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

1 Introduction

Let Ω ⊂ R 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 the parameter α satisfies 1 / 2 ≤ α ≤ 1 , Δ is the Laplace operator, and the fractional Laplace operator ( − Δ ) β β = α or β = α 2 is defined as

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, g ∈ L 2 ( Ω ) , and ( ⋅ , ⋅ ) and ‖ ⋅ ‖ stand for the inner product and norm of L 2 ( Ω ) , respectively.

Problem (1.1) models several interesting phenomena studied in mathematical physics. One-dimensional model (1.1) with no damping term ( − Δ ) α u 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, 0 < x < L 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 ∫ 0 L ( u x ) 2 d x 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.

System (1.1) is said to be non-degenerate if ϕ satisfies the strict hyperbolicity condition ϕ ( s ) ≥ a for some constant a > 0 , while degenerate if ϕ just satisfies the degenerate hyperbolicity condition ϕ ( s ) ≥ 0 for all s ≥ 0 . Moreover, the term ( − Δ ) α u t is called strong, structural, or weak damping if α = 1 , α ∈ ( 0 , 1 ) , or α = 0 , and the strong damping models the flow of viscoelastic fluids as well as in the theory of heat conduction, see [27,28,29,30]; the structural damping model waves propagate through a lossy media, for example, fractal rock layers, human tissues, and different biomedical materials, see [31,32]; the weak damping models various oscillatory processes in many areas of modern mathematical physics including electrodynamics, quantum mechanics, nonlinear elasticity, see [33,34,35]. Moreover, partial differential equations with fractional Laplace operator ( − Δ ) α , 0 < α < 1 has gained great attention. This type of operator arises in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes [36,37,38]. We refer the readers to [39,40,41,42,43] for some books on this topic, [41,44] for the fractional Sobolev spaces, [45,46,47,48,49,50] for fractional Laplace and p -Laplace equations, [19,25,51,52,53,54] for fractional Kirchhoff equations, and [26,55,56] for fractional Choquard equations .

Now we review some related studies. First we recall the following Kirchhoff wave model with damping

where θ ∈ [ 0 , 1 ] is a constant. Well-posedness and global attractor for the aforementioned problem have been studied intensively in the past years, we refer to [6,13,16, 57,58,59, 60,61] and references therein, but most of the papers dealt with the case θ = 0 or θ = 1 , σ ( s ) ≡ σ 0 > 0 is a constant, and ϕ ( s ) = ϕ 0 + ϕ 1 s γ , where γ > 0 and ϕ 1 > 0 are constants, and ϕ 0 ≥ 0 (degenerate and non-degenerate cases) for well-posedness, ϕ 0 > 0 (non-degenerate case) for global attractor. For θ ∈ ( 0 , 1 ] , problem (1.2) was studied in [63] with σ ( s ) ≡ σ 0 > 0 is a constant and f ( u ) ≡ 0 . The main result in that article states the existence of weak solutions for θ ∈ ( 0 , 1 / 2 ] with initial data u 0 ∈ H max { 1 / 4 , θ / 2 } ( Ω ) and u 1 ∈ L 2 ( Ω ) and their uniqueness for θ ∈ [ 1 / 2 , 1 ] . For θ < 1 and f ( u ) ≢ 0 , recently, Li and Yang [64] considered (1.2) for non-degenerate case for θ ∈ [ 1 / 2 , 1 ) , and they proved the well-posedness and longtime dynamics such as optimal global attractor, optimal exponential attractor, and upper semi-continuous of the global attractors with respect to θ .

For degenerate case, Chueshov [65] considered problem (1.2) with θ = 1 , i.e.,

with the assumptions that

( ϕ σ [ 65 ] ) The damping σ and the stiffness ϕ are C 1 functions on the semi-axis R + = [ 0 , ∞ ) . Moreover, σ ( s ) > 0 for all R + and there exist η 0 ≥ 0 and c i ≥ 0 ( i = 1 , 2 ) such that

and

( f [ 65 ] ) f is a C 1 ( R + ) function such that f ( 0 ) = 0 (without loss of generality),

and the following properties are valid:

1. if d = 1 , then f is arbitrary;

2. if d = 2 , then there exist constants C > 0 and p ≥ 1 such that

   (1.7) ∣ f ′ ( s ) ∣ ≤ C ( 1 + ∣ s ∣ p − 1 ) ;

3. if d ≥ 3 , then there exist constants C > 0 and p ∈ 1 , d + 2 d − 2 such that

   (1.8) ∣ f ′ ( s ) ∣ ≤ C ( 1 + ∣ s ∣ p − 1 ) ,

   or else there exist constants C i > 0 ( i = 0 , 1 , 2 ) and p ∈ d + 2 d − 2 , d + 4 ( d − 4 ) + such that

   (1.9) C 0 ∣ u ∣ p − 1 − C 1 ≤ f ′ ( u ) ≤ C 2 ( 1 + ∣ u ∣ p − 1 ) ,

   where s + = ( s + ∣ s ∣ ) / 2 .

With the assumptions ( ϕ σ [ 65 ] ) and ( f [ 65 ] ), the author got the following results:

( [ 65 ] wp ) Existence and uniqueness of weak solutions u ∈ C ( 0 , T ; H 0 1 ( Ω ) ∩ L p + 1 ( Ω ) ) with u t ∈ C ( 0 , T ; L 2 ( Ω ) ) ;

( [ 65 ] ga ) Existence of a global attractor with finite fractal dimension in ( H 0 1 ( Ω ) ∩ L p + 1 ( Ω ) ) × L 2 ( Ω ) 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] considered problem (1.3) with σ ( x ) ≡ σ 0 > 0 as a constant, i.e.,

with assumption that

( ϕ [ 66 ] ) the function ϕ ∈ C 1 ( R + ) possessing the following properties:

1. ϕ ( s ) ≥ min { L 1 s γ , L 2 } , where γ ≥ 0 , L 1 , L 2 > 0 are constants;

2. μ ˆ ϕ λ 1 + μ f > 0 , where μ ˆ ϕ = liminf s → ∞ ϕ ( s ) > 0 and μ f is defined in (1.6).

By using the method of Condition (C) (see [67]) and Z 2 index (see [68, Section 2.5]), the authors got the following results:

( [ 66 ] ga ) Existence of a global attractor in H 0 1 ( Ω ) × L 2 ( Ω ) ;

( [ 66 ] 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:

1. First, by comparing the assumptions ( ϕ [ 66 ] ) and ( ϕ σ [ 65 ] ) with σ ( x ) ≡ σ 0 > 0 is a constant, we know that there are much more restrictions on ϕ . Can we weaken those restrictions make it closer to the assumption ( ϕ σ [ 65 ] )?

2. Second, when d ≥ 3 , in assumption ( f [ 66 ] ), f satisfying (1.8) with p ∈ 1 , d + 2 d − 2 was assumed, the main reason is that when verifying Condition (C) the embedding H 1 ( Ω ) ↪ L p + 1 ( Ω ) 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 :

  1. ϕ is a C 1 ( R + ) function. Besides, for every s ∈ R + , there exist η 0 ≥ 0 and c i ≥ 0 ( i = 1 , 2 ) such that

     (1.12) ∫ 0 s ϕ ( ξ ) d ξ + η 0 s → ∞ as s → ∞

     and

     (1.13) ϕ ( s ) s + c 1 s ≥ − c 2 for s ∈ R + .

  2. f is a C 1 ( R + ) function such that f ( 0 ) = 0 (without loss of generality) and (1.6) holds, and the following properties are valid:

     1. if d < 2 α , then f is arbitrary;

     2. if d = 2 α , then there exist constants C > 0 and p ≥ 1 such that

        (1.14) ∣ f ′ ( s ) ∣ ≤ C ( 1 + ∣ s ∣ p − 1 ) ;

     3. if d > 2 α , then there exist constants C > 0 and p ∈ 1 , d + 2 α d − 2 α such that

        (1.15) ∣ f ′ ( s ) ∣ ≤ C ( 1 + ∣ s ∣ p − 1 ) .

• Second, for uniqueness of weak solutions, we add the following further assumptions on ϕ :

  1. ϕ is nonnegative and ∫ a 1 a 2 ϕ ( s ) d s > 0 for any 0 ≤ a 1 < a 2 < ∞ .

• Third, to obtain the existence of global attractor, we make the following additional assumptions on the function ϕ :

  1. lim r → ∞ ϕ ( r ) = ∞ or lim r → ∞ ϕ ( r ) = ς and

     ς > max ∣ μ f − 2 ∣ λ 1 − α , 1 2 ( ∣ μ f − 2 ∣ λ 1 − α + η λ 1 − α + η ) ,

     where μ f is the constant given in (1.6), λ 1 is the first eigenvalue of − Δ with homogeneous Dirichlet boundary and

     (1.16) η ≔ λ 1 2 α 2 ( ∣ μ f − 2 ∣ + λ 1 2 α ) ∈ ( 0 , 1 ) .

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 ] ⊂ R , and m = 0 , 1 , 2 , … . We denote by C m ( a , b ; X ) ≡ C m ( [ a , b ] ; X ) the space of m -differentiable (in the norm topology) functions on [ a , b ] with values in X . If [ a , b ] is a finite interval, then C m ( a , b ; X ) equipped with the norm

  ‖ u ‖ C m ( a , b ; X ) = max { ∥ u ( k ) ( t ) ∥ X : t ∈ [ a , b ] , k = 0 , 1 , … , m }

  becomes a Banach space. Here u ( k ) ( t ) = ∂ t k u ( t ) is the strong derivative of u of order k . We denote by C m ( [ a , b ) ; X ) the space of functions u : [ a , b ) → X such that u ∈ C m ( [ a , b ′ ] ; X ) for any b ′ ∈ ( a , b ) . A similar meaning has the notation C m ( ( a , b ] ; X ) and C m ( ( a , b ) ; X ) . We also use the notation C w ( a , b ; X ) for the space of the functions on [ a , b ] that are continuous with respect to weak topology on X . L p ( a , b ; X ) , 1 ≤ p ≤ ∞ 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 ‖ f ( ⋅ ) ‖ X ∈ L p ( a , b ; R ) . Each L p ( a , b ; X ) is a Banach space with the norm

  ‖ f ‖ L p ( a , b ; X ) = ∫ a b ‖ f ( t ) ‖ X p d t 1 / p , if 1 ≤ p < ∞ , esssup { ‖ f ( t ) ‖ X : t ∈ [ a , b ] } , if p = ∞ .

• For brevity, we use the following abbreviations:

  L p = L p ( Ω ) , H s = H s ( Ω ) , H = L 2 , V 1 = H 0 1 , V − 1 = H − 1 , ‖ ⋅ ‖ = ‖ ⋅ ‖ L 2 , ‖ ⋅ ‖ p = ‖ ⋅ ‖ L p

  with p ≥ 1 and s ∈ R , H 0 = H , H s is the L 2 -based Sobolev spaces for s > 0 , H 0 s are the completion of C 0 ∞ ( Ω ) in H s for s > 0 , and H − s = ( H 0 s ) ′ for s > 0 .

• ( ⋅ , ⋅ ) 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.

Let { e j } j = 1 ∞ ⊂ C 0 ∞ ( Ω ) be a complete orthonormal family of H , which is made of eigenfunctions of A :

Following [69, Section 2.2.1], we can define the powers A k of A for all k ∈ R :

• For every k > 0 , A k is an unbounded self-adjoint operator in H with a dense domain

  V 2 k = D ( A k ) = { ϕ ∈ H : A k ϕ ∈ H } ,

  and

  A k ϕ = ∑ j = 1 ∞ λ j k ( ϕ , e j ) e j .

  The operator A k is strictly positive and injective. The space V k is endowed with the scalar product and the norm

  (1.18) ( ϕ , ψ ) V k = A k 2 u , A k 2 v = ∑ j = 1 ∞ λ j k ( ϕ , e j ) ( ψ , e j ) , ‖ ϕ ‖ V k = A k 2 ϕ = ∑ j = 1 ∞ λ j k ( ϕ , e j ) 2 1 2 ,

  which makes V k a Hilbert space and A k is an isomorphism from V 2 k onto H . Obviously,

  ‖ ϕ ‖ V 2 = ‖ A ϕ ‖ = ‖ Δ ϕ ‖ , ‖ ϕ ‖ V 1 = A 1 2 ϕ = ‖ ∇ ϕ ‖ .

• 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 − 2 k . 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 ∈ R , and

  V k 1 ↪ V k 2 for all k 1 , k 2 ∈ R and k 1 ≥ k 2 , V k 1 ↪ ↪ V k 2 for all k 1 , k 2 ∈ R and k 1 > k 2 .

  Each space is dense in the following one, the injection is continuous, and A k 1 − k 2 is an isomorphism of V 2 k 1 into V 2 k 2 for all k 1 , k 2 ∈ R satisfying k 1 > k 2 .

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.

2 Well-posedness

In this section, we study the well-posedness of solutions to problem (1.1), and the following lemmas are used frequently.

In this article, we mainly concern the weak solutions of problem (1.20), which are defined as follows:

The main result of this section is the following theorem.