ROBUST ATTRACTORS FOR A KIRCHHOFF-BOUSSINESQ TYPE EQUATION

. The paper studies the existence of the pullback attractors and robust pullback exponential attractors for a Kirchhoﬀ-Boussinesq type equation: We show that when the growth exponent p of the nonlinearity g ( u ) is up to the critical range: 1 , (i) the IBVP of the equation is well-posed, and its solution has additionally global regularity when t > τ ; (ii) the related dynamical process { U f ( t,τ ) } has a pullback attractor; (iii) in particular, when 1 ≤ p < p ∗ , the process { U f ( t,τ ) } has a family of pullback exponential attractors, which is stable with respect to the perturbation f ∈ Σ (the sign space).

1. Introduction. In this paper, we are concerned with the existence of the pullback attractors and robust pullback exponential attractors for a Kirchhoff-Boussinesq type equation: where Ω is a bounded domain in R N (1 ≤ N ≤ 6) with the smooth boundary ∂Ω, and the assumptions on the nonlinear feedback force g(u) and the time-dependent external force f (x, t) will be specified later. Chueshov and Lasiecka [6,8] proposed 2D Kirchhoff-Boussinesq model: where k > 0 is the damping parameter, the mapping g 0 : R 2 → R 2 and the sufficiently smooth functions g 1 and g 2 represent nonlinear feedback forces acting upon the plate. They showed that the model (3) arises as the limit of the Midlin-Timoshenko equations which describe the dynamics of a plate that accounts for transverse shear effects (cf. [21,20]). When the terms div[g 0 (∇u)] and g 2 (u) are absent, Eq. (3) becomes "good" Boussinesq equation with viscous damping (cf. [1,22,27]). While the nonlinear restoring force g 0 (∇u) naturally arises in the Kirchhoff plate model (cf. [7]), this is why Eq. (3) is called Kirchhoff-Boussinesq model. One typical example of physical interests is g 0 (∇u) = |∇u| p−1 ∇u (p-Laplacian) (cf. [16,17,23]); another is g 0 (∇u) = ∇u √ 1+|∇u| 2 (cf. [16,18,24,31]). By taking g 0 (∇u) = |∇u| 2 ∇u, g 1 (u) = u 2 + u, Chueshov and Lasiecka [6,8] studied the well-posedness of the corresponding 2D model (3), and they also established the existence of global attractor provided that the damping parameter k is sufficiently large. These investigations are of particular difficulties because both the Kirchhoff nonlinearity g 0 (∇u) and the Boussinesq nonlinearity ∆[g 1 (u)] appear at the same time, which leads to that the techniques used to deal with the longtime dynamics of both Kirchhoff plate model (cf. [23,29]) and the Boussinesq model (cf. [14,15,30,32,34,35]) fail.
Replacing the weak damping ku t in Eq. (1) by the strong structural damping −∆u t , which plays a regularizing effect for the solutions and may make that the corresponding equation is of some parabolic-like properties, taking g 0 (∇u) = ∇u √ 1+|∇u| 2 and taking account of the effect of the time-dependent external force, we get model equation (1).
Pullback attractor and pullback exponential attractor are two key concepts to study the longtime dynamics of non-autonomous infinite dimensional dynamical system, the theory on them have been extensively developed and applied to many model equations arising from mathematical physics (cf. [2,3,26,28] and references therein). In the concrete, a process acting on the Banach space E is a twoparametrical family of operators {U (t, τ ) : E → E|t ≥ τ, t, τ ∈ R} (or t, τ ∈ Z for discrete time) satisfying A family of nonempty compact subsets {A(t)} t∈R in E is said to be a pullback attractor of the process U (t, τ ) if it is invariant, i.e., U (t, s)A(s) = A(t), t ≥ s, and it pullback attracts all bounded subsets of E, i.e., for every bounded subset D ⊂ E and t ∈ R, However, the pullback attractor may have two drawbacks: (i) the rate of convergence in (5) may be slow, which leads to that it is difficult to estimate the pullback attracting rate in term of the physical parameters of the system; (ii) in many situations, one cannot show the finite dimensionality of the sections of the pullback attractor, which results in that the pullback attractor may be unobservable in experiments or in numerical simulations.
In order to overcome these drawbacks, Efendiev, Miranville and Zelik [12] further proposed the concept of pullback exponential attractor (see Def. 4.1 below) and established its existence and stability criterion for the discrete dynamical process. Later, Langa, Miranville and Real [19], Czaja and Efendiev [10] extended the existence results in [13] to the continuous process. Carvalho and Sooner [4,5] further improved the results in [10,19] and give some applications to the hyperbolic equations. More recently, Yang and Li [33] further established two criteria on the existence of robust pullback exponential attractors.
In the present paper, the Kirchhoff type nonlinearity g 0 (∇u) = ∇u √ 1+|∇u| 2 is another kind of nonlinear restoring force, whose role is different from that in (4) for it can not provide useful L 4 -estimate for the gradient of the weak solutions (cf. [6,8]), so considering this kind of Kirchhoff nonlinearity is also challenging.
By assuming strong damping instead of weak one, several subtle issues (e.g. uniqueness of solutions, the sufficiently large damping parameter k, the restriction N = 2 for the space dimension) dealt in the past for this kind of model (cf. [6,8]) are no longer obstacle because the strong damping provides the helpful regularity effects and increases the expectation of nice results for this kind of model.
But even in this case, the techniques of dealing with the damped Boussineaq type equations used in [30,32,34] are still useless for the appearance of the Kirchhoff type nonlinearity. However, we can use the delicate multiplier technology to get the Lipschitz stability of the weak solutions in weaker space, and then concentrate our attention on the robust attractors of the nonautonomous dynamical systems. The adding of nonautonomous term makes that the Kirchhoff-Boussinesq model is of more suitability and stimulates us to investigate the robustness of the related pullback exponential attractors on its perturbations, which is completely different from the upper semicontinuity of the global attractors on the damping parameters (dissipative index) as done for damped Boussinesq model in [32,34].
The research on the well-posedness and longtime dynamics of this mix-and-match model is interesting not only for its physical background but also for its complexity of overcoming both the the Kirchhoff nonlinearity and the Boussinesq nonlinearity. Though the model is originally a specifically 2D plate model, it is interesting to investigate it in more general ND (N ≥ 2) case just as done by mathematicians dealing with the Kirchhoff plate model [9]: Indeed, it is just Chueshov's work on the ND Kirchhoff plate model (6) and creatively using the multiplier method which contains many delicate techniques that make him find a supercritical index of the nonlinear source term f (u): p * * = N +4 and break though the longtime existed restriction of the critical index p * = N +2 and show that when the growth exponent p of the source term is up to the supercritical range p * < p < p * * (N ≥ 3), the IBVP of Eq. (6) is still well-posed and the related solution semigroup has a partially strong global attractor (cf. [9]). These new interesting phenomena are interested by PDE/dynamical systems researchers and readers and stimulate then a further study on the longtime dynamics for this model, one can see [11,28,33] and references therein. It is just relying on the continuous research from various angles for a model equation (e.g. Kirchhoff wave model, Boussinesq model, Kirchhoff-Boussinesq model and so on) that promotes the appearance of good mathematical idea and technology and deepens the understanding for the science laws behind it.
The present paper devotes to investigate the well-posedness and the existence of pullback attractor and robust pullback exponential attractors for problem (1)- (2).
We show that that when the growth exponent p of the nonlinearity g(u) is up to the critical range: 1 ≤ p ≤ p * ≡ N +2 (N −2) + , (i) problem (1)-(2) is well-posed, and its solution has additionally global regularity when t > τ ; (ii) the related dynamical process {U f (t, τ )} has a pullback attractor; (iii) in particular, when 1 ≤ p < p * , by using an abstract criteria recently established in [33] we show that the process {U f (t, τ )} has a family of pullback exponential attractors, which is stable with respect to the perturbation f ∈ Σ (the sign space).
The paper is organized as follows. In Section 2, we discuss the well-posedness and some parabolic-like properties of the weak solutions. In Section 3, we investigate the existence of pullback attractor. In Section 4, we study the existence and stability of pullback exponential attractors.
2. Well-posedness. We begin with the following abbreviations: The notation (·, ·) for the L 2 -inner product will also be used for the notation of duality pairing between dual spaces, C(· · · ) denotes positive constants depending on the quantities appearing in the parenthesis.
We define the operator A : Then, A is self-adjoint and strictly positive on V 2 , and 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 We define the operator B : Obviously, the operator B is bounded, We denote the phase spaces which are equipped with the usual graph norms, for example, Obviously, they are the Hilbert spaces and Rewriting Eq. (1) as an operator equation and applying A − 1 2 to the resulting expression, we get the equivalent problem: where λ 1 (> 0) is the first eigenvalue of the operator A, and where G(s) = s 0 g(r)dr.
By energy identity (14), we have that for any t 0 ≥ τ , lim sup where we have used in the last inequality the facts: (u, u t ) ∈ C w (R τ ; X 1 ), Remark 2.2 and the Fatou Lemma. Therefore, By the uniform convexity of the Banach space X 1 , (u, u t ) ∈ C(R τ ; X 1 ).
Remark 2.4. We see from formulas (23) and (27) that the restriction for the space dimension: 1 ≤ N ≤ 6 is indispensable for the proof of Theorem 2.3.
Then the process {U f (t, τ )} possesses a pullback attractor A = {A(t)} t∈R , where is bounded in X 2 for each t ∈ R, and where the set D(t) is as shown in (35).
Estimate (10) implies that the family B = {B(t)} t∈R is a pullback absorbing family of {U f (t, τ )} in X 1 . So for any bounded set . By the arbitrariness of ϕ t−τ ∈ B(t − τ ), we have that for any t ∈ R, there exists a T t > 0 such that By the pullback absorbing property of B = {B(t)} t∈R , for any bounded set B ⊂ X 1 , there exists a τ B > 0 such that Therefore, which means that D = {D(t)} t∈R is a pullback absorbing family of the process that is, the family D is also pullback D-absorbing. By formula (34), Hence, By estimate (13)    [33] Let Σ be a symbol space or an index set, M be a bounded closed sunbset of the Banach space E, which is equipped with the distance d(x, y) = x − y E , and (U σ (t, τ ), M, E) be a non-autonomous dynamical system for each σ ∈ Σ. Assume that: (i) there exist constants T > 0, L T > 0, such that, for any τ ∈ R,

the set A(t) is as shown in (32) (cf. Theorem 4.2 in [28]), and A(t) is bounded in
(ii) there exist a Banach space Z and a compact seminorm n Z (·) on Z, and there exists a mapping K σ n : M → Z for each σ ∈ Σ, n ∈ Z such that for any x, y ∈ M , sup , where η : 0 < η < 1, L > 0 are constants independent of σ and n.
Obviously, Σ is invariant with respect to {T (h)} h∈R , i.e., . Theorem 4.4. Let Assumptions 2.1 be valid, and f ∈ Σ. Then the weak solutions of problem (7)-(8) possesses the following Lipschitz stability where K 1 = C(R, t − τ, f 0 L 2 b (R;V−1) ), which is monotone on the second variable t − τ , z = u − v, u and v are two weak solutions of problem (7)-(8) corresponding to initial data (u τ 0 , u τ 1 ), (v τ 0 , v τ 1 ) ∈ X 1 and symbols f 1 , f 2 ∈ Σ. In particular, when 1 ≤ p < p * , the following Lipschitz stability and quasistability hold in weaker space X 0 , , and κ is a small positive constant. Proof. Obviously, the function z solves Similar to the proof on the stability of weak solutions in X 1 in Theorem 2.3, one easily obtains estimate (37) by using the multiplier z t in Eq. (40). Thus we only prove (38)-(39) here.
Using the multiplier A − 1 2 z t + z in Eq. (40) and making use of the Sobolev embedding V 1−δ → L p+1 for δ : 0 < δ 1 and the interpolation theorem, we have where z η is as shown in (26) and for > 0 suitably small. Hence, Applying the Gronwall inequality to (41) gives (39) and (38) (by using the fact Theorem 4.4 shows that the family of processes {U f (t, τ )}, f ∈ Σ, is (X 1 × Σ, X 1 ) continuous, and the following translation identity holds: Theorem 4.5. Let Assumptions 2.1 be valid, with f ∈ Σ and 1 ≤ p < p * . Then (i) (Existence) for each f ∈ Σ, the non-autonomous dynamical system (U f (t, τ ), X 1 ) has a pullback exponential attractor M f = {M f (t)} t∈R , and the sections M f (t) are uniformly (w.r.t. f ∈ Σ and t ∈ R) bounded in X 2 ; (ii) (Stability) there exists a δ > 0 such that for any where C > 0, 0 < ν < 1 are some constants independent of f .
Proof. By estimate (10), We see from (44) that the ball is a uniformly (w.r.t. f ∈ Σ and τ ∈ R) pullback absorbing ball of the process {U f (t, τ )}. So for every bounded set B ⊂ X 1 , there exists a T 1 = T (B) > 0 such that In particular, there exists a T 0 > 0 such that For every τ ∈ R, f ∈ Σ, translation identity (42) implies that there exist f 1 , f 2 ∈ Σ such that So for any bounded set B ⊂ X 1 , that is, B is a uniformly (w.r.t. f ∈ Σ and τ ∈ R) absorbing set of the dynamical system (U f (t, τ ), For every f ∈ Σ, By estimate (13) (taking τ = t − 1, By the lower semi-continuity of weak limit we obtain that B is bounded in X 2 . Obviously, B is a metric space equipped with the distance d(x, y) = x − y X0 , we show that the dynamical system (U f (t, τ ), B, X 0 ) has a robust pullback exponential attractor. By estimate (44), For every ϕ, ψ ∈ B, f 1 , f 2 ∈ Σ and τ ∈ R, let (z(τ ), z t (τ )) = ϕ − ψ.