Abstract and Applied AnalysisVolume 2012 (2012), Article ID 450984, 21 pagesdoi:10.1155/2012/450984

Research Article

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

Pavlo O. Kasyanov,¹ Luisa Toscano,² and Nina V. Zadoianchuk¹

¹Institute for Applied System Analysis NAS of Ukraine, Peremogy Avenue 37, Building 35, Kyiv 03056, Ukraine²Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Via Claudio 21, 80125 Naples, Italy

Received 2 December 2011; Accepted 27 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Pavlo O. Kasyanov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let a viscoelastic body occupies a bounded domain 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 of is supposed to be Lipschitz continuous, and it is partitioned into two disjoint measurable parts and such that means . We consider the process of evolution of the mechanical state on the interval . The body is clamped on and thus the displacement vanishes there. The forces field of density act in , the surface tractions of density are applied on . We denote by the displacement vector, by the stress tensor and by the linearized (small) strain tensor , where .

The mechanical problem consists in finding the displacement field such that where and are given linear constitutive functions and being the outward unit normal vector to .

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 and 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 consists of two parts: which is prescribed external loading and which is the reaction of constrains introducing the skin effects, that is, . Here, is a possibly multivalued function of the displacement . We consider the reaction-displacement law of the form: where is locally Lipschitz function in , and represents the Clarke subdifferential with respect to . Let be the space of second-order symmetric tensors on .

We consider the following problem: examine the long-time (as ) 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 , , , and be the closed subspace of defined by

On we consider the inner product and the corresponding norm given by

From the Korn inequality for with , it follows that and are the equivalent norms on . Identifying with its dual, we have an evolution triple (see, e.g., [8]) with dense and compact embeddings. We denote by the duality of and its dual , by the norm in . We have for all and .

We admit the following hypotheses. the linear symmetric viscosity operator satisfies the Carathéodory condition (i.e., is measurable on for all , and is continuous on for a.e. ) and the elasticity operator is of the form (Hooke's law) with a symmetric elasticity tensor , that is, with . Moreover, is a function such that(i) is measurable for all and ;(ii) is locally Lipschitz and regular [5] for all ;(iii) for all with ;(iv) for all , with , where is the directional derivative of at the point in the direction . and .

Next we need the spaces , and , where the time derivative involved in the definition of is understood in the sense of vector-valued distributions, . Endowed with the norm , the space becomes a separable reflexive Banach space. We also have . The duality for the pair is denoted by . It is well known (cf. [8]) that the embedding and are continuous. Next we define by Taking into account the condition (1.6), we obtain the following variational formulation of our problem: We define the operators and by Obviously, the bilinear forms (1.13) are symmetric, continuous and coercive.

Let us introduce the functional defined by From [5, Chapter II] under Assumptions , the functional defined by (1.14) satisfies is a functional such that:(i) is well defined, locally Lipschitz (in fact, Lipschitz on bounded subsets of ) and regular;(ii) implies for with ;(iii) for with , where denotes the directional derivative of at a point in the direction .

We can now formulate the second-order evolution inclusions associated with the variational form of our problem. In this paper, we study a general autonomous evolution inclusion which includes (1.15).

2. Setting of the Problem

Let and be real separable Hilbert spaces such that with compact and dense embedding. Let be the dual space of . We identify with (dual space of ). For the linear operators , , and locally Lipschitz functional we consider a problem of investigation of dynamics for all weak solutions defined for of nonlinear second order autonomous differential-operator inclusion: We need the following hypotheses is a linear symmetric such that there exists for all ; is linear, symmetric, and there exists for all (we remark that operators and are continuous on [8, Chapter III]); is a function such that(i) is locally Lipschitz and regular [5, Chapter II], that is, (a)for any , the usual one-sided directional derivative exists,(b)for all , , where ;(ii)there exists : for all ;(iii)there exists : where for all denotes the Clarke subdifferential of at a point (see [5] for details), , : for all .

We note that in (1.15) we can consider . Indeed, let , then . If is a weak solution of (1.15), then is a weak solution of where satisfies 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: where with . Let .

Definition 2.1. The function with is called a weak solution for (2.1) on , if there exists , for a.e. , such that for all , for all :

Evidently, if is a weak solution of (2.1) on , then and .

We consider the class of functions . Further , and we recall parameters of Problem (2.1). The main purpose of this work is to investigate the long-time behavior (as ) of all weak solutions for the problem (2.1) on .

3. Preliminaries

To simplify our conclusions, since condition , we suppose that Lebourg mean value theorem [5, Chapter 2] provides the existence of constants and :

Lemma 3.1. Let be a locally Lipschitz and regular functional, . Then for , there exists for all . Moreover, .

Proof. Since , then is strictly differentiable at the point for any . Hence, taking into account the regularity of and [5, Theorem 2.3.10], we obtain that the functional is regular one at and . On the other hand, since is locally Lipschitz then is globally Lipschitz, and therefore it is absolutely continuous. Hence for a.e. there exists , and for all . At that taking into account the regularity of , note that for a.e. , and all . This implies that for .

At the inclusion (2.1) on we associate the conditions where and . From [37, Theorem 1] we get the following lemma.

Lemma 3.2. For any the Cauchy problem (2.1), (3.3) has a weak solution . Moreover, each weak solution of the Cauchy problem (2.1), (3.3) on the interval belongs to the space and .

4. Properties of Solutions

Let us consider the next denotations: for all we set is a weak solution of (2.1) on . From Lemma 3.2, it follows that .

Let us check that translation and concatenation of weak solutions are weak solutions too.

Lemma 4.1. If , then for all . If and , then belongs to .

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 as soon as , and , where

Let and

Lemma 4.2. Let . Then is absolutely continuous and for , where depends only on parameters of Problem (2.1) (we remark that is equivalent norm on , generated by inner product ).

Proof. Let be an arbitrary weak solution of (2.1) on . From [8, Chapter IV] we get that the function is absolutely continuous and for : where for and depends only on parameters of Problem (2.1), in virtue of is equivalent norm on . As and is regular and locally Lipschitz, according to Lemma 3.1 we obtain that for a.e. there exists . Moreover, and for a.e. and all we have . In particular, for a.e. . Taking into account (4.4), we finally obtain the necessary statement.The lemma is proved.

Lemma 4.3. Let . If is a weak solution of (2.1) on , then there exists its extension on which is weak solution of (2.1) on , that is, with for all and there exists for a.e. , such that for all , for all :

Proof. The statement of this lemma follows from Lemmas 3.2–4.2, Conditions (3.1), (3.2) and from the next estimates: for all , for all , for all for all .The lemma is proved.

For an arbitrary , let be the set of all weak solutions (defined on ) of problem (2.1) with initial data . We remark that from the proof of Lemma 4.3 we obtain the next corollary.

Corollary 4.4. For any and the next inequality is fulfilled:

From Corollary 4.4 and Conditions , , in a standard way we obtain such proposition.

Theorem 4.5. Let be an arbitrary sequence of weak solutions of (2.1) on such that weakly in . Then there exist and such that weakly in uniformly on , that is, weakly in for any with .

Theorem 4.6. Let be an arbitrary sequence of weak solutions of (2.1) on such that strongly in . Then there exists such that up to a subsequence in .

Proof. Let be an arbitrary sequence of weak solutions of (2.1) on such that Let and 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)): we establish that Let us prove that By contradiction suppose the existence of and subsequence such that for all for all . Without loss of generality we suppose that . Therefore, by virtue of the continuity of , we have On the other hand, we prove that Firstly, we remark that (Theorem 4.5) Secondly, let us prove that Since is sequentially weakly continuous on is sequentially weakly lower semicontinuous on . Hence, we obtain and hence We remark that the last inequality in (4.15) follows from weak convergence of to in and because of the functional is sequentially weakly lower semi-continuous on .Since the energy equation and (4.7) both sides of (4.16) are equal to (see Lemma 4.2), it follows that , 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 the -semiflow as . Denote the set of all nonempty (nonempty bounded) subsets of by ). We remark that the multivalued map is strict -semiflow, that is, (see Lemma 4.1) (the identity map), for all . Further, will mean that for some .

Definition 4.7 (see [41, page 35]). The -semiflow is called asymptotically compact, if for any sequence with bounded, and for any sequence , the sequence has a convergent subsequence.

Theorem 4.8. The -semiflow is asymptotically compact.

Proof. Let . Let us check the precompactness of in . In order to do that without loss of the generality it is sufficiently to extract a convergent in subsequence from . From Corollary 4.4 we obtain that there exist and such that weakly in , , . We show that . Let us fix an arbitrary . Then for rather big . Hence , where and (see Corollary 4.4). From Theorem 4.5 for some , we obtain From the definition of we set for all , , , , where , , , , Now we fix an arbitrary . Let for each , : Then, in virtue of [8, Chapter IV], where Thus, for any and From (4.6), (4.18) and Lemma 4.2 we have where is a constant. Moreover, Therefore, The last inequality holds because of the functional is sequentially weakly lower semicontinuous on . Furthermore, from which, by Corollary 4.4, we have Thus, and, due to (4.25), for any and Hence, for all we have Thus, .The theorem is proved.

Let us consider the family of all weak solutions of the inclusion (2.1), defined on . Note that is translation invariant one, that is, for all and all we have , where , . On , we set the translation semigroup , , , . In view of the translation invariance of , we conclude that as .

On , we consider the topology induced from the Fréchet space . Note that where is the restriction operator to the interval [42, page 179]. We denote the restriction operator to by .

Let us consider the autonomous inclusion (2.1) on the entire time axis. Similarly to the space , the space is endowed with the topology of local uniform convergence on each interval (cf. [42, page 180]). A function is said to be a complete trajectory of the inclusion (2.1), if for all [42, page 180]. Let be a family of all complete trajectories of the inclusion (2.1). Note that for all , for all . We say that the complete trajectory is stationary if for all for some (rest point). Following [43, page 486], we denote the set of rest points of by . We remark that .

From Conditions and and [39, Chapter 2], the following follows.

Lemma 4.9. The set is nonempty and bounded in .

From Lemma 4.2, the existence of Lyapunov function (see [43, page 486]) for follows.

Lemma 4.10. The functional , defined by (4.3), is a Lyapunov function for .

5. Existence of the Global Attractor

First we consider constructions presented in [43, 44]. We recall that the set is said to be a global attractor for , if(1) is negatively semi-invariant (i.e., for all );(2) is attracting set, that is, where is the Hausdorff semidistance;(3)for any closed set , satisfying (5.1), we have (minimality).

The global attractor is said to be invariant, if for all .

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. The -semiflow has the invariant compact in the phase space global attractor . For each the limit sets: are connected subsets of on which is constant. If is totally disconnected (in particular, if is countable) the limits: exist; and are rest points; furthermore, tends to a rest point as for every .

Proof. The existence of the global attractor with required properties directly follows from Lemmas 3.2, 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 (Section 4):

Lemma 6.1. The set is nonempty, and where is the global attractor from Theorem 5.1.

Proof. Let us show that . Note that in virtue of Lemma 4.9, the set is nonempty and bounded in . Let . We set for all . Then .Let us prove (6.2). For any there exists : for all . We set . Note that for all for all . From Theorem 5.1 and from (5.1), it follows that for all there exists : for all . Hence taking into account the compactness of in , it follows that for any .

Lemma 6.2. The set is compact in and bounded in .

Proof. The boundedness of in follows from (6.2) and from the boundedness of in .Let us check the compactness of in . In order to do that it is sufficient to check the precompactness and completeness.Step 1. Let us check the precompactness of in . If it is not true then in view of (6.1), there exists : is not precompact set in . Hence there exists a sequence , that has not a convergent subsequence in . On the other hand , where , , . Since is compact set in (see Theorem 5.1), then in view of Theorem 4.6, there exists , there exists , there exists : in , in , . We obtained a contradiction.Step 2. Let us check the completeness of in . Let , : in , . From the boundedness of in , it follows that . From Theorem 4.6 we have that for all the restriction to the interval belongs to . Therefore, is a complete trajectory of the inclusion (2.1). Thus, .

Lemma 6.3. Let be the global attractor from Theorem 5.1. Then

Proof. Let , . From Theorem 5.1   for all . Therefore, For any , we set Note that , for all (hence ) and in view of Lemma 4.1, . Moreover .

7. Existence of the Trajectory Attractor

We shall construct the attractor of the translation semigroup , acting on . We recall that the set is said to be an attracting one for the trajectory space of the inclusion (2.1) in the topology of , if for any bounded (in ) set and an arbitrary number the next relation: holds.

A set is said to be trajectory attractor in the trajectory space with respect to the topology of (cf. [42], Definition 1.2, page 179), if(i) is a compact set in and bounded in ;(ii) is strictly invariant with respect to , that is, for all ;(iii) is an attracting set in the trajectory space in the topology .

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. Let be the global attractor from Theorem 5.1. Then there exists the trajectory attractor in the space , and we have

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 Lemmas 6.1 and 6.2 and the continuity of the operator it follows that the set is nonempty, compact in and bounded in . Moreover, the second equality in (7.2) holds (Lemma 6.1 and the proof of Lemma 6.3). The strict invariance of follows from the autonomy of the inclusion (2.1).Let us prove that is the attracting set for the trajectory space in the topology of . Let be a bounded set in , . Let us suppose . Let us check (7.1). If it is not true, then there exist , the sequences , such that On the other hand, from the boundedness of in it follows that there exists : for all , for all . Hence, taking into account (5.1) and the asymptotic compactness of -semiflow (Theorem 4.8), we obtain that there exists , there exists there exists : in , . Further, for all we set , . Note that for all . Then from Theorem 4.6 there exists a subsequence and an element : Moreover, taking into account the invariance of (see Theorem 5.1) for all . From Lemma 6.3, there exist : . For any we set In view of Lemma 4.1  . Therefore, from (7.3) we obtain that contradicts with (7.4). We reason in the same way when .Thus, the set constructed in (7.2) is the trajectory attractor in the trajectory space with respect to the topology of .

8. Auxiliary Properties of the Global and Trajectory Attractors

Let be the global attractor from Theorem 5.1, and the trajectory attractor from Theorem 7.1. From previous sections results we have where is the family of all complete trajectories of the inclusion (2.1), is the restriction operator on . Moreover,

8.1. “Translation Compactness” of the Trajectory Attractor

For any let us set Such family is said to be the hull of function in .

Definition 8.1. The function is said to be translation-compact (tr.-c.) in if the hull is compact in .

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 family is said to be translation-compact if is a compact in .

From the autonomy of problems (2.1) and (8.2), the following follows.

Corollary 8.3. is translation-compact set in .

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 :(a)the set is a compact in for all ;(b)there exists a positive function such that

Similarly, if we set , we obtain the following.

Corollary 8.4. is translation-compact set in .

8.2. Stability

Definition 8.5 (see [43, page 487]). The subset is Lyapunov stable if for a given there exists such that if with then for all .

From [43, page 487], it follows that is Lyapunov stable if and only if given with and we have .

Definition 8.6 (see [43, page 487]). The subset is uniformly asymptotically stable if is Lyapunov stable and it is locally attracting, that is, if it attracts a neighborhood of (see [43, page 482]).

Note that bounded attracting set is locally attracting one.

Corollary 8.7. The global attractor from Theorem 5.1 is uniformly asymptotically stable.

Proof. The proof follows from the definition of , [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 sets and in corresponding extended phase spaces.

8.3. Connectedness

Definition 8.8 (see [43, page 485]). The -semiflow has Kneser's property, if is connected for each .

Corollary 8.9. If has Kneser's property, then is connected.

Proof. The proof follows from [43, Corollary 4.3], Lemma 3.2, and from the connectedness of the phase space .

Note that Kneser's property for 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. If is connected, then is connected, and 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. The -semiflow constructed on all generalized solutions of (1.1)–(1.6) has the compact invariant global attractor . For all generalized solutions of (1.1)–(1.6), defined on , there exists the trajectory attractor , and we have where is the family of all complete trajectories of corresponding differential-operator inclusion (2.1) in . 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 each , the limit sets: are connected subsets of on which 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 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.

Acknowledgments

The authors would like to express their gratitude to Professor M. Z. Zgurovsky, Rector of NTUU “KPI”, Director of IASA NAS of Ukraine for providing excellent research facilities.

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