The Global and Exponential Attractors for the Higher-order Kirchho ff-type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: utt + (−∆)ut + ( α + β ∥∇mu∥2 )q (−∆)mu + g(u) = f (x). At first, we do priori estimation for the equations to obtain two lemmas and prove the existence and uniqueness of the solution by the lemmas and the Galerkin method. Then, we obtain to the existence of the global attractor in Hm 0 (Ω) × L2(Ω) according to some of the attractor theorem. In this case, we consider that the estimation of the upper bounds of Hausdorff for the global attractors are obtained. At last, we also establish the existence of a fractal exponential attractor with the non-supercritical and critical cases.


Introduction
In this paper, we are concerned with the existence of global attractor for the following nonlinear Higher-order Kirchhofftype equations: where m > 1 is an integer constant, α > 0, β > 0 are constants and q is a real number.Moreover, Ω is a bounded domain in R n with the smooth boundary ∂Ω and v is the unit outward normal on ∂Ω.g(u) is a nonlinear function specified later.
It is known that Kirchhoff (1883) first investigated the following nonlinear vibration of an elastic string for δ = f = 0: (1.4) where u = u(x, t) is the lateral displacement at the space coordinate x and the time t, ρ the mass density, h the cross-section area, L the length, E the Young modulus, p0 the initial axial tension, δ the resistance modulus, and f the external force.
When α = 0, β = 1 and q > 0 are real number, Yunlong Gao, Yuting Sun and Guoguang Lin (2016) studied existence of weak solutions for degenerate High-order Kirchhoff equations: where m > 1 is an integer constant.Ω is a bounded domain in R n with the smooth boundary ∂Ω and v is the unit outward normal on ∂Ω.g(u) is a nonlinear function specified later.
When α = 0, β = 1, m = 1, g(u) = −|u| α u and q = γ > 0 is real number, Kosuke Ono (1997) had studied global existence, asymptotic stability and blowing up of solutions for Some Degenerate Non-linear Wave Equations: where Ω is a bounded domain in R n with the smooth boundary ∂Ω. When = 0 and no linear damping, Marina Ghisi and Massimo Gobbino (2009) studied spectral gap global solutions for degenerate Kirchhoff equations.Given a continuous function m : [0, +∞) → [0, +∞), they consider the Cauchy problem: where Ω ⊆ R n is an open set and ∇u and ∆u denote the gradient and the Laplacian of u with respect to the space variables.They prove that for such initial data (u 0 , u 1 ) there exist two pairs of initial data (ū 0 , ū1 ), (û 0 , û1 ) for which the solution is global, and such that u 0 = ū0 + û0 , u 1 = ū1 + û1 .
When m = 1, Yang Zhijian, Ding Pengyan and Lei Li (2016) studied Longtime dynamics of the Kirchhoff equations with fractional damping and supercritical nonlinearity: where α ∈ ( 1 2 , 1), Ω is a bounded domain R N with the smooth boundary ∂Ω , and the nonlinearity f (u) and external force term g will be specified.The main results are focused on the relationships among the growth exponent p of the nonlinearity f (u) and well-posedness.They show that (i)even if p is up to the supercritical range, that is, 1 ≤ p < N+4α (N−4α) + , the wellposedness and the longtime behavior of the solutions of the equation are of the characters of the parabolic equation; (ii) when N+4α (N−4α) + ≤ p < N+4 (N−4) + , the corresponding subclass G of the limit solutions exists and possesses a weak global attractor.
When m = 1, ) q is replaced σ(∥∆u∥ 2 ), Yang Zhijian, I.Chueshov ( Yang, Z. J. & et al., 2014;Zhijian Yang & Zhiming Liu., 2015;Igor Chueshov., 2012) studied the Global attractor and exponential attractors for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity: where Ω is a bounded domain in R N with the smooth boundary ∂Ω, σ(s), ϕ(s) and f (s) are nonlinear functions, and h(x) is an external force term.They prove that in strictly positive stiffness factors and supercritical nonlinearity case, there exists a global finite-dimensional attractor in the natural energy space endowed with strong topology.
For the most of the scholars represented by Yang Zhijian have studied all kinds of low order Kirchhoff equations and only a small number of scholars have studied the blow-up and asymptotic behavior of solutions for higher-order Kirchhoff equation.So, in this context, we study the high-order Kirchhoff equation is very meaningful.In order to study the high-order nonlinear Kirchhoff equation with the damping term, we borrow some of Li Yan's (Ball, J. M., 1997) partial assumptions (2.1)-(2.3)for the nonlinear term g in the equation.In order to prove that the lemma 2.4, we have improved the results from assumptions (2.1)-( 2.3) such that 0 < C 2 ≤ 1 2 .Then, under all assumptions, we prove that the equation has a unique smooth solution (u, u t ) ∈ L ∞ ((0, +∞); H 2m (Ω) ∩ H m 0 (Ω) × H m 0 (Ω)) and obtain the solution semigroup S (t) : has global attractor A and the upper bounds of Hausdorff dimensions.At last, we get the exponential attractor by strong quasi-stability.

Preliminaries
In this section, we introduce material needed in the proof our main result.We use the standard Lebesgue space L p (Ω) and Sobolev space H m (Ω) with their usual scalar products and norms.Meanwhile we define and introduce the following abbreviations: for any real number p > 1.

The Existence and Uniqueness of Solution
Theorem 3.1.Assume (H 1 ) − (H 4 ) hold, and (u 0 , u 1 ) ) . (3.1) Proof.By the Galerkin method, Lemma 2.4.and Lemma 2.5., we can easily obtain the existence of Solutions.Next, we prove the uniqueness of Solutions in detail.
By (H 4 ), Lemma 2.2., Lemma 2.4.and Lemma 2.5., we obtain where C 8 > 0 is constant.From the above, we have d dt (3.10)By using Gronwall's inequality for (3.10), we obtain (3.11)where Hence , we can get So, we get the uniqueness of the solution.
3) When t > 0, S (t) is a completely continuous operator.Therefore, the semigroup operator S(t) exists a compact global attractor A.
Theorem 3.3.Under the assume of Lemma 2.4., Lemma 2.5.and Theorem 3.1.,equations have global attractor (3.17)where Proof.Under the conditions of Theorem 3.1., it exists the solution semigroup S(t),S (t) : (1) From Lemma 2.4. to Lemma 2.5., we can get that ∀B ⊂ H 2m (Ω) So we get B 0 is the bounded absorbing set. ( is compact embedded, which means that the bounded set in E 1 is the compact set in E 0 , so the semigroup operator S(t) exists a compact global attractor A.

Differentiability of the Semigroup
In order to estimate dimensions, we suppose: (H 5 ) for every M > 0, there exist k = k(M), such that: The inner product and the norm in E 0 space are defined as follows: where (4.5) Proof.By (4.2)-(4.6),we get ) . (4.8) By using hölder inequality, Young's inequality and Poincaré inequality, we deal with the terms in (4.8) by as follows: and substituting (4.9)-(4.10)into (4.8),we obtain The proof of Lemma 4.1.1 is completed.
We can get the Lipchitz property of S (t) on the bounded sets of E 0 , that is Taking the scalar product of each side of (4.16) with θ t .Because of By (H 5 ), we have By (4.19)-(4.20)and Young's inequality, we have d dt ) . (4.21) By the Gronwall's inequality and (4.15), we get where C 16 , C 17 , C 18 , C 19 > 0.

The Upper Bounds of Hausdorff Dimensions for the Global Attractor
Consider the first variation of (4.4) with initial condition: where Ψ = (U, V) T ∈ E 0 , V = U t + εU and φ = (u, v) T ∈ E 0 is a solution of (4.3), It is easy to show from Lemma 4.1.2that (4.24) is a well-posed problem in E 0 , the mapping S ε (τ) : } is Fréchet differentiable on E 0 for any t ≥ 0, its differential at φ = (u 0 , u 1 + εu 0 ) T is the linear operator on E 0 , (ξ, ζ) T → (U(t), V(t)) T , where (U(t), V(t)) T is the solution of (4.24).
Lemma 4.2.1. (T eman,R.,1998)For any orthonormal family of elements of where Proof.This is a direct consequence of Lemma VI 6.3 of [17].