Dimension Estimate for the Global Attractor of an Evolution Equation

Abstract

We estimate the dimension of the global attractor of an evolution equation by the study of the evolution of the n-dimensional volumes under the flow. We compare these results with the estimate of the dimension of the inertial manifold.

1. Introduction

One of the most interesting problems in the analysis of partial differential equations is the study of the asymptotic behavior of solutions. Guided by the finite-dimensional case, in the last decades, the concept of global attractor was introduced in order to study the long-term dynamics of dissipative equations (see, e.g., [1, 2] and the references therein). This theory has been generalized to the case on non-autonomous systems by introducing the definition of uniform attractor; see, for example, the interesting article [3] in which a reaction diffusion equation is studied with nonlinear boundary conditions in competitions with the dissipative terms. However, the existence of uniform attractors fails in many cases and that is the reason why the new definition of Pullback Attractor is introduced (see [4]). This new concept works for both the non-autonomous (see [5]) and the random systems, for parabolic (see [6]), and hyperbolic stochastic equations (see [7]) in bounded and in unbounded domains. A further generalization of the concept of attractor to the setting of multivalued processes can be seen in [8], and an application to the case of 3D Navier Stokes systems can be seen in [9].

Once the existence of a global attractor has been established, in order to analyze its structure, it is a common question to investigate the (fractal) dimension of it. The classical method to do that consists in the study of the contraction of an 𝑛-dimensional volume element under the action of the flow (see [1, 2]). This is a very sharp method that works also in the case of non-autonomous systems; see, for instance, [5] in which also is pointed out the influence of the geometry of the domain in determining the dimension of attractors.

This method also works in numerical context; in fact there are also numerical schemes (see, e.g., [10]) that preserve important properties of the solution and that preserve the dimension of the attractor (see, for instance, [11]).

The disadvantage of the study of volume contraction method is that it requires higher regularity of the data and the solutions this can be avoided by using the theory of exponential attractors (see [12]) or the so-called method of l-trajectories (see, e.g., [13]). Both methods require less regularity and prove finite dimension of the attractor.

In this paper, we will study the dimension of the global attractor of an autonomous evolution equation by using different methods. The autonomous evolution equation considered here is a simple model of a dynamical system which presents different time scales in which the dynamics presents different features. We consider the following fourth-order evolution equation:𝑢_𝑡=−𝜀²𝑢_{𝑥𝑥𝑥𝑥}+12Φ^′′𝑢_𝑥𝑢_𝑥𝑥,(𝑥,𝑡)∈𝐼×ℝ₊,𝑢(0,𝑥)=𝑢₀(𝑥),in𝐼,𝑢=𝑢_𝑥𝑥=0,in𝜕𝐼×ℝ₊, (1.1) where the function Φ(𝑝)=(𝑝²−1)² is the so-called double well potential and 𝐼⊂ℝ is an open interval such that its length |𝐼|≤1. In particular, (1.1) is the 𝐿²-gradient flow associated to the functional:𝐹_𝜀(𝑢)=12𝜀²_𝐼𝑢²_𝑥𝑥1𝑑𝑥+2_𝐼Φ𝑢_𝑥𝑑𝑥. (1.2) In [14] it is studied the global dynamics of an evolution equation like (1.1) with a more general nonconvex function Φ. They use numerical experiments to discuss the dynamical behavior of the solutions for small values of 𝜀. In particular it was pointed out the existence of three different time scales with peculiar dynamic behavior.

In a first time scale of order 𝑡>𝑇_𝜀=𝑂(𝜀²), there is a drastic reduction of the energy 𝐹_𝜀(𝑢₀) of the initial data 𝑢₀ and a formation of microstructure in the region where Φ(⋅) is not convex.

In the second time scale of order 𝑡>𝑇=𝑂(1), we have that the equation exhibits a heat equation-like behavior in the convex regions while slow motion in the nonconvex ones.

In the third time scale of order 𝑡>1/𝑇_𝜀=𝑂(𝜀⁻²), the infinite dynamical system is reduced to a finite one. The solution is approximately the union of consecutive segments. The dynamic is slow and the number of segments decreases letting decrease the number of freedom degrees of the problem.

We use the following notation throughout the paper. We develop the analysis in the following spaces:𝐻=𝐿²(𝐼),𝑉=𝐻²(𝐼). (1.3)

In [15] the asymptotic behavior of (1.1) was studied and it was proven the existence of a global attractor for the semigroup associated to (1.1), that is, the semigroup of operators defined as𝑆(𝑡)∶𝑢₀∈𝐻⟶𝑢(𝑡)∈𝐻, (1.4) The main tool in proving the existence of a global attractor is to show the existence of an absorbing set in 𝐻²(𝐼) (see Theorem  1.1 in [1]).

Theorem 1.1 (see [15]). There exist positive constants 𝑟_𝑖, 𝜏_𝑖 with 𝑖=0,1,2 such that (i)the semigroup 𝑆(𝑡) posses an absorbing set in 𝐻: ‖𝑆(𝑡)𝑢₀‖≤𝑟₀,if𝑡≥𝜏₀, (1.5) where 𝑟₀=|𝐼|1+𝜀^21/2,𝜏₀=1𝜀²‖𝑢₀‖; (1.6)(ii)the semigroup 𝑆(𝑡) posses an absorbing set in 𝐻¹₀(𝐼): ‖𝑆(𝑡)𝑢₀‖_𝐻¹_0(𝐼)≤𝑟₁,if𝑡≥𝜏₁, (1.7) where 𝑟₁=𝑟²₀11+2𝜀⁴+|𝐼|𝜀^21/2𝑒,𝜏₁=𝜏₀+𝜀²; (1.8)(iii)the semigroup 𝑆(𝑡) posses an absorbing set in 𝐻²(𝐼): 𝑆(𝑡)𝑢_0𝐻2(𝐼)≤𝑟²₀+𝑟²₁+𝑟²₂^1/2,if𝑡≥𝜏₂, (1.9) where𝑟₂=𝑒²𝜀²180𝑟⁶₁+𝑟²₀2+|𝐼|𝜀^21/2,𝜏₂=𝜏₁+𝜀². (1.10)

Theorem 1.1 let us conclude that the setℬ_𝜀=𝑢∈𝐻∶𝑢_𝑥≤𝑟₁,𝑢_𝑥𝑥≤𝑟₂ (1.11) is absorbing for all the bounded sets of 𝐻. In particular the global attractor is given as the omega limit set of the absorbing set, 𝒜_𝜀=𝜔(ℬ). The authors also obtained in [15] a bound for the dimension of 𝒜_𝜀 by proving the existence of an exponential attractor ℐ_𝜀 whose dimension is of order 𝑂(𝜀⁻¹⁰). Moreover, in ℬ_𝜀 we have the following estimates for 𝑡>𝜏₂:‖𝑢‖_∞≤‖𝑢‖^1/2𝑢_𝑥^1/2≤𝑟₀𝑟₁^1/2,𝑢_𝑥∞≤𝑢_𝑥^1/2𝑢_𝑥𝑥^1/2≤𝑟₁𝑟₂^1/2. (1.12)(1.13) In accordance with the numerical experiments, the time in which the bounded solutions enter the absorbing set is of order 𝑂(𝜀⁻²) (see [14, 15]).

We are interested in finding estimates for the dimension of 𝒜_𝜀. In Section 2 we find an estimate of the fractal and Hausdorff dimension of 𝒜_𝜀 by studying the evolution of 𝑛-dimension volume elements, this following a classical theory by Temam (see [1]). In Section 3 we prove the existence of an inertial manifold ℳ_𝜀 (see [16]) for the system (1.1), that is, a smooth manifold that attracts all the orbit at an exponential rate and that contains the global attractor. By estimating the dimension of ℳ_𝜀, we obtain, in another way, a different estimate of dim(𝒜_𝜀). In Section 4 we study the regularity of the attractor while in the last section we prove the existence of an absorbing set in 𝐿^∞(𝐼). This is fundamental information in order to study the approximation of the attractor (see [17]).

We define the linear operator 𝐴=𝜕⁴/𝜕𝑥⁴. The set 𝐷(𝐴)={𝑢∈𝐻⁴(𝐼)∶𝑢=𝑢_𝑥𝑥=0in𝜕𝐼} is its domain. Also, then (𝐴𝑢,𝑢)=‖𝑢_𝑥𝑥‖² where ‖⋅‖∶=‖⋅‖_𝐿2(𝐼) and the operator is self-adjoint in 𝐷(𝐴). From classical results on solutions of evolution equations (see [18] or see Theorems 3.1 and 3.3 on [1]), we have the following:

Theorem 1.2. Problem (1.1) admits a unique solution 𝑢 such that if𝑢₀∈𝐻⟹𝑢∈𝐶⁰([0,𝑇],𝐻)∩𝐿²(0,𝑇;𝑉),∀𝑇>0;if𝑢₀∈𝑉⟹𝑢∈𝐶([0,𝑇],𝑉)∩𝐿²(0,𝑇;𝐷(𝐴)),∀𝑇>0. (1.14)

2. Dimension of the Attractor

Following [1, 2] we give an estimate of the dimension of the attractor 𝒜_𝜀 by estimating how the 𝑛-dimensional volume elements are distorted by the flow. We will show the following theorem

Theorem 2.1. The global attractor 𝒜_𝜀 of the semigroup has fractal dimension 𝑑_𝐹(𝒜_𝜀)<2𝑛 where 𝑛∶𝑂(𝜀⁻¹).

Proof. In general if we consider an evolution equation of the type 𝑢_𝑡(𝑢),𝑢(0)=𝐹=𝑢₀, (2.1) then the idea is to investigate the evolution of 𝑛-dimensional infinitesimal volumes and find the smallest 𝑛 such that all the 𝑛-infinitesimal volumes contract exponentially. Then one may guess that the attractor does not contain these elements and so that the dimension of 𝒜_𝜀 is less than or equal to 𝑛. We study the evolution of a set of infinitesimal displacement 𝛿𝑥^(𝑖) about a trajectory 𝑣(𝑡). In order to do that, we linearize the equation around the solution 𝑣(𝑡)=𝑆(𝑡)𝑢₀: 𝑈_𝑡=𝐹^′𝑆(𝑡)𝑢₀𝑈∶=𝐿𝑡;𝑢₀𝑈(0)𝑈,=𝜉, (2.2) where 𝐹′ is the Fráchet derivative of 𝐹. Each displacement evolves 𝑑𝑑𝑡𝛿𝑥^(𝑖)=𝐿𝑡;𝑢₀𝛿𝑥^(𝑖). (2.3) The volume is given by 𝑉_𝑛(𝑡)=|𝛿𝑥⁽¹⁾∧⋯∧𝛿𝑥^(𝑛)| where 𝛿𝑥⁽¹⁾∧⋯∧𝛿𝑥^(𝑛)2=det𝛿𝑥^(𝑖),𝛿𝑥^(𝑗)_𝑖,𝑗∶=det𝑀(𝑡), (2.4) then (see, e.g., [2] for details) 𝑑𝑉𝑑𝑡^𝑛(𝑡)=12𝑇𝑅𝑀⁻¹𝑑𝑀𝑑𝑡. (2.5) Then if 𝑃_𝑛(𝑡) is the projector on the space spanned by the displacements 𝛿𝑥^(𝑖) with base 𝜑₁(𝑡),…,𝜑_𝑛(𝑡), we have 𝑉_𝑛(𝑡)=𝑉_𝑛(0)exp^𝑡₀𝑇𝑅𝐿𝑠;𝑢₀𝑃_𝑛(𝑠)𝑑𝑠, (2.6) and consequently the asymptotic growth rate is given by lim_𝑡→∞1𝑡^𝑡₀𝑇𝑅𝐿𝑠;𝑢₀𝑃_𝑛(𝑠)𝑑𝑠. (2.7) As we need the maximal growth rate, we compute the sup over the 𝑢₀∈𝒜_𝜀 and all the 𝑛-dimensional projectors: sup_{𝑢0∈𝒜𝜀}sup_𝑃𝑛limsup_𝑡→+∞1𝑡^𝑡₀𝑇𝑅𝐿𝑠;𝑢₀𝑃_𝑛(𝑠)𝑑𝑠∶=𝑇𝑅_𝑛𝒜_𝜀. (2.8) In order to have exponential decay, we look for the smallest 𝑛 such that the number 𝑇𝑅_𝑛(𝒜_𝜀) is negative. Thus, using theorem 13.16 page 341 in [2] let us conclude that 𝑑_𝐹𝒜_𝜀≤2𝑛. (2.9) In order to estimate 𝑇𝑅_𝑛(𝒜_𝜀), we look for an uniform bound of the following expression: sup_𝑢0∈𝒜sup_𝑃𝑛𝑇𝑅𝐿𝑠;𝑢₀𝑃_𝑛(𝑠)𝑑𝑠≤𝐶,∀𝑠, (2.10) from which we conclude that 𝑇𝑅_𝑛(𝒜)≤𝐶. In our case the linearized equation is 𝑈_𝑡=−𝜀²𝑈_{𝑥𝑥𝑥𝑥}−2𝑈_𝑥𝑥+6𝑣²_𝑥𝑈_𝑥𝑥+12𝑣_𝑥𝑣_𝑥𝑥𝑈_𝑥.𝑈(0)=𝜉. (2.11) In details let 𝜑₁(𝑡),…,𝜑_𝑛(𝑡) be an orthonormal base for 𝑃_𝑛𝐿²(𝐼) then 𝑇𝑅𝐹^′(𝑣)𝑃_𝑛=^𝑛_𝑖=1𝐹^′(𝑣)𝜑_𝑖,𝜑_{𝑖𝐿2(𝐼)}, (2.12) with 𝑣=𝑆(𝑡)𝑢₀, 𝐹^′(𝑣)=−𝜀²𝐷_{𝑥𝑥𝑥𝑥}−2𝐷_𝑥𝑥+6𝑣²_𝑥𝐷_𝑥𝑥+12𝑣_𝑥𝑣_𝑥𝑥𝐷_𝑥. (2.13) We estimate the terms separately. The first term −𝜀^2𝑛_𝑖=1𝐷_{𝑥𝑥𝑥𝑥}𝜑_𝑖,𝜑_𝑖=−𝜀^2𝑛_𝑖=1𝜑_𝑖𝑥𝑥². (2.14) The second term −2𝐷_𝑥𝑥𝜑_𝑖,𝜑_𝑖=2𝜑_𝑖𝑥²_𝐿2(𝐼), (2.15) from which by interpolation we get −2^𝑛_𝑖=1𝐷_𝑥𝑥𝜑_𝑖,𝜑_𝑖=2^𝑛_𝑖=1𝜑_𝑖𝑥²_𝐿2(𝐼)≤2^𝑛_𝑖=1𝜑_𝑖𝜑_𝑖𝑥𝑥≤2^𝑛_𝑖=1𝜑_𝑖^21/2𝑛_𝑖=1𝜑_𝑖𝑥𝑥^21/2=2𝑛^1/2𝑛_𝑖=1𝜑_𝑖𝑥𝑥^21/2≤𝜀²2^𝑛_𝑖=1𝜑_𝑖𝑥𝑥²+2𝑛𝜀². (2.16) The last term ^𝑛_𝑖=16_𝐼𝑣²_𝑥𝜑_𝑖𝑥𝑥𝜑_𝑖𝑑𝑥+12_𝐼𝑣_𝑥𝑣_𝑥𝑥𝜑_𝑖𝑥𝜑_𝑖𝑑𝑥=−^𝑛_𝑖=16_𝐼𝑣²_𝑥𝜑²_𝑖𝑥𝑑𝑥 (2.17) is negative and consequently it can be neglected in the computation. Then we have the following estimate: 𝑇𝑅𝐹^′(𝑣)𝑃_𝑛≤−𝜀^2𝑛_𝑖=1𝜑_𝑖𝑥𝑥²+𝜀²2^𝑛_𝑖=1𝜑_𝑖𝑥𝑥²+2𝑛𝜀²𝜀=−²2^𝑛_𝑖=1𝜑_𝑖𝑥𝑥²+2𝑛𝜀²𝜀≤−²2𝜆₁𝑛⁵+2𝑛𝜀², (2.18) where we have used that −^𝑛_𝑖=1𝐷_{𝑥𝑥𝑥𝑥}𝜑_𝑖,𝜑_𝑖≤−𝜆₁𝑛⁵ (2.19) and where 𝜆_𝑗=((2𝜋)⁴/|𝐼|⁴)𝑗⁴=𝜆₁𝑗⁴ are the eigenvalues of the operator 𝐷_{𝑥𝑥𝑥𝑥}. Since from (2.18) we have that 𝑇𝑅(𝐹^′(𝑣)𝑃_𝑛) is uniformly bounded in time, then we conclude that 𝑇𝑅_𝑛𝒜_𝜀≤−𝑘₁𝑛⁵+𝑘₂, (2.20) where we have used a Young inequality and the constants 𝑘₁, 𝑘₂ are given by 𝑘₁=𝜀²𝜆₁4,𝑘₂=2^15/4𝜀³𝜆₁^1/45^5/4. (2.21) This (see [1]) also gives an estimate for the Lyapunov exponents 𝜇_𝑖: 𝜇₁+⋯+𝜇_𝑛≤𝑇𝑅_𝑛𝒜_𝜀≤−𝑘₁𝑛⁵+𝑘₂,∀𝑛∈𝒩. (2.22) Moreover (see [1], Theorem 2.2 pag. 396), we obtain an estimate on the dimension of 𝒜_𝜀.

Theorem 2.2. Let 𝑛∈𝒩 such that 𝑛−1<2𝑘₂𝑘₁^1/5≤𝑛, (2.23) then the 𝑛-dimensional volume element in the phase space is exponentially decaying as 𝑡→∞. Moreover, the global attractor 𝒜_𝜀 has finite dimension: 𝑑_𝐻(𝒜_𝜀)≤𝑛, 𝑑_𝐹(𝒜_𝜀)≤2𝑛.
Then we have that 2𝑛∼^7/20|𝐼|𝜋5^1/4𝜀⁻¹. (2.24) From the above expression we have that 𝑛=𝑂(𝜀⁻¹). This provides a much better estimate than the one obtained in [15] by the fractal dimension of the exponential attractor 𝑛_∗=𝑂(𝜀⁻¹⁰).

3. Estimate via the Existence of the Inertial Manifold

In this section we will show the existence of the inertial manifold ℳ_𝜀 for system (1.1). Then the dimension of ℳ_𝜀 will provide a further estimate of the dimension of the global attractor since 𝒜_𝜀⊂ℳ_𝜀. In order to carry on our analysis, we follow the strategy explained in [2].

Theorem 3.1. The system (1.1) possesses an inertial manifold ℳ_𝜀 having dimension of order 𝑂(𝜀⁻¹⁹).

One first restricts (1.1) to the absorbing set ℬ_𝜀. Then in order to prove the existence of ℳ_𝜀 one has to check the fulfillment of two conditions. The first condition consists of proving that the nonlinear term 𝑅(𝑢)∶=(−1/2)Φ^′′(𝑢_𝑥)𝑢_𝑥𝑥 of (1.1) is Lipschitz from 𝐻²(𝐼) to 𝐿²(𝐼):𝑅𝑢₁−𝑅𝑢₂≤𝐶_𝐿𝑢₁−𝑢_2𝐻2(𝐼), (3.1)∀𝑢,𝑣∈ℬ_𝜀∩𝐻²(𝐼).In particular the constant 𝐶_𝐿=𝐶_𝐿(ℬ_𝜀) depends on the radius of ℬ_𝜀.

The second condition is the so-called strong squeezing property.

Definition 3.2. Let 𝑢₁, 𝑢₂ two solutions of (1.1), the strong squeezing property holds if for some 𝑁 and 𝐾𝑄_𝑁𝑢₁(0)−𝑢₂(0)_𝐻2(𝐼)≤𝐾𝑃_𝑁𝑢₁(0)−𝑢₂(0)_𝐻2(𝐼) (3.2) implies that 𝑄_𝑁𝑢₁(𝑡)−𝑢₂(𝑡)_𝐻2(𝐼)≤𝐾𝑃_𝑁𝑢₁(𝑡)−𝑢₂(𝑡)_𝐻2(𝐼),∀𝑡≥0, (3.3) and furthermore that if 𝑄_𝑁𝑢₁𝑡₀−𝑢₂𝑡_0𝐻2(𝐼)≥𝐾𝑃_𝑁𝑢₁𝑡₀−𝑢₂(𝑡)_0𝐻2(𝐼), (3.4) then 𝑄_𝑁𝑢₁(𝑡)−𝑢₂(𝑡)_𝐻2(𝐼)≤𝑄_𝑁𝑢₁(0)−𝑢₂(0)_𝐻2(𝐼)𝑒⁻𝑘_𝑡, (3.5) for 0≤𝑡≤𝑡₀ and some 𝑘>0 and where 𝑄_𝑁=𝐼−𝑃_𝑁.

Again following [2] we have that a sufficient condition to prove that the strong squeezing property holds is the so-called spectral gap condition:𝜆_𝑁+1−𝜆_𝑁>2𝐶_𝐿𝜆^1/2_𝑁+1+𝜆_𝑁^1/2, (3.6) for 𝑁 sufficiently large and where 𝜆_𝑘 are the eigenvalues of the operator 𝐴=𝜀²𝐷_{𝑥𝑥𝑥𝑥}. The spectral gap condition will be the second property we will show holds. We divide the proof into two steps.

3.1. Lipschitz Property for 𝑅

Theorem 3.3. Given solutions 𝑢₁ and 𝑢₂, then there exists a constant 𝐶_𝐿>0 such that 𝑅𝑢₁−𝑅𝑢_2𝐿2(𝐼)≤𝐶_𝐿𝑢₁−𝑢_1𝐻2(𝐼), (3.7) where 𝑅 denotes the nonlinear part of (1.1), that is, 𝑅(𝑢)1∶=−2𝜙^′′𝑢_𝑥𝑢_𝑥𝑥. (3.8)

Proof. First, we suppose that 𝑢 is a solution of (1.1), then, for any unitary vector 𝑣 in 𝐿²(𝐼), we have that ⟨𝑅(𝑢),𝑣⟩=_𝐼−12𝜙^′′𝑢_𝑥(𝑠)𝑢_𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠=2_𝐼1−3𝑢²_𝑥(𝑠)𝑢_𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠=2_𝐼𝑢_𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠−6_𝐼𝑢²_𝑥(𝑠)𝑢_𝑥𝑥(𝑠)𝑣(𝑠)𝑢𝑑𝑠≤2_{𝑥𝑥𝐿2(𝐼)}‖𝑣‖_𝐿2(𝐼)+6‖𝑢_𝑥‖²_∞𝑢_{𝑥𝑥𝐿2(𝐼)}‖𝑣‖_𝐿2(𝐼)=2+6‖𝑢_𝑥‖²_∞‖𝑣‖_𝐿2(𝐼)𝑢_{𝑥𝑥𝐿2(𝐼)}≤2+6‖𝑢_𝑥‖²_∞‖𝑣‖_𝐿2(𝐼)‖𝑢‖_𝐻2(𝐼), (3.9) and consequently the nonlinear operator restricted on ℬ_𝜀 is bounded from 𝐻²(𝐼)→𝐿²(𝐼): ‖𝑅(𝑢)‖_𝐿2(𝐼)≤2+6𝑟₁𝑟₂‖𝑢‖_𝐻2(𝐼). (3.10)
Now, if 𝑢₁ and 𝑢₂ are solutions of (1.1), let 𝑢∶=𝑢₁−𝑢₂, then we proceed in a similar manner as before to get that if 𝑣∈𝐿²(𝐼), then 𝑅𝑢₁−𝑅𝑢₂,𝑣=_𝐼−12𝜙^′′𝑢_1𝑥(𝑠)𝑢_1𝑥𝑥(𝑠)+12𝜙^′′𝑢_2𝑥(𝑠)𝑢_2𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠=2_𝐼1−3𝑢²_1𝑥(𝑠)𝑢_1𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠−2_𝐼1−3𝑢²_2𝑥(𝑠)𝑢_2𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠=2_𝐼𝑢_𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠+6_I𝑢²_1𝑥(𝑠)𝑢_𝑥𝑥(𝑠)𝑣(𝑠)𝑑𝑠+6_𝐼𝑢_𝑥(𝑠)𝑢_2𝑥𝑥(𝑠)𝑣(𝑠)𝑢_1𝑥(𝑠)+𝑢_2𝑥(𝑠𝑢)𝑑𝑠≤2_{𝑥𝑥𝐿2(𝐼)}‖𝑣‖_𝐿2(𝐼)+6𝑢_{𝑥𝑥𝐿2(𝐼)}‖𝑣‖_𝐿2(𝐼)𝑢_1𝑥²_{𝐿∞(𝐼)}+6𝑢_{2𝑥𝑥𝐿2(𝐼)𝐼}𝑢²_𝑥𝑣²𝑢_1𝑥+𝑢_2𝑥²𝑑𝑠^1/2≤2+6𝑟₁𝑟₂^1/2‖𝑢‖_𝐻2(𝐼)‖𝑣‖_𝐿2(𝐼)+12𝑟₂𝑟₁𝑟₂^1/2𝑢_{𝑥𝐿∞(𝐼)}‖𝑣‖_𝐿2(𝐼)≤2+6+12𝑟₂𝑟₁𝑟₂^1/2‖𝑢‖_𝐻2(𝐼)‖𝑣‖_𝐿2(𝐼), (3.11) where the last inequality follows from (see [19]) 𝑢_{𝑥𝐿∞(𝐼)}≤𝑢_{𝑥𝐻1(𝐼)}≤𝐶‖𝑢‖_𝐻2(𝐼). (3.12) Thus, if we take 𝐶_𝐿∶=2+6+12𝑟₂𝑟₁𝑟₂^1/2, (3.13) then we have that 𝑅𝑢₁−𝑅𝑢₂,𝑣≤𝐶_𝐿𝑢₁−𝑢_2𝐻2(𝐼)‖𝑣‖_𝐻2(𝐼), (3.14) and it follows that the nonlinear functional 𝑅 is Lipschitz continuous. From the expression of the radii 𝑟₁,𝑟₂, it results that the Lipschitz constant is of order 𝑂(𝜀⁻¹⁸).

3.2. Spectral Gap Condition

Theorem 3.4. There exists a natural number 𝑁 sufficiently large so that 𝜆_𝑁+1−𝜆_𝑁>2𝐶_𝐿𝜆^1/2_𝑁+1+𝜆_𝑁^1/2. (3.15)

Proof. The eigenvalues of the operator 𝐴 are given by 𝜆₁=(2𝜋)⁴|𝐼|⁴,…,𝜆_𝑁=𝜆₁𝑁⁴, (3.16) while the eigenvalues of 𝐴=𝜀²𝐴 are 𝜆_𝑖=𝜀²𝜆_𝑖. Then the spectral gap condition is 𝜀²𝜆_𝑁+1−𝜆_𝑁>2𝐶_𝐿𝜀𝜆^1/2_𝑁+1+𝜆_𝑁^1/2. (3.17) In details we have 𝜀𝜆₁(𝑁+1)⁴−𝑁⁴>2𝐶_𝐿𝜆₁^1/2(𝑁+1)²+𝑁², (3.18) that is, 2𝑁+1>2𝐶_𝐿𝜀𝜆₁^1/2=𝐶_𝐿|𝐼|²2𝜀𝜋². (3.19) Then if 𝑁 satisfies the previous inequality, we have that 𝜆_𝑁+1−𝜆_𝑁>2𝐶_𝐿𝜆^1/2_𝑁+1+𝜆_𝑁^1/2. (3.20)

Thus, the spectral gap condition is satisfied and the existence of the inertial manifold is assured. This proves Theorem 3.1.

We remark that the dimension of ℳ_𝜀 is𝑁∼𝐶_𝐿|𝐼|²𝜀⁻¹=𝑂𝜀⁻¹⁹. (3.21) Then comparing the estimates obtained, we have that we get 𝑂(𝜀⁻¹) by the method of 𝑛-dimensional volume evolution and, respectively, 𝑂(𝜀⁻¹⁰) and 𝑂(𝜀⁻¹⁹) by the existence of the inertial set ℐ_𝜀 and the inertial manifold ℳ_𝜀. It is not a surprise that the dimension of the inertial manifold is much bigger than that of the inertial set and global attractor since we are requiring the existence of a smooth structure.

4. Regularity of Attractor and Dimension

In this section we show a result that gives further regularity of the attractor. By a classic theorem (see [17, 20]), we have the following theorem.

Theorem 4.1. If 𝐼 is a bounded 𝐶^∞ domain and if 𝑢₀ and Φ are 𝐶^∞ functions, then the global attractor 𝒜_𝜀 is a bounded subset of 𝐻^𝑘(𝐼) for every 𝑘≥0. In particular, if 𝑢∈𝒜_𝜀, then 𝑢∈𝐶^∞(𝐼).

This regularity result let us conclude by theorem  15.1 in [17] that for 𝑘≥16𝑑_𝑓(𝒜_𝜀)+1, for almost every set 𝑥=(𝑥₁,…,𝑥_𝑘) of 𝑘 points in 𝐼, the mapping𝑢→𝑢𝑥₁,…,𝑢𝑥_𝑘 (4.1) is an embedding of 𝒜_𝜀 into ℝ^𝑘. This gives a parametrization of the attractor 𝒜_𝜀 and gives an estimate of the degree of freedom of the system: 𝑘=𝑂(𝜀⁻¹). Consequently if we want to parameterize the attractor by 𝑘 equally spaced points in 𝐼, it results that each subinterval of 𝐼 is of order |𝐼|/𝑘=𝑂(𝜀) (see [17] for a more detailed discussion). This is in full accordance with the numerical experiments proposed in [14] in which it was pointed out that the wave length of microstructure is of order 𝑂(𝜀).

This result has a connection with the definition of determining nodes (see [21]): a set of points {𝑥₁,…,𝑥_𝑘} in 𝐼 is said to be asymptotically determining if for two solutions 𝑢₁(𝑥,𝑡) and 𝑢₂(𝑥,𝑡)max_{𝑖=1,…,𝑘}𝑢₁𝑥_𝑖,𝑡−𝑢₂𝑥_𝑖,𝑡⟶0as𝑡⟶∞ (4.2) implies thatsup_𝑥∈𝐼𝑢₁(𝑥,𝑡)−𝑢₂(𝑥,𝑡)⟶0as𝑡⟶∞. (4.3) From a theorem in [17], if 𝒜_𝜀 attracts all the solutions in the norm of 𝐿^∞(𝐼) then almost every set of 𝑘 nodes {𝑥₁,…,𝑥_𝑘} in 𝐼 is asymptotically determining. In the last section we will show that this is true for (1.1); that is, there exists an absorbing set in 𝐿^∞(𝐼).

We conclude this section by proving and giving explicit bounds for 𝒜_𝜀 in 𝐻⁴(𝐼). In order to do that, we follow a strategy suggested by Robinson in [2].

Theorem 4.2. The global attractor 𝒜_𝜀 of (1.1) is bounded in 𝐻⁴(𝐼).

Proof. Multiply the equation by 𝑢_𝑡 and integrate over 𝐼: 𝑢_𝑡²+𝜀²2𝑑𝑢𝑑𝑡^𝑥𝑥2+12_𝐼Φ^′𝑢_𝑥𝑢_𝑡𝑥𝑑𝑥=0. (4.4) By integrating the previous equation, with respect to time, over (0,𝑡) we get ^𝑡₀𝑢_𝑡²𝜀𝑑𝑠+²2𝑢_𝑥𝑥(𝑡)²−𝜀²2𝑢_𝑥𝑥(0)²+12^𝑡_0𝐼Φ^′𝑢_𝑥𝑢_𝑡𝑥𝑑𝑥𝑑𝑠=0. (4.5) The integrand of the last term of the previous equality can be written as (𝑑/𝑑𝑡)Φ(𝑢_𝑥), and this yields to ^𝑡₀𝑢_𝑡²𝜀𝑑𝑠=²2𝑢_𝑥𝑥(0)²−𝜀²2𝑢_𝑥𝑥(𝑡)²+12_𝐼Φ𝑢_𝑥(0)−Φ𝑢_𝑥(𝑡)𝑑𝑥. (4.6) Since the attractor is bounded in 𝐿^∞ and 𝐻²(𝐼), we have ^𝑡₀𝑢_𝑡²𝑑𝑠≤𝜀²𝑟²₂+2|𝐼|𝑢_𝑥²_{𝐿∞(𝐼)}𝑢_𝑥²_{𝐿∞(𝐼)}+1≤𝜀²𝑟²₂+2|𝐼|𝑟₁𝑟₂𝑟₁𝑟₂+1∶=𝑅. (4.7) We will prove that 𝑢_𝑡 is uniformly bounded in 𝐿²(𝐼). We derive (1.1) with respect to time, multiply it by 𝑡²𝑢_𝑡, and integrate over 𝐼: 12𝑑𝑑𝑡𝑡𝑢_𝑡²−𝑡𝑢_𝑡²+𝑡²𝜀²𝑢_𝑥𝑥𝑡²=𝑡²2_𝐼Φ^′𝑢_𝑥𝑥𝑡𝑢_𝑡𝑡𝑑𝑥=−²2_𝐼Φ^′′𝑢_𝑥𝑢²_𝑡𝑥𝑑𝑥=−6𝑡²_𝐼𝑢²_𝑥𝑢²_𝑡𝑥𝑑𝑥+2𝑡²_𝐼𝑢²_𝑡𝑥𝑑𝑥. (4.8) Then, using interpolation we get 12𝑑𝑑𝑡𝑡𝑢_𝑡²−𝑡𝑢_𝑡²+𝑡²𝜀²𝑢_𝑥𝑥𝑡²≤2𝑡²𝑢_𝑡𝑥²≤𝑡²𝜀²𝑢_𝑥𝑥𝑡²+𝑡²𝜀²𝑢_𝑡², (4.9) from which 12𝑑𝑑𝑡𝑡𝑢_𝑡²≤𝑡²𝜀²𝑢_𝑡²+𝑡𝑢_𝑡². (4.10) Integrating the previous inequality, with respect to time, in (0,𝑡), we have 𝑡𝑢_𝑡²≤2^𝑡₀𝑠𝑠+²𝜀²𝑢_𝑡²𝑑𝑠. (4.11) Since the term (𝑠+𝑠²/𝜀²) is bounded in [0,1], we obtain the following inequality by setting 𝑡=1: 𝑢_𝑡(1)²1≤21+𝜀^2𝑡₀𝑢_𝑡²1𝑑𝑠≤2𝑅1+𝜀². (4.12) This gives an uniform 𝐿² bound for 𝑢_𝑡(1), and since 𝑢∈𝒜_𝜀, it is sufficient to conclude that 𝑢_𝑡∈𝐿^∞(ℝ,𝐿²(𝐼)). In particular, let 𝑢∈𝒜_𝜀 and 𝑠≥0, we set 𝑢(𝑡,𝑥)∶=𝑢(𝑠+𝑡−1,𝑥) for any 𝑡≥1. Then 𝑢_𝑡(𝑡,𝑥)=𝑢_𝑡(𝑠+𝑡−1,𝑥) and so 𝑢_𝑡(1,𝑥)=𝑢_𝑡(𝑠,𝑥), from which we have that 𝑢_𝑡(𝑠)²1≤2𝑅1+𝜀²,∀𝑠>0. (4.13) Now we consider the 𝐿² norm of the nonlinear term (1/2)Φ^′′(𝑢_𝑥)𝑢_𝑥𝑥: 14_𝐼Φ^′′𝑢_𝑥²𝑢²_𝑥𝑥𝑑𝑥≤4_𝐼√3𝑢_{𝑥𝐿∞(𝐼)}+1²𝑢²_𝑥𝑥𝑑𝑥≤46‖𝑢_𝑥‖²_{𝐿∞(𝐼)}+1𝑢_𝑥𝑥²≤46𝑟₁𝑟₂+1𝑟²₂, (4.14) where we have used (1.8), (1.10), and [15]. Then, from the previous inequality, we get that 1/2Φ^′′(𝑢_𝑥)𝑢_𝑥𝑥 is uniformly bounded in 𝐿²(𝐼). Then since 𝜀²𝑢_{𝑥𝑥𝑥𝑥}=−𝑢_𝑡+12Φ^′′𝑢_𝑥𝑢_𝑥𝑥, (4.15) we have that 𝑢_{𝑥𝑥𝑥𝑥} is uniformly bounded in 𝐿²(𝐼). Then from the following interpolating inequality 𝑢_𝑥𝑥𝑥≤𝑢_𝑥𝑥^1/2𝑢_{𝑥𝑥𝑥𝑥}^1/2, (4.16) we get that 𝒜_𝜀 is bounded in 𝐻⁴(𝐼).

5. Absorbing Set in 𝐿^∞(𝐼)

In this last section we show the existence of an absorbing set in 𝐿^∞(𝐼).

Theorem 5.1. The semigroup 𝑆(𝑡) defined in (1.4) posses an absorbing set in 𝐿^∞(𝐼): 𝑆(𝑡)𝑢_{0𝐿∞(𝐼)}≤𝑟_∞,if𝑡>𝜏𝑅_∞+𝜀², (5.1) where 𝑟_∞=11+𝜀^21/2|𝐼|^1/2𝜀^1/2𝜀²241+𝜀⁴+1𝜀^21/4,𝜏𝑅_∞=1𝜀²log𝑅_∞, (5.2) and 𝑅_∞ is a positive constant such that 𝑢_{0𝐿∞(𝐼)}≤𝑅_∞. (5.3)

Proof. Let 𝑢 be a solution of (1.1) such that 𝑢(0)=𝑢₀∈𝐿^∞(𝐼) and ‖𝑢₀‖_{𝐿∞(𝐼)}≤𝑅_∞.
Multiply (1.1) by 𝑢 and integrate on 𝐼, we obtain 12𝑑‖𝑢‖𝑑𝑡²+𝜀²𝑢_𝑥𝑥²+12_𝐼Φ^′𝑢_𝑥𝑢_𝑥𝑑𝑥=0, (5.4) from which 12𝑑‖𝑢‖𝑑𝑡²+𝜀²𝑢_𝑥𝑥²+2𝑢_𝑥⁴_𝐿4(𝐼)=2𝑢_𝑥²≤2𝑢_𝑥⁴_𝐿4(𝐼)|𝐼|^1/2≤𝑢_𝑥⁴_𝐿4(𝐼)+|𝐼|, (5.5) and then 12𝑑‖𝑢‖𝑑𝑡²+𝜀²𝑢_𝑥𝑥²≤|𝐼|. (5.6) By the boundary conditions, from the hypothesis that |𝐼|≤1 and from the interpolating inequality ‖𝑢_𝑥‖²≤(1/2){‖𝑢‖²+‖𝑢_𝑥𝑥‖²}, we have that ‖𝑢‖≤𝑢_𝑥𝑥. (5.7) Then we can rewrite (5.6) only in terms of the 𝐿²-norm of 𝑢: 12𝑑‖𝑢‖𝑑𝑡²+𝜀²‖𝑢‖²≤|𝐼|. (5.8) Integrating the previous inequality over (0,𝑡), we get ‖𝑢(𝑡)‖²≤𝑢₀²𝑒^{−2𝜀2𝑡}+|𝐼|𝜀²1−𝑒^{−2𝜀2𝑡}≤𝑅²_∞𝑒^{−2𝜀2𝑡}+1𝜀²|𝐼|, (5.9) where the last inequality follows from ‖𝑢‖≤|𝐼|^1/2‖𝑢‖_{𝐿∞(𝐼)}. Then if we set 𝜏𝑅_∞=1𝜀²log𝑅_∞, (5.10) we have that ‖𝑢(𝑡)‖²≤11+𝜀²|𝐼|,if𝑡≥𝜏𝑅_∞. (5.11) Now coming back to (5.5), we use a different estimate method 12𝑑‖𝑢‖𝑑𝑡²+𝜀²𝑢_𝑥𝑥²+2‖𝑢_𝑥‖⁴_𝐿4(𝐼)=2𝑢_𝑥²≤2‖𝑢‖𝑢_𝑥𝑥≤𝜀²2𝑢_𝑥𝑥²+2𝜀²‖𝑢‖², (5.12) from which 𝑑‖𝑢‖𝑑𝑡²+𝜀²𝑢_𝑥𝑥²≤4𝜀²‖𝑢‖². (5.13) Integrating the previous inequality with respect to time in (𝑡,𝑡+𝜌), with 𝜌≥𝜏(𝑅_∞), we get 𝜀²_𝑡^𝑡+𝜌𝑢_𝑥𝑥²𝑑𝑡≤4/𝜀²+2‖𝑢‖²≤4/𝜀²𝜌+21+1/𝜀²|𝐼|. (5.14) Moreover by an interpolating inequality, we have that _𝑡^𝑡+𝜌‖𝑢‖²1𝑑𝑡≤2_𝑡^𝑡+𝜌‖𝑢‖²+_𝑡^𝑡+𝜌𝑢_𝑥𝑥²≤11+𝜀²|𝐼𝜌|241+𝜀⁴+1𝜀²∶=𝑎₃. (5.15) We multiply (1.1) by 𝑢_𝑥𝑥 and integrate over 𝐼: 12𝑑𝑢𝑑𝑡^𝑥2+𝜀²𝑢_𝑥𝑥𝑥²+6𝑢²_𝑥𝑢_𝑥𝑥²=2𝑢_𝑥𝑥², (5.16) from which by using an interpolating inequality and neglecting the positive term 6‖𝑢²_𝑥𝑢_𝑥𝑥‖² we get 12𝑑𝑢𝑑𝑡^𝑥2+𝜀²𝑢_𝑥𝑥𝑥²≤1𝜀²𝑢_𝑥²+𝜀²𝑢_𝑥𝑥𝑥², (5.17) from which we conclude 𝑑𝑢𝑑𝑡^𝑥2≤2𝜀²𝑢_𝑥². (5.18) We apply the uniform Gronwall lemma (see [1]) with 𝑎₁=_𝑡^𝑡+𝜌2𝜀²𝑑𝑠=2𝜌𝜀²,𝑎₂=0, (5.19) from which we obtain 𝑢_𝑥𝑡+𝜌²≤𝑎₃𝜌𝑒^𝑎1,for𝑡≥𝜏𝑅_∞. (5.20) We fix 𝜌∶=𝜀², and then 𝑢_𝑥≤11+𝜀^21/2|𝐼|^1/2𝜀𝜀²241+𝜀⁴+1𝜀^21/2,for𝑡≥𝜏𝑅_∞+𝜀². (5.21) Then, using an interpolating inequality, we have that ‖𝑢‖_{𝐿∞(𝐼)}≤‖𝑢‖^1/2𝑢_𝑥^1/2≤𝑟_∞,for𝑡≥𝜏𝑅_∞+𝜀², (5.22) where 𝑟_∞=11+𝜀^21/2|𝐼|^1/2𝜀^1/2𝜀²241+𝜀⁴+1𝜀^21/4. (5.23) This completes the proof, the set 𝐵_∞=𝑢∈𝐿^∞(𝐼)∶‖𝑢‖_{𝐿∞(𝐼)}≤𝑟_∞ (5.24) is absorbing for all the bounded sets of 𝐿^∞(𝐼).

Acknowledgment

The first author would like to thank Professor Giorgio Fusco for suggesting the subject of research during his Ph.D. studies and for helpful hints and remarks.