Global attractors of the degenerate fractional Kirchho ﬀ wave equation with structural damping or strong damping

: This article deals with the degenerate fractional Kirchho ﬀ wave equation with structural damping or strong damping. The well - posedness and the existence of global attractor in the natural energy space by virtue of the Faedo - Galerkin method and energy estimates are proved. It is worth mentioning that the results of this article cover the case of possible degeneration ( or even negativity ) of the sti ﬀ ness coe ﬃ cient. Moreover, under further suitable assumptions, the fractal dimension of the global attractor is shown to be in ﬁ nite by using (cid:2) 2 index theory.


Introduction
Let ⊂ Ω d (d is a positive integer) be a bounded domain with smooth boundary ∂Ω, we consider the following fractional Kirchhoff wave equation with damping: where { } = ∞ λ j j 1 is the set of eigenvalues of −Δ with homogeneous Dirichlet boundary, e j is the corresponding eigenfunction to λ j such that (see (1.17) below), ϕ and f are nonlinear scalar functions to be specified later, , and ( ) ⋅ ⋅ , and ‖⋅‖ stand for the inner product and norm of ( ) L Ω 2 , respectively.Problem (1.1) models several interesting phenomena studied in mathematical physics.One-dimensional model (1.1) with no damping term ( ) − u Δ α t and source term ( ) f u was introduced by Kirchhoff [1] to describe the transversal vibrations of a stretched string, and the model reads as follows: where L is the length of the string, < < x L 0 is the space coordinate, ≥ t 0 is the time, ( ) = u u x t , denotes the transverse displacement of the point x at the instant t, E is related to the intrinsic properties of the string (such as Young's modulus, the string cross-sectional area, and some of the other physical quantities), ρ is the mass density, h is the cross-section area, a 0 is the initial axial tension, and g is the external force.In this model, Kirchhoff used the integral term ( ) to present the average change in tension along the vibrating string taking into account the change of the string's length.Moreover, such models can be used in tension modulations for the sound synthesis and the control practice of mechanical systems, see, for example, [2] and [3].Further details and physical models described by Kirchhoff's classical theory can be found in [4][5][6][7][8][9][10].Since the previous study [1], the fitness and asymptotic properties of this type of models with different types of linear dissipation have been extensively studied, we refer to [11,12] for asymptotic stability, [12][13][14][15][16][17][18] for global existence and decay rate estimation, [12,16,[19][20][21][22] for blow-up of solutions, [12,23,24] for global attractors, and [12,25,26] for steady-state solutions.
With the assumptions ( [ ] ϕσ 65 ) and ( [ ] f 65 ), the author got the following results: ([ ] ) 65 wp Existence and uniqueness of weak solutions 65 ga Existence of a global attractor with finite fractal dimension in ( ( ) endowed with a partially strong topology with additional assumption that i.e., the system is non-degenerate.
In [66], the authors showed that the method used in study the asymptotically smooth in [65] can only be used when (1.10) is true; however, for the degenerate case, the method cannot apply (see [66, page 3] for details).
To study the degenerate case, Ma et al. [66] Global attractors of the degenerate fractional Kirchhoff wave equation  995 with assumption that possessing the following properties: and μ f is defined in (1.6).
and (1.8) holds.By using the method of Condition (C) (see [67]) and 2 index (see [68,Section 2.5]), the authors got the following results: ([ ] 66 ga ) Existence of a global attractor in ( ) ( ) fr ) The fractal dimension of the global attractor is infinite with ( ) ≡ g x 0 and suitable additional assumptions on ϕ and f .In view of papers [65] and [66], for the existence of global attractor, we have the following concerns: (P1) First, by comparing the assumptions is a constant, we know that there are much more restrictions on ϕ.Can we weaken those restrictions make it closer to the assumption ( [ ] ϕσ 65 )?
was assumed, the main reason is that when verifying Condition (C) the embedding ( ) ( ) needs to be compact.Can we weaken this assumption?
In view of the aforementioned introductions, in this article, we consider problem (1.3) with −Δ replacing by ( ) −Δ α and ( ) ≡ σ s 1, i.e., problem (1.1).We mainly deal with the problems (P1) and (P2) listed above: • First, for the existence of weak solutions, we make the following assumptions on ϕ and f , which is the counterpart of the assumptions in [65] with −Δ replaced by ( ) −Δ α and ( ) ≡ σ s 1: function such that ( ) = f 0 0 (without loss of generality) and (1.6) holds, and the following properties are valid: • Second, for uniqueness of weak solutions, we add the following further assumptions on ϕ: .
• Third, to obtain the existence of global attractor, we make the following additional assumptions on the function ϕ: where μ f is the constant given in (1.6), λ 1 is the first eigenvalue of −Δ with homogeneous Dirichlet boundary and Remark 2. When = α 1, comparing the aforementioned assumptions with the assumptions in [66], i.e., the assumptions ( [ ] ϕ 66 ) and ( [ ] f 66 ), our assumption is much more weaker, especially, the assumption (ii) of ) is eliminated and p can be taken equal to for ≥ d 3 in ( [ ] f 66 ).So problems (P1) and (P2) are partially solved, and the results of this article can be regarded as a generalization of the results obtained in [65] and [66].
At the end of this section, we introduce some notations, which will be used in this article.• Let X and Y be two Banach spaces, the notation ↪ X Y means X is continuously embedding in Y , and the notation ↪↪ X Y means X is compactly embedding in Y .The norm of a general Banach space X is denoted by ‖⋅‖ X .
• Let X be a Banach space, we denote by ′ X the dual space of X. • Let X be a Banach space with norm ‖⋅‖ X , [ ] ⊂ a b , , and = … m 0, 1, 2, .We denote by is the strong derivative of u of order k.We denote by , that are continuous with respect to weak topology on X. ( ) are classical L p spaces defined as sets of (classes of almost everywhere equal) strong Bochner- measurable functions ( ) f t with values in X such that ( ) p is a Banach space with the norm • For brevity, we use the following abbreviations: , is used for the notation of the H -inner product, and it is also used for the notation of duality pairing between dual spaces.• ⋯ C stands for positive constants depending on the quantities appearing in the bottom right corner.
Then A is a closed positive self-adjoint operator in H with domain Global attractors of the degenerate fractional Kirchhoff wave equation  997 be a complete orthonormal family of H , which is made of eigenfunctions of A: Ae λ e j e e δ i j i j (1.17) Following [69, Section 2.2.1], we can define the powers A k of A for all ∈ k : • For every > k 0, A k is an unbounded self-adjoint operator in H with a dense domain The operator A k is strictly positive and injective.The space V k is endowed with the scalar product and the norm which makes V k a Hilbert space and A k is an isomorphism from V k 2 onto H . Obviously, • For every > k 0, we define − V k as the dual of V k and can extend A k as an isomorphism from H onto − V k 2 .Alternatively, − V k can be endowed with the scalar product and the norm in (1.18) where k is replaced by −k.
• We obtain, finally, an increasing family of spaces V k , ∈ k , and for all , and , for all , and .
Each space is dense in the following one, the injection is continuous, and By using the aforementioned notations, we define the following fundamental Hilbert space: By using the notations introduced above, problem (1.1) can be equivalently rewritten as the following abstract form: The remaining of this article is organized as follows.In Section 2, we study the well-posedness and the additional regularity of the weak solution.In Section 3, we discuss the existence of global attractors.In Section 4, we investigate the fractal dimension of the global attractor.

Well-posedness
In this section, we study the well-posedness of solutions to problem (1.1), and the following lemmas are used frequently.where : 0, ; 0, ; : 0, ; .
x n n 1 be a bounded sequence and ( ) ∈ f C be a monotone increasing function.Then In this article, we mainly concern the weak solutions of problem (1.20), which are defined as follows: , and ∈ g H .A function u such that (2.4) for > d α 2 (see assumption (f ass1 )), we obtain ( ) for > d α 2 .So it follows from the above analysis that Global attractors of the degenerate fractional Kirchhoff wave equation  999 (ii) For any make sense.
The main result of this section is the following theorem.
Theorem 1.Let assumptions (ϕ ass1 ) and (f ass1 ) be in force, , where > R 0 is a constant.Then for every > T 0, problem (1.20) has a weak solution , and (2.3) holds.Moreover, there exists a constant C R T , such that 0, ; 0, ; 0, ; 0, ; , (2.5) , ; , and there exists a constant C R T , such that , ; , ; , ; , (2.6) and it holds where Here, s s 0 0 (2.10) In addition, (i) if ϕ is nonnegative the "≤" in (2.8) can be changing to "=", which, together with (2.7), implies the following energy equality: , where > R 0 is a constant, then for every > t 0 and (2.12) Proof.For every > η 0 we introduce the following energy-type function ( ) Let η 0 be the nonnegative constant given in (ϕ ass1 ), then it follows from (1.12) that the infimum exists, which is denoted by ϱ, i.e., Then, For all ∈ + ν and > η 0, let For every ≥ η η 0 , we can find ≥ ν 0, which depends on η and μ f , a positive constant a 1 , and a monotonic positive function ( ) G s such that where μ f is the constant defined in (1.6).We refer to the Appendix for the proof of (2.18).
In the following, we divide the proof into several steps.We denote by C a positive constants indepen- dent of m, which may change from line to line.
Step 1: Existence of weak solutions.Let { } = ∞ e i i 1 be defined in Section 2, we seek for the approximate solutions of the form ( ) ( ) where , . In view of ∈ g H , and the assumptions (ϕ ass1 ) and (f ass1 ), by standard theory of ordinary differential equations,

21)
Global attractors of the degenerate fractional Kirchhoff wave equation  1001 where , it is easy to see Then, in view of ( ) , we can assume that for some positive constant C R depending only on R.
By multiplying both sides of (2.21) by u mt and integrating over Ω, we obtain where ( ) ⋅ ⋅ E , is defined in (2.9).By multiplying both sides of (2.21) by u m and integrating over Ω, we obtain . .
Due to (2.9), (2.23), (2.24), (A.8), and (2.29), we obtain Then the above analysis and (2.20) imply , In view of (2.30), integrating the above inequality from 0 to T , it follows By multiplying both sides of (2.21) by A u α m and integrating over Ω, it follows from the definition of operator A and (2.29) that . (2.32) Global attractors of the degenerate fractional Kirchhoff wave equation  1003 For the term |( ( ) m , in view of the definition of the operator A, it holds (2.33) Then we have the following estimate (note ≤ ≤ α 1 1 2 ): (2.29)), we obtain from (A.4) and Hölder's inequality that , we can choose > ε 0 small enough such that , then we obtain ), we obtain from (1.15) and (2.33) that , we obtain from (1.15) and (2.33) that (see the assumptions on p in (f ass1 )), it follows that , we obtain from In view of the above analysis, ∈ g H , and (2.32), we obtain . (2.34) .
Then it follows from (2.42) that .
By multiplying both sides of (2.41) by − A v α mt , we obtain Then, by Cauchy-Schwartz's inequality, we obtain By similar proof to (2.43), we have 2, , it follows from the above analysis and Cauchy's inequality that We have, by Cauchy-Schwartz's inequality and Young's inequality, Using (2.44) and (2.45) we have that (2.47) Then, by similar proof of (2.39), it follows from (2.46) that and For any ( ) ∈ a T 0, , it follows from (2.47), (2.46), (2.49) , Then, in view of (2.49), we obtain that  for any > q 1.
Then we obtain from (10) (2.51) Moreover, we obtain from Then we obtain Similarly, it follows from (4) and (2.29) that Therefore, we obtain from the above two inequalities, weak lower semicontinuity of norms, (1), ( 5), ( 7 In fact, by (2.29), (2.5), and (10), it follows from Hölder's inequality that With the above preparations, now we show the function u got above is a weak solution.For any

ψ t e ϕ u A u ψ t e f u ψ t e t g ψ t e t i m
, , , , .
, we obtain from the above analysis that , by similar proof to Remark 3, it follows, ).Then integrating the above equality from s to t, it follows So, in order to show (2.8), we only need to show (2.57) For any < < s T 0 , by (2.24), we have Consequently, we obtain from (2.9) and (2.59) that , by the same proof as above, we obtain (2.61) (12)) and ( ) , and which are denoted by σ 1 , σ 2 , and σ 3 , respectively, i.e., By the property of the limit inferior, there exists a sequence (2.66) x Ω.

Existence of the global attractor
In this section, we will study the existence of global attractor for problem (1.20).First, we recall some definitions and results related to the global attractor.More details can be found in [69].
Definition 2. The pair ( ) X S , t with X being a complete metric space is said to be a continuous dynamical system if , t be a continuous dynamical system, where X is complete metric space with metric d.
, t is said to be (bounded) dissipative if it possesses a bounded absorbing set B, and a closed set ⊂ B X is said to be absorbing for S t if for any bounded set ⊂ D X there exists ( ) , t is said to be asymptotically smooth if for every bounded set D such that ⊂ S D D t for > t 0 there exists a compact set K in the closure D of D, such that S D t converges uniformly to K in the sense that The following theorem is used to study the dissipativity.
, t be a continuous dynamical system in some Banach space X with norm ‖⋅‖ X .Assume that (i) there exists a continuous function ( ) W x on X possessing the properties where ϕ i are continuous functions on where the constant * R depends on the functions ϕ 1 and ϕ 2 and the χ only.
By Theorem 1, we obtain the following corollary.
Corollary 1.Let the assumptions (ϕ ass1 ), (ϕ ass2 ), (ϕ ass3 ), and , where > R 0 is a constant.Then for every > T 0, problem (1.20) has a unique weak solution u on [ ] T 0, such that , and there exists a constant C R T , such that , ; , ; , ; , and it holds the following energy equality where E is defined in (2.9) and (2.10).Moreover, if ( ) u t and ( ) w t are two weak solutions with initial data , where > R 0 is a constant, then for every > t 0 and Proof.We will prove this theorem by using Theorem 2, and the proof is divided into two steps.

ϕσ 65 )
The damping σ and the stiffness ϕ are C 1 functions on the semi-axis [ )

1 . 2
10) and note ≡ g 0 and Assumption 5.1) be given in (1.17).Then, by(1.18),Because V α m is m-dimensional, all norms on V α m are equivalent.In particular, there exists a constant > ϖ 0 such that m is the boundary of the unit ball B m of m , i.e., is a bounded symmetric neighborhood of the origin in m .By Proposition 1, we obtain B

3 ) 5 )
On the other hand, by assumption (f ass1 ), we obtain when ≥ d 2, there exist positive constants C ˆ1 and C ˆ1Then there exist positive constants C ˜1 and C ˜2 such that
t 0 is a family of continuous mappings of X into itself and it satisfies the semigroup property: = S I [58]following theorem is used to study the existence of global attractor, which can be found in[58].Global attractors of the degenerate fractional Kirchhoff wave equation  1015 and bar over a set means the closure in X.Furthermore, ∈ ω if and only if there exist sequences { } ⊂ 2 2.
In view of Theorem 3, it follows from (3.11), (3.12), and the above ordinary differential inequality, the dynamical system ( ) S , t is dissipative and has a bounded absorbing set * holds for all > ≥ t s 0, where (see (2.9) and (2.