Hyperbolic Relaxation of a Fourth Order Evolution Equation

Abstract

We propose a hyperbolic relaxation of a fourth order evolution equation, with an inertial term 𝜂𝑢_𝑡𝑡, where 𝜂∈(0,1]. We prove the existence of several absorbing sets having different regularities and the existence of a global attractor that is bounded in 𝐻⁴(𝐼)×{𝐻²(𝐼)∩𝐻¹₀(𝐼)}.

1. Introduction

Leting 𝐼⊂ℝ be an open interval, with |𝐼|≤1, we consider the following initial-boundary value problem for 𝑢∶𝐼×ℝ⁺→ℝ: 𝜂𝑢_𝑡𝑡+𝑢_𝑡=−𝜀²𝑢_{𝑥𝑥𝑥𝑥}+12𝑊^′′𝑢_𝑥𝑢_𝑥𝑥,(𝑥,𝑡)∈𝐼×ℝ⁺,𝑢(𝑥,0)=𝑢₀(𝑥)𝑢,in𝐼,_𝑡(𝑥,0)=𝑢₁(𝑥),in𝐼,𝑢=𝑢_𝑥𝑥=0,in𝜕𝐼,𝑡≥0, (1) where the function 𝑊(𝑝)=(𝑝²−1)² is the so-called double-well potential, 0<𝜀≪1 and 𝜂∈(0,1] are positive parameters.

Problem (1), with 𝜂=0, was proposed in [1] where the global dynamics was studied. In particular, the dynamical behavior of the solutions for small values of the parameter 𝜀 was studied by means of numerical experiments. The existence of three well-differentiated time scales with peculiar dynamical behavior was showen. In the first time scale of order 𝑂(𝜀²) there is the formation of microstructure (see [2]) in the region where the gradient of the initial datum falls in the nonconvex region of 𝑊; this phenomenon produces a drastic reduction of the energy of the initial datum. In the second time scale of order 𝑂(1) the equation exhibits a heat equation-like behavior in the convex regions while slow motion in the nonconvex ones. In the last time scale of order 𝑂(𝜀⁻²) the equation shows a finite-dimensional behavior: the solution is approximately the union of consecutive segments and the dynamic is slow.

In [3], the third time scale was studied; the authors proved the existence of a global attractor 𝒜_𝜀⊂𝐿²(𝐼) (see [4]) that is bounded in 𝐻²(𝐼). The time for which the solutions enter the absorbing set ℬ_𝜀 is of order 𝑂(𝜀⁻²) and it is consistent with the estimates found in [1]. Moreover the authors proved the existence of an exponential attractor ℐ_𝜀 with finite fractal dimension of order 𝑂(𝜀⁻¹⁰). In [5] the authors proved the existence of an inertial manifold ℳ_𝜀 (see [6]) whose dimension is of order 𝑂(𝜀⁻¹⁹), and by the 𝑛-dimensional volume elements methods (see [7]) an estimate of the dimension of the global attractor of order 𝑂(𝜀⁻¹) was found. This estimate is also consistent with the numerical experiments developed in [1]; in fact it was found that the wave length of the microstructure is of order 𝑂(𝜀⁻¹).

In the last years the viscous and no viscous hyperbolic relaxation of the Cahn-Hilliard equation has been extensively investigated. The model was proposed in [8] while in [9] the existence of a family of exponential attractors was proved. The viscous and nonviscous perturbation has been studied in [10] where the existence of a family of global attractors that are upper semicontinuous with respect to the vanishing of perturbations parameters was proved. These results have been extended in 2 and 3 dimensions; see for example [11, 12] and the references therein.

Due to the similarity of problem (1) to the Cahn-Hilliard equation we consider it interesting to study the hyperbolic relaxation of the fourth order evolution equation proposed in [1]. In particular if 𝑣 is the solution of the Cahn-Hilliard equation 𝑣_𝑡+Δ𝜀²Δ𝑣−𝑊^′(𝑣)=0, (2) with Neumann boundary conditions: 𝜕𝜕𝜕𝑛𝑣={},𝜕𝑛Δ𝑣=0,𝑥∈0,1 (3) then 𝑢(𝑥)=^𝑥₀𝑣(𝑠)𝑑𝑠, (4) is the solution of (1), with 𝜂=0, with the corresponding boundary conditions: 𝑢=𝑢_𝑥𝑥=0,𝑥∈{0,1}. (5) In the present work we put the problem in the correct mathematical framework and prove the existence of a global attractor 𝒜_𝜂,𝜀 while we have left the proof of the existence of exponential attractors for a forthcoming paper. In Sections 2 and 3 we define the solution semigroup in the appropriate spaces and present some important energy estimates. In Section 4 we prove the existence of several absorbing sets with different regularities while in the last section we prove the existence of the global attractor.

2. Preliminaries

We begin this section by defining the following Hilbert spaces that will be helpful for our analysis: ℋ_𝜂,𝜀=𝐻²(𝐼)∩𝐻¹₀(𝐼)×𝐿²(𝐼),𝒱_𝜂,𝜀=𝐷(𝐴)×𝐻²(𝐼), (6)(7) where 𝐷(𝐴)=𝑢∈𝐻⁴(𝐼)∶𝑢=𝑢_𝑥𝑥=0in𝜕𝐼 (8) is the domain of the differential operator 𝐴=𝜕⁴/𝜕𝑥⁴. The above spaces are equipped with the norms ‖(𝑢,𝑣)‖²_ℋ𝜂,𝜀=𝜀²2𝑢_𝑥𝑥²+𝜂2‖𝑣‖²,‖(𝑢,𝑣)‖²_{𝒱𝜂,𝜀}=𝜀²2𝑢_{x𝑥𝑥𝑥}²+𝜂2𝑣_𝑥𝑥², (9)(10) where ‖⋅‖ represents the 𝐿² norm. We will denote by ⟨⋅,⋅⟩ the inner product in 𝐿²(𝐼). We recall that for all 𝑢∈𝐻²(𝐼)∩𝐻¹₀(𝐼) with |𝐼|≤1, we have ‖𝑢‖≤𝑢_𝑥≤𝑢_𝑥𝑥. (11) Throughout the paper we will use two norms that are equivalent to (9) and (10) in order to simplify the computation. Given functions 𝑢∈𝐻²(𝐼)∩𝐻¹₀(𝐼) and 𝑣∈𝐿²(𝐼) we define the function 𝐸 on ℋ_𝜂,𝜀 by 𝐸(𝑢,𝑣)=𝜂‖𝑣‖²+12𝜀²𝑢_𝑥𝑥²+𝛽𝜂⟨𝑣,𝑢⟩, (12) where 𝛽∈[0,𝜀²].

Proposition 1. For all 𝛽∈[0,𝜀²] the function 𝐸(⋅,⋅) induces a norm equivalent to the norm on ℋ_𝜂,𝜀.

Proof. By Schwartz inequality (11) and using the fact that 𝛽≤𝜀²<1, 𝜂∈(0,1], we get ⟨⟩‖𝑣𝑢‖≤1𝛽𝜂𝑣,𝑢≤𝛽𝜂‖‖2𝜂²‖𝑣‖²+12𝜀⁴𝑢_𝑥𝑥²≤𝜂2‖𝑣‖²+𝜀²2𝑢_𝑥𝑥²=‖(𝑢,𝑣)‖²_ℋ𝜀,𝜂. (13) From the previous inequality and by definition of 𝐸(𝑢,𝑣) we get 𝐸(𝑢,𝑣)≤3‖(𝑢,𝑣)‖²_ℋ𝜀,𝜂. (14) By a different application of Schwartz inequality and from (11) we get 𝛽𝜂⟨𝑣,𝑢⟩≤𝛽𝜂‖𝑣‖‖𝑢‖≤𝜂²‖𝑣‖²+𝛽²4‖𝑢‖²≤𝜂‖𝑣‖²+𝜀⁴4‖𝑢‖²≤𝜂‖𝑣‖²+𝜀²4𝑢_𝑥𝑥². (15) Combining (13) and (15) we get 𝐸()𝜂𝑢,𝑣≥max2‖𝑣‖²,14𝜀²𝑢_𝑥𝑥², (16) and as a consequence 𝐸()≥1𝑢,𝑣3‖(𝑢,𝑣)‖²_ℋ𝜀,𝜂. (17)

The proof of the following theorem follows from classical applications of the Faedo-Galerkin method. We will only show a Lipschitz estimate that will be needed for further computations.

Theorem 2 2. For every (𝑢₀,𝑢₁)∈ℋ_𝜀,𝜂 there exists a unique solution 𝑢(𝑡) for the initial value problem (1) such that 𝑢∈𝐶_𝑏ℝ⁺;𝐻²(𝐼)∩𝐻¹₀(𝐼)∩𝐶¹_𝑏ℝ⁺;𝐿²(𝐼). (18) If, moreover, (𝑢₀,𝑢₁)∈𝒱_𝜀,𝜂, then: 𝑢∈𝐶_𝑏ℝ⁺;𝐷(𝐴)∩𝐶¹_𝑏ℝ⁺;𝐻²(𝐼)∩𝐻¹₀(𝐼)∩𝐶²_𝑏ℝ⁺;𝐿²(𝐼). (19)

Proposition 3 3. For any constant 𝑅≥0 there exists a positive constant 𝐾=𝐾(𝑅) such that, for any initial data 𝑢₁(0), 𝑢₂(0) with ‖𝑢_𝑖(0)‖_{𝐻𝜂,𝜀}≤𝑅, 𝑖=1,2 one has 𝑆_𝜀,𝜂(𝑡)𝑢₁(0)−𝑆_𝜀,𝜂(𝑡)𝑢₂(0)_{𝐻𝜂,𝜀}≤𝑒^{(𝐾2/𝜀2)𝑡}𝑢₁(0)−𝑢₂(0)_{𝐻𝜂,𝜀}, (20) where 𝑆_𝜀,𝜂(𝑡) is the solution semigroup of the problem (1).

Proof. Let 𝑢₁, 𝑢₂, two solutions of (1) with initial data 𝑢₁(0) and 𝑢₂(0). Let, 𝑤=𝑢₁−𝑢₂ then we write 𝜂𝑤_𝑡𝑡+𝑤_𝑡+𝜀²𝑤_{𝑥𝑥𝑥𝑥}=12𝑑𝑊𝑑𝑥^′𝑢_1𝑥−𝑊^′𝑢_2𝑥. (21) We multiply the above equation by 𝑤_𝑡 in 𝐿²(𝐼); then 𝜂2𝑑𝑤𝑑𝑡^𝑡2+𝜀²2𝑑𝑤𝑑𝑡^𝑥𝑥2+𝑤_𝑡²=−2_𝐼𝑤_𝑥𝑥𝑤_𝑡𝑑𝑥+6_𝐼𝑢²_1𝑥𝑤_𝑥𝑥+𝑢_2𝑥x𝑤_𝑥𝑢_1𝑥+𝑢_2𝑥𝑤_𝑡𝑤𝑑𝑥≤2_𝑥𝑥𝑤_𝑡+6𝑢_1𝑥²_∞𝑤_𝑥𝑥𝑤_𝑡+12𝑢_{2𝑥𝑥𝐼}𝑤²_𝑥𝑤²_𝑡𝑢²_1𝑥+𝑢²_2𝑥𝑑𝑥^1/2≤21+18𝐶²𝑢_1𝑥𝑥²+12𝑢_2𝑥𝑥×𝑢_1𝑥𝑥²+𝑢_2𝑥𝑥^21/2𝑤_𝑥𝑥𝑤_𝑡≤21+36𝐶²+24√2𝑅²𝜀²𝑤_𝑥𝑥𝑤_𝑡≤2𝐾𝑤_𝑥𝑥𝑤_𝑡≤𝐾²𝑤_𝑥𝑥²+𝑤_𝑡², (22) where we have used the following inequality (see [13] and inequality (11)): 𝑢_𝑥²_∞≤𝐶²𝑢_𝑥²_𝐻1(𝐼)≤2𝐶²𝑢_𝑥𝑥². (23) The constant 𝐾 can be explicitly computed using the estimates of the absorbing set presented in the next section.
From the last inequality we get 𝑑𝑑𝑡(𝑤,𝑤_𝑡)²_ℋ𝜀,𝜂≤2𝐾²𝜀²(𝑤,𝑤_𝑡)²_ℋ𝜀,𝜂, (24) and then using Gronwall's lemma we get (𝑤,𝑤_𝑡)_ℋ𝜀,𝜂≤𝑒^{(𝐾2/𝜀2)𝑡}(𝑤(0),𝑤_𝑡(0))_ℋ𝜀,𝜂. (25)

3. A Priori Estimates

In this section we provide useful a priori estimates of energy type. Equation (1) admits a Liapunov functional of the form: 𝐸𝑢,𝑢_𝑡=𝜂𝑢_𝑡²+12𝜀²𝑢_𝑥𝑥²+12_𝐼𝑊𝑢_𝑥𝑑𝑥, (26) that is not increasing along the solutions; in fact if we multiply (1) by 𝑢_𝑡 and integrate over 𝐼 we obtain 𝑑𝐸𝑑𝑡𝑢,𝑢_𝑡=−𝑢_𝑡². (27) Moreover, integrating the previous inequality on (0,𝑡) with respect to time we get ^𝑡₀𝑢_𝑡²𝑑𝑠+𝐸𝑢,𝑢_𝑡=𝐸𝑢₀,𝑢₁. (28) Then for any fixed initial data (𝑢₀,𝑢₁)∈ℋ_𝜀,𝜂 we have that the corresponding solution satisfies sup_𝑡≥0𝜂2𝑢_𝑡²+𝜀²2𝑢_𝑥𝑥²=sup_𝑡≥0𝑢,𝑢_𝑡²_ℋ𝜀,𝜂≤𝐸𝑢₀,𝑢₁. (29) Moreover if we integrate over ℝ⁺ we get the integral control ^∞₀𝑢_𝑡²𝑑𝑡≤𝐸𝑢₀,𝑢₁. (30) We consider an important estimate that will be useful later. Let 𝛽>0 be a parameter to be determined later; if we multiply (1) by 𝛽𝑢 we obtain 𝛽𝜂_𝐼𝑢_𝑡𝑡𝛽𝑢𝑑𝑥+2𝑑‖𝑢‖𝑑𝑡²+𝛽𝜀²𝑢_𝑥𝑥²+2𝛽𝑢_𝑥⁴₄=2𝛽𝑢_𝑥². (31) By summing (27) and (31) we get 𝑑𝑑𝑑𝑡𝐸+𝛽𝜂𝑢𝑑𝑡^𝑡,𝑢+1−𝛽𝜂𝑢_𝑡²+𝛽2𝑑‖𝑢‖𝑑𝑡²+𝛽𝜀²𝑢_𝑥𝑥²+2𝛽𝑢_𝑥⁴₄=2𝛽𝑢_𝑥². (32) Using the expression of the energy we can rewrite the previous in the following way: 𝑑𝑑𝑑𝑡𝐸+𝛽𝜂𝑢𝑑𝑡^𝑡,𝑢+1−3𝛽𝜂𝑢_𝑡²+𝛽⟨𝑢_𝑡𝑢,𝑢⟩+2𝛽𝐸+^𝑥442−|𝐼|2=0. (33) We will estimate some of the terms of the previous inequality in the following lemma.

Lemma 4. Fix 𝜂≤1 then for all 𝛽∈(0,1/3) one has 2𝛽²𝜂⟨𝑢_𝑡,𝑢⟩≤1−3𝛽𝜂𝑢_𝑡²+𝛽𝑢_𝑥⁴₄+𝛽|𝐼|+𝛽⟨u_𝑡,𝑢⟩. (34)

Proof. By Holder's inequality we get 2𝑢_𝑥²≤2𝑢_𝑥²₄|𝐼|^1/2≤𝑢_𝑥⁴₄+|𝐼|, (35) then we use Poincaré's inequality and the fact that |𝐼|≤1 to conclude that ‖𝑢‖²≤𝑢_𝑥⁴₄2+|𝐼|2. (36) Therefore, for a positive constant 𝑐₁ to be determined later, we have that 𝑢_𝑡,𝑢≤𝑢_𝑡𝑢‖≤𝑐‖¹2𝑢_𝑡²+12𝑐₁‖𝑢‖²≤𝑐₁2𝑢_𝑡²+12𝑐₁𝑢_𝑥⁴₄2+|𝐼|2. (37) Consequently, by choosing the parameters 𝑐₁=1/12 and 𝛽<1/3 we have that 2𝛽²𝜂−𝛽𝑢_𝑡≤,𝑢2𝛽²𝑐𝜂−𝛽¹2𝑢_𝑡²+2𝛽²𝜂−𝛽4𝑐₁𝑢_𝑥⁴₄+2𝛽²𝜂−𝛽4𝑐₁|𝐼|≤1−3𝛽𝜂𝑢_𝑡²+𝛽𝑢_𝑥⁴₄+𝛽|𝐼|. (38)

Now, using the previous lemma and (33) we conclude that 𝑑𝑑𝑑𝑡𝐸+𝛽𝜂𝑢𝑑𝑡^𝑡,𝑢+2𝛽²𝜂𝑢_𝑡,𝑢+2𝛽𝐸≤2𝛽|𝐼|. (39) If we set Φ𝑢,𝑢_𝑡=Φ(𝑡)=𝐸𝑢,𝑢_𝑡+𝛽𝜂⟨𝑢_𝑡,𝑢⟩, (40) then from (39) we get 𝑑Φ(𝑡)(𝑡)𝑑𝑡+2𝛽Φ≤2𝛽. (41) Now, integrating we have that Φ(𝑡)≤Φ𝑢₀,𝑢₁−1𝑒^−2𝛽𝑡+1. (42) The previous inequality will be used in the next section to show the existence of absorbing sets for the problem (1).

We remark that the following inequality holds for the functions 𝐸, 𝐸, and Φ: Φ(𝑡)=𝐸(𝑡)+12_𝐼𝑊𝑢_𝑥𝑑𝑥≥𝐸≥0. (43)

4. Absorbing Sets

In this section we will show the existence of several absorbing sets for the solution semigroup of (1) in the space ℋ_𝜀,𝜂. Also, assuming further regularity of the initial data, the existence of a more regular absorbing sets is also shown.

First, notice the following estimate of the nonlinear term of the energy functional defined in (26). Using (23) we get 12_𝐼𝑊𝑢_𝑥1𝑑𝑥≤2𝑢_𝑥²𝑢_𝑥²_{𝐿∞(𝐼)}−𝑢_𝑥²+|𝐼|2≤12𝑢_𝑥²𝑢_𝑥𝑥²+|𝐼|2≤12𝑢_𝑥𝑥⁴+|𝐼|2. (44) We will use this inequality several times in what follows.

4.1. Absorbing Set in ℋ_𝜀,𝜂

Proposition 5. For all 𝑅₀>1 the set 𝐵_Φ∶=𝑢,𝑢_𝑡∈ℋ_𝜀,𝜂∶Φ𝑢,𝑢_𝑡≤𝑅₀ (45) is bounded, absorbing, and positively invariant for the semigroup 𝑆_𝜀,𝜂(𝑡) in ℋ_𝜀,𝜂.

Proof. The set is positively invariant, in fact if (𝑢₀,𝑢₁)∈ℋ_𝜀,𝜂 with Φ(𝑢₀,𝑢₁)≤𝑅₀ we have, from (42), that Φ𝑢,𝑢_𝑡≤𝑅₀−1𝑒^−2𝛽𝑡+1≤𝑅₀,∀𝑡>0. (46) The boundness of 𝐵_Φ follows directly from (42). Now suppose that (𝑢₀,𝑢₁) are such that Φ(𝑢₀,𝑢₁)≤𝑅, with 𝑅>𝑅₀ then again from (42) we get Φ𝑢,𝑢_𝑡≤𝑅₀,∀𝑡≥𝜏_Φ, (47) where 𝜏_Φ1∶=2𝛽log𝑅−1𝑅₀.−1 (48) Then 𝐵_Φ is absorbing for the semigroup 𝑆_𝜀,𝜂(𝑡).

Proposition 6. For all 𝑅₁>1 one has that the set 𝐵_𝐸∶=𝑢,𝑢_𝑡∈ℋ_𝜀,𝜂∶𝐸𝑢,𝑢_𝑡≤𝑅₁ (49) is absorbing for the semigroup 𝑆_𝜀,𝜂(𝑡).

Proof. We have from (42) and (43) that 𝐸𝑢,𝑢_𝑡≤Φ𝑢₀,𝑢₁−1𝑒^−2𝛽𝑡+1. (50) Now, take (𝑢₀,𝑢₁) such that 𝐸(𝑢₀,𝑢₁)≤𝑅, with 𝑅>𝑅₁. Then, by (42) and (44) we get that 𝐸(𝑡)(𝑡)≤𝐸(𝑡)+1≤Φ2𝑢_𝑥⁴₄+|𝐼|2≤𝐸(𝑡)+12𝑢_𝑥𝑥⁴+|𝐼|2≤𝐸(𝑡)+4𝜀²𝐸(𝑡)²+|𝐼|2. (51) Therefore, 𝐸𝑢,𝑢_𝑡≤𝐸𝑢₀,𝑢₁+4𝜀²𝐸𝑢₀,𝑢₁²+|𝐼|2−1𝑒^−2𝛽𝑡≤4+1𝑅+𝜀²𝑅²+|𝐼|2−1𝑒^−2𝛽𝑡+1. (52) Consequently, we have that 𝐸𝑢,𝑢_𝑡≤𝑅₁,∀𝑡≥𝜏_𝐸, (53) where 𝜏_𝐸=12𝛽log𝑅+4/𝜀⁴𝑅²+(|𝐼|/2)−1𝑅₁.−1 (54)

Now using the equivalence of the norm on ℋ_𝜀,𝜂 and (52) we get 𝑢,𝑢_𝑡²_ℋ𝜀,𝜂≤33(𝑢₀,𝑢₁)²_ℋ𝜀,𝜂+36𝜀²(𝑢₀,𝑢₁)⁴_ℋ𝜀,𝜂+|𝐼|2−1𝑒^−2𝛽𝑡+3. (55) Let 𝑅₂>√3. If (𝑢₀,𝑢₁)∈ℋ_𝜀,𝜂 is such that (𝑢₀,𝑢₁)_ℋ𝜀,𝜂≤𝑅, (56) then we have that (𝑢,𝑢_𝑡)_ℋ𝜀,𝜂≤𝑅₂,∀𝑡≥𝜏_𝐻, (57) where 𝜏_𝐻=132𝛽log3𝑅²+36/𝜀²𝑅⁴+|𝐼|/2−1𝑅²₂.−3 (58) Thus, we have proved the following.

Proposition 7 7. For all 𝑅₂>√3 one has that the ball 𝐵_ℋ𝜀,𝜂∶=𝑢,𝑢_𝑡∈ℋ_𝜀,𝜂∶𝑢,𝑢_{𝑡ℋ𝜀,𝜂}≤𝑅₂ (59) is an absorbing set for the semigroup 𝑆_𝜀,𝜂(𝑡) in ℋ_𝜀,𝜂.

4.2. Absorbing Set in 𝒰_𝜀,𝜂

Now suppose that the initial data has some additional regularity. Then we can prove the existence of more regular absorbing sets. Let us define the following space: 𝒰_𝜀,𝜂=𝐻³(𝐼)∩𝐻²(𝐼)∩𝐻¹₀(𝐼)×𝐻¹(𝐼), (60) equipped with the norm (𝑢,𝑢_𝑡)²_{𝒰𝜀,𝜂}=𝜂2𝑢_𝑡𝑥²+𝜀²2𝑢_𝑥𝑥𝑥². (61) Then we have the following.

Proposition 8 8. There exists 𝑅₃>0 such that the closed ball ℬ₃=𝑢,𝑢_𝑡∈𝒰_𝜀,𝜂∶𝑢,𝑢_{𝑡𝒰𝜀,𝜂}≤𝑅₃, (62) is a bounded absorbing set for 𝑆_𝜀,𝜂(𝑡) in 𝒰_𝜀,𝜂.

Proof. If we multiply (1) by 𝑢_𝑥𝑥𝑡+𝑢_𝑥𝑥 and integrate over 𝐼 we obtain 𝑑𝜂𝑑𝑡2𝑢_𝑡𝑥²+𝜀²2𝑢_𝑥𝑥𝑥²+12𝑢_𝑥²+𝜂⟨𝑢_𝑡𝑥,𝑢_𝑥⟩+1−𝜂𝑢_𝑡𝑥²+𝜀²𝑢_𝑥𝑥𝑥²1=−2𝑊^′𝑢_𝑥𝑥,𝑢_𝑥𝑥𝑡−12𝑊^′𝑢_𝑥𝑥,𝑢_𝑥𝑥. (63) Let us denote the differential term of the previous inequality as Ψ₀(𝑡)=𝜂2𝑢_𝑡𝑥²+𝜀²2𝑢_𝑥𝑥𝑥²+𝜂𝑢_𝑡𝑥,𝑢_𝑥+12𝑢_𝑥². (64) We will estimate the right hand side of (63). The first term can be estimated as follows: −12𝑊^′𝑢_𝑥𝑥,𝑢_𝑥𝑥≤12𝑊^′′(𝑢_𝑥)_∞𝑢_𝑥𝑥²≤23𝑢_𝑥²_∞+1𝑢_𝑥𝑥²≤23𝑢_𝑥𝑥²+1𝑢_𝑥𝑥²≤4𝜀²𝑅²₂6𝑅²₂𝜀²+1∶=𝐶₁. (65) Let us define Ψ₁=14_𝐼𝑊^′′𝑢_𝑥𝑢²_𝑥𝑥𝑑𝑥; (66) then we can rewrite the the second term of r.h.s of (63) −12𝑊^′𝑢_𝑥𝑥,𝑢_𝑥𝑥𝑡𝑑=−Ψ𝑑𝑡¹+14_𝐼𝑊^′′′𝑢_𝑥𝑢_𝑥𝑡𝑢²_𝑥𝑥, (67) and estimate 14_𝐼𝑊^′′′𝑢_𝑥𝑢_𝑥𝑡𝑢²_𝑥𝑥≤6𝑢_{𝑥𝑥∞𝐼}𝑢_𝑥𝑢_𝑥𝑡𝑢_𝑥𝑥𝑢𝑑𝑥≤6_𝑥𝑥∞𝑢_𝑥𝑡𝑢_𝑥𝑢_𝑥𝑥≤6𝑢_𝑥𝑥∞𝑢_𝑥𝑡𝑢_𝑥𝑥²≤6𝑢_𝑥𝑥^5/2𝑢_𝑥𝑥𝑥^1/2𝑢_𝑥𝑡√≤62𝜀𝑅₂^5/2𝑢_𝑥𝑥𝑥^1/2𝑢_𝑥𝑡≤𝜂2𝑢_𝑥𝑡²+18𝜂√2𝜀𝑅₂⁵𝑢_𝑥𝑥𝑥≤𝜂2𝑢_𝑥𝑡²+1218√𝜂𝜀2𝜀𝑅₂⁵²+𝜀²2𝑢_𝑥𝑥𝑥²𝜂∶=2𝑢_𝑥𝑡²+𝜀²2𝑢_𝑥𝑥𝑥²+𝐶₂. (68) Then if 𝜂∈(0,1/2) and setting Ψ∶=Ψ₀+Ψ₁, 𝐶∶=𝐶₁+𝐶₂ we have 𝑑𝜂𝑑𝑡Ψ+2𝑢_𝑥𝑡²+𝜀²2𝑢_𝑥𝑥𝑥²≤𝐶. (69) To conclude the proof we note that Ψ≥𝑢,𝑢_𝑡²_{𝒰𝜀,𝜂}+12𝑢_𝑥𝑥²−𝜂𝑢_𝑡𝑥𝑢_𝑥−𝑢_𝑥𝑥²≥(𝑢,𝑢_𝑡)²_{𝒰𝜀,𝜂}−𝜂4𝑢_𝑡𝑥²−32𝑢_𝑥𝑥²≥12𝑢,𝑢_𝑡²_{𝒰𝜀,𝜂}−3𝜀²𝑅²₂1∶=2(𝑢,𝑢_𝑡)²_{𝒰𝜀,𝜂}−𝑐₁, (70) and that Ψ≤𝑢,𝑢_𝑡²_{𝒰𝜀,𝜂}+𝜂2𝑢_𝑡𝑥²+𝑢_𝑥𝑥²+𝑢_𝑥𝑥²3𝑢_𝑥²_∞+1≤2𝑢,𝑢_𝑡²_{𝒰𝜀,𝜂}+2𝜀²𝑅²₂6𝜀²𝑅²₂+2∶=2(𝑢,𝑢_𝑡)²_{𝒰𝜀,𝜂}+𝑐₂. (71) From the previous inequalities we get 12(𝑢,𝑢_𝑡)²_{𝒰𝜀,𝜂}−𝑐₁≤Ψ≤2(𝑢,𝑢_𝑡)²_{𝒰𝜀,𝜂}+𝑐₂. (72) Then from (72) we get 𝑑1𝑑𝑡Ψ+2Ψ≤𝐶+𝑐₂∶=𝐶, (73) and by Gronwall's lemma we obtain Ψ(𝑡)≤Ψ(0)−2𝐶𝑒^−(1/2)𝑡+2𝐶,𝑢,𝑢_𝑡²_{𝒰𝜀,𝜂}≤2𝑢₀,𝑢₁²_{𝒰𝜀,𝜂}+𝑐₂−2𝐶𝑒^−(1/2)𝑡+4𝐶+2𝑐₁. (74) Then if (𝑢₀,𝑢₁)∈𝒰_𝜀,𝜂 such that (𝑢₀,𝑢₁)_{𝒰𝜀,𝜂}≤𝑅, (75) we have that (𝑢,𝑢_𝑡)_{𝒰𝜀,𝜂}≤𝑅₃,∀𝑡≥𝜏_𝑈, (76) where 𝜏_𝑈∶=2log2𝑅²+2𝑐₂−4𝐶𝑅²₃−4𝐶−2𝑐₁, (77) and this concludes the proof.

4.3. Absorbing Set in 𝒱_𝜀,𝜂

We consider the space 𝒱_𝜀,𝜂 as defined in (7), equipped with the norm defined in (10). By considering more regular initial conditions we can prove the following result.

Proposition 9 9. There exists 𝑅₄>0 such that the closed ball ℬ₄=𝑢,𝑢_𝑡∈𝒱_𝜀,𝜂∶(𝑢,𝑢_𝑡)_{𝒱𝜀,𝜂}≤𝑅₄ (78) is a bounded absorbing set for the semigroup 𝑆_𝜀,𝜂(𝑡) in 𝒱_𝜀,𝜂.

Proof. Multiply (1) by 𝑢_{𝑥𝑥𝑥𝑥𝑡}+𝑢_{𝑥𝑥𝑥𝑥}; then we obtain 𝑑𝜂𝑑𝑡2𝑢_𝑡𝑥𝑥²+𝜀²2𝑢_{𝑥𝑥𝑥𝑥}²+𝜂𝑢_𝑡𝑥𝑥,𝑢_𝑥𝑥+12𝑢_𝑥𝑥𝑥²+1−𝜂𝑢_𝑥𝑥𝑡+𝜀²𝑢_{𝑥𝑥𝑥𝑥}²=12𝑊^′𝑢_𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑥𝑡}+𝑢_{𝑥𝑥𝑥𝑥}. (79) We call 𝜃₀(𝑡) the differential term of the previous inequality and we estimate the r.h.s.: 12𝑊^′𝑢_𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑥}1=−2𝑊^′𝑢_𝑥𝑥𝑥,𝑢_𝑥𝑥𝑥1=−2𝑊^′′′𝑢_𝑥𝑢²_𝑥𝑥,𝑢_𝑥𝑥𝑥−12𝑊^′′𝑢_𝑥𝑥𝑥,𝑢²_𝑥𝑥𝑥=−12_𝐼𝑢_𝑥𝑢²_𝑥𝑥𝑢_𝑥𝑥𝑥𝑑𝑥−2_𝐼3𝑢²_𝑥−1𝑢²_𝑥𝑥𝑥𝑢𝑑𝑥≤2_𝑥𝑥𝑥²+12𝑢_𝑥𝑥²_∞𝑢_𝑥𝑢_𝑥𝑥𝑥≤2𝑢_𝑥𝑥𝑥²6𝑢_𝑥𝑥²+1≤𝜀²𝑅²₃12𝜀²𝑅²₂+1∶=𝐴₁. (80) Let us define 𝜃₁(𝑡)1∶=4_𝐼𝑊^′′𝑢_𝑥𝑢²_𝑥𝑥𝑥𝑑𝑥; (81) then 12𝑊^′𝑢_𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑥𝑡}1=−2𝑊^′𝑢_𝑥𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑡}1=−2𝑊^′′′𝑢_𝑥𝑢²_𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑡}−12𝑊^′′𝑢_𝑥𝑢_𝑥𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑡}𝑑=−𝜃𝑑𝑡¹+12𝑊^′′′𝑢_𝑥𝑢²_𝑥𝑥𝑥,𝑢_𝑥𝑥𝑡+14_𝐼𝑊^′′′𝑢_𝑥𝑢_𝑥𝑡𝑢²_𝑥𝑥𝑥𝑑𝑥. (82) We estimate the last two terms of the previous equality: 14_𝐼𝑊^′′′𝑢_𝑥𝑢_𝑥𝑡𝑢²_𝑥𝑥𝑥𝑑𝑥=6_𝐼𝑢_𝑥𝑢_𝑥𝑡𝑢²_𝑥𝑥𝑥𝑢𝑑𝑥≤6_{𝑥𝑥𝑥∞}𝑢_𝑥∞𝑢_𝑥𝑡𝑢_𝑥𝑥𝑥≤6𝑢_𝑥𝑥𝑥^3/2𝑢_{𝑥𝑥𝑥𝑥}^1/2𝑢_𝑥𝑡𝑢_𝑥𝑥²≤12⋅2^3/4𝑅²₂𝑅₃^3/2𝜀^7/2𝑢_{𝑥𝑥𝑥𝑥}^1/2𝑢_𝑥𝑡≤12⋅2^3/4𝑅²₂𝑅₃^3/2𝜀^7/22𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑥𝑡²≤12𝜀²12⋅2^3/4𝑅²₂𝑅₃^3/2𝜀^7/24+2𝜂𝑅²₃+𝜀²2𝑢_{𝑥𝑥𝑥𝑥}²𝜀∶=²2𝑢_{𝑥𝑥𝑥𝑥}²+𝐴₂,12𝑊^′′′𝑢_𝑥𝑢²_𝑥𝑥𝑥,𝑢_𝑥𝑥𝑡=12𝑊^𝑖𝑣𝑢_𝑥𝑢³_𝑥𝑥+𝑊^′′′𝑢_𝑥2𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥,𝑢_𝑥𝑥𝑡=12_𝐼𝑢³_𝑥𝑥𝑢_𝑥𝑥𝑡𝑑𝑥+24_𝐼𝑢_𝑥𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥𝑢_𝑥𝑥𝑡𝑢𝑑𝑥≤12_𝑥𝑥²_∞𝑢_𝑥𝑥+2𝑢_𝑥∞𝑢_𝑥𝑥∞𝑢_𝑥𝑥𝑥𝑢_𝑥𝑥𝑡≤12𝑢_𝑥𝑥²𝑢_𝑥𝑥𝑥+2𝑢_𝑥𝑥^3/2𝑢_𝑥𝑥𝑥^3/2𝑢_𝑥𝑥𝑡≤𝜂2𝑢_𝑥𝑥𝑡²+72𝑢_𝑥𝑥²𝑢_𝑥𝑥𝑥+2𝑢_𝑥𝑥^3/2𝑢_𝑥𝑥𝑥^3/22≤𝜂2𝑢_𝑥𝑥𝑡²2√+722𝑅²₂𝑅₃𝜀³+22𝑅₂𝑅₃𝜀^23/22𝜂∶=2𝑢_𝑥𝑥𝑡²+𝐴₃. (83) Then if we set 𝜃(𝑡)=𝜃₀(𝑡)+𝜃₁(𝑡) we get 𝑑𝜃(𝑡)+𝑑𝑡(𝑢,𝑢_𝑡)²_{𝒱𝜀,𝜂}≤𝐴₁+𝐴₂+𝐴₃∶=𝐴. (84) Moreover we have 𝜃(𝑡)≥(𝑢,𝑢_𝑡)²_{𝒱𝜀,𝜂}−𝜂4𝑢_𝑥𝑥𝑡²−12𝑢_𝑥𝑥²≥12𝑢,𝑢_𝑡²_{𝒱𝜀,𝜂}−12𝑅²₂1∶=2(𝑢,𝑢_𝑡)²_{𝒱𝜀,𝜂}−𝑎₁,𝜃(𝑡)≤𝑢,𝑢_𝑡²_{𝒱𝜀,𝜂}+𝜂𝑢_𝑥𝑥𝑡𝑢_𝑥𝑥+12𝑢_𝑥𝑥²+14𝑊^′′(𝑢_𝑥)_∞𝑢_𝑥𝑥𝑥³≤2𝑢,𝑢_𝑡²_{𝒱𝜀,𝜂}+𝑢_𝑥𝑥²+3𝑢_𝑥²_∞+1𝑢_𝑥𝑥𝑥²≤2𝑢,𝑢_𝑡²_{𝒱𝜀,𝜂}+2𝜀²𝑅²₂+6𝜀²𝑅²₂2+1𝜀²𝑅²₃∶=2(𝑢,𝑢_𝑡)²_{𝒱𝜀,𝜂}+𝑎₂. (85) Then putting all together we get 12(𝑢,𝑢_𝑡)²_{𝒱𝜀,𝜂}−𝑎₁≤𝜃(𝑡)≤2(𝑢,𝑢_𝑡)²_{𝒱𝜀,𝜂}+𝑎₂. (86) From the previous inequality we get 𝑑𝜃(𝑡)+1𝑑𝑡2𝜃(𝑡)𝑎≤𝐴+²2∶=𝐴 (87) and by Gronwall's lemma 𝜃(𝑡)≤𝜃(0)−2𝐴𝑒^−(1/2)𝑡+2𝐴. (88) Using again (86) we get 𝑢,𝑢_𝑡²_{𝒱𝜀,𝜂}≤22𝑢₀,𝑢₁²_{𝒱𝜀,𝜂}+𝑎₂−2𝐴𝑒^−(1/2)𝑡+4𝐴+2𝑎₁. (89) Then if (𝑢₀,𝑢₁)∈𝒱_𝜀,𝜂 such that (𝑢₀,𝑢₁)_{𝒱𝜀,𝜂}≤𝑅, (90) then there exists 𝑅₄>0: (𝑢,𝑢_𝑡)_{𝒱𝜀,𝜂}≤𝑅₄,∀𝑡≥𝜏_𝑉, (91) where 𝜏_𝑉2∶=2log2𝑅²+𝑎₂−2𝐴𝑅²₄−4𝐴−2𝑎₁, (92) and this concludes the proof.

5. Global Attractor

In this section we will show the existence of a global attractor for the semigroup 𝑆_𝜀,𝜂(𝑡) in ℋ_𝜀,𝜂. Since we have already proved the existence of the absorbing set in ℋ_𝜀,𝜂, then it is sufficient (see, e.g., [4] or [14] for general results or [15] for a recent application on a weakly damped wave equation) to prove that, for any fixed bounded set ℬ⊂ℋ_𝜀,𝜂, the solution semigroup 𝑆_𝜀,𝜂(𝑡) admits the decomposition: 𝑆_𝜀,𝜂(𝑡)=𝐿_𝜀,𝜂(𝑡)+𝑁_𝜀,𝜂(𝑡), (93) such that (𝐶1)sup_𝑡≥0sup_{𝑧∈ℋ𝜀,𝜂}𝐿_𝜀,𝜂(𝑡)𝑧_{𝒱𝜀,𝜂}()<∞,𝐶2lim_𝑡→∞sup_{𝑧∈ℋ𝜀,𝜂}𝑁_𝜀,𝜂(𝑡)𝑧_ℋ𝜀,𝜂=0. (94)(95)

Let ℬ⊂ℋ_𝜀,𝜂 be a fixed bounded set and let (𝑢₀,𝑢₁)∈ℬ⊂ℋ_𝜀,𝜂. We will define the decomposition of 𝑆_𝜀,𝜂 as follows: 𝑆_𝜀,𝜂(𝑡)𝑢₀,𝑢₁=𝑢,𝑢_𝑡,𝐿_𝜀,𝜂(𝑡)𝑢₀,𝑢₁=ℎ,ℎ_𝑡,𝑁_𝜀,𝜂(𝑡)𝑢₀,𝑢₁=𝑣,𝑣_𝑡, (96) where ℎ and 𝑣 are solutions of the following problems: 𝜂ℎ_𝑡𝑡+ℎ_𝑡+𝜀²ℎ_{𝑥𝑥𝑥𝑥}−𝛼ℎ_𝑥𝑥=12𝑊^′′ℎ_𝑥ℎ_𝑥𝑥−𝛼𝑢_𝑥𝑥,ℎ()ℎ𝑥,0=0,_𝑡(𝑥,0)=0,ℎ=ℎ_𝑥𝑥=0,in𝜕𝐼,𝜂𝑣_𝑡𝑡+𝑣_𝑡+𝜀²𝑣_{𝑥𝑥𝑥𝑥}−𝛼𝑣_𝑥𝑥=12𝑊^′′𝑢_𝑥𝑢_𝑥𝑥−12𝑊^′′ℎ_𝑥ℎ_𝑥𝑥,𝑣(0)=𝑢₀,𝑣_𝑡(0)=𝑢₁,𝑣=𝑣_𝑥𝑥=0,in𝜕𝐼, (97)(98) where 6√𝛼≥2𝜀𝑅₂>6√6. (99) Before showing that the semigroups 𝐿_𝜀,𝜂(𝑡) and 𝑁_𝜀,𝜂(𝑡) satisfy the conditions (94) and (95), respectively, we consider the following lemma (see [9]) that will be useful for the sequel.

Lemma 10 10. Let 𝜓∶ℝ⁺→ℝ be an absolutely continuous function which fulfills, for some 𝜈>0 and almost every 𝑡≥0, the differential inequality 𝑑𝜓(𝑡)(𝑡)(𝑡)𝜓(𝑡),𝑑𝑡+2𝜈𝜓≤𝑓 (100) where 𝑓 is a positive function satisfying ^𝑡₀𝑓𝑦𝑑𝑦≤𝜈𝑡+𝑐,∀𝑡≥0. (101) Then 𝜓(𝑡)≤𝑐𝜓(0)𝑒^−𝜈𝑡,𝑡≥0. (102)

5.1. The Semigroup 𝐿_𝜀,𝜂(𝑡)

Due to the similarity of problem (97) to (1), Proposition 9 still holds in this setting. We will omit the proof. Consequently we get that condition (94) holds by noting that (ℎ(0),ℎ_𝑡(0))=(0,0).

Moreover by (97), multiplying the equation by ℎ_𝑡 and integrating over 𝐼, we get that there exists a constant 𝑐>0 such that 𝑑𝜂𝑑𝑡2ℎ_𝑡²+𝜀²2ℎ_𝑥𝑥²+12_𝐼𝑊ℎ_𝑥𝛼𝑑𝑥+2ℎ_𝑥²+𝛼⟨𝑢_𝑥𝑥+ℎ,ℎ⟩_𝑡2=𝛼⟨𝑢_𝑡,ℎ_𝑥𝑥⟩≤𝑐𝑢_𝑡. (103) Then integrating the above equation on (0,𝑡) and using (30) we get that there exist constants 𝐴>0 and 𝐵>0 such that ^𝑡₀ℎ_𝑡²𝑑𝑠≤𝐴+𝑐^𝑡₀𝑢_𝑡𝑑𝑠≤𝐴+𝐵𝑡^1/2. (104) Consequently, for any 𝛾>0 there exists a constant 𝐶_𝛾≥0 such that ^𝑡₀ℎ_𝑡²𝑑𝑠≤𝛾𝑡+𝐶_𝛾. (105)

5.2. The Semigroup 𝑁_𝜀,𝜂(𝑡)

Let 𝛽>0 be a parameter to be determined later and let us multiply, in 𝐿²(𝐼), (98) by 2𝑣_𝑡+𝛽𝑣. Then we get 𝑑𝜂𝑣𝑑𝑡^𝑡2+𝜂𝛽⟨𝑣_𝑡,𝑣⟩+𝜀²𝑣_𝑥𝑥²+𝛽2‖𝑣‖²+𝛼𝑣_𝑥²+⟨𝑊𝑢_𝑥−𝑊ℎ_𝑥−𝑊^′ℎ_𝑥𝑣_𝑥+𝑣,1⟩2−𝜂𝛽_𝑡²+𝜀²𝛽𝑣_𝑥𝑥²−𝛽2𝑊^′𝑢_𝑥−𝑊^′ℎ_𝑥,𝑣_𝑥=ℎ_𝑡𝑥,𝑊^′𝑢_𝑥−𝑊^′ℎ_𝑥−𝑊^′′ℎ_𝑥𝑣_𝑥. (106) We estimate the term involving 𝑊 and its derivatives: −𝛽2𝑊^′𝑢_𝑥−𝑊^′ℎ_𝑥,𝑣_𝑥𝛽=−2⟨𝑊^′′𝑘_𝑥𝑣_𝑥,𝑣_𝑥⟩=2𝛽_𝐼1−3𝑘²_𝑥𝑣²_𝑥𝑣𝑑𝑥≥−6𝛽_𝑥∞𝑣_𝑥²√≥−6𝛽2𝜀𝑅₂𝑣_𝑥², (107) where 𝑘_𝑥(𝑥) is between ℎ_𝑥(𝑥) and 𝑢_𝑥(𝑥) and satisfies ‖𝑘_𝑥‖_∞≤‖𝑣_𝑥‖_∞. Moreover we have 𝑊𝑢_𝑥−𝑊ℎ_𝑥−𝑊^′ℎ_𝑥𝑣_𝑥=𝑊,1_′ℎ_𝑥+𝑘_𝑥−𝑊^′ℎ_𝑥,𝑣_𝑥=𝑊^′′𝑠_𝑥𝑘_𝑥,𝑣_𝑥=12_𝐼𝑠²_𝑥𝑘_𝑥𝑣_𝑥𝑑𝑥−4_𝐼𝑘_𝑥𝑣_𝑥𝑣𝑑𝑥≥−4_𝑥², (108) where 𝑘_𝑥(𝑥) is between 0 and 𝑣_𝑥(𝑥), 𝑠_𝑥(𝑥) is between 0 and 𝑘_𝑥(𝑥) and ‖𝑠_𝑥‖_∞≤‖𝑘_𝑥‖_∞≤‖𝑣_𝑥‖_∞. Moreover we have used that 𝑣²_𝑥≥𝑘_𝑥𝑣_𝑥≥0.

We denote as ℱ the differential term of (106) and prove that it induces a norm that is equivalent to that of ℋ_𝜀,𝜂. Then from (99) and (107) we have that ℱ(𝑡)≥𝜂𝑣_𝑡²−𝜂2𝑣_𝑡²−𝛽2‖𝑣‖²+𝜀²𝑣_𝑥𝑥²+𝛼𝑣_𝑥²−6𝛽√2𝜀𝑅₂𝑣_𝑥²≥𝜂2𝑣_𝑡²+𝜀²𝑣_𝑥𝑥²≥(𝑣,𝑣_𝑡)²_ℋ𝜀,𝜂. (109) Moreover we have ℱ(𝑡)≤𝜂𝑣_𝑡²+𝜂2𝑣_𝑡²+𝛽2‖𝑣‖²+𝜀²𝑣_𝑥𝑥²+𝛽2‖𝑣‖²+𝛼𝑣_𝑥²+𝑊^′′(𝑘_𝑥)_∞𝑣_𝑥²≤32𝜂𝑣_𝑡²+𝛽𝑣_𝑥𝑥²+𝜀²𝑣_𝑥𝑥²+𝛼𝑣_𝑥𝑥²+43𝑢_𝑥²_∞+1𝑣_𝑥𝑥²≤32𝜂𝑣_𝑡²+𝛽+𝜀²√+𝛼+122𝑅₂𝜀+4𝑣_𝑥𝑥²3∶=2𝜂𝑣_𝑡²𝜀+𝐴²2𝑣_𝑥𝑥²≤𝑐(𝑣,𝑣_𝑡)²_ℋ𝜀,𝜂, (110) where we have used the same argument as (107) and (108) and where 2𝐴=𝜀²𝛽+𝜀²√+𝛼+122𝑅₂𝜀+4,𝑐=max3,𝐴. (111) Then we have (𝑣,𝑣_𝑡)²_ℋ𝜀,𝜂≤ℱ(𝑡)≤𝑐(𝑣,𝑣_𝑡)²_ℋ𝜀,𝜂. (112) We estimate the right hand side of (106): ℎ_𝑡𝑥,𝑊^′𝑢_𝑥−𝑊^′ℎ_𝑥−𝑊^′′ℎ_𝑥𝑣_𝑥=ℎ_𝑡,𝑊^′′′ℎ_𝑥ℎ_𝑥𝑥𝑣_𝑥+𝑊^′′ℎ_𝑥−𝑊^′′𝑢_𝑥𝑢_𝑥𝑥=ℎ_𝑡,24ℎ_𝑥ℎ_𝑥𝑥𝑣_𝑥+12ℎ²_𝑥−𝑢²_𝑥𝑢_𝑥𝑥=ℎ_𝑡,12ℎ_𝑥𝑣_𝑥2ℎ_𝑥𝑥−𝑢_𝑥𝑥−12𝑣_𝑥𝑢_𝑥𝑢_𝑥𝑥=ℎ_𝑡,12𝑣_𝑥ℎ_𝑥ℎ_𝑥𝑥−𝑣_𝑥𝑥−𝑢_𝑥𝑢_𝑥𝑥=ℎ_𝑡,12𝑣_𝑥−2𝑣_𝑥𝑥ℎ_𝑥−𝑣_𝑥𝑢_𝑥𝑥≤24ℎ_𝑡𝐼𝑣²_𝑥𝑣²_𝑥𝑥ℎ²_𝑥^1/2+12ℎ_𝑡𝐼𝑢²_𝑥𝑥𝑣⁴_𝑥^1/2≤24ℎ_𝑡𝑣_𝑥∞ℎ_𝑥∞𝑣_𝑥𝑥+12ℎ_𝑡𝑣_𝑥∞𝑢_𝑥𝑥∞𝑣_𝑥≤12ℎ_𝑡𝑣_𝑥𝑥²2ℎ_𝑥𝑥+𝑢_𝑥𝑥𝑥≤𝜎ℎ_𝑡𝑣_𝑥𝑥²≤2𝜎²𝛽𝜀²ℎ_𝑡²+𝛽𝜀²2𝑣_𝑥𝑥², (113) where √𝜎=122𝜀2𝑅₂+𝑅₃. (114) Then it remains to estimate the following term: 2−𝜂𝛽𝑣_𝑡²+𝛽𝜀²𝑣_𝑥𝑥²+𝛼𝛽𝑣_𝑥²−𝛽2⟨𝑊^′𝑢_𝑥−𝑊^′ℎ_𝑥,𝑣_𝑥⟩𝜂≥𝛽2𝑣_𝑡²+𝛽𝜀²𝑣_𝑥𝑥²+𝛼𝛽𝑣_𝑥²√−6𝛽2𝜀𝑅₂𝑣_𝑥²𝜂≥𝛽2𝑣_𝑡²+𝛽𝜀²𝑣_𝑥𝑥². (115) Then, using the above estimates, we can rewrite (106) in the following way: 𝑑ℱ(𝑡)+𝛽𝑑𝑡𝑐ℱ(𝑡)≤𝜎²2𝛽𝜀²ℎ_𝑡²𝑣_𝑥𝑥²≤𝜎²𝛽𝜀⁴ℎ_𝑡²𝑣,𝑣_𝑡²_ℋ𝜀,𝜂≤𝜎²𝛽𝜀⁴ℎ_𝑡²ℱ(𝑡). (116) Then by Lemma 10 and from (105) we conclude that there exists a positive constant 𝐾 such that ℱ(𝑡)≤𝐾ℱ(0)𝑒^{−(𝛽/2𝑐)𝑡}, (117) and consequently from (112) we conclude (𝑢,𝑢_𝑡)²_ℋ𝜂,𝜀≤𝑐𝐾(𝑢₀,𝑢₁)²_ℋ𝜂,𝜀𝑒^{−(𝛽/2𝑐)𝑡}. (118) This shows that the semigroup 𝑁_𝜀,𝜂(𝑡) satisfies condition (95) and hence we have proven the following.

Theorem 11 11. The semigroup 𝑆_𝜀,𝜂(𝑡) possesses a connected global attractor 𝒜_𝜀,𝜂⊂ℋ_𝜀,𝜂 that is bounded in 𝒱_𝜀,𝜂.

Acknowledgment

The authors would like to thank the editor and the referees for their helpful suggestions and remarks. This work was partially supported by projects PPTA 4476 and PPTA 5571 of Pontificia Universidad Javeriana, Bogotá.