Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping

In this paper, we obtain the existence of pullback attractors for nonautonomous Kirchhoff equations with strong damping, which covers the case of possible generation of the stiffness coefficient. For this purpose, a necessary method via “the measure of noncompactness” is established.


Introduction
Let Ω ⊂ ℝ n be a bounded domain with smooth boundary ∂Ω. We consider the following Kirchhoff wave model with strong damping: where hðx, tÞ is a time-dependent external force term, u 0 τ and u 1 τ are initial data, and ϕ and f are nonlinear functions specified later.
To describe small vibrations of an elastic stretched string, Kirchhoff [1] introduced the equation where u = uðx, tÞ is the lateral deflection, 0 < x < L the space coordinate, t ≥ 0 the time, E the Young's modulus, ρ the mass density, h the cross-section area, L the length, p 0 the initial axial tension, and g the external force. It has been called the Kirchhoff equation since then. In general, we call the Kirchhoff equation nondegenerate if the stiffness ϕ sat-isfies the strict hyperbolicity condition ϕðsÞ ≥ c > 0 and degenerate if ϕðsÞ ≥ 0 on ℝ + . Obviously, the degenerate stiffness coefficient ϕðsÞ in (1) corresponds to the case that the initial axial tension equals zero.
Introducing the strong damping term −Δu t provides an additional a priori estimate. Certainly, from the physical point of view, the dissipative plays an important spreading role for the energy gathered arising from the nonlinearity in a real process. Concerning Kirchhoff equations with strong dissipation, the first result on the well-posedness we are aware of was obtained by Nishihara [16]. He proved the global existence of the solution for the model u tt − Δu t − mðk∇ukÞΔu = 0. In recent years, many mathematicians and physicists paid their attentions to this type of problem and obtained the well-posedness under different types of hypotheses, such as the absent source term [17] and the subcritical source term [18][19][20][21][22][23]. In general, the exponent p * = n + 2/ðn − 2Þ + is called to be critical when someone studies the problem in H 1 0 ðΩÞÞ × L 2 ðΩÞ. Assuming the stiffness factor is nondegenerate (ϕðsÞ ≥ ϕ 0 > 0), References [18][19][20][21][22][23][24] also proved the existence of the attractor. In the case of possible degeneration of the stiffness coefficient and the case of supercritical source term (p * < p < ðn + 4Þ/ðn − 4Þ + ), the first result on the well-posedness we are aware of is given by Chueshov [25]. However, when he proved the existence of a global attractor for problem (1) in the natural energy space ðH 1 0 ðΩÞ ∩ L p+1 ðΩÞÞ × L 2 ðΩÞ endowed with a partially strong topology (in the sense, if ðu n 0 , u n 1 Þ ⟶ ðu 0 , u 1 Þ with a partially strong topology, then ðu n 0 , u n 1 Þ ⟶ ðu 0 , u 1 Þ strongly in H 1 0 × L 2 and u n 0 ⇀ u 0 weakly in L p+1 ), he assumed that Under this condition, one can conclude that ϕð k∇uðtÞk 2 Þ ≥ c 0 > 0 if ∥∇uðtÞ∥ is bounded for t ∈ ℝ + . Recently, Ma et al. [26] proved the existence of the global attractor in the case of degeneration for the autonomous Kirchhoff system.
In this paper, we consider the problem (1) under the degenerate hyperbolicity condition ϕðsÞ ≥ 0. We do not assume that ϕ is monotone and allow ϕð0Þ = 0, such as ϕðs Þ = bs γ (degenerate and monotone) or ϕðsÞ = ð1 + sin 2 sÞs γ (degenerate and nonmonotone) with γ ≥ 1. Based on the result in [25,26], we prove the existence of pullback attractors in H 1 0 ðΩÞÞ × L 2 ðΩÞ if ϕ is really degenerate. To overcome the difficulties caused by the degeneration, we first established a method (condition (D-PC)) via "the measure of noncompactness" (some ideas come from [35,36]) to prove that the process is pullback D-asymptotically compact.
The paper is organized as follows. In Section 2, we introduce some preliminaries and establish a necessary abstract result (see Theorem 5). In Section 3, we discuss the existence of pullback attractors for the equation (1) (see Theorem 12).

Preliminaries
In this section, we will give some notations and results. As usual, we denote by ∥·∥ and ð·, · Þ the norm and the inner product in L 2 ðΩÞ, respectively. Let H = H 1 0 ðΩÞ × L 2 ðΩÞ. We define the norms in H by ku 0 , u 1 k 2 H = k∇u 0 k 2 + ku 1 k 2 : Let X be a Banach space and Uðt, τÞ be a process acting on X. In the following, we recall some definitions and results related to the pullback attractors; more details can be found in [27,29,33]. Definition 3. A process Uð·, · Þ is said to be pullback D -asymptotically compact in X, if for any t ∈ ℝ, any sequences τ n ⟶ ∞ and x n ∈ Dðt − τ n Þ; the sequence fUðt, t − τ n Þx n g n∈ℕ is relatively compact in X.
To verify the pullback D-asymptotically compact property in X, it suffices to check the following condition.

Theorem 5.
Let the family D = fDðtÞg t∈ℝ be a pullback D -absorbing family of the process Uðt, τÞ. If the D-pullback condition (D -PC) holds, then Uð·, · Þ is pullback D-asymptotically compact in X.
Proof. By Definition 3, the result will be proven if we can show that for any t ∈ ℝ, any sequences τ n ⟶ ∞ and x n ∈ Dðt − τ n Þ, fUðt, t − τ n Þx n g n∈ℕ is relatively compact in X.

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where γ is the measure of noncompactness defined as γ B ð Þ = inf δ > 0 | B admits a finite cover by sets whose diameter ≤ δ f g : On the other hand, the properties of D give that there Then, we can find N 0 , such that γð S n>N 0 Uðt, t − τ n Þx n ÞÞ ≤ 2δ, which means that fUðt, t − τ n Þx n g n∈ℕ has a finite 4δ -net for any δ > 0. The proof is complete.

Existence of Pullback Attractors
In this section, we will prove the existence of the pullback attractor when ϕðsÞ is really degenerate and f ðuÞ is subcritical. We assume that f , ϕ, and h satisfy the following conditions.
Assumption 7. f ðuÞ is a C 1 function, f ð0Þ = 0, f ′ðsÞ ≥ −c 1 , and s ∈ ℝ, and the following properties hold: where c 1 and C are positive constants and λ 1 is the first eigenvalue of −Δ.
(1) ϕðsÞ = L 1 s α or e ϕðsÞ = ð1 + sin 2 sÞs α ðα ≥ 1Þ satisfies Assumption 6. It indicates that we include into the consideration the case of possibly degenerate ϕ since If α = 0, then ϕðsÞ is a constant, and equation (1) is the nonlinear wave equation with strong damping.
(2) Assumptions 6 and 7 imply that there exist constants where FðsÞ = Ð s 0 f ðtÞdt. The well-posedness of the problem has been established by Chueshov [25] in the autonomous case. Noticing that the conditions of ϕ, f are more general than the above Assumptions 6-8, we can obtain the following Proposition 10 by a similar argument as in [25], except for the treatment of hðx, tÞ. The reader is referred to the Appendix for a detailed proof of these facts. Proposition 10. Let Assumptions 6-8 be in force. Then, for τ, T ∈ ℝðτ < TÞ and ðu 0 τ , u 1 τ Þ ∈ H , problem (1) has a unique weak solution u with ðu, u t Þ ∈ Cð½τ, T ; H Þ and (1) for every t ∈ ½τ, T, there exists C = C R,τ,T > 0 such that where

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(3) the Lipschitz stability holds for zðtÞ = u 1 ðtÞ − u 2 ðtÞ, where u 1 , u 2 are two weak solutions of problem (1) We define the solution operator Uðt, τÞ: H ⟶ H associated to problem (1) as where u is the weak solution of problem (1) corresponding to initial data ðu 0 τ , u 1 τ Þ ∈ H . Then, we know from Proposition 10 that Uðt, τÞ: H ⟶ H is a continuous evolution process. For convenience, we denote by ξ u ðtÞ = ðuðtÞ, u t ðtÞÞ for any function uðtÞ. As ðuðτÞ, u t ðτÞÞ = ðu 0 τ , u 1 τ Þ, we also denote ðu 0 τ , u 1 τ Þ by ξ u ðτÞ. Proof. As usual, the argument below can be justified by considering Galerkin approximations. Using the multiplier u t + ηu in Equation (1), we have that where for η > 0 which is small enough, κ > 0 is a positive constant, and κ, C 3 are independent of ξ u ðtÞ.
Case 2. There exist ðu 0 t 0 −τ 0 , u 1 t 0 −τ 0 Þ ∈ Dðt 0 − τ 0 Þ and t 1 ∈ ½t 0 − τ 0 + 1, t 0 such that In this case, we claim that the following inequality is true, i.e., for every t 1 ≤ t ≤ t 0 , In fact, if this claim is not true, the continuity of k∇ u 2 ðtÞk gives that is not an empty set. Let t 3 = inf E. It is easy to prove that k∇u 2 ðt 3 Þk = 2ε. Moreover, by the definition of t 3 , we have that According to the intermediate value theorem, we know that the set 6 Advances in Mathematical Physics is not empty. Denoting t 2 = sup E 1 , we can conclude from the definition of supremum that Thus, Notice that k∇uk ≥ k∇u 2 k and k∇u 2 k ≤ 1 for t ∈ ½t 2 , t 3 ; we have that ϕðk∇uk 2 Þ ≥ k∇u 2 k 2α . Then, integrating (43) on ðt 2 , t 3 Þ, we have that Combing (40), (41), and (59),we get