ATTRACTORS FOR WEAKLY DAMPED BEAM EQUATIONS WITH p -LAPLACIAN

. This paper is concerned with a class of weakly damped one-dimen-sional beam equations with lower order perturbation of p -Laplacian type ,L ) , where σ ( z ) = | z | p − 2 z , p ≥ 2, k > 0 and f ( u ) and h ( x ) are forcing terms. Well-posedness, exponential stability and existence of a ﬁnite-dimensional attractor are proved.

Our main result establishes the existence of a finite-dimensional global attractor to the system (1)-(3) with a weak damping. For the N -dimensional problem a corresponding result with strong damping −∆u t was proved by Yang [14]. We also present a complete proof of uniqueness for weak solutions which seems to be new in the context of p-Laplacian wave equations. In addition, with respect to global existence, we notice the Yang [12] studied problem (1)-(3) with weak damping u t by assuming σ(0) = σ (0) = σ (0) = 0 and σ is locally Lipschitz, which implies that p ≥ 5. In our case, we prove uniqueness and continuous dependence of initial data for p ≥ 3. However, for the existence of global attractors we assume p ≥ 4. This restriction can be weakened to p ≥ 2 if a strong damping −u txx is added in the system.
Our work is organized as follows. In Section 2 we present the assumptions and the main results. In Section 3 we prove the existence of weak and stronger solutions. In Section 4 we prove the existence of a finite dimensional attractor.

Assumptions and main results.
Let us assume f ∈ C 1 (R) and that there exists a constant ρ > 0 such that wheref (s) = s 0 f (τ ) dτ . In this paper we use standard notations for Sobolev Spaces as in the book by Lions [10]. In L p (0, L), we denote the usual norm by u p p = L 0 |u| p dx. Then the energy of the system (1)-(3) is written as For notation convenience, let us define (ii) If the initial data (u 0 , is well-posed with respect to weak solutions. The well-posedness of weak solutions asserted in Theorem 2.1 shows that problem (1)-(3) corresponds to a nonlinear C 0 -semigroup S(t) on the phase space where u is the corresponding weak solution with initial data (u 0 , u 1 ). Then we can consider problem (1)-(3) as an infinite dimensional dynamical system. Theorem 2.2. Assume the hypotheses of Theorem 2.1 hold with p ≥ 4. Then the corresponding dynamical system (H, S(t)) has a compact attractor with finite fractal dimension.
3. Well-posedness. In this Section we prove Theorem 2.1. The existence of a global solution is given by using Faedo-Galerkin method. The uniqueness of weak solutions is proved with a regularization argument. Proof of Theorem 2.1. This will be done through several steps. The proof of existence of weak solutions is presented briefly.
Step 1. Approximate Problem. Let {w j } j∈N be a Galerkin basis given by the eigenfunctions of u = λu, u(0) = u(L) = u (0) = u (L) = 0, and let 1 ≤ j ≤ m , with initial condition where u 0m and u 1m are chosen such that By standard ODE theory, problem (10)-(12) has a local solution u m (t). The estimate below will allow the local solutions be extended to an interval [0, T ], for any given T > 0.
Step 2. A Priori Estimate. Replacing w j by u m t (t) in (10) we get by integration where E m (t) is the approximate energy defined from (6) with u replaced by u m . Now, noting that for u ∈ H 2 (0, L) ∩ H 1 0 (0, L) one has u 2 ≤ L 2 π −2 u xx 2 , assumption (5) implies where C hρ = ρL + L 2 π −2 h 2 2 . Then integrating (13) from 0 to t we conclude that Step 3. Passage to the Limit. From estimate (15), going to a subsequence if necessary, we infer that ). Then we obtain from Aubin-Lions theorem, , as proved, for instance, in Kim [8], Lemma 1.4. Then we can pass to the limit the approximate problem (10)- (11) in order to get a weak solution of problem (1)-(3). This essentially proves Theorem 2.1 (i).
Step 5. Uniqueness of Weak Solutions. We apply the classical regularization method of Vishik-Ladyzenskaya [10]. Let u, v be two weak solutions of problem (1)-(3). Then w = u − v is a weak solution of with boundary condition w(0, t) = w(L, t) = w xx (0, t) = w xx (L, t) = 0 and null initial condition. For a fixed s ∈ [0, T ] we define Then we see that , ψ(s) = 0. With the above regularity for ψ(t), we can multiply equation (17) by ψ(t) and integrate with respect to x and t. We have Denoting w 1 (t) = t 0 w(ξ)dξ we have ψ(t) = w 1 (t)−w 1 (s), 0 < t < s. Then, arguing as is in Lions [10], we obtain the following standard estimates: where C 0 > 0 denotes a generic constant depending on the initial data. It remains to estimate the term with p-Laplacian. First we note that we get from a priori estimate (15) that z(t) ∞ ≤ C 0 . In addition, since p ≥ 3 implies that σ is locally bounded, we also get z Then we infer that Therefore combining the above estimates with (19) yields Since w(0) 2 = w 1xx (0) 2 = 0, Gronwall inequality implies that w(s) = 0 in L 2 (0, L), ∀ s ∈ (0, T ), and therefore u = v.
Step 6. Continuous Dependence on Initial Data. Since we have proved the uniqueness of weak solutions, we can work on stronger solutions satisfying (8) and then extend the conclusion to weak solutions by standard density arguments. Given initial data (u 0 , u 1 ) and (v 0 , v 1 ) in H 3 b (0, L) × H 1 0 (0, L), we write w = u − v where u, v are the corresponding regular solutions. Then w satisfies equation (17) with initial data w(0) = u 0 − v 0 and w t (0) = u 1 − v 1 . In addition, Now, Using again that σ is locally bounded (p ≥ 3), we get where C 0 > 0 is a constant depending on the initial data. Then as before see that Hence, given T > 0, there exists a constant α 0 > 0 such that 4. Global attractors. The definitions and classical results to global attractors of infinite dimensional dynamical systems can be found, e.g., in the books by Babin & Vishik [3], Hale [7], Ladyzhenskaya [9] and Teman [11]. We follow closely the book by Chueshov & Lasiecka [4], Chapter 7.
Our framework is that of dissipative dynamical systems. One says that (H, S(t)) is dissipative if it possesses an absorbing set, that is, a bounded set B ⊂ H such that for any bounded set B ⊂ H there exists t B ≥ 0 satisfying Let H = X × Y with X compactly embedded in Y . Suppose that (H, S(t)) is a dynamical system given by an evolution operator where the function u has regularity We recall that a seminorm n X (x) defined on X is compact if any sequence such that x j 0 weakly in X implies that n X (x j ) → 0. Then one says that (H, S(t)) is quasistable 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 w 1 , w 2 ∈ B. The inequality (25) is often called stabilizability inequality. In this context, the following result is proved in Chueshov & Lasiecka [4], Corollary 7.9.5 and Theorem 7.9.6.
Theorem 4.1. Assume that (H, S(t)) is a dissipative dynamical system of the form (22) and satisfying (23). Assume in addition that the system is quasi-stable on any bounded positively invariant set. Then (H, S(t)) has compact a global attractor with finite fractal dimension.
In order to apply Theorem 4.1 we prove the following two lemmas. Proof. Given ε > 0 let From (14) we see that Then there exists ε 0 > 0 such that Next we show that there exists ε 1 > 0 such that By density arguments, we can assume that solutions are regular. Then using equation (1) and adding and subtracting E(t) we get Since (ku(t), u t ) ≤ 1 2 u xx (t) 2 2 + C 0 u t (t) 2 2 , we conclude in view of (5) that Now choosing ε < min{ε 0 , ε 1 } such that ε 3 2 + C 0 ≤ k, the above inequality implies (28). Then combining (28) with (27) yields From estimate (26) we see that Taking R > (12C hρ ) 1/2 the ball B(0, R) ⊂ H is an absorbing set.

Remark 1.
From the proof of the Lemma 4.2 we see that if h = ρ = 0 then the system's energy E(t) decays exponentially. More precisely, from (29), where u, v are the corresponding weak solutions of (1)-(3) and C B > 0 is a constant depending on B but not on t.
From definition of F (t) we obtain (30) with µ = 2η/3. Proof of Theorem 2.2. Our dynamical system is defined by the evolution operator (9) and has regularity (7). Therefore (22) and (23) hold. We also see that (H, S(t)) is quasi-stable in bounded positively invariant sets. Indeed, condition (24) comes from (21) and condition (25) follows promptly from (30) with n X (u) = u x 2 . Then taking into account that Lemma 4.2 implies that (H, S(t)) is dissipative, we conclude from Theorem 4.1 that our system has a compact global attractor with finite fractal dimension. This ends the proof of Theorem 2.2.