Long-time dynamics in plate models with strong nonlinear damping

We study long-time dynamics of a class of abstract second order in time evolution equations in a Hilbert space with the damping term depending both on displacement and velocity. This damping represents the nonlinear strong dissipation phenomenon perturbed with relatively compact terms. Our main result states the existence of a compact finite dimensional attractor. We study properties of this attractor. We also establish the existence of a fractal exponential attractor and give the conditions that guarantee the existence of a finite number of determining functionals. In the case when the set of equilibria is finite and hyperbolic we show that every trajectory is attracted by some equilibrium with exponential rate. Our arguments involve a recently developed method based on the"compensated"compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, Karman and Berger plate models with different types of boundary conditions and strong damping terms. Our results can be also applied to the nonlinear wave equations.


Introduction
We study a class of plate models with the strong nonlinear damping which abstract form is the following Cauchy problem in a separable Hilbert space H: ∂ tt u + D(u, ∂ t u) + Au + F (u) = 0, t > 0; u| t=0 = u 0 , ∂ t u| t=0 = u 1 . (1) We impose the following set of hypotheses: The operator A is a linear self-adjoint positive operator densely defined on a separable Hilbert space H (we denote by | · | and (·, ·) the norm and the scalar product in this space). We assume that the resolvent of A is compact in H. We also denote by H s (with s > 0) the domain D(A s/2 ) equipped with the graph norm | · | s = |A s/2 · |. In this case H −s denotes the completion of H with respect to the norm | · | −s = |A −s/2 · |. Below we denote by {e k } the orthonormal basis in H consisting of the eigenfunctions of the operator A : Ae k = λ k e k , 0 < λ 1 ≤ λ 2 ≤ · · · , lim k→∞ λ k = ∞, and by P N the orthoprojector onto Span{e k : k = 1, 2, . . . , N }.
Remark 1.2 Our conditions can be relaxed in different directions. For instance, for the wellposedness in Theorem 2.1 we can assume that δ = 0 in (3) and (6), and require F to map H 1−σ into H −l continuously for some σ > 0 and l > 0. In a similar way, instead of (Diii) we can assume other continuity properties of the damping operator D that will allow us to perform the limit transition in the corresponding Galerkin approximations. To obtain the global attractor existence, we can also relax (3) by including into its latter term the expression ε|u 1 − u 2 | 1 with a small parameter ε (see, e.g., Assumption 3.21 in [10] for a similar requirement in the case of the monotone damping). Moreover, instead of (3) we can assume that for every ε > 0. However, we do not pursue these possible generalizations because our abstract hypotheses are motivated by the plate models described below. We note that in this paper we concentrate on the case when θ is positive. For some results for the case when θ = 0 and D(u, u t ) is linear with respect to u t , we refer to [8]. It is also worth mentioning that our damping operator D is positive (see the first relation in (2)) but not monotone (see (3)) in general. Thus, we cannot apply here the theory developed in [10].
• The damping operator D(u, u t ) may have the form where σ 0 (s 1 ), σ 1 (s 1 , s 2 , s 3 ) and g(s 1 , s 2 ) are locally Lipschitz functions of s i ∈ R, i = 1, 2, 3, such that σ 0 (s 1 ) > 0, σ 1 (s 1 , s 2 , s 3 ) ≥ 0 and g(s 1 , s 2 )s 2 ≥ 0. Also the functions σ 1 and g satisfy some growth conditions (for a more detailed discussion of properties of the damping functions we refer to Section 5 below). We note that every term in (9) represents a different type of damping mechanisms. The first one is the so-called viscoelastic Kelvin-Voight damping, the second one represents the structural damping and the term g(u, u t ) is the dynamical friction (or viscous damping). We refer to [20,Chapter 3] and to the references therein for a discussion of stability properties caused by each type of the damping terms in the case of linear systems.
• The nonlinear feedback (elastic) force F (u) may have one of the following forms (which represent different plate models): (a) Kirchhoff model: F (u) is the Nemytskii operator where κ ≥ 0, q > r ≥ 0, µ ∈ R are parameters, p ∈ L 2 (Ω) and ϕ ∈ Lip loc (R) fulfills the condition where λ 1 is the first eigenvalue of the Laplacian with the Dirichlet boundary conditions.
(Ω) and p ∈ L 2 (Ω) are given functions, the von Karman bracket [u, v] is given by and the Airy stress function v(u) solves the following elliptic problem Von Karman equations are well known in nonlinear elasticity and constitute a basic model describing nonlinear oscillations of a plate accounting for large deflections, see [21,11] and the references therein.
Long-time dynamics of second order equations with a nonlinear damping was studied by many authors. We refer to [2,15,24,25,8,18] for the case of a damping with a nonlinear displacement-dependent coefficient and to [9,10,11] and to the references therein for the velocitydependent damping. Models with different types of a strong (linear) damping in wave equations were considered in [3,4,17,23,29], see also the literature quoted in these references.
The main novelty of the current paper is the following: (i) we can consider the strong nonlinear displacement-and velocity-dependent damping of a general structure (thus we cannot use analyticity of the corresponding model with a zero source term which takes place in the case when D(u, u t ) = Bu t , where B is a self-adjoint operator satisfying (2) with θ ∈ [1/2, 1], see, e.g., [20,Chapter 3] and the references therein); (ii) this damping can be perturbed by low order terms.
Our main result (see Theorem 3.1) states the existence of a compact global attractor and describes other asymptotic (long-term) properties of the system generated by (1). To establish this result we use the recently developed approach (see [9] and also [10] and [11,Chapters 7,8]). We first prove that the corresponding system is quasi-stable in the sense of the definition given in [11,Section 7.9], and then we apply general theorems on properties of quasi-stable systems. In the same framework we also establish a result on the rate of stabilization (see Theorem 3.2) which states that under some additional conditions every solution is attracted by an equilibrium with an exponential rate. To obtain this result we rely on some type of the observability inequality and use the same idea as in [10,Section 4.3] (see also [9] and [11]).
The paper is organized as follows. In the preliminary Section 2 we discuss well-posedness of our abstract model and the dynamical system generation. We also recall several notions and results from the theory of dissipative dynamical systems. Our main results on the global attractor for (1) and on asymptotic properties of individual trajectories are stated in Section 3. The proofs based on the quasi-stability property of the corresponding system are given in Section 4. In Section 5 we discuss some applications.
Below constants denoted by the same symbol may vary from line to line.

Preliminaries
In this section we show that problem (1) generates a dynamical system.

Well-posedness
We first prove the existence and uniqueness of weak solutions to problem (1). We recall that a function u(t) is a weak solution to (1) and (1) is satisfied in the sense of distributions.
The main statement of the section is the following assertion which also contains some auxiliary solution properties needed for the results on asymptotic dynamics.
holds for every t > 0, where the energy E is defined by the formula Moreover, this solution u(t) satisfies the estimate 2. If u 1 (t) and u 2 (t) are two weak solutions such that for some constants a R , b R , c R > 0.
Proof. To prove the existence of solutions, we use the standard Galerkin method of seeking for approximations of the form that solve the finite-dimensional projections of (1). Such solutions exist, and after multiplication of the corresponding projection of (1) by ∂ t u N (t) we get that u N (t) satisfies the energy relation (13). By (7) we obtain that Therefore, by (2) the energy relation for u N (t) yields estimate (14) for approximate solutions with the constant C(R) independent of N . Using the equation for u N (t) and also the conditions (2) and (6), it can be shown in the standard way that Thus the Aubin-Dubinsky theorem (see [26,Corollary 4] for every ε > 0. These compactness properties make it possible to show the existence of weak solutions satisfying (14). For the limit transition in the nonlinear terms we use the property (Diii) in Assumption 1.1 and relation (6). It is also clear that t → (u(t); ∂ t u(t)) is a weakly continuous function in H = D(A 1/2 ) × H. To obtain the energy relation in (13), we note that the function u n (t) = P n u(t) solves an equation of the form with some h ∈ L 2 (0, T ; H). This makes it possible to obtain a certain energy relation for u n which gives us (13) after the limit transition n → ∞. This also allows us to obtain the strong continuity properties of t → (u(t); ∂ t u(t)) in H by the standard method.
To prove (15), we note that z(t) = u 1 (t) − u 2 (t) solves the equation Thus, multiplying this equation by ∂ t z and integrating from s to t we have for any 0 ≤ s < t, where E z (t) = E(z(t), ∂ t z(t)). Therefore, using (3), (6) and (14) we obtain that for some c R > 0. Now we can apply the Gronwall lemma to obtain (15) which, in particular, implies the uniqueness of weak solutions.

Generation of a dynamical system
We recall that a dynamical system (see, e.g., [6,16,27]) is a pair X, S(t) of a complete metric space X and a family of continuous mappings S(t) : X → X, t ≥ 0, such that (i) t → S(t)y is continuous in X for every y ∈ X and (ii) the semigroup property is satisfied, i.e., S(t + τ ) = S(t) • S(τ ) for any t, τ ≥ 0 and S(0) is the identity operator. We also recall that the system X, S(t) is gradient if it possesses a strict Lyapunov function, i.e., there exists a continuous functional Φ(y) on X such that (i) Φ S(t)y ≤ Φ(y) for all t ≥ 0 and y ∈ X; (ii) the equality Φ(y) = Φ(S(t)y) may take place for all t > 0 if only y is a stationary point of S(t).
Applying Theorem 2.1 we obtain the following assertion.
This system is gradient with the full energy E(u 0 ; u 1 ) as a strict Lyapunov function (this follows from the energy relation in (13)).
We also recall that a system X, S(t) is called asymptotically smooth (see [16]) if for any closed The global attractor (see, e.g., [1,6,16,27]) of a dynamical system X, S(t) is defined as a bounded closed set A ⊂ X which is invariant (S(t)A = A for all t > 0) and uniformly attracts all other bounded sets: lim t→∞ sup{dist X (S(t)y, A) : y ∈ B} = 0 for any bounded set B in X.
In this paper we use the following criterion of the global attractor existence for gradient systems (see, e.g., [28,Theorem 4.6]): 3 Let X, S(t) be an asymptotically smooth gradient system such that for any bounded set B ⊂ X there exists τ > 0 such that γ τ (B) ≡ t≥τ S(t)B is bounded. If the set N of stationary points is bounded, then X, S(t) has a compact global attractor A which coincides with the unstable set M + (N ) emanating from N , i.e., A = M + (N ).
We recall (see, e.g., [1]) that the unstable set M + (N ) emanating from N is a subset of X such that for each z ∈ M + (N ) there exists a full trajectory {y(t) : t ∈ R} satisfying u(0) = z and dist X (y(t), N ) → 0 as t → −∞.

Main results
Our first main result is the following theorem. (2) This attractor has a finite fractal dimension.
(3) Any trajectory γ = {(u(t); ∂ t u(t)) : t ∈ R} from the attractor A possesses the property and there is R > 0 such that (4) The system (H, S(t)) possesses a (generalized) fractal exponential attractor A exp whose dimension is finite in the spaceH = H θ × H −1 .
Then there exists ε 0 > 0 such that under the condition ε L ≤ ε 0 the set L is (asymptotically) determining in the sense that the property implies that lim t→∞ |S(t)y 1 − S(t)y 2 | H = 0. Here above S(t)y i = (u i (t); ∂ t u i (t)), i = 1, 2.
We recall that the fractal dimension dim X f M of a compact set M in a complete metric space X is defined as where N (M, ε) is the minimal number of closed sets of diameter 2ε in X needed to cover the set M . We also recall (see, e.g., [13] and also [10,22] and the references therein) that a compact set A exp ⊂ H is said to be a (generalized) fractal exponential attractor for the dynamical system (H, S(t)) iff A exp is a positively invariant set of finite fractal dimension (in some extended spacẽ H) and for every bounded set D ⊂ H there exist positive constants t D , C D and γ D such that As for the determining functionals, we mention that this notion goes back to the papers by Foias and Prodi [14] and by Ladyzhenskaya [19] for the 2D Navier-Stokes equations. For the further development of the theory we refer to [12] and to the survey [5] and to the references quoted therein (see also [6,Chap.5]). We note that for the first time determining functionals for second order (in time) evolution equations with a nonlinear damping was considered in [7], see also a discussion in [11,Section 8.9] We also refer to [5,6] for a description of sets of functionals with a small completeness defect. Determining modes and nodes are among them.
Using the same idea as in [9,10,11] we can establish the following result on convergence of individual solutions to equilibria with an exponential rate.
Theorem 3.2 In addition to Assumption 1.1 we assume that F (u) is Fréchet differentiable and its derivative F ′ (u) possesses the properties and for any u, v ∈ H 1 such that |u| 1 ≤ R and |v| 1 ≤ R with δ > 0. Here F ′ (u), w is the value of F ′ (u) on the element w. Let the set N be finite and all equilibria be hyperbolic in the sense that the equation Au + F ′ (φ), u = 0 has only a trivial solution for each (φ; 0) ∈ N . Then for any y ∈ H there exists an equilibrium e = (φ; 0) ∈ N and constants γ > 0, C > 0 such that We note that this type of stabilization theorems is well-known in literature for different classes of gradient systems, and several approaches to the question on stabilization rates are available (see, e.g., [1] and also [9,10,11] and the references therein). The approach presented in [1] relies on the analysis of linearized dynamics near each equilibrium and requires the hyperbolicity condition in a dynamical form. Here we use the method developed in [9,10] (see also a discussion in [11]), and we need this condition in a weaker form.

Proofs
As it was already mentioned, the main ingredient of the proof of Theorem 3.1 is a quasi-stability property of the dynamical system (H, S(t)) generated by (1).

Quasi-stability
We show that under the conditions listed in Assumption 1.1 the system (H, S(t)) is quasi-stable in the sense of the definition given in [11,Section 7.9]. Namely, we prove the following proposition.
Proposition 4.1 Let Assumption 1.1 hold. Assume that u i (t), i = 1, 2 are two weak solutions to problem (1) with initial data y i = (u i 0 ; u i 1 ) such that A 1/2 u i 0 2 + |u i 1 | 2 ≤ R 2 for some R > 0. We denote S(t)y i = (u i (t); ∂ t u i (t)), i = 1, 2. Then there exist C(R), γ(R) > 0 such that This type of estimates was originally introduced in [9] and related to a decomposition of the evolution operator S(t) into uniformly exponentially stable and compact parts, see also a discussion in [10] and [11,Section 7.9]. We start with two preliminary lemmas.
, and the functionals E z , D and Ψ T are defined as , z t (t)) + (Az(t), z(t))) , Proof. We use the standard arguments involving the multipliers z t and z for (16). We refer to the proof of Lemma 3.23 in [10] and also [11,Lemma 8.3.1], where this lemma is proved under another set of hypotheses concerning the damping operator. However the corresponding argument does not depend on a structure of the damping operator.

Lemma 4.3
Let u 1 and u 2 be two solutions to (1) with the initial data (u i 0 ; u i 1 ). We assume that where E z and D(t) are the same as in Lemma 4.2.
Proof. This follows from (17), the Lipschitz property for F and the uniform estimate (14) for u i (t), i = 1, 2. We refer to [10, Lemma 3.25] for a similar assertion. Now we complete the proof of Proposition 4.1. Using (4) with θ ∈ (0, 1] we obtain that for any ε > 0. We also have from (4) for θ < 1 and from (5) for θ = 1 that The subcritical estimate in (6) yields Therefore, Lemma 4.2 implies (26) for every T ≥ T 0 . By Lemma 4.3 using (14) and (25) we have that where the constants c 0 and C R are independent of T . Therefore, from (26) which, after an appropriate choice of ε, implies that for every T ≥ T 0 . Using (3), (6) and (17) we conclude that there existsγ R > 0 such that Consequently, choosing ε small enough, by (27) we have that with κ R < 1 and T ≥ T 0 . Now the standard argument (cf., e.g., [10, p. 62] or [11, p.414]) leads to (24). This concludes the proof of Proposition 4.1.

Completion of the proof of Theorem 3.1
1. Proposition 4.1 means that the system (H, S(t)) is quasi-stable in the sense of Definition 7.9.2 [11]. Therefore, by Proposition 7.9.4 [11] (H, S(t)) is asymptotically smooth. By Proposition 2.2 (H, S(t)) is a gradient system. Thus, Remark 2.4 and Theorem 2.3 imply that there exists a compact global attractor. By the standard results on gradient systems with compact attractors (see, e.g., [1,6,27]) we have that A = M + (N ) and (18) holds.

2.
Since (H, S(t)) is quasi-stable, the finiteness of the fractal dimension dim f A follows from Theorem 7.9.6 [11].

4.
One can see from (1) and Theorem 2.1 that any weak solution u(t) possesses the property This implies that t → S(t)y is a 1/2-Hölder continuous function with values inH = H θ × H −1 for every y ∈ B R . Therefore, the existence of a fractal exponential attractor follows from Theorem 7.9.9 [11].

5.
To prove the statement concerning determining functionals, we use the same idea as in the proof of Theorem 8.9.3 [11], see also Theorem 7.9.11 [11].

Proof of Theorem 3.2
We use the same idea as in [9,10,11]. Since N is finite by (18) in Theorem 3.1 we have that for any y ∈ H there exists an equilibrium e = (φ; 0) ∈ N such that Thus we need only to prove that S(t)y tends to e with the stated rate. Let S(t)y = (u(t); u t (t)). We can assume that sup t≥0 |S(t)y| H ≤ R, for some R > 0. The function z(t) = u(t) − φ satisfies the following equation Let E(t) = E z (t) + Φ(t), where E z (t) is the same as in Lemma 4.2 and One can see that In particular, we have that E(t) is non-increasing. Moreover, since (z; z t ) → 0 in H 1 × H as t → +∞, we have that E(t) → 0 when t → +∞. Thus E(t) ≥ 0 for all t ≥ 0. It is also clear from Applying (27) for the case when u 1 (t) = u(t) and u 2 (t) = φ we obtain that for T ≥ T 0 with some T 0 > 0. Now we prove the following lemma.
Lemma 4.4 Let z(t) be a weak solution to (29) such that with some δ, ̺ > and T > 1. Then there exists δ 0 > 0 such that for every 0 < δ ≤ δ 0 , where the constant C may depend on δ, ̺ and T .

Applications
As it was mentioned in the introduction, our main applications are plate models.

Plate models
For the definiteness, we concentrate on the hinged boundary conditions (the results remain true with other types of self-adjoint boundary conditions). Below · s is the norm in the Sobolev space H s (Ω) of order s.
Forcing term: We first check that the forcing term F satisfies Assumption 1.1(F) for all cases described above.
We can also consider more general (anisotropic) damping operators D(u, u t ) which are defined variationally by the formula for every ψ ∈ (H 2 ∩ H 1 0 )(Ω) under appropriate positivity and smoothness hypotheses concerning the coefficients.

Wave equation with strong damping
As an example, we can also consider the following wave equation on a bounded domain Ω in R 3 with a nonlocal damping coefficient: u tt − σ 0 ( u η )∆u t + σ 1 (u)u t − ∆u + ϕ(u) = f (x), u ∂Ω = 0.
We also assume that the source term ϕ ∈ C 1 (R) possesses the properties lim inf |s|→∞ ϕ(s)s −1 > −λ 1 , |ϕ ′ (s)| ≤ C(1 + |s| q ), s ∈ R, q < 4, where λ 1 is the first eigenvalue of the Laplacian with the Dirichlet boundary conditions. In this case we can apply Theorem 3.1 for the model in (44) with θ = 1. We also note that basing on a requirement like (8) we can also cover the case when η = 1 in (44). As for the case of the critical growth exponents (q 1 = 3 and q = 4) of the damping coefficient σ 1 and the force ϕ, we cannot apply here our abstract approach. This case requires a separate consideration involving a specific structure of the model.
In a similar way, we can also consider the wave model (44) in arbitrary dimension d and with another structure of the damping operator.