Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping *

In this paper we consider new results on well-posedness and long-time dynamics for a class of extensible beam/plate models whose dissipative effect is given by the product of two nonlinear terms. The addressed model contains a nonlocal nonlinear damping term which generalizes some classes of dissipations usually given in the literature, namely, the linear, the nonlinear and the nonlocal frictional ones. A first mathematical analysis of such damping term is presented and represents the main novelty in our approach.


1.
Introduction. This article addresses global well-posedness and long-time dynamics to the following extensible beam model subject to a nonlocal nonlinear damping u tt + ∆ 2 u − κM ( ∇u 2 2 )∆u + N ( ∇u 2 2 )g(u t ) + f (u) = h in Ω × (0, ∞), (1) where Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, M ( ∇u 2 2 ) and N ( ∇u 2 2 ) represent nonlocal coefficients, where · 2 stands for the norm in L 2 (Ω), g and f are real nonlinear functions, h is an external force and the parameter κ ≥ 0 is related to the extensibility of the beam. The assumptions on the nonlinear terms M , N , g and f will be given in Sections 2 and 3. We consider two types of boundary conditions, namely, clamped boundary condition where ν is the unit exterior normal to ∂Ω, or hinged boundary condition The initial conditions associated with displacement u are given by u(·, 0) = u 0 (·), u t (·, 0) = u 1 (·) in Ω. (4) As reported by the authors in [15,16], equation (1) is an n-dimensional abstract version related to vibrations of nonlinear plate and extensible beam models. In [16] it is provided an extensive survey on references concerning to extensible models closely to (1)- (4) and their results on existence, uniqueness, asymptotic stability and long-time behavior.
In this work our main goal is to present a first analysis on well-posedness and long-time behavior when problem (1)- (4) is under the influence of the following nonlocal nonlinear damping term N ( ∇u 2 2 )g(u t ), (5) which is given by the product of two nonlinear components. Indeed, N is assumed to be any positive C 1 -function and g is a nonlinear C 1 -function with polynomial growth, for instance g(u t ) ≈ |u t | γ u t , γ > 0. See Assumptions (A 2 )-(A 3 ) in Section 2. This kind of nonlocal dissipative effect constitutes a generalization of nonlocal weak damping N ( ∇u 2 2 )u t which was first introduced by Lange and Perla Menzala [23] with respect to plate models and subsequently studied by Cavalcanti et al. [3] in a viscoelastic context. In that occasion, the authors in [23] derived the plate model by taking the imaginary part of a Schrödinger equation with nonlocal term a(u(t)) := N ( ∇u(·, t) 2 2 ) and, consequently, the term N ( ∇u 2 2 )u t arises as a type of nonlocal Kirchhoff damping. Moreover, the damping (5) can be seen as generalization of a dissipation studied more recently in [15,16]. Indeed, when the growth of g is linear (g(u t ) ≈ u t ), then the damping term in (5) turns into the simpler case N ( ∇u 2 2 )u t , which makes all computations easier than the case treated in the present paper. We also refer to [8,20,21] where it is studied long-time dynamics of a class of plate models with state-dependent nonlinear weak damping σ(u)u t .
As far as we know, another kind of nonlocal fractional damping is given by Chueshov and Kolbasin [9] studied long-time behavior for a class of abstract equations encompassing 2-dimensional Berger plate models (M being linear in (1)) with damping given by (6) and power θ covering the range 0 < θ ≤ 1. In [9] the chosen structure of the damping also approaches viscoelastic Kelvin-Voigt, structural and viscous damping. When N is constant, for example N ≡ 1, then Coti Zelati [13] considered a structural damping A θ u t with 0 ≤ θ ≤ 1 and Biazutti and Crippa [2] assumed power 0 < θ ≤ 1. In the latter, the condition θ > 0 is crucial since the multiplier A θ/4 u t is required (see [2,Lemma 2.2]). But these two last papers are concerned with fully linear damping which goes away from the scope of this work. To our best knowledge the only paper which deals with damping like (6) by moving θ uniformly on [0, 1] and, at the same time, considering a nonlocal damping coefficient N > 0 is given in [16]. In what concerns beam models in a 1-dimensional framework, replacing the gradient by the Laplacian operator as the argument of the function N , then the case θ = 1 in (6) can be also seen as a kind of nonlocal damping related to flight structures, see e.g. Balakrishnan and Taylor [1]. Although nonlocal, we observe that the damping term (6) is just nonlinear in one of its components which does not address nonlinearities like (5). It is worth noting that the case θ = 0 in (6) characterizes a particular case of the damping (5) when g(u t ) ≈ u t .
We also refer to [6,7,24,29] for results on existence and long-time behavior for a class Kirchhoff wave models with nonlocal fractional damping (6). On the other hand, a more natural damping term in the existing literature for beam models like (1) is given by g(u t ), say N ≡ 1 in (5), as a generalization of linear frictional dissipations u t . For this kind of damping there are a lot of papers which provide studies on well-posedness and long-time dynamics for beam/plate models related to (1). See for instance [4,5,12,22,17,18,19,27,28,35,36,37] and reference therein. However, the class of functions N allowed in (5) encompasses the constant case N ≡ 1 and, therefore, our results are given in a more general framework. An example of this situation appears when we deal with the difference of two trajectories u 1 − u 2 of problem (1)- (4). In such case the following term arises whose its controllability is worse than the standard case g(u 1 t ) − g(u 2 t ) appearing in the above mentioned works. Hence, a new mathematical analysis is necessary for the term (7) in all crucial results of the this paper, see e.g. the proofs of Theorem 2.1-(iv), Proposition 2 and Proposition 4.
The main results of this paper are Theorem 2.1 (see Section 2) which ensures the well-posedness of problem (1)-(4) and Theorems 3.1 and 3.2 (stated at the end of Section 3) which establish the existence of attractors to the dynamical system associated with problem (1)-(4) as well as their qualitative properties with respect to geometrical characterization, finite dimension, regularity of trajectories and exponential attractor. To their proofs new mathematical arguments (estimates) are provided with respect to nonlocal nonlinear damping term (5). When compared with the existing literature, our results complement and generalize all the above mentioned works dealing with beam/plate models under frictional dissipations. In other words, the nonlocal nonlinear damping term generalizes (at least) three classes of usual damping in the literature, namely, the linear u t , the nonlinear g(u t ), and the nonlocal frictional one N ( ∇u 2 2 )u t . The remaining paper is organized as follows. In Section 2 we first provide the notations and assumptions used throughout the paper and then the well-posedness result to problem (1)-(4) with respect to strong and weak solutions. In Section 3 we consider the dynamical system corresponding to problem (1)-(4) and several results on its long-time dynamic properties up to the main results on attractors. A final Appendix A ends this work with examples of functions f , g and M satisfying their respective assumptions required a priori.
2. Well-posedness. We begin this section with the introduction of the function spaces which shall be used throughout this paper: (3), and for γ ≥ 0, When γ = 0 note that W 4,0 = W 4 . Here the notation ( · , · ) stands for L 2 -inner product and · p denotes L p -norm. Thus, ∇ · 2 and ∆ · 2 represent the norms in W 1 and W 2 , respectively. When there is no possibility of confusion we shall use the same notation ( · , · ) to represent the duality pairing between any Banach space W and its dual W . Denoting by λ 1 > 0 the first eigenvalue of the bi-harmonic operator ∆ 2 with boundary condition (2) or (3), then We also consider the following phase space where the analysis of the asymptotic behavior of solutions shall be done. Now we introduce the assumptions on the functions f, g, M, and N as follows: where we consider c f > 0, c f ≥ 0, 0 ≤ α 1 < λ 1 , and where we assume c g , C g > 0, and where 0 ≤ α 2 < λ 1/2 1 ; (A 4 ) α 1 and α 2 are chosen so that Remark 1. Assumptions (A 1 )-(A 4 ) deserve some a priori comments as follows.
The last term in the above expression is nonnegative since both terms within parentheses always have the same sign, which can be easily verified analyzing the cases |s| > |r| and |s| < |r|. Thus, inequality (18) holds true. Such inequality will be a critical estimate in order to conclude the continuous dependence of solutions as well as in the subsequent section dealing with long time behavior.
(iii) (Energy inequality) Both strong and weak solutions satisfy the following energy inequality for some constant stands for the energy solution (iv) (Continuous dependence) Both strong and weak solutions depend continuously on the initial data in H. More precisely, if z 1 = (u, u t ), z 2 = (v, v t ) are two solutions corresponding to initial data z 1 0 = (u 0 , u 1 ), z 2 0 = (v 0 , v 1 ), respectively, then γ+2 . In particular, problem (1)-(4) has uniqueness of solution in both cases.
The proof on existence relies on the Faedo-Galerkin method. The nonlocal nonlinear damping in (1) needs a careful analysis during the whole proof as we shall see in the sequel. Theorem 2.1 is proved in some steps as follows.
for j = 1, . . . , m, which has a local solution given by a standard method in ODE theory, where (ω j ) j∈N is an orthonormal basis in W 0 given by eigenfunctions of ∆ 2 with boundary condition (2) or (3). In what follows a priori estimates are shown in order to extend the local solution to the interval [0, T ] and then conclude the existence of strong and weak solutions for (1)-(4). We first consider the problem (25)- (26) with A Priori Estimate I. Replacing ω j by u m t (t) in the approximate equation (25) yields where E m κ (t) is the functional energy (23) for Galerkin's solutions. Besides, using (17) we obtain Combining (28) with (29) and integrating the resulting expression from 0 to t ≤ T m , we have On the other hand, from assumptions (10), (14)- (15), observing (8) and (16), and applying Young inequality with = αλ 1 /4, then From (30)- (31), since N > 0 on [0, +∞), and E m κ (0) is bounded when assumptions (10)- (14) are taken in place, then Going back to (30) one has Combining (31) and (33) we conclude for any t ∈ [0, T ] and m ∈ N, and some constant C 1 > 0 depending on initial data in H. Therefore, from (34) we obtain A Priori Estimate II. Differentiating the approximate equation (25) with respect to time and taking the multiplier u m tt (t) in the resulting expression we have 1 2 In addition, assumption (12) implies Returning to (38), using that M, N ∈ C 1 ([0, ∞)), the estimates (32) and (34), and also Cauchy-Schwarz and Young inequalities, we obtain 1 2 for some constant C > 0 depending on the initial data in H. Let us estimate the right hand side of (39). From now on we use the same parameter C to designate different positive constants which may depend on weak initial data but not on t and m.
Remark 2. With respect to boundary condition (3) we can also make another a priori estimate by replacing ω j by −∆u m t in the approximate equation (25). Then performing analogous calculations as given by the authors in [16] it is possible to find that strong solutions also satisfy The only limit we need to watch out is about the nonlocal nonlinear term given by the product of two nonlinearities. Actually, such limit needs to be clarified since it is not provided by previous literature so far. Thereby, a more accurate analysis of this convergence is made below for weak solutions and so it holds true in the stronger case.
In addition, for each regular initial data (u m 0 , u m 1 ), there exists a solution in the class to the following problem Taking the multiplier u m t in (46) and proceeding similar to a priori estimate I, observing (45), then estimate (34) remains true, namely, Besides, as we shall see in the proof of Theorem 2.1 (iv), the difference of strong Applying Gronwall's inequality we obtain, in view of (48), that where (47) and (45) it follows that (u m , u m t ) is a Cauchy sequence in C([0, T ], H). Thereby, applying (48) and (49) there exists a subsequence (u k ) of (u m ) such that The above limits (50)-(54) along with (45) are enough to pass the limit on the system (46)-(47) and show that (20)-(21) are satisfied with with initial conditions (4). In accordance with Remark 3 it remains to analyze the limit of the nonlocal nonlinear term N ( ∇u k (t) 2 2 )g(u k t (t)). This is done as follows. Analysis on the nonlocal nonlinear term N ( ∇u k (t) 2 2 )g(u k t (t)). From Aubin-Lions compactness theorem, see e.g. [25], there exists a subsequence of (u k ), still denoted by (u k ), such that Since N ∈ C 1 ([0, ∞)) we get, from the Main Value Theorem, that On the other hand, using again the Mean Value Theorem and assumption (12) with g(0) = 0, we obtain (Ω)).
Then, using that g ∈ C 1 (R) and uniqueness of the weak limit we infer In particular, taking w = ω j θ with θ ∈ L γ+2 (0, T ) and j ∈ N, j ≤ k, it follows that Thus, Hence, combining (56) and (57) with classical results in functional analysis we are able to pass the limit on the nonlocal nonlinear term in (46) to achieve the desired weak solution (and also in (25) for regular solutions).

Proof Theorem 2.1 (iii):
Taking the multiplier u t in (1), then a straightforward computation implies that E κ (t) given in (23) satisfies for strong solutions, where the derivative is understood in the distributional sense, see for instance Yang [37,Lemma 3.2]. Regarding the same arguments as in (29) and (32), and denoting by C N = Therefore, the energy inequality (22) holds true by integrating (58) on [s, t] for any 0 ≤ s < t. In addition, (22) is also ensured for weak solutions using standard density arguments.
2.4. Proof Theorem 2.1 (iv): Let z 1 = (u, u t ) and z 2 = (v, v t ) be two strong (or weak) solutions of (1)-(4) with initial data z 1 0 = (u 0 , u 1 ) and z 2 0 = (v 0 , v 1 ), respectively. Setting w = u − v, the difference z 1 − z 2 = (w, w t ) solves the following problem in the strong (or weak) sense We first consider the estimates below for strong solutions. Taking the multiplier w t in (59) we infer where In addition, we note that estimate (34), which holds for any strong (or weak) solution, implies that N ∇u(t) 2 2 ≥ c N > 0 for every t > 0, where c N = c N ( z 1 0 H ) > 0. From this and condition (18) we get Inserting (61) in (60) and denoting by c 1 = c N c g 2(γ+1) > 0 we deduce Now let us estimate the right-hand side of (62). For simplicity, the parameter C > 0 shall denote several constants which depend on the initial data in H, but not on t > 0. First of all, since M, N ∈ C 1 ([0, ∞)), we have from (34), the Main Value Theorem, and embedding W 2 → W 1 , that and Using Young inequality it follows directly , for some C > 0. Applying assumption (12), embedding W 2 → W 1 , Hölder and Young inequalities, one has Further, using assumption (9), Hölder inequality with ρ 2(ρ+1) + 1 2(ρ+1) + 1 2 = 1, embedding W 2 → L 2(ρ+1) (Ω), and Young inequality, one gets Replacing these last four estimates in (62) results for any t ∈ [0, T ] and some C = C( γ+2 has the property β ∈ L 1 (0, T ). Thus, integrating (65) on [0, t] and applying Gronwall's inequality we arrive at for some constant C = C( z 1 0 H , z 2 0 H ) > 0. Thereby, estimate (24) follows for strong solutions keeping in mind that (w, w t ) = z 1 − z 2 and taking C 0 = C/2. In particular, we have uniqueness of strong solution by taking z 1 0 = z 2 0 . Employing density arguments the same conclusion also holds true for the difference of weak solutions since they were obtained by the limit transition from strong solutions. Indeed, given weak initial conditions Since (24) holds for strong solutions we conclude γ+2 . Therefore, (24) is ensured for weak solutions after passing the limit in (69) and using (67)-(68). In addition, since the energy identity (60) remains valid for weak solutions using smooth mollifiers, see e.g. Lions and Magenes [26,Lemma 8.3] (see also Yang [37, Lemma 3.2]), then we also have uniqueness of weak solution performing the same estimates as above.
The proof of Theorem 2.1 is now complete.
Remark 4. The same conclusions (i)-(iv) of Theorem 2.1 can be shown by changing assumption (A 1 ) to the following: where we consider c f > 0, 0 ≤ α 1 < λ 1 , and Condition (72) shows that we can extend the growth ρ of the source f (u) up to the growth γ corresponding to damping term g(u t ). However, in spite of providing a better range for ρ, it is worth noting that (A 5 ) requires hypotheses on the second derivative of f with growth dominated by the growth of g. Summarizing, assumptions (A 2 )-(A 5 ) are sufficient to conclude well-posedness to problem (1)-(4) as stated in Theorem 2.1. The proof can be done under minor modifications only in the proof of A Priori Estimate II and item (iv) of Theorem 2.1. The precise details are similar to those ones given by Yang [37, Theorem 2.1].

Generation of a dynamical system. Let us define the family of nonlinear evolution operators S κ (t) : H → H given by
where (u, u t ) is the unique weak solution of (1)-(4) given by Theorem 2.1. Thus, in view of assumptions (A 1 )-(A 4 ) or (A 2 )-(A 5 ) we have a well defined family of dynamical systems (H, S κ (t)), κ ≥ 0, possessing the following properties: • (Gradient System) the energy relation (22) implies that (H, S κ (t)), κ ≥ 0, is a gradient dynamical system with strict Lyapunov functional given by the corresponding energy E κ (t) defined in (23); • (Lipschitz Property) from (24) the evolution semigroup S κ (t) satisfies a locally Lipschitz property on the phase space H for every parameter κ ≥ 0.
In what follows our main goal is to provide additional properties to the dynamical system defined in (73). Then we establish our results on attractors and their qualitative properties. To do so we first show some technical results on stability to the solution of problem (1)-(4).

3.2.
Technical results. We also assume the following assumption on the nonlocal coefficient M of extensibility.
If we consider a weak solution z = (u, u t ) of (1)-(4) corresponding to initial data (u 0 , u 1 ) ∈ B, where B ⊂ H is an arbitrary bounded set, then there exists a constant µ = µ B > 0 (depending on the size of B) such that and for any t > 0, where we denote s + = (s + |s|)/2 and θ = ln 1+µ µ > 0.

Corollary 2. (Absorbing set)
Under assumptions of Proposition 1, let us consider any bounded set B ⊂ H. If (u 0 , u 1 ) ∈ B, then there exists t B > 0 such that where (u(t), u t (t)) = S κ (t)(u 0 , u 1 ) is the weak solution of problem (1)-(4) and r > 0 is a constant independent of (u 0 , u 1 ). In particular, the set is a bounded absorbing set for S κ (t) defined in (73). In other words, the dynamical system (H, S κ (t)) is dissipative.
Proof. For initial data (u 0 , u 1 ) lying in B we obtain from estimates (76) and (77) that there exists t B > 0 dependent of B ⊂ H such that From (78) one sees that (94) follows by taking r = 2 2R α 1/2 > 0.
Remark 5. From (20) and (94) one sees that the solution (u, u t ) of (1)-(4) corresponding to an initial data (u 0 , u 1 ) lying in bounded sets B ⊂ H is globally bounded in H, that is, for some constant C B > 0 depending on B. Moreover, going back to (22) we infer lim sup for some constant C B > 0.
for any sequence (z n ) in B. Then (H, S(t)) is an asymptotically smooth dynamical system.
Proof. Given a bounded positively invariant set B ⊂ H, we denote by C B > 0 different constants depending on the size of B but not on t. For z 1 , z 2 ∈ B we need to show that S κ (t)z i = (u i (t), u i t (t)), i = 1, 2, satisfies (115)-(116). Indeed, given ε > 0, from Proposition 2, inequalities (98)-(99), we can choose T > 0 large enough such that for some constant C B > 0 depending on B. Let us estimate the right-hand side of (117). Applying interpolation theorem and (95) it follows that for some constant C B > 0 and θ 1 = 1/2. Also, taking θ 2 = n 4 1 − 1 ρ+1 , 2 . Taking ϑ = min{θ 1 , θ 2 }, and noting that u 1 (t) 2 and u 2 (t) 2 are uniformly bounded, there exists a constant C B > 0 such that (118) Replacing (118) in (117) we get where Ψ T : H × H → R is given by Now let us show that Ψ T satisfies (116). In fact, given a sequence of initial data z m = (u m 0 , u m 1 ) ∈ B, as before, we write S κ (t)z m = (u m (t), u m t (t)). Since B is invariant by S κ (t), t ≥ 0, it follows that (u m (t), u m t (t)) are uniformly bounded in H = W 2 × W 0 . Hence, Since W 2 is compactly embedded in W 0 , then in view of Aubin's Lemma (see Simon [34,Corollary 4]) there exists a subsequence of (u m ), still denoted by (u m ), such that (u m ) converges strongly in C([0, T ], W 0 ), T > 0, which is enough to conclude lim k→∞ lim m→∞ Ψ T (z k , z m ) = 0.
Besides, from Young inequality and second part of (129) it is easy to check that and from the choice of ε > 0, we get Combining (142) and (143), we have where we define Applying Gronwall inequality in (144) we deduce Now, given η > 0, we apply proper Hölder and Young inequalities three times and take into account (96) to deduce for some constants C η , C B > 0. Then, once fixed ε > 0, we choose 0 < η ≤ αε 12C B so that Finally, combining (146) with (129)-(130), we conclude that the stability inequality (119) holds true. Therefore, the proof of Proposition 4 is complete.
The above Proposition 4 shall allow us to achieve richer qualitative properties to the dynamical system (H, S κ (t)), κ ≥ 0, defined in (73). In fact, in Chueshov and Lasiecka [11] it is introduced concept of quasi-stability property that, in particular, also gives another way to conclude asymptotic smoothness property. In order to make this paper more self-contained, we introduce the concept of a quasi-stable dynamical system.
We recall that a seminorm n X (·) defined on a Banach space X is compact if whenever a sequence x j 0 weakly in X one has n X (x j ) → 0. Let X, Y, Z be three reflexive Banach spaces with X compactly embedded in Y and put H = X × Y × Z, where the case with trivial space Z = {0} is allowed. Consider the dynamical system (H, S(t)) given by an evolution operator S(t)z = (u(t), u t (t), ζ(t)), z = (u 0 , u 1 , ζ 0 ) ∈ H, where the functions u and ζ have regularity u ∈ C(R + ; X) ∩ C 1 (R + ; Y ), ζ ∈ C(R + ; Z).
Then one says that (H, S(t)) is quasi-stable on a set B ⊂ H if there exists a compact seminorm n X on X and nonnegative scalar functions a(t) and c(t) locally bounded in [0, ∞), and b(t) ∈ L 1 (R + ) with lim t→∞ b(t) = 0, such that, and for any z 1 , z 2 ∈ B. Inequality (150) is called stabilizability inequality.
Corollary 4 (Quasi-stability property). Let assumptions of Proposition 4 be in force. Then the dynamical system (H, S κ (t)) is quasi-stable on any bounded positively invariant set B ⊂ H. In particular, it is also asymptotically smooth.
Proof. We need to show that (H, S κ (t)) satisfies (147)-(150). Since (H, S κ (t)) is defined in (73), then Theorem 2.1 implies that conditions (147)  then, in view of (119), the semigroup solution S κ (t)z i 0 = (u i (t), u i t (t)), i = 1, 2, satisfies for any z i 0 ∈ B and t > 0. Since B is bounded one sees that b ∈ L 1 (R + ) with lim t→+∞ b(t) = 0 and c(t) is locally bounded on [0, +∞). Moreover, form definition of c(t) it is ensured that In addition, from compact embedding W 2 → W 1 , we conclude that n W2 (·) is a compact seminorm on the space W 2 . Therefore, the stabilizability estimate (150) also holds true. This completes the proof of the quasi-stability property to the dynamical system (H, S κ (t)). In particular, from [11,Proposition 7.9.4], (H, S κ (t)) is asymptotically smooth.

3.3.
Main results on attractors. Now we are in conditions to state our main results on attractors with respect to the dynamical system (H, S κ (t)), κ ≥ 0, defined in (73). In particular, for every κ ≥ 0, the set of stationary solutions N κ constitutes a global minimal attractor to the dynamical system (H, S κ (t)).