# Dimension Estimate for the Global Attractor of an Evolution Equation

Authors: Renato Colucci; Gerardo R. Chacón
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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


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## Body

## 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 ofglobal 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 ann-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:(1.1)ut=-ε2uxxxx+12Φ′′(ux)uxx,(x,t)∈I×R+,u(0,x)=u0(x),inI,u=uxx=0,in∂I×R+,
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:(1.2)Fε(u)=12ε2∫Iuxx2dx+12∫IΦ(ux)dx.
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 afirst time scale of order t>Tε=O(ε2), there is a drastic reduction of the energy Fε(u0) of the initial data u0 and a formation of microstructure in the region where Φ(·) is not convex.In thesecond 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 thethird time scale of order t>1/Tε=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:(1.3)H=L2(I),V=H2(I).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 as(1.4)S(t):u0∈H⟶u(t)∈H,
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 constantsri,  τi with  i=0,1,2 such that (i)
the semigroupS(t) posses an absorbing set in H:
(1.5)∥S(t)u0∥≤r0,ift≥τ0,
where
(1.6)r0=(1+|I|ε2)1/2,τ0=1ε2∥u0∥;(ii)
the semigroupS(t) posses an absorbing set in H01(I):
(1.7)∥S(t)u0∥H01(I)≤r1,ift≥τ1,
where
(1.8)r1={r02(1+12ε4)+|I|ε2}1/2e,τ1=τ0+ε2;(iii)
the semigroupS(t) posses an absorbing set in H2(I):
(1.9)‖S(t)u0‖H2(I)≤(r02+r12+r22)1/2,ift≥τ2,
where(1.10)r2=e2ε2{180r16+r022+|I|ε2}1/2,τ2=τ1+ε2.Theorem1.1 let us conclude that the set(1.11)Bε={u∈H:‖ux‖≤r1,‖uxx‖≤r2}
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, 𝒜ε=ω(ℬ). The authors also obtained in [15] a bound for the dimension of 𝒜ε by proving the existence of an exponential attractor ℐε whose dimension is of order O(ε-10). Moreover, in ℬε we have the following estimates for t>τ2:(1.12)‖u‖∞≤‖u‖1/2‖ux‖1/2≤(r0r1)1/2,(1.13)‖ux‖∞≤‖ux‖1/2‖uxx‖1/2≤(r1r2)1/2.
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 of𝒜ε. In Section 2 we find an estimate of the fractal and Hausdorff dimension of 𝒜ε 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 ℳε (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 ℳε, we obtain, in another way, a different estimate of dim(𝒜ε). 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 operatorA=∂4/∂x4. The set D(A)={u∈H4(I):u=uxx=0in∂I} is its domain. Also, then (Au,u)=∥uxx∥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
(1.14)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.


## 2. Dimension of the Attractor

Following [1, 2] we give an estimate of the dimension of the attractor 𝒜ε by estimating how the n-dimensional volume elements are distorted by the flow. We will show the following theoremTheorem 2.1.
The global attractor𝒜ε of the semigroup has fractal dimension  dF(𝒜ε)<2n where  n:O(ε-1).Proof.
In general if we consider an evolution equation of the type(2.1)ut=F(u),u(0)=u0,
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 𝒜ε 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:
(2.2)Ut=F′(S(t)u0)U:=L(t;u0)U,U(0)=ξ,
where F′ is the Fráchet derivative of F. Each displacement evolves
(2.3)ddtδx(i)=L(t;u0)δx(i).
The volume is given by Vn(t)=|δx(1)∧⋯∧δx(n)| where
(2.4)|δx(1)∧⋯∧δx(n)|2=det⁡((δx(i),δx(j)))i,j:=det⁡M(t),
then (see, e.g., [2] for details)
(2.5)ddtVn(t)=12TR[M-1ddtM].
Then if Pn(t) is the projector on the space spanned by the displacements δx(i) with base φ1(t),…,φn(t), we have
(2.6)Vn(t)=Vn(0)exp⁡(∫0tTR(L(s;u0)Pn(s))ds),
and consequently the asymptotic growth rate is given by
(2.7)lim⁡t→∞1t∫0tTR(L(s;u0)Pn(s))ds.
As we need the maximal growth rate, we compute the sup over the u0∈𝒜ε and all the n-dimensional projectors:
(2.8)sup⁡u0∈Aεsup⁡Pnlimsup⁡t→+∞1t∫0tTR(L(s;u0)Pn(s))ds:=TRn(Aε).
In order to have exponential decay, we look for the smallest n such that the number TRn(𝒜ε) is negative. Thus, using theorem 13.16 page 341 in [2] let us conclude that
(2.9)dF(Aε)≤2n.
In order to estimate TRn(𝒜ε), we look for an uniform bound of the following expression:
(2.10)sup⁡u0∈Asup⁡PnTR(L(s;u0)Pn(s))ds≤C,∀s,
from which we conclude that TRn(𝒜)≤C. In our case the linearized equation is
(2.11)Ut=-ε2Uxxxx-2Uxx+6vx2Uxx+12vxvxxUx.U(0)=ξ.
In details let φ1(t),…,φn(t) be an orthonormal base for PnL2(I) then
(2.12)TR(F′(v)Pn)=∑i=1n(F′(v)φi,φi)L2(I),
with v=S(t)u0,
(2.13)F′(v)=-ε2Dxxxx-2Dxx+6vx2Dxx+12vxvxxDx.
We estimate the terms separately. The first term
(2.14)-ε2∑i=1n(Dxxxxφi,φi)=-ε2∑i=1n‖φixx‖2.
The second term
(2.15)-2(Dxxφi,φi)=2‖φix‖L2(I)2,
from which by interpolation we get
(2.16)-2∑i=1n(Dxxφi,φi)=2∑i=1n‖φix‖L2(I)2≤2∑i=1n‖φi‖‖φixx‖≤2(∑i=1n‖φi‖2)1/2(∑i=1n‖φixx‖2)1/2=2n1/2(∑i=1n‖φixx‖2)1/2≤ε22∑i=1n‖φixx‖2+2nε2.
The last term
(2.17)∑i=1n{6∫Ivx2φixxφidx+12∫Ivxvxxφixφidx}=-∑i=1n{6∫Ivx2φix2dx}
is negative and consequently it can be neglected in the computation. Then we have the following estimate:
(2.18)TR(F′(v)Pn)≤-ε2∑i=1n‖φixx‖2+ε22∑i=1n‖φixx‖2+2nε2=-ε22∑i=1n‖φixx‖2+2nε2≤-ε22λ1n5+2nε2,
where we have used that
(2.19)-∑i=1n(Dxxxxφi,φi)≤-λ1n5
and where λj=((2π)4/|I|4)j4=λ1j4 are the eigenvalues of the operator Dxxxx. Since from (2.18) we have that TR(F′(v)Pn) is uniformly bounded in time, then we conclude that
(2.20)TRn(Aε)≤-k1n5+k2,
where we have used a Young inequality and the constants k1, k2 are given by
(2.21)k1=ε2λ14,k2=215/4ε3λ11/455/4.
This (see [1]) also gives an estimate for the Lyapunov exponents μi:
(2.22)μ1+⋯+μn≤TRn(Aε)≤-k1n5+k2,∀n∈N.
Moreover (see [1], Theorem 2.2 pag. 396), we obtain an estimate on the dimension of 𝒜ε.Theorem 2.2.
Letn∈𝒩 such that
(2.23)n-1<(2k2k1)1/5≤n,
then the n-dimensional volume element in the phase space is exponentially decaying as t→∞. Moreover, the global attractor 𝒜ε has finite dimension: dH(𝒜ε)≤n,  dF(𝒜ε)≤2n.
Then we have that(2.24)n~[27/20|I|π51/4]ε-1.
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 manifoldℳε for system (1.1).  Then the dimension of ℳε will provide a further estimate of the dimension of the global attractor since 𝒜ε⊂ℳε. 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  ℳε having dimension of order  O(ε-19).One first restricts (1.1) to the absorbing set ℬε. Then in order to prove the existence of ℳε one has to check the fulfillment of two conditions. The first condition consists of proving that the nonlinear term R(u):=(-1/2)Φ′′(ux)uxx of (1.1) is Lipschitz from H2(I) to L2(I):(3.1)‖R(u1)-R(u2)‖≤CL‖u1-u2‖H2(I),∀u,v∈ℬε∩H2(I).In particular the constant CL=CL(ℬε) depends on the radius of ℬε.The second condition is the so-called strong squeezing property.Definition 3.2.
Letu1, u2 two solutions of (1.1), the strong squeezing property holds if for some N and K(3.2)‖QN(u1(0)-u2(0))‖H2(I)≤K‖PN(u1(0)-u2(0))‖H2(I)
implies that
(3.3)‖QN(u1(t)-u2(t))‖H2(I)≤K‖PN(u1(t)-u2(t))‖H2(I),∀t≥0,
and furthermore that if
(3.4)‖QN(u1(t0)-u2(t0))‖H2(I)≥K‖PN(u1(t0)-u2(t)0)‖H2(I),
then
(3.5)‖QN(u1(t)-u2(t))‖H2(I)≤‖QN(u1(0)-u2(0))‖H2(I)e-k̃t,
for 0≤t≤t0 and some  k̃>0 and where QN=I-PN.Again following [2] we have that a sufficient condition to prove that the strong squeezing property holds is the so-called spectral gap condition:(3.6)λ̃N+1-λ̃N>2CL(λ̃N+11/2+λ̃N1/2),
for N sufficiently large and where λ̃k are the eigenvalues of the operator Ã=ε2Dxxxx. The spectral gap condition will be the second property we will show holds. We divide the proof into two steps.


### 3.1. Lipschitz Property forR

Theorem 3.3.
Given solutionsu1 and  u2, then there exists a constant  CL>0 such that
(3.7)‖R(u1)-R(u2)‖L2(I)≤CL‖u1-u1‖H2(I),
where R denotes the nonlinear part of (1.1), that is,
(3.8)R(u):=-12ϕ′′(ux)uxx.Proof.
First, we suppose thatu is a solution of (1.1), then, for any unitary vector v in L2(I), we have that
(3.9)〈R(u),v〉=∫I-12ϕ′′(ux(s))uxx(s)v(s)ds=2∫I(1-3ux2(s))uxx(s)v(s)ds=2∫Iuxx(s)v(s)ds-6∫Iux2(s)uxx(s)v(s)ds≤2‖uxx‖L2(I)‖v‖L2(I)+6∥ux∥∞2‖uxx‖L2(I)‖v‖L2(I)=(2+6∥ux∥∞2)‖v‖L2(I)‖uxx‖L2(I)≤(2+6∥ux∥∞2)‖v‖L2(I)‖u‖H2(I),
and consequently the nonlinear operator restricted on ℬε is bounded from H2(I)→L2(I):
(3.10)‖R(u)‖L2(I)≤(2+6r1r2)‖u‖H2(I).
Now, ifu1 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
(3.11)〈R(u1)-R(u2),v〉=∫I[-12ϕ′′(u1x(s))u1xx(s)+12ϕ′′(u2x(s))u2xx(s)]v(s)ds=2∫I[1-3u1x2(s)]u1xx(s)v(s)ds-2∫I[1-3u2x2(s)]u2xx(s)v(s)ds=2∫I(uxx(s))v(s)ds+6∫I‍u1x2(s)uxx(s)v(s)ds+6∫Iux(s)u2xx(s)v(s)[u1x(s)+u2x(s)]ds≤2‖uxx‖L2(I)‖v‖L2(I)+6‖uxx‖L2(I)‖v‖L2(I)‖u1x‖L∞(I)2+6‖u2xx‖L2(I)(∫Iux2v2(u1x+u2x)2ds)1/2≤[2+6(r1r2)1/2]‖u‖H2(I)‖v‖L2(I)+12r2(r1r2)1/2‖ux‖L∞(I)‖v‖L2(I)≤[2+(6+12r2)(r1r2)1/2]‖u‖H2(I)‖v‖L2(I),
where the last inequality follows from (see [19])
(3.12)‖ux‖L∞(I)≤‖ux‖H1(I)≤C‖u‖H2(I).
Thus, if we take
(3.13)CL:=2+(6+12r2)(r1r2)1/2,
then we have that
(3.14)〈R(u1)-R(u2),v〉≤CL‖u1-u2‖H2(I)‖v‖H2(I),
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 numberN sufficiently large so that
(3.15)λ̃N+1-λ̃N>2CL(λ̃N+11/2+λ̃N1/2).Proof.
The eigenvalues of the operatorA are given by
(3.16)λ1=(2π)4|I|4,…,λN=λ1N4,
while the eigenvalues of Ã=ε2A are λ̃i=ε2λi. Then the spectral gap condition is
(3.17)ε2(λN+1-λN)>2CLε(λN+11/2+λN1/2).
In details we have
(3.18)ελ1[(N+1)4-N4]>2CLλ11/2[(N+1)2+N2],
that is,
(3.19)2N+1>2CLελ11/2=CL|I|22επ2.
Then if N satisfies the previous inequality, we have that
(3.20)λ̃N+1-λ̃N>2CL(λ̃N+11/2+λ̃N1/2).Thus, thespectral gap condition is satisfied and the existence of the inertial manifold is assured. This proves Theorem 3.1.We remark that the dimension ofℳε is(3.21)N~CL|I|2ε-1=O(ε-19).
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 ℐε and the inertial manifold ℳε. 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.


## 3.1. Lipschitz Property forR

Theorem 3.3.
Given solutionsu1 and  u2, then there exists a constant  CL>0 such that
(3.7)‖R(u1)-R(u2)‖L2(I)≤CL‖u1-u1‖H2(I),
where R denotes the nonlinear part of (1.1), that is,
(3.8)R(u):=-12ϕ′′(ux)uxx.Proof.
First, we suppose thatu is a solution of (1.1), then, for any unitary vector v in L2(I), we have that
(3.9)〈R(u),v〉=∫I-12ϕ′′(ux(s))uxx(s)v(s)ds=2∫I(1-3ux2(s))uxx(s)v(s)ds=2∫Iuxx(s)v(s)ds-6∫Iux2(s)uxx(s)v(s)ds≤2‖uxx‖L2(I)‖v‖L2(I)+6∥ux∥∞2‖uxx‖L2(I)‖v‖L2(I)=(2+6∥ux∥∞2)‖v‖L2(I)‖uxx‖L2(I)≤(2+6∥ux∥∞2)‖v‖L2(I)‖u‖H2(I),
and consequently the nonlinear operator restricted on ℬε is bounded from H2(I)→L2(I):
(3.10)‖R(u)‖L2(I)≤(2+6r1r2)‖u‖H2(I).
Now, ifu1 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
(3.11)〈R(u1)-R(u2),v〉=∫I[-12ϕ′′(u1x(s))u1xx(s)+12ϕ′′(u2x(s))u2xx(s)]v(s)ds=2∫I[1-3u1x2(s)]u1xx(s)v(s)ds-2∫I[1-3u2x2(s)]u2xx(s)v(s)ds=2∫I(uxx(s))v(s)ds+6∫I‍u1x2(s)uxx(s)v(s)ds+6∫Iux(s)u2xx(s)v(s)[u1x(s)+u2x(s)]ds≤2‖uxx‖L2(I)‖v‖L2(I)+6‖uxx‖L2(I)‖v‖L2(I)‖u1x‖L∞(I)2+6‖u2xx‖L2(I)(∫Iux2v2(u1x+u2x)2ds)1/2≤[2+6(r1r2)1/2]‖u‖H2(I)‖v‖L2(I)+12r2(r1r2)1/2‖ux‖L∞(I)‖v‖L2(I)≤[2+(6+12r2)(r1r2)1/2]‖u‖H2(I)‖v‖L2(I),
where the last inequality follows from (see [19])
(3.12)‖ux‖L∞(I)≤‖ux‖H1(I)≤C‖u‖H2(I).
Thus, if we take
(3.13)CL:=2+(6+12r2)(r1r2)1/2,
then we have that
(3.14)〈R(u1)-R(u2),v〉≤CL‖u1-u2‖H2(I)‖v‖H2(I),
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 numberN sufficiently large so that
(3.15)λ̃N+1-λ̃N>2CL(λ̃N+11/2+λ̃N1/2).Proof.
The eigenvalues of the operatorA are given by
(3.16)λ1=(2π)4|I|4,…,λN=λ1N4,
while the eigenvalues of Ã=ε2A are λ̃i=ε2λi. Then the spectral gap condition is
(3.17)ε2(λN+1-λN)>2CLε(λN+11/2+λN1/2).
In details we have
(3.18)ελ1[(N+1)4-N4]>2CLλ11/2[(N+1)2+N2],
that is,
(3.19)2N+1>2CLελ11/2=CL|I|22επ2.
Then if N satisfies the previous inequality, we have that
(3.20)λ̃N+1-λ̃N>2CL(λ̃N+11/2+λ̃N1/2).Thus, thespectral gap condition is satisfied and the existence of the inertial manifold is assured. This proves Theorem 3.1.We remark that the dimension ofℳε is(3.21)N~CL|I|2ε-1=O(ε-19).
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 ℐε and the inertial manifold ℳε. 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.
IfI is a bounded  C∞ domain and if  u0 and  Φ are  C∞ functions, then the global attractor  𝒜ε is a bounded subset of  Hk(I) for every  k≥0. In particular, if  u∈𝒜ε, then  u∈C∞(I).This regularity result let us conclude by theorem  15.1 in [17] that for k≥16df(𝒜ε)+1, for almost every set x=(x1,…,xk) of k points in I, the mapping(4.1)u→(u(x1),…,u(xk))
is an embedding of 𝒜ε into ℝk. This gives a parametrization of the attractor 𝒜ε 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 ofdetermining nodes (see [21]): a set of points {x1,…,xk} in I is said to be asymptotically determining if for two solutions u1(x,t) and u2(x,t)(4.2)max⁡i=1,…,k|u1(xi,t)-u2(xi,t)|⟶0ast⟶∞
implies that(4.3)sup⁡x∈I|u1(x,t)-u2(x,t)|⟶0ast⟶∞.
From a theorem in [17], if 𝒜ε attracts all the solutions in the norm of L∞(I) then almost every set of k nodes {x1,…,xk} 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 for𝒜ε in H4(I). In order to do that, we follow a strategy suggested by Robinson in [2].Theorem 4.2.
The global attractor𝒜ε of (1.1) is bounded in  H4(I).Proof.
Multiply the equation byut and integrate over I:
(4.4)‖ut‖2+ε22ddt‖uxx‖2+12∫IΦ′(ux)utxdx=0.
By integrating the previous equation, with respect to time, over (0,t) we get
(4.5)∫0t‖ut‖2ds+ε22‖uxx(t)‖2-ε22‖uxx(0)‖2+12∫0t(∫IΦ′(ux)utxdx)ds=0.
The integrand of the last term of the previous equality can be written as (d/dt)Φ(ux), and this yields to
(4.6)∫0t‖ut‖2ds=ε22‖uxx(0)‖2-ε22‖uxx(t)‖2+12∫I[Φ(ux(0))-Φ(ux(t))]dx.
Since the attractor is bounded in L∞ and H2(I), we have
(4.7)∫0t‖ut‖2ds≤ε2r22+2|I|‖ux‖L∞(I)2(‖ux‖L∞(I)2+1)≤ε2r22+2|I|r1r2(r1r2+1):=R.
We will prove that ut is uniformly bounded in L2(I). We derive (1.1) with respect to time, multiply it by t2ut, and integrate over I:
(4.8)12ddt‖tut‖2-t‖ut‖2+t2ε2‖uxxt‖2=t22∫I[Φ′(ux)]xtutdx=-t22∫IΦ′′(ux)utx2dx=-6t2∫Iux2utx2dx+2t2∫Iutx2dx.
Then, using interpolation we get
(4.9)12ddt‖tut‖2-t‖ut‖2+t2ε2‖uxxt‖2≤2t2‖utx‖2≤t2ε2‖uxxt‖2+t2ε2‖ut‖2,
from which
(4.10)12ddt‖tut‖2≤t2ε2‖ut‖2+t‖ut‖2.
Integrating the previous inequality, with respect to time, in (0,t), we have
(4.11)‖tut‖2≤2∫0t(s+s2ε2)‖ut‖2ds.
Since the term (s+s2/ε2) is bounded in [0,1], we obtain the following inequality by setting t=1:
(4.12)‖ut(1)‖2≤2(1+1ε2)∫0t‖ut‖2ds≤2R(1+1ε2).
This gives an uniform L2 bound for ut(1), and since u∈𝒜ε, it is sufficient to conclude that ut∈L∞(ℝ,L2(I)). In particular, let u∈𝒜ε and s≥0, we set û(t,x):=u(s+t-1,x) for any t≥1. Then ût(t,x)=ut(s+t-1,x) and so ût(1,x)=ut(s,x), from which we have that
(4.13)‖ut(s)‖2≤2R(1+1ε2),∀s>0.
Now we consider the L2 norm of the nonlinear term (1/2)Φ′′(ux)uxx:
(4.14)14∫I(Φ′′(ux))2uxx2dx≤4∫I(3‖ux‖L∞(I)+1)2uxx2dx≤4(6∥ux∥L∞(I)2+1)‖uxx‖2≤4(6r1r2+1)r22,
where we have used (1.8), (1.10), and [15]. Then, from the previous inequality, we get that 1/2Φ′′(ux)uxx is uniformly bounded in L2(I). Then since
(4.15)ε2uxxxx=-ut+12Φ′′(ux)uxx,
we have that uxxxx is uniformly bounded in L2(I). Then from the following interpolating inequality
(4.16)‖uxxx‖≤‖uxx‖1/2‖uxxxx‖1/2,
we get that 𝒜ε is bounded in H4(I).


## 5.  Absorbing Set inL∞(I)

In this last section we show the existence of an absorbing set inL∞(I).Theorem 5.1.
The semigroupS(t) defined in (1.4) posses an absorbing set in  L∞(I):
(5.1)‖S(t)u0‖L∞(I)≤r∞,ift>τ(R∞)+ε2,
where
(5.2)r∞=(1+1ε2)1/2|I|1/2ε1/2{ε22(1+4ε4)+1ε2}1/4,τ(R∞)=1ε2log⁡R∞,
and R∞ is a positive constant such that
(5.3)‖u0‖L∞(I)≤R∞.Proof.
Letu 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
(5.4)12ddt‖u‖2+ε2‖uxx‖2+12∫IΦ′(ux)uxdx=0,
from which
(5.5)12ddt‖u‖2+ε2‖uxx‖2+2‖ux‖L4(I)4=2‖ux‖2≤2‖ux‖L4(I)4|I|1/2≤‖ux‖L4(I)4+|I|,
and then
(5.6)12ddt‖u‖2+ε2‖uxx‖2≤|I|.
By the boundary conditions, from the hypothesis that |I|≤1 and from the interpolating inequality ∥ux∥2≤(1/2){∥u∥2+∥uxx∥2}, we have that
(5.7)‖u‖≤‖uxx‖.
Then we can rewrite (5.6) only in terms of the L2-norm of u:
(5.8)12ddt‖u‖2+ε2‖u‖2≤|I|.
Integrating the previous inequality over (0,t), we get
(5.9)‖u(t)‖2≤‖u0‖2e-2ε2t+|I|ε2(1-e-2ε2t)≤(R∞2e-2ε2t+1ε2)|I|,
where the last inequality follows from ∥u∥≤|I|1/2∥u∥L∞(I). Then if we set
(5.10)τ(R∞)=1ε2log⁡R∞,
we have that
(5.11)‖u(t)‖2≤(1+1ε2)|I|,ift≥τ(R∞).
Now coming back to (5.5), we use a different estimate method
(5.12)12ddt‖u‖2+ε2‖uxx‖2+2∥ux∥L4(I)4=2‖ux‖2≤2‖u‖‖uxx‖≤ε22‖uxx‖2+2ε2‖u‖2,
from which
(5.13)ddt‖u‖2+ε2‖uxx‖2≤4ε2‖u‖2.
Integrating the previous inequality with respect to time in (t,t+ρ), with ρ≥τ(R∞), we get
(5.14)ε2∫tt+ρ‖uxx‖2dt≤(4/ε2+2)‖u‖2≤(4/ε2ρ+2)(1+1/ε2)|I|.
Moreover by an interpolating inequality, we have that
(5.15)∫tt+ρ‖u‖2dt≤12{∫tt+ρ‖u‖2+∫tt+ρ‖uxx‖2}≤(1+1ε2)|I|{ρ2(1+4ε4)+1ε2}:=a3.
We multiply (1.1) by uxx and integrate over I:
(5.16)12ddt‖ux‖2+ε2‖uxxx‖2+6‖ux2uxx‖2=2‖uxx‖2,
from which by using an interpolating inequality and neglecting the positive term 6∥ux2uxx∥2 we get
(5.17)12ddt‖ux‖2+ε2‖uxxx‖2≤1ε2‖ux‖2+ε2‖uxxx‖2,
from which we conclude
(5.18)ddt‖ux‖2≤2ε2‖ux‖2.
We apply the uniform Gronwall lemma (see [1]) with
(5.19)a1=∫tt+ρ2ε2ds=2ρε2,a2=0,
from which we obtain
(5.20)‖ux(t+ρ)‖2≤a3ρea1,fort≥τ(R∞).
We fix ρ:=ε2, and then
(5.21)‖ux‖≤(1+1ε2)1/2|I|1/2ε{ε22(1+4ε4)+1ε2}1/2,fort≥τ(R∞)+ε2.
Then, using an interpolating inequality, we have that
(5.22)‖u‖L∞(I)≤‖u‖1/2‖ux‖1/2≤r∞,fort≥τ(R∞)+ε2,
where
(5.23)r∞=(1+1ε2)1/2|I|1/2ε1/2{ε22(1+4ε4)+1ε2}1/4.
This completes the proof, the set
(5.24)B∞={u∈L∞(I):‖u‖L∞(I)≤r∞}
is absorbing for all the bounded sets of L∞(I).



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*Source: 541426-2012-02-20.xml*