1.Introduction RenatoColucciandGerardoR.Chacón HyperbolicRelaxationofaFourthOrderEvolutionEquation ResearchArticle

Volume 2013, Article ID 372726,11pages http://dx.doi.org/10.1155/2013/372726

Research Article

Hyperbolic Relaxation of a Fourth Order Evolution Equation

Renato Colucci and Gerardo R. Chacón

Departamento de Matem´aticas, Pontificia Universidad Javeriana, Cra. 7 No. 43-82, Bogot´a, Colombia

Correspondence should be addressed to Renato Colucci; [email protected] Received 26 November 2012; Revised 30 January 2013; Accepted 3 February 2013 Academic Editor: Juan J. Nieto

Copyright © 2013 R. Colucci and G. R. Chac´on. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a hyperbolic relaxation of a fourth order evolution equation, with an inertial term𝜂𝑢_𝑡𝑡, where𝜂 ∈ (0, 1]. We prove the existence of several absorbing sets having different regularities and the existence of a global attractor that is bounded in𝐻⁴(𝐼) × {𝐻²(𝐼) ∩ 𝐻₀¹(𝐼)}.

1. Introduction

Leting𝐼 ⊂Rbe an open interval, with|𝐼| ≤ 1, we consider the following initial-boundary value problem for𝑢 : 𝐼 ×R⁺ → R:

𝜂𝑢_𝑡𝑡+ 𝑢_𝑡= − 𝜀²𝑢_{𝑥𝑥𝑥𝑥} +1

2𝑊^󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑥, (𝑥, 𝑡) ∈ 𝐼 ×R⁺, 𝑢 (𝑥, 0) = 𝑢₀(𝑥) , in 𝐼,

𝑢_𝑡(𝑥, 0) = 𝑢₁(𝑥) , in 𝐼, 𝑢 = 𝑢_𝑥𝑥= 0, in 𝜕𝐼, 𝑡 ≥ 0,

(1)

where the function𝑊(𝑝) = (𝑝²− 1)²is the so-called double- well potential, 0 < 𝜀 ≪ 1 and 𝜂 ∈ (0, 1] are positive parameters.

Problem (1), with𝜂 = 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 order𝑂(𝜀²)there is the formation of microstructure (see [2]) in the region where the gradient of the initial datum falls in the nonconvex region of𝑊; this phenomenon produces a drastic reduction of the energy of the initial datum. In the second time scale of order𝑂(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 in𝐻²(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 attractorI_𝜀 with finite fractal dimension of order 𝑂(𝜀⁻¹⁰). In [5] the authors proved the existence of an inertial manifoldM_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 order𝑂(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 order𝑂(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 ifVis the solution of the Cahn-Hilliard equation

V_𝑡+ Δ [𝜀²ΔV− 𝑊^󸀠(V)] = 0, (2) with Neumann boundary conditions:

𝜕

𝜕𝑛V= 𝜕

𝜕𝑛ΔV= 0, 𝑥 ∈ {0, 1} , (3)

then

𝑢 (𝑥) = ∫^𝑥

0 V(𝑠) 𝑑𝑠, (4)

is the solution of (1), with𝜂 = 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 attractorA_𝜂,𝜀while we have left the proof of the existence of exponential attractors for a forthcoming paper. In Sections 2and3we define the solution semigroup in the appropriate spaces and present some important energy estimates. In Section 4we 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) ∩ 𝐻₀¹(𝐼)] × 𝐿²(𝐼) , (6)

V_𝜂,𝜀= 𝐷 (𝐴) × 𝐻²(𝐼) , (7)

where

𝐷 (𝐴) = { 𝑢 ∈ 𝐻⁴(𝐼) : 𝑢 = 𝑢_𝑥𝑥= 0 in 𝜕𝐼} (8) is the domain of the differential operator𝐴 = 𝜕⁴/𝜕𝑥⁴. The above spaces are equipped with the norms

‖(𝑢,V)‖²_H𝜂,𝜀= 𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+𝜂

2‖V‖², (9)

‖(𝑢,V)‖²_V𝜂,𝜀= 𝜀²

2󵄩󵄩󵄩󵄩𝑢^{x𝑥𝑥𝑥}󵄩󵄩󵄩󵄩²+𝜂

2󵄩󵄩󵄩󵄩V_𝑥𝑥󵄩󵄩󵄩󵄩², (10) where‖ ⋅ ‖represents the𝐿²norm. We will denote by⟨⋅, ⋅⟩the inner product in𝐿²(𝐼). We recall that for all𝑢 ∈ 𝐻²(𝐼)∩𝐻₀¹(𝐼) with|𝐼| ≤ 1, we have

‖𝑢‖ ≤ 󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩. (11) Throughout the paper we will use two norms that are equiv- alent to (9) and (10) in order to simplify the computation.

Given functions𝑢 ∈ 𝐻²(𝐼) ∩ 𝐻¹₀(𝐼)andV∈ 𝐿²(𝐼)we define the functioñ𝐸onH_𝜂,𝜀by

̃𝐸 (𝑢,V) = 𝜂‖V‖²+1

2𝜀²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+ 𝛽𝜂⟨V, 𝑢⟩, (12) where𝛽 ∈ [0, 𝜀²].

Proposition 1. For all𝛽 ∈ [0, 𝜀²]the functioñ𝐸(⋅, ⋅)induces a norm equivalent to the norm onH_𝜂,𝜀.

Proof. By Schwartz inequality (11) and using the fact that𝛽 ≤ 𝜀²< 1,𝜂 ∈ (0, 1], we get

𝛽𝜂 ⟨V, 𝑢⟩ ≤ 𝛽𝜂 ‖V‖‖𝑢‖ ≤ 1

2𝜂²‖V‖²+1

2𝜀⁴󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≤ 𝜂

2‖V‖²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²= ‖(𝑢,V)‖²_H𝜀,𝜂.

(13)

From the previous inequality and by definition of̃𝐸(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+𝛽² 4 ‖𝑢‖²

≤ 𝜂‖V‖²+𝜀⁴

4‖𝑢‖²≤ 𝜂‖V‖²+𝜀² 4󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩².

(15)

Combining (13) and (15) we get

̃𝐸 (𝑢,V) ≥max{𝜂 2‖V‖²,1

4𝜀²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²} , (16) and as a consequence

̃𝐸 (𝑢,V) ≥ 1

3‖(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. For every(𝑢₀, 𝑢₁) ∈ H_𝜀,𝜂there exists a unique solution𝑢(𝑡)for the initial value problem(1)such that

𝑢 ∈ 𝐶_𝑏(R⁺; 𝐻²(𝐼) ∩ 𝐻₀¹(𝐼)) ∩ 𝐶¹_𝑏(R⁺; 𝐿²(𝐼)) . (18) If, moreover,(𝑢₀, 𝑢₁) ∈V_𝜀,𝜂, then:

𝑢 ∈ 𝐶_𝑏(R⁺; 𝐷 (𝐴)) ∩ 𝐶¹_𝑏(R⁺; 𝐻²(𝐼) ∩ 𝐻₀¹(𝐼))

∩ 𝐶²_𝑏(R⁺; 𝐿²(𝐼)) . (19)

Proposition 3. For any constant𝑅 ≥ 0there exists a positive constant𝐾 = 𝐾(𝑅)such that, for any initial data𝑢₁(0),𝑢₂(0) with‖𝑢_𝑖(0)‖_{𝐻𝜂,𝜀} ≤ 𝑅,𝑖 = 1, 2one 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) where𝑆_𝜀,𝜂(𝑡)is the solution semigroup of the problem(1).

Proof. Let𝑢₁,𝑢₂, two solutions of (1) with initial data𝑢₁(0) and𝑢₂(0). Let,𝑤 = 𝑢₁− 𝑢₂then we write

𝜂𝑤_𝑡𝑡+ 𝑤_𝑡+ 𝜀²𝑤_{𝑥𝑥𝑥𝑥}=1 2

𝑑

𝑑𝑥{𝑊^󸀠(𝑢_1𝑥) − 𝑊^󸀠(𝑢_2𝑥)} . (21) We multiply the above equation by𝑤_𝑡in𝐿²(𝐼); then

𝜂 2

𝑑

𝑑𝑡󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩²+𝜀² 2

𝑑

𝑑𝑡󵄩󵄩󵄩󵄩𝑤^𝑥𝑥󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩²

= −2 ∫

𝐼𝑤_𝑥𝑥𝑤_𝑡𝑑𝑥 + 6 ∫

𝐼{𝑢²_1𝑥𝑤_𝑥𝑥+ 𝑢_2𝑥x𝑤_𝑥[𝑢_1𝑥+ 𝑢_2𝑥]} 𝑤_𝑡𝑑𝑥

≤ 2 󵄩󵄩󵄩󵄩𝑤^𝑥𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩 + 6󵄩󵄩󵄩󵄩𝑢^1𝑥󵄩󵄩󵄩󵄩²∞󵄩󵄩󵄩󵄩𝑤^𝑥𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩

+ 12 󵄩󵄩󵄩󵄩𝑢^2𝑥𝑥󵄩󵄩󵄩󵄩(∫_𝐼𝑤²_𝑥𝑤²_𝑡[𝑢²_1𝑥+ 𝑢²_2𝑥] 𝑑𝑥)^1/2

≤ 2 {1 + 18𝐶²󵄩󵄩󵄩󵄩𝑢^1𝑥𝑥󵄩󵄩󵄩󵄩²+ 12 󵄩󵄩󵄩󵄩𝑢^2𝑥𝑥󵄩󵄩󵄩󵄩

×(󵄩󵄩󵄩󵄩𝑢^1𝑥𝑥󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩𝑢^2𝑥𝑥󵄩󵄩󵄩󵄩²)^1/2} 󵄩󵄩󵄩󵄩𝑤^𝑥𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩

≤ 2 {1 + (36𝐶²+ 24√2)𝑅²

𝜀²} 󵄩󵄩󵄩󵄩𝑤^𝑥𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩

≤ 2𝐾 󵄩󵄩󵄩󵄩𝑤^𝑥𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩 ≤ 𝐾²󵄩󵄩󵄩󵄩𝑤^𝑥𝑥󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩𝑤^𝑡󵄩󵄩󵄩󵄩²,

(22)

where we have used the following inequality (see [13] and inequality (11)):

󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²∞≤ 𝐶²󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²𝐻¹(𝐼)≤ 2𝐶²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩². (23) The constant𝐾 can be explicitly computed using the esti- mates of the absorbing set presented in the next section.

From the last inequality we get 𝑑

𝑑𝑡󵄩󵄩󵄩󵄩(𝑤,𝑤^𝑡)󵄩󵄩󵄩󵄩²H𝜀,𝜂 ≤2𝐾²

𝜀² 󵄩󵄩󵄩󵄩(𝑤,𝑤^𝑡)󵄩󵄩󵄩󵄩²H𝜀,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:

𝐸 (𝑢, 𝑢_𝑡) = 𝜂󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²+1

2𝜀²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+1 2∫

𝐼𝑊 (𝑢_𝑥) 𝑑𝑥, (26) that is not increasing along the solutions; in fact if we multiply (1) by𝑢_𝑡and integrate over𝐼we obtain

𝑑

𝑑𝑡𝐸 (𝑢, 𝑢_𝑡) = −󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩². (27) Moreover, integrating the previous inequality on(0, 𝑡)with respect to time we get

∫^𝑡

0󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²𝑑𝑠 + 𝐸 (𝑢, 𝑢_𝑡) = 𝐸 (𝑢₀, 𝑢₁) . (28) Then for any fixed initial data(𝑢₀, 𝑢₁) ∈H_𝜀,𝜂we have that the corresponding solution satisfies

sup𝑡≥0{𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²} =sup

𝑡≥0󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²H𝜀,𝜂

≤ 𝐸 (𝑢₀, 𝑢₁) .

(29)

Moreover if we integrate overR⁺we get the integral control

∫^∞

0 󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²𝑑𝑡 ≤ 𝐸 (𝑢₀, 𝑢₁) . (30) We consider an important estimate that will be useful later.

Let𝛽 > 0be a parameter to be determined later; if we multiply (1) by𝛽𝑢we obtain

𝛽𝜂 ∫𝐼𝑢_𝑡𝑡𝑢𝑑𝑥 +𝛽 2

𝑑

𝑑𝑡‖𝑢‖²+ 𝛽𝜀²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩² + 2𝛽󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4= 2𝛽󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩².

(31)

By summing (27) and (31) we get 𝑑

𝑑𝑡𝐸 + 𝛽𝜂𝑑

𝑑𝑡⟨𝑢_𝑡, 𝑢⟩ + (1 − 𝛽𝜂) 󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩² +𝛽

2 𝑑

𝑑𝑡‖𝑢‖²+ 𝛽𝜀²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+ 2𝛽󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4

= 2𝛽󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩².

(32)

Using the expression of the energy we can rewrite the previous in the following way:

𝑑

𝑑𝑡𝐸 + 𝛽𝜂𝑑

𝑑𝑡⟨𝑢_𝑡, 𝑢⟩ + (1 − 3𝛽𝜂) 󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩² + 𝛽⟨𝑢_𝑡, 𝑢⟩ + 2𝛽 (𝐸 +󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4

2 − |𝐼|

2) = 0.

(33)

We will estimate some of the terms of the previous inequality in the following lemma.

Lemma 4. Fix𝜂 ≤ 1then for all𝛽 ∈ (0, 1/3)one has 2𝛽²𝜂⟨𝑢_𝑡, 𝑢⟩ ≤ (1 − 3𝛽𝜂) 󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²+ 𝛽󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4+ 𝛽 |𝐼| + 𝛽⟨u_𝑡, 𝑢⟩.

(34) Proof. By Holder’s inequality we get

2󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²≤ 2󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²4 |𝐼|^1/2≤ 󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4+ |𝐼| , (35) then we use Poincar´e’s inequality and the fact that|𝐼| ≤ 1to conclude that

‖𝑢‖²≤ 󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4

2 + |𝐼|

2 . (36)

Therefore, for a positive constant𝑐₁to be determined later, we have that

⟨𝑢_𝑡, 𝑢⟩ ≤ 󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩‖𝑢‖ ≤ 𝑐2¹󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²+ 1 2𝑐₁‖𝑢‖²

≤ 𝑐₁

2󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²+ 1

2𝑐₁(󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4

2 + |𝐼|

2 ) .

(37)

Consequently, by choosing the parameters𝑐₁= 1/12and𝛽 <

1/3we have that

(2𝛽²𝜂 − 𝛽) ⟨𝑢_𝑡, 𝑢⟩

≤ 󵄨󵄨󵄨󵄨󵄨2𝛽²𝜂 − 𝛽󵄨󵄨󵄨󵄨󵄨𝑐₁ 2󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩² +󵄨󵄨󵄨󵄨󵄨2𝛽²𝜂 − 𝛽󵄨󵄨󵄨󵄨󵄨

4𝑐₁ 󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4+󵄨󵄨󵄨󵄨󵄨2𝛽²𝜂 − 𝛽󵄨󵄨󵄨󵄨󵄨

4𝑐₁ |𝐼|

≤ (1 − 3𝛽𝜂) 󵄩󵄩󵄩󵄩𝑢^𝑡󵄩󵄩󵄩󵄩²+ 𝛽󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴⁴+ 𝛽 |𝐼| .

(38)

Now, using the previous lemma and (33) we conclude that 𝑑

𝑑𝑡𝐸 + 𝛽𝜂𝑑

𝑑𝑡⟨𝑢_𝑡, 𝑢⟩ + 2𝛽²𝜂 ⟨𝑢_𝑡, 𝑢⟩ + 2𝛽𝐸 ≤ 2𝛽 |𝐼| . (39) If we set

Φ (𝑢, 𝑢_𝑡) = Φ (𝑡) = 𝐸 (𝑢, 𝑢𝑡) + 𝛽𝜂⟨𝑢_𝑡, 𝑢⟩, (40) then from (39) we get

𝑑

𝑑𝑡Φ (𝑡) + 2𝛽Φ (𝑡) ≤ 2𝛽. (41)

Now, integrating we have that

Φ (𝑡) ≤ [Φ (𝑢₀, 𝑢₁) − 1] 𝑒^−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 functions𝐸,̃𝐸, andΦ:

Φ (𝑡) = ̃𝐸 (𝑡) +1 2∫

𝐼𝑊 (𝑢_𝑥) 𝑑𝑥 ≥ ̃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 spaceH_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

1 2∫

𝐼𝑊 (𝑢_𝑥) 𝑑𝑥 ≤ 1

2󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²𝐿^∞(𝐼)− 󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²+ |𝐼|

2

≤ 1

2󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+ |𝐼|

2

≤ 1

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩⁴+ |𝐼|

2 .

(44)

We will use this inequality several times in what follows.

4.1. Absorbing Set inH_𝜀,𝜂

Proposition 5. For all𝑅₀> 1the set

𝐵_Φ:= {(𝑢, 𝑢_𝑡) ∈H_𝜀,𝜂: Φ (𝑢, 𝑢_𝑡) ≤ 𝑅₀} (45) is bounded, absorbing, and positively invariant for the semi- group𝑆_𝜀,𝜂(𝑡)inH_𝜀,𝜂.

Proof. The set is positively invariant, in fact if(𝑢₀, 𝑢₁) ∈H_𝜀,𝜂 withΦ(𝑢₀, 𝑢₁) ≤ 𝑅₀we have, from (42), that

Φ (𝑢, 𝑢_𝑡) ≤ [𝑅₀− 1] 𝑒^−2𝛽𝑡+ 1 ≤ 𝑅₀, ∀𝑡 > 0. (46) The boundness of𝐵_Φfollows directly from (42). Now suppose that(𝑢₀, 𝑢₁)are such thatΦ(𝑢₀, 𝑢₁) ≤ 𝑅, with𝑅 > 𝑅₀then again from (42) we get

Φ (𝑢, 𝑢_𝑡) ≤ 𝑅₀, ∀𝑡 ≥ 𝜏_Φ, (47) where

𝜏_Φ:= 1

2𝛽log{𝑅 − 1

𝑅₀− 1} . (48)

Then𝐵_Φis absorbing for the semigroup𝑆_𝜀,𝜂(𝑡).

Proposition 6. For all𝑅₁> 1one has that the set

𝐵_̃𝐸:= {(𝑢, 𝑢_𝑡) ∈H_𝜀,𝜂: ̃𝐸 (𝑢, 𝑢_𝑡) ≤ 𝑅₁} (49) is absorbing for the semigroup𝑆_𝜀,𝜂(𝑡).

Proof. We have from (42) and (43) that

̃𝐸 (𝑢, 𝑢_𝑡) ≤ [Φ (𝑢₀, 𝑢₁) − 1] 𝑒^−2𝛽𝑡+ 1. (50) Now, take(𝑢₀, 𝑢₁)such that̃𝐸(𝑢₀, 𝑢₁) ≤ 𝑅, with𝑅 > 𝑅₁. Then, by (42) and (44) we get that

̃𝐸 (𝑡) ≤ Φ (𝑡) ≤ ̃𝐸 (𝑡) + 12󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩⁴4+ |𝐼|

2

≤ ̃𝐸 (𝑡) +1

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩⁴+ |𝐼|

2

≤ ̃𝐸 (𝑡) + 4

𝜀²̃𝐸(𝑡)²+ |𝐼|

2 .

(51)

Therefore,

̃𝐸 (𝑢, 𝑢_𝑡) ≤ {̃𝐸 (𝑢₀, 𝑢₁) + 4

𝜀²̃𝐸(𝑢₀, 𝑢₁)²+ |𝐼|

2 − 1} 𝑒^−2𝛽𝑡+ 1

≤ {𝑅 + 4

𝜀²𝑅²+ |𝐼|

2 − 1} 𝑒^−2𝛽𝑡+ 1.

(52) Consequently, we have that

̃𝐸 (𝑢, 𝑢_𝑡) ≤ 𝑅₁, ∀𝑡 ≥ 𝜏_̃𝐸, (53) where

𝜏_̃𝐸= 1

2𝛽log{𝑅 + (4/𝜀⁴) 𝑅²+ (|𝐼| /2) − 1

𝑅₁− 1 } . (54)

Now using the equivalence of the norm onH_𝜀,𝜂and (52) we get

󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²H𝜀,𝜂 ≤ 3 {3󵄩󵄩󵄩󵄩(𝑢⁰, 𝑢₁)󵄩󵄩󵄩󵄩²H𝜀,𝜂

+36

𝜀²󵄩󵄩󵄩󵄩(𝑢⁰, 𝑢₁)󵄩󵄩󵄩󵄩⁴H_𝜀,𝜂+ |𝐼|

2 − 1} 𝑒^−2𝛽𝑡+ 3.

(55) Let𝑅₂> √3. If(𝑢₀, 𝑢₁) ∈H_𝜀,𝜂is 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= 1

2𝛽log3 { 3𝑅²+ (36/𝜀²) 𝑅⁴+ |𝐼| /2 − 1}

𝑅²₂− 3 . (58)

Thus, we have proved the following.

Proposition 7. For all𝑅₂> √3one has that the ball

𝐵_H𝜀,𝜂:= {(𝑢, 𝑢_𝑡) ∈H_𝜀,𝜂: 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩H_𝜀,𝜂≤ 𝑅₂} (59) is an absorbing set for the semigroup𝑆_𝜀,𝜂(𝑡)inH_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) ∩ 𝐻₀¹(𝐼)} × 𝐻¹(𝐼) , (60) equipped with the norm

󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂 =𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑡𝑥󵄩󵄩󵄩󵄩²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩². (61) Then we have the following.

Proposition 8. There exists𝑅₃> 0such that the closed ball

B₃= {(𝑢, 𝑢_𝑡) ∈U_𝜀,𝜂: 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩U𝜀,𝜂≤ 𝑅₃} , (62)

is a bounded absorbing set for𝑆_𝜀,𝜂(𝑡)inU_𝜀,𝜂.

Proof. If we multiply (1) by𝑢_𝑥𝑥𝑡+ 𝑢_𝑥𝑥and integrate over𝐼we obtain

𝑑 𝑑𝑡{𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑡𝑥󵄩󵄩󵄩󵄩²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²+1

2󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²+ 𝜂⟨𝑢_𝑡𝑥, 𝑢_𝑥⟩}

+ (1 − 𝜂) 󵄩󵄩󵄩󵄩𝑢^𝑡𝑥󵄩󵄩󵄩󵄩²+ 𝜀²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²

= −1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥, 𝑢_𝑥𝑥𝑡⟩ −1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥, 𝑢_𝑥𝑥⟩ . (63) Let us denote the differential term of the previous inequality as

Ψ₀(𝑡) = 𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑡𝑥󵄩󵄩󵄩󵄩²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²+ 𝜂 ⟨𝑢_𝑡𝑥, 𝑢_𝑥⟩ +1

2󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩².

(64)

We will estimate the right hand side of (63). The first term can be estimated as follows:

−1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥, 𝑢_𝑥𝑥⟩ ≤ 1

2󵄩󵄩󵄩󵄩󵄩𝑊^󸀠󸀠(𝑢_𝑥)󵄩󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≤ 2 (3󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²∞+ 1) 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≤ 2 (3󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+ 1) 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≤ 4

𝜀²𝑅²₂(6𝑅²₂

𝜀² + 1) := 𝐶₁. (65)

Let us define

Ψ₁= 1 4∫

𝐼𝑊^󸀠󸀠(𝑢_𝑥) 𝑢²_𝑥𝑥𝑑𝑥; (66) then we can rewrite the the second term of r.h.s of (63)

−1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥, 𝑢_𝑥𝑥𝑡⟩ = −𝑑 𝑑𝑡Ψ₁+1

4∫

𝐼𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑡𝑢²_𝑥𝑥, (67) and estimate

1 4∫

𝐼𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑡𝑢²_𝑥𝑥

≤ 6󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩∞∫

𝐼𝑢_𝑥𝑢_𝑥𝑡𝑢_𝑥𝑥𝑑𝑥

≤ 6󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑢^𝑥𝑢_𝑥𝑥󵄩󵄩󵄩󵄩

≤ 6󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≤ 6󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩^5/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩^1/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩

≤ 6(√2 𝜀 𝑅₂)

5/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩^1/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩

≤𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩²+18 𝜂(√2

𝜀 𝑅₂)

5󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩

≤𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩²+1 2[18

𝜂𝜀(√2 𝜀 𝑅₂)

5

]

2

+𝜀² 2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩² := 𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²+ 𝐶₂.

(68)

Then if𝜂 ∈ (0, 1/2)and settingΨ := Ψ₀+ Ψ₁,𝐶 := 𝐶₁+ 𝐶₂ we have

𝑑 𝑑𝑡Ψ +𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²≤ 𝐶. (69) To conclude the proof we note that

Ψ ≥ 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²U_𝜀,𝜂+1 2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

− 𝜂 󵄩󵄩󵄩󵄩𝑢^𝑡𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≥ 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²U_𝜀,𝜂−𝜂

4󵄩󵄩󵄩󵄩𝑢^𝑡𝑥󵄩󵄩󵄩󵄩²−3 2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≥ 1

2󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂− 3 𝜀²𝑅²₂ := 1

2󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂− 𝑐₁,

(70)

and that

Ψ ≤ 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²U_𝜀,𝜂+𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑡𝑥󵄩󵄩󵄩󵄩²+ 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩² + 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²(3󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩^2∞+ 1)

≤ 2󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂+ 2 𝜀²𝑅²₂(6

𝜀²𝑅²₂+ 2) := 2󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂+ 𝑐₂.

(71)

From the previous inequalities we get 1

2󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂− 𝑐₁≤ Ψ ≤ 2󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂+ 𝑐₂. (72)

Then from (72) we get 𝑑 𝑑𝑡Ψ +1

2Ψ ≤ 𝐶 + 𝑐₂:= ̃𝐶, (73)

and by Gronwall’s lemma we obtain

Ψ (𝑡) ≤ (Ψ (0) − 2̃𝐶) 𝑒^−(1/2)𝑡+ 2̃𝐶,

󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²U𝜀,𝜂 ≤ 2 (󵄩󵄩󵄩󵄩(𝑢⁰, 𝑢₁)󵄩󵄩󵄩󵄩²U𝜀,𝜂+ 𝑐₂− 2̃𝐶) 𝑒^−(1/2)𝑡 + 4̃𝐶 + 2𝑐₁.

(74)

Then if(𝑢₀, 𝑢₁) ∈U_𝜀,𝜂such that

󵄩󵄩󵄩󵄩(𝑢⁰, 𝑢₁)󵄩󵄩󵄩󵄩U𝜀,𝜂≤ 𝑅, (75) we have that

󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩U𝜀,𝜂≤ 𝑅₃, ∀𝑡 ≥ 𝜏_𝑈, (76) where

𝜏_𝑈:= 2log(2𝑅²+ 2𝑐₂− 4̃𝐶

𝑅²₃− 4̃𝐶 − 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. There exists𝑅₄> 0such 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 semigroup𝑆_𝜀,𝜂(𝑡)inV_𝜀,𝜂. Proof. Multiply (1) by𝑢_{𝑥𝑥𝑥𝑥𝑡}+ 𝑢_{𝑥𝑥𝑥𝑥}; then we obtain

𝑑 𝑑𝑡{𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑡𝑥𝑥󵄩󵄩󵄩󵄩²+𝜀²

2󵄩󵄩󵄩󵄩𝑢^{𝑥𝑥𝑥𝑥}󵄩󵄩󵄩󵄩² +𝜂 ⟨𝑢_𝑡𝑥𝑥, 𝑢_𝑥𝑥⟩ +1

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²} + (1 − 𝜂) 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩 + 𝜀²󵄩󵄩󵄩󵄩𝑢^{𝑥𝑥𝑥𝑥}󵄩󵄩󵄩󵄩²

= 1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥, 𝑢_{𝑥𝑥𝑥𝑥𝑡}+ 𝑢_{𝑥𝑥𝑥𝑥}⟩ .

(79)

We call𝜃₀(𝑡)the differential term of the previous inequality and we estimate the r.h.s.:

1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥, 𝑢_{𝑥𝑥𝑥𝑥}⟩

= −1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥𝑥, 𝑢_𝑥𝑥𝑥⟩

= −1

2⟨𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢²_𝑥𝑥, 𝑢_𝑥𝑥𝑥⟩ −1

2⟨[𝑊^󸀠󸀠(𝑢_𝑥)]_𝑥𝑥, 𝑢²_𝑥𝑥𝑥⟩

= −12 ∫

𝐼𝑢_𝑥𝑢²_𝑥𝑥𝑢_𝑥𝑥𝑥𝑑𝑥 − 2 ∫

𝐼(3𝑢²_𝑥− 1) 𝑢²_𝑥𝑥𝑥𝑑𝑥

≤ 2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²+ 12󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²∞󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩

≤ 2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²(6󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+ 1)

≤ 𝜀²𝑅²₃(12

𝜀²𝑅²₂+ 1) := 𝐴₁.

(80) Let us define

𝜃₁(𝑡) :=1 4∫

𝐼𝑊^󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑥𝑥² 𝑑𝑥; (81) then

1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥, 𝑢_{𝑥𝑥𝑥𝑥𝑡}⟩

= −1

2⟨[𝑊^󸀠(𝑢_𝑥)]_𝑥𝑥, 𝑢_{𝑥𝑥𝑥𝑡}⟩

= −1

2⟨𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢²_𝑥𝑥, 𝑢_{𝑥𝑥𝑥𝑡}⟩

−1

2⟨𝑊^󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑥𝑥, 𝑢_{𝑥𝑥𝑥𝑡}⟩

= −𝑑 𝑑𝑡𝜃₁+1

2⟨[𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢²_𝑥𝑥]_𝑥, 𝑢_𝑥𝑥𝑡⟩ +1

4∫

𝐼𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑡𝑢²_𝑥𝑥𝑥𝑑𝑥.

(82)

We estimate the last two terms of the previous equality:

1 4∫

𝐼𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑡𝑢²_𝑥𝑥𝑥𝑑𝑥

= 6 ∫

𝐼𝑢_𝑥𝑢_𝑥𝑡𝑢²_𝑥𝑥𝑥𝑑𝑥

≤ 6󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩

≤ 6󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩^3/2󵄩󵄩󵄩󵄩𝑢^{𝑥𝑥𝑥𝑥}󵄩󵄩󵄩󵄩^1/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≤ (12 ⋅ 2^3/4𝑅²₂𝑅^3/2₃

𝜀^7/2 ) 󵄩󵄩󵄩󵄩𝑢^{𝑥𝑥𝑥𝑥}󵄩󵄩󵄩󵄩^1/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩

≤ (12 ⋅ 2^3/4𝑅²₂𝑅^3/2₃ 𝜀^7/2 )

2󵄩󵄩󵄩󵄩𝑢^{𝑥𝑥𝑥𝑥}󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑢^𝑥𝑡󵄩󵄩󵄩󵄩²

≤ 1

2𝜀²(12 ⋅ 2^3/4𝑅²₂𝑅^3/2₃ 𝜀^7/2 )

4

+2 𝜂𝑅²₃+𝜀²

2󵄩󵄩󵄩󵄩𝑢^{𝑥𝑥𝑥𝑥}󵄩󵄩󵄩󵄩² := 𝜀²

2󵄩󵄩󵄩󵄩𝑢^{𝑥𝑥𝑥𝑥}󵄩󵄩󵄩󵄩²+ 𝐴₂, 1

2⟨[𝑊^󸀠󸀠󸀠(𝑢_𝑥) 𝑢²_𝑥𝑥]_𝑥, 𝑢_𝑥𝑥𝑡⟩

= 1

2⟨𝑊^𝑖V(𝑢_𝑥) 𝑢³_𝑥𝑥+ 𝑊^󸀠󸀠󸀠(𝑢_𝑥) 2𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥, 𝑢_𝑥𝑥𝑡⟩

= 12 ∫

𝐼𝑢³_𝑥𝑥𝑢_𝑥𝑥𝑡𝑑𝑥 + 24 ∫

𝐼𝑢_𝑥𝑢_𝑥𝑥𝑢_𝑥𝑥𝑥𝑢_𝑥𝑥𝑡𝑑𝑥

≤ 12 (󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩

+2󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩)󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩

≤ 12 (󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩

+2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩^3/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩^3/2) 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩

≤ 𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩²+ 72 (󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩

+2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩^3/2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩^3/2)²

≤ 𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩²+ 72[2√2𝑅²₂𝑅₃

𝜀³ + 2(2𝑅₂𝑅₃ 𝜀² )^3/2]

2

:= 𝜂

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩²+ 𝐴₃.

(83)

Then if we set𝜃(𝑡) = 𝜃₀(𝑡) + 𝜃₁(𝑡)we get

𝑑

𝑑𝑡𝜃 (𝑡) + 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²V_𝜀,𝜂 ≤ 𝐴₁+ 𝐴₂+ 𝐴₃:= 𝐴. (84)

Moreover we have

𝜃 (𝑡) ≥ 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂−𝜂

4󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩²−1 2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²

≥ 1

2󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂−1 2𝑅₂² := 1

2󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂− 𝑎₁,

𝜃 (𝑡) ≤ 󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂+ 𝜂 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑡󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩

+1

2󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩²+1

4󵄩󵄩󵄩󵄩󵄩𝑊^󸀠󸀠(𝑢_𝑥)󵄩󵄩󵄩󵄩󵄩∞󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩³

≤ 2󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂+ 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥󵄩󵄩󵄩󵄩² + (3󵄩󵄩󵄩󵄩𝑢^𝑥󵄩󵄩󵄩󵄩²∞+ 1) 󵄩󵄩󵄩󵄩𝑢^𝑥𝑥𝑥󵄩󵄩󵄩󵄩²

≤ 2󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂+ 2

𝜀²𝑅²₂+ (6

𝜀²𝑅²₂+ 1) 2 𝜀²𝑅²₃ := 2󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂+ 𝑎₂.

(85) Then putting all together we get

1

2󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂− 𝑎₁≤ 𝜃 (𝑡) ≤ 2󵄩󵄩󵄩󵄩(𝑢, 𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂+ 𝑎₂. (86) From the previous inequality we get

𝑑

𝑑𝑡𝜃 (𝑡) + 1

2𝜃 (𝑡) ≤ 𝐴 +𝑎₂

2 := ̃𝐴 (87)

and by Gronwall’s lemma

𝜃 (𝑡) ≤ [𝜃 (0) − 2̃𝐴] 𝑒^−(1/2)𝑡+ 2̃𝐴. (88) Using again (86) we get

󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩²V𝜀,𝜂 ≤ 2 [2󵄩󵄩󵄩󵄩(𝑢⁰, 𝑢₁)󵄩󵄩󵄩󵄩²V𝜀,𝜂+ 𝑎₂− 2̃𝐴] 𝑒^−(1/2)𝑡 + 4̃𝐴 + 2𝑎₁.

(89)

Then if(𝑢₀, 𝑢₁) ∈V_𝜀,𝜂such that

󵄩󵄩󵄩󵄩(𝑢⁰, 𝑢₁)󵄩󵄩󵄩󵄩V𝜀,𝜂 ≤ 𝑅, (90) then there exists𝑅₄> 0:

󵄩󵄩󵄩󵄩(𝑢,𝑢^𝑡)󵄩󵄩󵄩󵄩V𝜀,𝜂 ≤ 𝑅₄, ∀𝑡 ≥ 𝜏_𝑉, (91) where

𝜏_𝑉:= 2log[2 (2𝑅²+ 𝑎₂− 2̃𝐴)

𝑅₄²− 4̃𝐴 − 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 inH_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 setB⊂H_𝜀,𝜂, the solution semigroup𝑆_𝜀,𝜂(𝑡)admits the decomposition:

𝑆_𝜀,𝜂(𝑡) = 𝐿_𝜀,𝜂(𝑡) + 𝑁_𝜀,𝜂(𝑡) , (93)

such that

(𝐶1) sup

𝑡≥0 sup

𝑧∈H𝜀,𝜂󵄩󵄩󵄩󵄩󵄩𝐿^𝜀,𝜂(𝑡)𝑧󵄩󵄩󵄩󵄩󵄩V𝜀,𝜂 < ∞, (94)

(𝐶2) lim

𝑡 → ∞{sup

𝑧∈H𝜀,𝜂󵄩󵄩󵄩󵄩󵄩𝑁^𝜀,𝜂(𝑡) 𝑧󵄩󵄩󵄩󵄩󵄩H𝜀,𝜂} = 0. (95) LetB ⊂ H_𝜀,𝜂be a fixed bounded set and let(𝑢₀, 𝑢₁) ∈ B⊂H_𝜀,𝜂. We will define the decomposition of𝑆_𝜀,𝜂as 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)

whereℎandVare solutions of the following problems:

𝜂ℎ_𝑡𝑡+ ℎ_𝑡+ 𝜀²ℎ_{𝑥𝑥𝑥𝑥}− 𝛼ℎ_𝑥𝑥

= 1

2𝑊^󸀠󸀠(ℎ_𝑥) ℎ_𝑥𝑥− 𝛼𝑢_𝑥𝑥, ℎ (𝑥, 0) = 0,

ℎ_𝑡(𝑥, 0) = 0, ℎ = ℎ_𝑥𝑥= 0, in𝜕𝐼,

(97)

𝜂V_𝑡𝑡+V_𝑡+ 𝜀²V_{𝑥𝑥𝑥𝑥}− 𝛼V_𝑥𝑥

= 1

2𝑊^󸀠󸀠(𝑢_𝑥) 𝑢_𝑥𝑥−1

2𝑊^󸀠󸀠(ℎ_𝑥) ℎ_𝑥𝑥, V(0) = 𝑢₀,

V_𝑡(0) = 𝑢₁, V=V_𝑥𝑥= 0, in 𝜕𝐼,

(98)

where

𝛼 ≥ 6√2

𝜀 𝑅₂> 6√6. (99)

Before showing that the semigroups𝐿_𝜀,𝜂(𝑡)and𝑁_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. Let𝜓 : R⁺ → Rbe an absolutely continuous function which fulfills, for some]> 0and almost every𝑡 ≥ 0, the differential inequality

𝑑

𝑑𝑡𝜓 (𝑡) + 2]𝜓 (𝑡) ≤ 𝑓 (𝑡) 𝜓 (𝑡) , (100) where𝑓is a positive function satisfying

∫^𝑡

0𝑓 (𝑦) 𝑑𝑦 ≤]𝑡 + 𝑐, ∀𝑡 ≥ 0. (101) Then

𝜓 (𝑡) ≤ 𝑐𝜓 (0) 𝑒^−]𝑡, 𝑡 ≥ 0. (102)