(x+tv)-J(x))/t exists,(b)
for allx,v∈H, J′(x;v)=J∘(x;v), where J∘(x;v)=lim̅y→x,t↘0(J(y+tv)-J(y))/t;(ii)
there existsc1>0: supd∈∂J(v)∥d∥H≤c1(1+∥v∥H) for all v∈H;(iii)
there existsc2>0:
(2.2)inf⁡d∈∂J(v)⁡(d,v)H≥-λ‖v‖H2-c2,∀v∈H,
where ∂J(v)={p∈H∣(p,w)H≤J∘(v;w)for all w∈H} denotes the Clarke subdifferential of J(·) at a point v∈H (see [5] for details), λ∈(0,λ1), λ1>0: ∥v∥V2≥λ1∥v∥H2  for all v∈V.We note that in (1.15) we can consider g=0¯. Indeed, let g∈V*, then u*=B-1g∈V⊂H. If u(·) is a weak solution of (1.15), then u¯(·)=u(·)-u* is a weak solution of
(2.3)u¯′′(t)+Au¯′(t)+Bu¯(t)+∂J1(u¯(t))∋0¯a.e.t>0,
where J1(·)=J(·+u*) satisfies H(J)̲ with respective parameters. Thus, to simplify our conclusions, without loss of generality, further we will consider problem (2.1).The phase space for Problem (2.1) is the Hilbert space:
(2.4)E={(a,b)T∣a∈V,b∈H},
where (a,b)T=(ab) with ∥(a,b)T∥E=(∥a∥V2+∥b∥H2)1/2. Let -∞<τ<T<+∞.Definition 2.1.
The function(u(·),u′(·))T∈L∞(τ,T;E) with u′(·)∈L2(τ,T;V) is called a weak solution for (2.1) on [τ,T], if there exists d(·)∈L2(τ,T;H), d(t)∈∂J(u(t)) for a.e. t∈(τ,T), such that  for all ψ∈V,  for all η∈C0∞(τ,T):
(2.5)-∫τT(u′(t),ψ)Hη′(t)dt+∫τT[〈Au′(t),ψ〉V+〈Bu(t),ψ〉V+(d(t),ψ)H]η(t)dt=0.Evidently, if(u(·),u′(·))T is a weak solution of (2.1) on [τ,T], then u′′(·)∈L2(τ,T;V*),(u(·),u′(·))T∈C([τ,T];E) and d(·)∈L∞(τ,T;H).We consider the class of functionsWτT=C([τ,T];E). Further c1,c2,λ,λ1,cA, and cB we recall parameters of Problem (2.1). The main purpose of this work is to investigate the long-time behavior (as t→+∞) of all weak solutions for the problem (2.1) on [0,+∞).


## 3. Preliminaries

To simplify our conclusions, since conditionH(B)̲, we suppose that
(3.1)(u,v)V=〈Bu,v〉V,‖v‖V2=〈Bv,v〉V,cB=1,∀u,v∈V.
Lebourg mean value theorem [5, Chapter 2] provides the existence of constants c3,c4>0 and μ∈(0,λ1):
(3.2)|J(u)|≤c3(1+‖u‖H2),J(u)≥-μ2‖u‖H2-c4,∀u∈H.Lemma 3.1.
LetJ:H→ℝ be a locally Lipschitz and regular functional, y∈C1([τ,T];H). Then for a.e.t∈(τ,T), there exists (d/dt)(J∘y)(t)=(p,y′(t)) for all p∈∂J(y(t)). Moreover, (d/dt)(J∘y)(·)∈L1(τ,T).Proof.
Sincey∈C1([τ,T];H), then y is strictly differentiable at the point t0 for any t0∈(τ,T). Hence, taking into account the regularity of J and [5, Theorem 2.3.10], we obtain that the functional J∘y is regular one at t0∈(τ,T) and ∂(J∘y)(t0)={(p,y′(t0))∣p∈∂J(y(t0))}. On the other hand, since y∈C1([τ,T];H),J:H→ℝ is locally Lipschitz then J∘y:[τ,T]→ℝ is globally Lipschitz, and therefore it is absolutely continuous. Hence for a.e. t∈(τ,T) there exists d(J∘y)(t)/dt,d(J∘y)(·)/dt∈L1(τ,T), and ∫st(d/dξ)(J∘y)(ξ)dξ=(J∘y)(t)-(J∘y)(s) for all τ≤s<t≤T. At that taking into account the regularity of J∘y, note that (J∘y)∘(t0,v)=(J∘y)′(t0,v)=d(J∘y)(t0)/dt·v for a.e. t0∈(τ,T), and all v∈ℝ. This implies that for a.e.t0∈(τ,T)∂(J∘y)(t0)={d(J∘y)(t0)/dt}.At the inclusion (2.1) on [τ,T] we associate the conditions
(3.3)u(τ)=a,u′(τ)=b,
where a∈V and b∈H. From [37, Theorem 1] we get the following lemma.Lemma 3.2.
For anyτ<T,a∈V,b∈H the Cauchy problem (2.1), (3.3) has a weak solution (y,y′)T∈L∞(τ,T;E). Moreover, each weak solution (y,y′)T of the Cauchy problem (2.1), (3.3) on the interval [τ,T] belongs to the space C([τ,T];E) and y′∈L2(τ,T;V);y′′∈L2(τ,T;V*).


## 4. Properties of Solutions

Let us consider the next denotations:  for allφτ=(a,b)T∈E we set 𝒟τ,T(φτ)={(u(·),u′(·))T∣(u,u′)T is a weak solution of (2.1) on [τ,T],u(τ)=a,u′(τ)=b}. From Lemma 3.2, it follows that 𝒟τ,T(φτ)⊂C([τ,T];E)=WτT.Let us check that translation and concatenation of weak solutions are weak solutions too.Lemma 4.1.
Ifτ<T,φτ∈E,φ(·)∈𝒟τ,T(φτ), then ψ(·)=φ(·+s)∈𝒟τ-s,T-s(φτ) for all s. If τ<t<T,φτ∈E,φ(·)∈𝒟τ,t(φτ) and ψ(·)∈𝒟t,T(φ(t)), then
(4.1)θ(s)={φ(s),s∈[τ,t],ψ(s),s∈[t,T]
belongs to 𝒟τ,T(φτ).Proof.
The first part of the statement of this lemma follows from the autonomy of the inclusion (2.1). The proof of the second part follows from the definition of the solution of (2.1) and from that fact that z∈Wτ,T as soon as v∈Wτ,t, u∈Wt,T and v(t)=u(t), where
(4.2)z(s)={v(s),s∈[τ,t]u(s),s∈[t,T].Letφ=(a,b)T∈E and(4.3)V(φ)=12‖φ‖E2+J(a).Lemma 4.2.
Letτ<T,φτ∈E,φ(·)=(y(·),y′(·))T∈𝒟τ,T(φτ). Then 𝒱∘φ:[τ,T]→ℝ is absolutely continuous and for a.e.t∈(τ,T)d𝒱(φ(t))/dt=-〈Ay′(t),y′(t)〉V≤-γ∥y′(t)∥V2, where γ>0 depends only on parameters of Problem (2.1) (we remark that 〈Au,u〉V is equivalent norm on V, generated by inner product 〈Au,v〉V).Proof.
Let-∞<τ<T<+∞,φ(·)=(y(·),y′(·))T∈WτT be an arbitrary weak solution of (2.1) on (τ,T). From [8, Chapter IV] we get that the function t→∥y′(t)∥H2+∥y(t)∥V2 is absolutely continuous and for a.e.t∈(τ,T):
(4.4)12ddt[‖y′(t)‖H2+‖y(t)‖V2]=〈y′′(t)+By(t),y′(t)〉V=-〈Ay′(t),y′(t)〉V-(d(t),y′(t))H≤-γ‖y′(t)‖V2-(d(t),y′(t))H,
where d(t)∈∂J(y(t)) for a.e.t∈(τ,T),d(·)∈L2(τ,T;H) and γ>0 depends only on parameters of Problem (2.1), in virtue of u→〈Au,u〉V is equivalent norm on V. As y(·)∈C1([τ,T];H) and J:H→ℝ is regular and locally Lipschitz, according to Lemma 3.1 we obtain that for a.e. t∈(τ,T) there exists (d/dt)(J∘y)(t). Moreover, (d/dt)(J∘y)(·)∈L1(τ,T) and for a.e. t∈(τ,T) and all p∈∂J(y(t)) we have (d/dt)(J∘y)(t)=(p,y′(t))H. In particular, for a.e. t∈(τ,T)(d/dt)(J∘y)(t)=(d(t),y′(t))H. Taking into account (4.4), we finally obtain the necessary statement.
The lemma is proved.Lemma 4.3.
LetT0>0. If (u(·),u′(·))T is a weak solution of (2.1) on [0,T0], then there exists its extension on [0,+∞)(u¯(·),u¯′(·))T which is weak solution of (2.1) on [0,+∞), that is, (u¯(·),u¯′(·))T∈C(ℝ+;E)∩L∞(ℝ+;E) with u¯′(·)∈L2(0,T;V) for all T>0 and there exists d(·)∈L∞(0,+∞;H),d(t)∈∂J(u¯(t)) for a.e. t∈(0,+∞), such that  for all ψ∈V,  for all η∈C0∞(0,+∞):
(4.5)-∫0+∞(u¯′(t),ψ)Hη′(t)dt+∫0+∞[〈Au¯′(t),ψ〉V+〈Bu¯(t),ψ〉V+(d(t),ψ)H]η(t)dt=0.Proof.
The statement of this lemma follows from Lemmas3.2–4.2, Conditions (3.1), (3.2) and from the next estimates:  for all τ<T,  for all φτ∈E,  for all φ(·)=(y(·),y′(·))T∈𝒟τ,T(φτ) for all t∈[τ,T]2c3+(1+2c3/λ1)∥y(τ)∥V2+∥y′(τ)∥H2≥2𝒱(φ(τ))≥2𝒱(φ(t))=∥y(t)∥V2+∥y′(t)∥H2+2J(y(ts))≥(1-μ/λ1)∥y(t)∥V2+∥y′(t)∥H2-2c4.
The lemma is proved.For an arbitraryφ0∈E, let 𝒟(φ0) be the set of all weak solutions (defined on [0,+∞)) of problem (2.1) with initial data φ(0)=φ0. We remark that from the proof of Lemma 4.3 we obtain the next corollary.Corollary 4.4.
For anyφ0∈E and φ∈𝒟(φ0) the next inequality is fulfilled:
(4.6)‖φ(t)‖E2≤λ1+2c3λ1-μ‖φ(0)‖E2+2(c3+c4)λ1λ1-μ∀t>0.From Corollary4.4 and Conditions H(A)̲, H(B)̲, H(J)̲ in a standard way we obtain such proposition.Theorem 4.5.
Letτ<T,{φn(·)}n≥1⊂WτT be an arbitrary sequence of weak solutions of (2.1) on [τ,T] such that φn(τ)→φτ weakly in E,n→+∞. Then there exist φ∈𝒟τ,T(φτ) and {φnk(·)}k≥1⊂{φn(·)}n≥1 such that φnk(·)→φ(·) weakly in E uniformly on [τ,T],k→+∞, that is, φnk(tk)→φ(t0) weakly in E,k→+∞ for any {tk}k≥1⊂[τ,T] with tk→t0,k→+∞.Theorem 4.6.
Letτ<T,{φn(·)}n≥1⊂WτT be an arbitrary sequence of weak solutions of (2.1) on [τ,T] such that φn(τ)→φτ strongly in E,n→+∞. Then there exists φ∈𝒟τ,T(φτ) such that up to a subsequence φn(·)→φ(·) in C([τ,T];E),n→+∞.Proof.
Let{φn(·)=(un(·),un′(·))T}n≥1⊂WτT be an arbitrary sequence of weak solutions of (2.1) on [τ,T] such that
(4.7)φn(τ)⟶φτstronglyinE,n→+∞.
Letφ=(u(·),u′(·))T∈𝒟τ,T(φτ) and {φnk(·)}k≥1⊆{φn(·)}n≥1 as in Theorem 4.5. It is important to remark that in the proof of Theorem 4.5, by using the inequality (Lemma 4.2, Corollary 4.4, (3.2)):
(4.8)γ‖un′(⋅)‖L2(τ,T;V)≤V(φn(τ))-V(φn(T))≤sup⁡n≥1[V(φn(τ))+μ2‖un(T)‖H2]+c4<+∞,
we establish that
(4.9)unk′(⋅)⟶u′(⋅)weaklyinL2(τ,T;V),k⟶+∞.
Let us prove that(4.10)φnk⟶φinWτT,k⟶+∞.
By contradiction suppose the existence of L>0 and subsequence {φkj}j≥1⊂{φnk}k≥1 such that  for all  for all j≥1maxt∈[τ,T]∥φkj(t)-φ(t)∥E=∥φkj(tj)-φ(tj)∥E≥L. Without loss of generality we suppose that tj→t0∈[τ,T],j→+∞. Therefore, by virtue of the continuity of φ:[τ,T]→E, we have
(4.11)lim⁡̲j→+∞‖φkj(tj)-φ(t0)‖E≥L.
On the other hand, we prove that(4.12)φkj(tj)⟶φ(t0)inE,j⟶+∞.
Firstly, we remark that (Theorem4.5)
(4.13)φkj(tj)⟶φ(t0)weaklyinE,j⟶+∞.
Secondly, let us prove that
(4.14)lim⁡j→+∞‖φkj(tj)‖E=‖φ(t0)‖E.
SinceJ is sequentially weakly continuous on V,𝒱 is sequentially weakly lower semicontinuous on E. Hence, we obtain
(4.15)V(φ(t0))≤lim⁡̲j→+∞V(φkj(tj)),∫τt0〈Au′(s),u′(s)〉Vds≤lim⁡̲j→+∞∫τtj〈Aukj′(s),ukj′(s)〉Vds,
and hence
(4.16)V(φ(t0))+∫τt0〈Au′(s),u′(s)〉Vds≤lim⁡̲j→+∞V(φkj(tj))+∫τt0〈Au′(s),u′(s)〉Vds≤lim⁡¯j→+∞V(φkj(tj))+∫τt0〈Au′(s),u′(s)〉Vds≤lim⁡¯j→+∞V(φkj(tj))+lim⁡̲j→+∞∫τtj〈Aukj′(s),ukj′(s)〉Vds≤lim⁡¯j→+∞(V(φkj(tj))+∫τtj〈Aukj′(s),ukj′(s)〉Vds).
We remark that the last inequality in (4.15) follows from weak convergence of unk′(·) to u′(·) in L2(τ,T;V) and because of the functional v→∫τT〈Av(s),v(s)〉Vds is sequentially weakly lower semi-continuous on L2(τ,T;V).
Since the energy equation and (4.7) both sides of (4.16) are equal to 𝒱(φ(τ)) (see Lemma 4.2), it follows that 𝒱(φkj(tj))→𝒱(φ(t0)), j→+∞ and then (4.14). Convergence (4.12) directly follows from (4.13) and (4.14). Finally, we remark that (4.12) contradicts (4.11). Therefore, (4.10) is true.
The theorem is proved.We define them-semiflow 𝒢 as 𝒢(t,ξ0)={ξ(t)ξ(·)∈𝒟(ξ0)},t≥0. Denote the set of all nonempty (nonempty bounded) subsets of E by P(E)(β(E)). We remark that the multivalued map 𝒢:ℝ+×E→P(E) is strictm-semiflow, that is, (see Lemma 4.1) 𝒢(0,·)=Id (the identity map), 𝒢(t+s,x)=𝒢(t,𝒢(s,x)) for all x∈E,t,s∈ℝ+. Further, φ∈𝒢 will mean that φ∈𝒟(ξ0) for some ξ0∈E.Definition 4.7 (see [41, page 35]).
Them-semiflow 𝒢 is called asymptotically compact, if for any sequence φj∈𝒢 with φj(0) bounded, and for any sequence tj→+∞, the sequence φj(tj) has a convergent subsequence.Theorem 4.8.
Them-semiflow 𝒢 is asymptotically compact.Proof.
Letξn∈𝒢(tn,vn),vn∈B∈β(E),n≥1,tn→+∞,n→+∞. Let us check the precompactness of {ξn}n≥1 in E. In order to do that without loss of the generality it is sufficiently to extract a convergent in E subsequence from {ξn}n≥1. From Corollary 4.4 we obtain that there exist {ξnk}k≥1 and ξ∈E such that ξnk→ξ weakly in E, ∥ξnk∥E→a≥∥ξ∥E, k→+∞. We show that a≤∥ξ∥E. Let us fix an arbitrary T0>0. Then for rather big k≥1𝒢(tnk,vnk)=𝒢(T0,𝒢(tnk-T0,vnk)). Hence ξnk∈𝒢(T0,βnk), where βnk∈𝒢(tnk-T0,vnk) and
(4.17)δ:=sup⁡k≥1‖βnk‖E<+∞,
(see Corollary 4.4). From Theorem 4.5 for some {ξkj,βkj}j≥1⊂{ξnk,βnk}k≥1, βT0∈E we obtain
(4.18)ξ∈G(T0,βT0),βkj⟶βT0weaklyinE,j⟶+∞.
From the definition of 𝒢 we set  for all j≥1ξkj=(yj(T0),yj′(T0))T, βkj=(yj(0),yj′(0))T, ξ=(y0(T0),y0′(T0))T, βT0=(y0(0),y0′(0))T, where φj=(yj,yj′)T∈C([0,T0];E), yj′∈L2(0,T0;V), yj′′∈L2(0,T0;V*), dj∈L∞(0,T0;H),
(4.19)yj′′(t)+Ayj′(t)+Byj(t)+dj(t)=0¯,dj(t)∈∂J(yj(t))fora.e.t∈(0,T0),∀j≥0.
Now we fix an arbitraryɛ>0. Let for each t∈[0,T0], j≥0:
(4.20)Iɛ(φj(t))≔12‖φj(t)‖E2+J(yj(t))+ɛ(yj′(t),yj(t))H.
Then, in virtue of [8, Chapter IV],
(4.21)dIɛ(φj(t))dt=-2ɛIɛ(φj(t))+2ɛHɛ(φj(t))-ɛ〈Ayj′(t),yj(t)〉V-〈Ayj′(t),yj′(t)〉V,fora.e.t∈(0,T0),
where
(4.22)Hɛ(φj(t))=J(yj(t))-12(dj(t),yj(t))H+‖yj′(t)‖H2+ɛ(yj′(t),yj(t))H,fora.e.t∈(0,T0).
Thus, for any j≥0 and t∈[0,T0](4.23)Iɛ(φj(T0))=Iɛ(φj(0))e-2ɛT0+2ɛ∫0T0Hɛ(φj(t))e-2ɛ(T0-t)dt-ɛ∫0T0〈Ayj(t),yj′(t)〉Ve-2ɛ(T0-t)dt-∫0T0〈Ayj′(t),yj′(t)〉Ve-2ɛ(T0-t)dt.
From (4.6), (4.18) and Lemma 4.2 we have
(4.24)‖yj′(t)‖H2+‖yj(t)‖V2+γ∫0t‖yj′(t)‖V2dt≤R¯,∀j≥0,∀t∈[0,T0],
where R¯>0 is a constant. Moreover,
(4.25)yj⟶y0weaklyinL2(0,T0;V),yj′⟶y0′weaklyinL2(0,T0;V),yj⟶y0stronglyinL2(0,T0;H),yj′⟶y0′stronglyinL2(0,T0;H),yj′′⟶y0′′weaklyinL2(0,T0;V*),dj⟶d0weaklyinL2(0,T0;H),∀t∈[0,T0]yj(t)⟶y0(t)inH,yj′(t)⟶y0′(t)weaklyinH,j⟶+∞.
Therefore,
(4.26)∫0T0Hɛ(φj(t))e-2ɛ(T0-t)dt⟶∫0T0Hɛ(φ0(t))e-2ɛ(T0-t)dt,j⟶+∞,lim⁡̲j→+∞∫0T0〈Ayj′(t),yj′(t)〉Ve-2ɛ(T0-t)dt≥∫0T0〈Ay0′(t),y0′(t)〉Ve-2ɛ(T0-t)dt.
The last inequality holds because of the functional v(·)→∫0T0〈Av(t),v(t)〉Ve-2ɛ(T0-t)dt is sequentially weakly lower semicontinuous on L2(τ,T0;V). Furthermore,
(4.27)ɛ∫0T0〈Ayj(t),yj′(t)〉Ve-2ɛ(T0-t)dt=ɛ2〈Ayj(T0),yj(T0)〉V-ɛ2〈Ayj(0),yj(0)〉Ve-2ɛT0-ɛ2∫0T0〈Ayj(t),yj(t)〉Ve-2ɛ(T0-t)dt∀j≥0,
from which, by Corollary 4.4, we have
(4.28)|ɛ∫0T0〈Ayj(t),yj′(t)〉Ve-2ɛ(T0-t)dt|≤2ɛγ[λ1+2c3λ1-μδ2+2(c3+c4)λ1λ1-μ],∀j≥0.
Thus,
(4.29)lim⁡¯j→+∞Iɛ(φj(T0))≤Iɛ(φ0(0))e-2ɛT0+[lim⁡¯j→+∞Iɛ(φj(0))-Iɛ(φ0(0))]e-2ɛT0+2ɛ∫0T0Hɛ(φ0(t))e-2ɛ(T0-t)dt-ɛ∫0T0〈Ay0(t),y0′(t)〉Ve-2ɛ(T0-t)dt-∫0T0〈Ay0′(t),y0′(t)〉Ve-2ɛ(T0-t)dt+4ɛγ[λ1+2c3λ1-μδ2+2(c3+c4)λ1λ1-μ]≤Iɛ(φ0(T0))+δ2e-2ɛT0+4ɛγ[λ1+2c3λ1-μδ2+2(c3+c4)λ1λ1-μ],
and, due to (4.25), for any T0>0 and ɛ>0(4.30)12‖ξ‖E2≤12a2=12lim⁡j→+∞‖ξkj‖E2≤12‖ξ‖E2+δ2e-2ɛT0+4ɛγ[λ1+2c3λ1-μδ2+2(c3+c4)λ1λ1-μ].
Hence, for all ɛ>0 we have
(4.31)12‖ξ‖E2≤12a2≤12‖ξ‖E2+4ɛγ[λ1+2c3λ1-μδ2+2(c3+c4)λ1λ1-μ].
Thus, a=∥ξ∥E.
The theorem is proved.Let us consider the family𝒦+=∪y0∈E𝒟(y0) of all weak solutions of the inclusion (2.1), defined on [0,+∞). Note that 𝒦+ is translation invariant one, that is, for all u(·)∈𝒦+ and all h≥0 we have uh(·)∈𝒦+, where uh(s)=u(h+s), s≥0. On 𝒦+, we set the translation semigroup{T(h)}h≥0, T(h)u(·)=uh(·), h≥0, u∈𝒦+. In view of the translation invariance of 𝒦+, we conclude that T(h)𝒦+⊂𝒦+ as h≥0.On𝒦+, we consider the topology induced from the Fréchet space Cloc(ℝ+;E). Note that
(4.32)fn(⋅)⟶f(⋅)inCloc(R+;E)⟺∀M>0ΠMfn(⋅)⟶ΠMf(⋅)inC([0,M];E),
where ΠM is the restriction operator to the interval [0,M] [42, page 179]. We denote the restriction operator to [0,+∞) by Π+.Let us consider the autonomous inclusion (2.1) on the entire time axis. Similarly to the space Cloc(ℝ+;E), the space Cloc(ℝ;E) is endowed with the topology of local uniform convergence on each interval [-M,M]⊂ℝ (cf. [42, page 180]). A function u∈Cloc(ℝ;E)∩L∞(ℝ;E) is said to be a complete trajectory of the inclusion (2.1), if  for all h∈ℝΠ+uh(·)∈𝒦+ [42, page 180]. Let 𝒦 be a family of all complete trajectories of the inclusion (2.1). Note that  for all h∈ℝ,  for all u(·)∈𝒦uh(·)∈𝒦. We say that the complete trajectory φ∈𝒦 is stationary if φ(t)=z for all t∈ℝ for some z=(u,0¯)T∈E (rest point). Following [43, page 486], we denote the set of rest points of 𝒢 by Z(𝒢). We remark that Z(𝒢)={(u,0¯)T∣u∈V,B(u)+∂J(u)∋0¯}.From ConditionsH(B)̲ and H(J)̲ and [39, Chapter 2], the following follows.Lemma 4.9.
The setZ(𝒢) is nonempty and bounded in E.From Lemma4.2, the existence of Lyapunov function (see [43, page 486]) for 𝒢 follows.Lemma 4.10.
The functional𝒱:E→ℝ, defined by (4.3), is a Lyapunov function for 𝒢.


## 5. Existence of the Global Attractor

First we consider constructions presented in [43, 44]. We recall that the set 𝒜⊂E is said to be a global attractor for 𝒢, if(1)
𝒜 is negatively semi-invariant (i.e., 𝒜⊂𝒢(t,𝒜)  for all t≥0);(2)
𝒜 is attracting set, that is,
(5.1)dist⁡(G(t,B),A)⟶0,t⟶+∞∀B∈β(E),
where dist(C,D)=supc∈Cinfd∈D∥c-d∥E is the Hausdorff semidistance;(3)
for any closed setY⊂E, satisfying (5.1), we have 𝒜⊂Y (minimality).The global attractor is said to beinvariant, if 𝒜=𝒢(t,𝒜)  for all t≥0.Note that from the definition of the global attractor it follows that it is unique.We prove the existence of the invariant compact global attractor.Theorem 5.1.
Them-semiflow 𝒢 has the invariant compact in the phase space E global attractor 𝒜. For each ψ∈𝒦 the limit sets:
(5.2)α(ψ)={z∈E∣ψ(tj)⟶zforsomesequencetj⟶-∞},ω(ψ)={z∈E∣ψ(tj)⟶zforsomesequencetj⟶+∞},
are connected subsets of Z(𝒢) on which 𝒱 is constant. If Z(𝒢) is totally disconnected (in particular, if Z(𝒢) is countable) the limits:
(5.3)z-=lim⁡t→-∞ψ(t),z+=lim⁡t→+∞ψ(t)
exist; z- and z+ are rest points; furthermore, φ(t) tends to a rest point as t→+∞ for every φ∈𝒦+.Proof.
The existence of the global attractor with required properties directly follows from Lemmas3.2, 4.1, 4.9, and 4.10, Theorems 4.5–4.8 and [41, Theorem 2.7].


## 6. Properties of Complete Trajectories

We remark in advance (Section4):(6.1)∀h∈R,∀u(⋅)∈Kuh(⋅)∈K.Lemma 6.1.
The set𝒦 is nonempty, and
(6.2)∀ξ(⋅)∈K,∀t∈Rξ(t)∈A,
where 𝒜 is the global attractor from Theorem 5.1.Proof.
Let us show that𝒦≠∅. Note that in virtue of Lemma 4.9, the set Z(𝒢) is nonempty and bounded in E. Let (v,0¯)T∈Z(𝒢). We set u(t)=v  for all t∈ℝ. Then (u,u′)T∈𝒦≠∅.
Let us prove (6.2). For any y∈𝒦 there exists d>0: ∥y(t)∥E≤d  for all t∈ℝ. We set B=∪t∈ℝ{y(t)}∈β(E). Note that  for all τ∈ℝ,for all t∈ℝ+y(τ)=yτ-t(t)∈𝒢(t,yτ-t(0))⊂𝒢(t,B). From Theorem 5.1 and from (5.1), it follows that  for all ɛ>0 there exists T>0:  for all τ∈ℝdist(y(τ),𝒜)≤dist(𝒢(T,B),𝒜)<ɛ. Hence taking into account the compactness of 𝒜 in E, it follows that y(τ)∈𝒜 for any τ∈ℝ.Lemma 6.2.
The set𝒦 is compact in Cloc(ℝ;E) and bounded in L∞(ℝ;E).Proof.
The boundedness of𝒦 in L∞(ℝ+;E) follows from (6.2) and from the boundedness of 𝒜 in E.
Let us check the compactness of𝒦 in Cloc(ℝ;E). In order to do that it is sufficient to check the precompactness and completeness.
Step 1.
Let us check the precompactness of𝒦 in Cloc(ℝ;E). If it is not true then in view of (6.1), there exists M>0: ΠM𝒦 is not precompact set in C([0,M];E). Hence there exists a sequence {vn}n≥1⊂ΠM𝒦, that has not a convergent subsequence in C([0,M];E). On the other hand vn=ΠMun, where un∈𝒦, vn(0)=un(0)∈𝒜, n≥1. Since 𝒜 is compact set in E (see Theorem 5.1), then in view of Theorem 4.6, there exists {vnk}k≥1⊂{vn}n≥1, there exists η∈E, there exists v(·)∈𝒟0,M(η): vnk(0)→η in E, vnk→v in C([0,T];E), k→+∞. We obtained a contradiction.Step 2.
Let us check the completeness of𝒦 in Cloc(ℝ;E). Let {vn}n≥1⊂𝒦, v∈Cloc(ℝ;E): vn→v in Cloc(ℝ;E), n→+∞. From the boundedness of 𝒦 in L∞(ℝ;E), it follows that v∈L∞(ℝ;E). From Theorem 4.6 we have that  for all M>0 the restriction v(·) to the interval [-M,M] belongs to 𝒟-M,M(v(-M)). Therefore, v(·) is a complete trajectory of the inclusion (2.1). Thus, v∈𝒦.Lemma 6.3.
Let𝒜 be the global attractor from Theorem 5.1. Then
(6.3)∀y0∈A∃y(⋅)∈K:y(0)=y0.Proof.
Lety0∈𝒜, u(·)∈𝒟(y0). From Theorem 5.1  𝒢(t,𝒜)=𝒜  for all t∈ℝ+. Therefore,
(6.4)u(t)∈A∀t∈R+,∀η∈A∃ξ∈A,∃φη(⋅)∈D0,1(ξ):φη(1)=η.
For any t∈ℝ, we set
(6.5)y(t)={u(t),t∈R+,φy(-k+1)(t+k),t∈[-k,-k+1),k∈N.
Note that y∈Cloc(ℝ;E), y(t)∈𝒜  for all t∈ℝ (hence y∈L∞(ℝ;E)) and in view of Lemma 4.1, y∈𝒦. Moreover y(0)=y0.


## 7. Existence of the Trajectory Attractor

We shall construct the attractor of the translation semigroup{T(h)}h≥0, acting on 𝒦+. We recall that the set 𝒫⊂Cloc(ℝ+;E)∩L∞(ℝ+;E) is said to be an attracting one for the trajectory space 𝒦+ of the inclusion (2.1) in the topology of Cloc(ℝ+;E), if for any bounded (in L∞(ℝ+;E)) set ℬ⊂𝒦+ and an arbitrary number M≥0 the next relation:(7.1)dist⁡C([0,M];E)(ΠMT(t)B,ΠMP)⟶0,t⟶+∞
holds.A set𝒰⊂𝒦+ is said to be trajectory attractor in the trajectory space 𝒦+ with respect to the topology of Cloc(ℝ+;E) (cf. [42], Definition 1.2, page 179), if(i)
𝒰 is a compact set in Cloc(ℝ+;E) and bounded in L∞(ℝ+;E);(ii)
𝒰 is strictly invariant with respect to {T(h)}h≥0, that is, T(h)𝒰=𝒰  for all h≥0;(iii)
𝒰 is an attracting set in the trajectory space 𝒦+ in the topology Cloc(ℝ+;E).Note that from the definition of the trajectory attractor it follows that it is unique.The existence of the trajectory attractor and its structure properties follow from such theorem.Theorem 7.1.
Let𝒜 be the global attractor from Theorem 5.1. Then there exists the trajectory attractor 𝒫⊂𝒦+ in the space 𝒦+, and we have
(7.2)P=Π+K={y∈K+∣y(t)∈A∀t∈R+}.Proof.
The proof intersects with proofs of corresponding statements from [14, 45] (see papers and references therein), and it is based on previous sections results.
From Lemmas6.1 and 6.2 and the continuity of the operator Π+:Cloc(ℝ;E)→Cloc(ℝ+;E) it follows that the set Π+𝒦 is nonempty, compact in Cloc(ℝ+;E) and bounded in L∞(ℝ+;E). Moreover, the second equality in (7.2) holds (Lemma 6.1 and the proof of Lemma 6.3). The strict invariance of Π+𝒦 follows from the autonomy of the inclusion (2.1).
Let us prove thatΠ+𝒦 is the attracting set for the trajectory space 𝒦+ in the topology of Cloc(ℝ+;E). Let B⊂𝒦+ be a bounded set in L∞(ℝ+;E), M≥0. Let us suppose M>0. Let us check (7.1). If it is not true, then there exist ɛ>0, the sequences tn→+∞, vn(·)∈B such that
(7.3)∀n≥1dist⁡C([0,T];E)(ΠMvn(tn+⋅),ΠMK)≥ɛ.
On the other hand, from the boundedness of B in L∞(ℝ+;E) it follows that there exists R>0:  for all v(·)∈B,  for all t∈ℝ+∥v(t)∥E≤R. Hence, taking into account (5.1) and the asymptotic compactness of m-semiflow 𝒢 (Theorem 4.8), we obtain that there exists {vnk(tnk)}k≥1⊂{vn(tn)}n≥1, there exists there exists z∈𝒜: vnk(tnk)→z in E, k→+∞. Further,  for all k≥1 we set φk(t)=vnk(tnk+t), t∈[0,M]. Note that  for all k≥1φk(·)∈𝒟0,M(vnk(tnk)). Then from Theorem 4.6 there exists a subsequence {φkj}j≥1⊂{φk}k≥1 and an element φ(·)∈𝒟0,M(z):
(7.4)φkj⟶φinC([0,M];E),j⟶+∞.
Moreover, taking into account the invariance of 𝒜 (see Theorem 5.1) for all t∈[0,M]φ(t)∈𝒜. From Lemma 6.3, there exist y(·),v(·)∈𝒦: y(0)=z,v(0)=φ(M). For any t∈ℝ we set
(7.5)ψ(t)={y(t),t≤0,φ(t),t∈[0,M],v(t-M),t≥M.
In view of Lemma 4.1  ψ(·)∈𝒦. Therefore, from (7.3) we obtain
(7.6)∀k≥1‖ΠMvnk(tnk+⋅)-ΠMψ(⋅)‖C([0,M];E)=‖φk-φ‖C([0,M];E)≥ɛ,
that contradicts with (7.4). We reason in the same way when M=0.
Thus, the set𝒫 constructed in (7.2) is the trajectory attractor in the trajectory space 𝒦+ with respect to the topology of Cloc(ℝ+;E).


## 8. Auxiliary Properties of the Global and Trajectory Attractors

Let𝒜 be the global attractor from Theorem 5.1, and 𝒫 the trajectory attractor from Theorem 7.1. From previous sections results we have(8.1)AisacompactsetinthespaceE;PisacompactsetinthespaceCloc(R+;E);P=Π+K={y∈K+∣y(t)∈A∀t∈R+}={y∈K+,∣y(0)∈A},
where 𝒦 is the family of all complete trajectories of the inclusion (2.1), Π+ is the restriction operator on ℝ+. Moreover,
(8.2)KisacompactinthespaceCloc(R;E);(8.3)∀ξ(⋅)∈K,∀t∈Rξ(t)∈A;(8.4)∀y0∈A,∀t0∈R∃y(⋅)∈K:y(t0)=y0.


### 8.1. “Translation Compactness” of the Trajectory Attractor

For anyy∈𝒦 let us set
(8.5)H(y)=clCloc(R;E){y(⋅+s)∣s∈R}⊂Cloc(R;E)∩L∞(R;E).
Such family is said to be the hull of function y(·) in Ξ=Cloc(ℝ;E).Definition 8.1.
The functiony(·)∈Ξ is said to be translation-compact (tr.-c.) in Ξ if the hull ℋ(y) is compact in Ξ.Similar constructions for the set of functional parameters that are responsible for nonautonomy of evolution equation are considered, for example, in [46, page 917].Definition 8.2.
The family𝒰⊂Ξ is said to be translation-compact if ℋ(𝒰)=clΞ{y(·+s)∣y(·)∈𝒰,s∈ℝ} is a compact in Ξ.From the autonomy of problems (2.1) and (8.2), the following follows.Corollary 8.3.
𝒦 is translation-compact set in Ξ.From autonomy of system (2.1), applying the Arzelá-Ascoli compactness criterion (see the proof of Proposition 6.1 from [46]), we obtain the translation compactness criterion for the family 𝒦:(a)
the set{y(t)∣y∈𝒦} is a compact in E  for all t∈ℝ;(b)
there exists a positive functionα(s)→0+(s→0+) such that
(8.6)‖y(t1)-y(t2)‖E≤α(|t1-t2|)∀y∈K,∀t1,t2∈R.Similarly, if we setΞ+=Cloc(ℝ+;E), we obtain the following.Corollary 8.4.
𝒫 is translation-compact set in Ξ+.


### 8.2. Stability

Definition 8.5 (see [43, page 487]).
The subsetF⊂E is Lyapunov stable if for a given ɛ>0 there exists δ>0 such that if D⊂E with dist(D,F)<δ then dist(𝒢(t,D),F)<ɛ for all t≥0.From [43, page 487], it follows that F is Lyapunov stable if and only if given {φj(·)}j≥1⊂𝒦+ with dist(φj(0),F)→0,j→+∞ and tj≥0 we have dist(φj(tj),F)→0,j→+∞.Definition 8.6 (see [43, page 487]).
The subsetF is uniformly asymptotically stable if F is Lyapunov stable and it is locally attracting, that is, if it attracts a neighborhood of F (see [43, page 482]).Note that bounded attracting set is locally attracting one.Corollary 8.7.
The global attractor𝒜 from Theorem 5.1 is uniformly asymptotically stable.Proof.
The proof follows from the definition of𝒢, [43, Theorem 6.1], properties of solutions from Lemma 3.2, Theorem 4.6, and from the autonomy of problem (2.1).Similar results are true for sets𝒫 and 𝒦 in corresponding extended phase spaces.


### 8.3. Connectedness

Definition 8.8 (see [43, page 485]).
Them-semiflow 𝒢 has Kneser's property, if 𝒢(t,z) is connected for each z∈E,t≥0.Corollary 8.9.
If𝒢 has Kneser's property, then 𝒜 is connected.Proof.
The proof follows from [43, Corollary 4.3], Lemma 3.2, and from the connectedness of the phase space E.Note that Kneser's property for𝒢 can be checked by different way (see, e.g., [35, 36, 47]). In order to do that as a rule, it is required an auxiliary regularity of interaction functions. In the general case, Kneser's property for problem (2.1) can be checked using the method of proof from [47, Theorem 5], where we can consider Yosida approximation instead the proposed approximation.Corollary 8.10.
If𝒦+ is connected, then 𝒦⊂Cloc(ℝ;E) is connected, and 𝒫⊂Cloc(ℝ+;E) is connected.Proof.
The proof is trivial.


## 8.1. “Translation Compactness” of the Trajectory Attractor

For anyy∈𝒦 let us set
(8.5)H(y)=clCloc(R;E){y(⋅+s)∣s∈R}⊂Cloc(R;E)∩L∞(R;E).
Such family is said to be the hull of function y(·) in Ξ=Cloc(ℝ;E).Definition 8.1.
The functiony(·)∈Ξ is said to be translation-compact (tr.-c.) in Ξ if the hull ℋ(y) is compact in Ξ.Similar constructions for the set of functional parameters that are responsible for nonautonomy of evolution equation are considered, for example, in [46, page 917].Definition 8.2.
The family𝒰⊂Ξ is said to be translation-compact if ℋ(𝒰)=clΞ{y(·+s)∣y(·)∈𝒰,s∈ℝ} is a compact in Ξ.From the autonomy of problems (2.1) and (8.2), the following follows.Corollary 8.3.
𝒦 is translation-compact set in Ξ.From autonomy of system (2.1), applying the Arzelá-Ascoli compactness criterion (see the proof of Proposition 6.1 from [46]), we obtain the translation compactness criterion for the family 𝒦:(a)
the set{y(t)∣y∈𝒦} is a compact in E  for all t∈ℝ;(b)
there exists a positive functionα(s)→0+(s→0+) such that
(8.6)‖y(t1)-y(t2)‖E≤α(|t1-t2|)∀y∈K,∀t1,t2∈R.Similarly, if we setΞ+=Cloc(ℝ+;E), we obtain the following.Corollary 8.4.
𝒫 is translation-compact set in Ξ+.


## 8.2. Stability

Definition 8.5 (see [43, page 487]).
The subsetF⊂E is Lyapunov stable if for a given ɛ>0 there exists δ>0 such that if D⊂E with dist(D,F)<δ then dist(𝒢(t,D),F)<ɛ for all t≥0.From [43, page 487], it follows that F is Lyapunov stable if and only if given {φj(·)}j≥1⊂𝒦+ with dist(φj(0),F)→0,j→+∞ and tj≥0 we have dist(φj(tj),F)→0,j→+∞.Definition 8.6 (see [43, page 487]).
The subsetF is uniformly asymptotically stable if F is Lyapunov stable and it is locally attracting, that is, if it attracts a neighborhood of F (see [43, page 482]).Note that bounded attracting set is locally attracting one.Corollary 8.7.
The global attractor𝒜 from Theorem 5.1 is uniformly asymptotically stable.Proof.
The proof follows from the definition of𝒢, [43, Theorem 6.1], properties of solutions from Lemma 3.2, Theorem 4.6, and from the autonomy of problem (2.1).Similar results are true for sets𝒫 and 𝒦 in corresponding extended phase spaces.


## 8.3. Connectedness

Definition 8.8 (see [43, page 485]).
Them-semiflow 𝒢 has Kneser's property, if 𝒢(t,z) is connected for each z∈E,t≥0.Corollary 8.9.
If𝒢 has Kneser's property, then 𝒜 is connected.Proof.
The proof follows from [43, Corollary 4.3], Lemma 3.2, and from the connectedness of the phase space E.Note that Kneser's property for𝒢 can be checked by different way (see, e.g., [35, 36, 47]). In order to do that as a rule, it is required an auxiliary regularity of interaction functions. In the general case, Kneser's property for problem (2.1) can be checked using the method of proof from [47, Theorem 5], where we can consider Yosida approximation instead the proposed approximation.Corollary 8.10.
If𝒦+ is connected, then 𝒦⊂Cloc(ℝ;E) is connected, and 𝒫⊂Cloc(ℝ+;E) is connected.Proof.
The proof is trivial.


## 9. Applications

As an application we consider the hemivariational inequality of hyperbolic type with multidimensional “reaction-displacement” law [19] (see Problem (1.1)–(1.6), undermentioned above conditions). From results of previous sections, it follows the next result.Corollary 9.1.
Them-semiflow 𝒢 constructed on all generalized solutions of (1.1)–(1.6) has the compact invariant global attractor 𝒜. For all generalized solutions of (1.1)–(1.6), defined on [0,+∞), there exists the trajectory attractor 𝒫, and we have
(9.1)A=P(0)={y(0)∣y∈K},P=Π+K,
where 𝒦 is the family of all complete trajectories of corresponding differential-operator inclusion (2.1) in Cloc(ℝ;E)∩L∞(ℝ;E). Moreover, global attractors are equal in the sense of [44, Definition 6, page 88] as well as in the sense of [42, Definition 2.2, page 182].
For eachψ∈𝒦, the limit sets:
(9.2)α(ψ)={z∈E∣ψ(tj)⟶zforsomesequencetj⟶-∞},ω(ψ)={z∈E∣ψ(tj)⟶zforsomesequencetj⟶+∞},
are connected subsets of Z(𝒢) on which 𝒱 is constant.Thus, all statements of previous sections are true for all generalized solutions of problem (1.1)–(1.6). In particular, all displacements and velocities are “attracted” as t→+∞ to all complete (defined on the entire time axis), globally bounded trajectories of corresponding “generalized” problem (2.1), which belong to compact sets in corresponding phase and extended phase spaces. Note that approaches proposed in works [42, 44] are based on properties of solutions of evolution objects. Our approaches are based on properties of interaction functions from (2.1) and properties of phase spaces.



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