ATTRACTORS AND THEIR STABILITY ON BOUSSINESQ TYPE EQUATIONS WITH GENTLE DISSIPATION

. The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation: u tt + ∆ 2 u + ( − ∆) α u t − ∆ f ( u ) = g ( x ), with α ∈ (0 , 1). For general bounded domain Ω ⊂ R N ( N ≥ 1), we show that there exists a critical exponent p α ≡ N +2(2 α − 1) ( N − 2) + depending on the dissipative index α such that when the growth p of the nonlinearity f ( u ) is up to the range: 1 ≤ p < p α , (i) the weak solutions of the equations are of additionally global smoothness when t > 0; (ii) the related dynamical system possesses a global attractor A α and an exponential attractor A αexp in natural energy space for each α ∈ (0 , 1), respectively; (iii) the family of global attractors {A α } is upper semicontinuous at each point α 0 ∈ (0 , 1], i.e., for any neighborhood U of A α 0 , A α ⊂ U when | α − α 0 | (cid:28) 1. These results extend those for structural damping case: α ∈ [1 , 2) in [31, 32].

For the physical background of them, one can see above-mentioned references in detail.
The study on the hyperbolic equations with fractional damping arises from [4]. In 1982, based on the results of empirical studies: there are always dissipative mechanisms acting within the linear elastic systems causing the energy to decrease during any positive time interval, Chen and Russell [4] presented an abstract mathematical model exhibiting the empirically observed damping rates in elastic systems: where A (the elastic operator) and B (the dissipation operator) are two positive, self-adjoint operators with domains D(A) and D(B) dense in the Hilbert space X satisfying ρ 1 A α ≤ B ≤ ρ 2 A α for some constants 0 < ρ 1 < ρ 2 < ∞ and 0 < α ≤ 1. More preciously, Bẋ = A αẋ is said to be structural damping when 1/2 ≤ α ≤ 1, gentle dissipation when 0 < α < 1/2 (cf. [5,6,7]). It is expected that this type of models will permit realistic simulation of various elastic systems wherein damping cannot be ignored. According to above classification, (−∆) α u t in Eq. (1) is gentle dissipation when α ∈ (0, 1), structural damping when α ∈ [1,2] for A = ∆ 2 (with hinged boundary condition (2)) and B = (−∆) α = A α/2 here, and α is said to be a dissipative index.
Obviously, Eq. (1) is hyperbolic and its solutions have no additional smoothness when t > 0 if α = 0. It is well-known that the dissipative term (−∆) α u t (α > 0) has a regularizing effect for the solutions of Eq. (1), i.e., it makes them be of additionally partial smoothness when t > 0. Recently, Yang et al. [31,32] showed that when α ∈ [1, 2) (structural damping case), Eq. (1) is like parabolic, the related solution semigroup has a global attractor and a "partially strong" exponential attractor in natural energy space when the growth exponent p of f (u) is up to the supercritical range: 1 ≤ p <p α ≡ N +2(2α−1) (N −2(2α−1)) + , where a + = max{a, 0}. Moreover, the family of global attractors {A α } is upper semicontinuous at the point α 0 ∈ [1, 2) in the following sense: for any neighborhood U of A α0 , A α ⊂ U when 0 < α − α 0 1. For the investigations on the existence of global and exponential attractors for the case of α = 1 in phase space H 2 ∩ H 1 0 × L 2 , one can see [22]. It is well known that the dissipative role of gentle dissipation is stronger than that of weak damping but weaker than that of structural damping. In this case, is the Eq. (1) like parabolic (as in the case of structural damping) or like hyperbolic (as in the case of weak damping)? what about the existence of its global and exponential attractors? what about the upper semicontinuity of perturbed attractors A α ? These questions are unsolved.
Here, the exponent p α is said to be critical because the uniqueness of solutions of problem (1)-(2) fails (or one can not get it) when p ≥ p α . Obviously, p α = p * when α = 1, which coincides with the critical exponent in [31].
We mention that there have been some recent researches on the well-posedness and longtime dynamics of nonlinear evolution equations with fractional damping, for example, the investigations on that of quasilinear wave equation with structural damping (cf. [13,14,20,36]) and that of the semilinear wave equation with structural damping or gentle dissipation (cf. [24,25,34,35]). And there have been extensive studies on the stability of exponential attractors when the perturbations are some coefficients of the evolution equations (cf. [17,18,19] and references therein). But it is challenging to construct robust families of exponential attractors when the perturbation is the dissipative index, which will be considered in our future work. This paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, we discuss the well-posedness and the like parabolic properties of problem (1)- (2). In Section 4, we establish the existence of global and exponential attractors in natural energy space and discuss some properties of the global attractor. In Section 5, we investigate the upper semicontinuity of the family of global attractors {A α }.

2.
Preliminaries. For brevity, we use the following abbreviations: , H k are the L 2 -based Sobolev spaces and H k 0 are the completion of C ∞ 0 (Ω) in H k for k > 0. The notation (·, ·) for the H-inner product will also be used for the notation of duality pairing between dual spaces, the sign H 1 → H 2 denotes that the functional space H 1 continuously embeds into H 2 and H 1 → → H 2 denotes that H 1 compactly embeds into H 2 , C(· · · ) denotes positive constants depending on the quantities appearing in the parenthesis.
(Au, v) = (∆u, ∆v) for any u, v ∈ V 2 . Then, A is self-adjoint in H and strictly positive on V 2 , so we can define the power A s of A (s ∈ R), and the spaces V s = D(A s 4 ) are Hilbert spaces with the scalar products and the norms Rewriting Eq. (1) at an abstract level and applying A − 1 2 to both sides, we obtain We denote the phase spaces with α ∈ (0, 1), which are equipped with usual graph norms, for example, (u, v) 2 Obviously, they are Hilbert spaces and X α → → X → → Y α .

ZHIJIAN YANG, PENGYAN DING AND XIAOBIN LIU
Assumption (H) (i) f ∈ C 1 (R), and when N ≥ 2, where λ 1 is the first eigenvalue of A, Remark 1. The first inequality in (6) implies that there exists a constant θ : where and in the following . Let X be a Banach space, the set Z ⊂ C(R + ; X), and Φ : X → R be a continuous functional satisfying for some η, K ≥ 0 and every z ∈ Z. In addition, assume that for every z ∈ Z the function t → Φ(z(t)) is continuously differentiable, and satisfies the differential inequality d dt Φ(z(t)) + δ z(t) 2 X ≤ k for some δ > 0 and k ≥ 0 independent of z ∈ Z. Then, for every γ > 0 there exists For simplicity, we restrict ourselves to the case N ≥ 3 for the cases N = 1, 2 are easy. But all the results hold for the cases N = 1, 2.

and the solutions are Lipschitz stable in weaker space
and quasi-stable in Y α , i.e., where k > 0 is a constant, κ denotes a small positive constant, z = u − v, u, v are two weak solutions of problem (4)-(5) corresponding to initial data (u 0 , u 1 ) and Proof. We first give some a priori estimates to the solutions of problem (4)- (5).
Based on estimates (7)-(8) (which obviously hold for the Galerkin approximations) and the Galerkin method, one can easily prove that problem (4)-(5) possesses a weak solution u, with (u, u t ) ∈ L ∞ (R + ; X) ∩ C w (R + ; X), and by the lower semicontinuity of weak limit, estimates (7)-(12) hold for u. We omit the process here.
(iii) (Strong continuity and stability) By using energy identity (13) and repeating the same proof as in [32], one easily gets that is, ξ u (t) X ∈ C[0, T ]. By the uniform convexity of the phase space X and (u, u t ) ∈ C w ([0, T ], X), we have (u, u t ) ∈ C([0, T ], X). Let z = u−v, where u, v are two weak solutions of problem (4)-(5) corresponding to initial data (u 0 , u 1 ) and (v 0 , v 1 ), respectively. Obviously, z solves
(iii) In order to investigate the upper semicontinuity of the global attractor A α on α, it is necessary to clarify the dependency and independency of the control constants on α in Theorem 3.1, respectively.
For simplicity, we denote S α (t) by S(t), (u α , u α t ) by (u, u t ) for each α ∈ (0, 1) in this section. Proof. When t = 0, the conclusion is obvious for S(0) = I. When t > 0, let (u n 0 , u n 1 ) → (u 0 , u 1 ) in X, and S(t)(u n 0 , u n 1 ) = ξ u n (t) = (u n (t), u n t (t)), S(t)(u 0 , u 1 ) = ξ u (t) = (u(t), u t (t)). Noticing that V α+1 → V 2−α for α ∈ (1/2, 1) (see (12)) and using the interpolation theorem, we get that for every α ∈ (0, 1), X . Therefore, S(t) is continuous in X for every t > 0. Theorem 4.2. Let Assumption (H) be valid, with 1 ≤ p < p α and g ∈ V −1 . Then the dynamical system (S(t), X) possesses a global attractor A α , which is bounded in X α , and where M + (N ) is the unstable manifold emanating from the set N , and N is the set of all the fixed points of S(t), that is, Moreover, (S(t), X) is a gradient system, and every full trajectory γ = {ξ u (t)|t ∈ R} from A α is of the following properties: Proof. Estimates (7) and (9)- (12) imply that the dynamical system (S(t), X) has a bounded absorbing set B 0 which is bounded in X α . Without loss of generality we assume that B 0 is forward invariant. Then the dynamical system (S(t), X) is dissipative and the semigroup S(t) is uniformly compact for X α → → X. Therefore, (S(t), X) possesses a global attractor A α = ω(B 0 ), which is bounded in X α for A α ⊂ B 0 . We infer from (13) that the functional E(ξ u ) defined there is continuous on X, and it is a strict Lyapunov function on X. Therefore, the dynamical system (S(t), X) is gradient, A α = M + (N ), and every full trajectory γ = {ξ u (t)|t ∈ R} from A α is of property (i)-(ii) (cf. Theorem 2.28 in [12] and [33]). Now, we investigate the exponential attractors. It follows from Theorem 4.2 that the dynamical system (S(t), X) has a forward invariant absorbing set B 0 , which is bounded in X α . Let where [ ] Yα denotes the closure in Y α . Obviously, B is also a forward invariant absorbing set of S(t), which is closed in Y α and bounded in X α . So the set B equipped with Y α -norm forms a complete metric space, the operator S(t) is Lipschitz continuous on B (w.r.t. Y α -topology), and (S(t), B) (equipped with Y α -topology) constitutes a dissipative dynamical system. We first give two lemmas which are indispensable for us to establish the existence of exponential attractors.
and a compact seminorm n Z (·) on Z such that where 0 < η < 1 is a constant. Then for any θ ∈ (η, 1), the discrete dynamical system (V k , M ) (M is equipped with the metric of X) has an exponential attractor A θ . Moreover, where m Z (R) is the maximal number of elements z i in the ball {z ∈ Z| z Z ≤ R} possessing the property n Z (z i − z j ) > 1 when i = j.
Theorem 4.5. Under the assumptions of Theorem 4.2, the dynamical system (S(t), X) has an exponential attractor A α exp . Proof. Define the operator We first show that the discrete system (V k , B) (equipped with Y α -topology) possesses an exponential attractor. We introduce the functional space Obviously, Z is a Banach space. Define the mapping where ξ u (·) means ξ u (t), t ∈ [0, T ]. By (14) and Lemma 4.4, [26]). Taking T : 0 < η T < 1, we infer from Lemma 4.3 that the dynamical system (V k , B) has an exponential attractor A. Let By the standard argument (cf. [33]) one easily knows that A α exp is an exponential attractor of the dynamical system (S(t), B) (equipped with Y α -topology). And by virtue of additional regularity of the weak solutions of Eq. (4), the interpolation theorem and the technique used in [32], we easily recover the topology of the phase space X and show that A α exp is just an exponential attractor of the dynamical system (S(t), X).
(i) Formally differentiating Eq. (4) with respect to t and using the multiplier A − γ+1 2 u α tt + A − 1 2 u α t ( 0 < 1) to the resulting expression, similar to the proof of (9) one easily gets that, for 1 ≤ p < p γ (≡ N +2(2γ−1) where C(R, g V−1 ) is a positive constant depending only on α 0 . Based on (48), repeating the proof of (11) (replacing α there by γ), one easily obtains Obviously, formulas (48) and (49) hold for the Galerkin approximations, and by the lower semicontinuity of weak limit and the uniqueness of the solutions (see (14)), estimates (48) and (49) uniformly (w.r.t. α ∈ Λ) hold for the weak solutions u α of problem (4)- (5). Estimate (7) shows that the ball is a common absorbing set of the dynamical system (S α (t), X) for all α ∈ Λ for suitably large R.
Therefore, by Lemma 5.2 and the arbitrariness of η, the family of attractors {A α } is upper semicontinuous at the point α 0 for 1 ≤ p < p α0 .