Singularly perturbed 1D Cahn–Hilliard equation revisited

Nonlinear Differ. Equ. Appl. 17 (2010), 663–695c© 2010 Springer Basel AG1021-9722/10/060663-33published online April 30, 2010DOI 10.1007/s00030-010-0075-0

Nonlinear Differential Equationsand Applications NoDEA

Singularly perturbed 1D Cahn–Hilliardequation revisited

Ahmed Bonfoh, Maurizio Grasselli and Alain Miranville

Abstract. We consider a singular perturbation of the one-dimensionalCahn–Hilliard equation subject to periodic boundary conditions. We con-struct a family of exponential attractors {Mε}, ε ≥ 0 being the perturba-tion parameter, such that the map ε �→ Mε is Holder continuous. Besides,the continuity at ε = 0 is obtained with respect to a metric independentof ε. Continuity properties of global attractors and inertial manifolds arealso examined.

Mathematics Subject Classification (2000). 35B25, 35B40, 37L25, 82C26.

Keywords. Cahn–Hilliard equations, Singular perturbations, Robustexponential attractors, Global attractors, Inertial manifolds.

1. Introduction

The long time behavior of solutions to many nonlinear evolution partial differ-ential equations is described by certain compact invariant subsets of the phasespace that attract uniformly the trajectories starting from bounded sets whentime goes to infinity. These sets are called global attractors and, in many cases,they have finite fractal dimension (see, e.g., [30,35,44]). However, we do notknow in general how trajectories approach the global attractor. In particular,the rate of attraction of the trajectories may be small and very sensitive toperturbations and approximations, including numerical simulations. Further-more, the known results on the continuity of global attractors with respectto perturbations make use of restrictive assumptions on the stationary solu-tions and their unstable sets (see, for instance, [35] and references therein).These drawbacks led Foias et al. to introduce a new object called exponentialattractor. This is a compact and positively invariant set with finite fractaldimension which attracts all the trajectories starting from bounded sets at auniform exponential rate (see [10]). Clearly, an exponential attractor containsthe global attractor. In addition, exponential attractors are more robust thanglobal attractors with respect to perturbations and approximations (see [35]and its references).

664 A. Bonfoh et al. NoDEA

The classical construction of exponential attractors makes use of orthog-onal projectors with finite rank and are valid in Hilbert spaces only (see, e.g.,[1,10]). Rather recently, this construction has been extended to Banach spaces(see [11], cf. also [9]). In particular, the new strategy devised in [11] allows toconstruct a robust (under perturbations) family of exponential attractors (seealso [12–14,24,34] and references therein). This family is characterized by anexplicit estimate on the symmetric distance between exponential attractors ofthe unperturbed and perturbed problems, and the continuity with respect tothe perturbations does not involve time shifts as in the previous results.

Here we want to use this machinery for refining some results obtained inthe literature for the singularly perturbed Cahn–Hilliard equation in one spa-tial dimension. We recall that the Cahn–Hilliard equation is central to Materi-als Science [4,39]. It is a conservation law (i.e., the spatial average of the orderparameter is conserved) and describes important qualitative features of phaseseparation processes, namely, the transport of atoms between unit cells. In theone-dimensional case, this equation reads

ρt − (−αρxx + f ′(ρ))xx = 0,

on some interval (0, L), L > 0, where α > 0 is given. Here ρ is the orderparameter (corresponding to a rescaled density of atoms) and f represents acoarse-grain free energy (e.g., a double-well potential) which accounts for thepresence of two different phases.

We want to compare the large time behavior of the above equation withthe one of its singular perturbation

ερtt + ρt − (−αρxx + f ′(ρ))xx = 0, (1.1)

for ε ∈ (0, ε0], 0 < ε0 ≤ 1 being fixed, which has been proposed to model theearly stages of spinodal decomposition in certain glasses (see [16–21,32]).

The first mathematical analysis of the longtime behavior of equation (1.1)with no-flux boundary conditions can be found in [7]. There the author provedthe existence of the (weak) global attractor and its upper semicontinuity withrespect to ε. These results were improved in [46], with different boundaryconditions, by establishing the existence of the (strong) global attractor andits smoothness, provided that ε is small enough. The same authors proved in[45] the existence of an exponential attractor and an inertial manifold, alwaysassuming ε small enough. We recall that inertial manifolds and their depen-dence on ε were also analyzed in [2,3] in the viscous case. The existence afamily of robust exponential attractors with no restriction on ε was proven in[22] by using the new construction mentioned above which is no longer basedon the squeezing property (see also [23] for a version with memory). How-ever, in [22], the convergence of the exponential attractors, as ε goes to 0, wasobtained with respect to a norm that depends on ε.

Inspired by the recent contributions [13] and [34], the main goal of thispaper is to construct a family of exponential attractors which is continuouswith respect to ε by using a metric which does not depend on ε as ε goes to 0.More precisely, in Sects. 2 and 3, we recall and demonstrate some basic resultson the perturbed problem and the unperturbed one. Then, in Sect. 4, using

Vol. 17 (2010) Singularly perturbed 1D Cahn–Hilliard equation revisited 665

a decomposition proposed in [41], we provide estimates of the differences oftrajectories associated with perturbation parameters ε1 and ε2 < ε1 (possiblyε2 = 0) with respect to a norm which does not depend on the parametersthemselves. Section 5 contains the main results. We first prove the existence ofa family of exponential attractors which is continuous with respect to ε. Moreprecisely, we first give explicit estimates of the Hausdorff symmetric distancebetween exponential attractors of two given perturbed problems on the onehand, and a proper exponential attractor of the perturbed problem and thenatural lifting of the corresponding exponential attractor in the unperturbedone on the other hand. These estimates are given with respect to a metricthat depends on the perturbation parameter. Then, we show the upper andlower semicontinuity of the exponential attractors at ε = 0 with respect to ametric independent of ε. Also, we establish the upper semicontinuity of theglobal attractor at any ε, including the limit case ε = 0. Here we report analternative proof along the lines of [17] (see also [2]) which differs from all theprevious ones based on a contradiction argument. Section 6 is dedicated toanalyze further properties of the global attractors under suitable assumptionson the nature of the stationary points. Finally, in Sect. 7, we discuss the exis-tence and stability of inertial manifolds. The results of this section completethe ones proven in [45] for Dirichlet-type boundary conditions; in particular,our construction allows us to establish some continuity properties of inertialmanifolds with respect to ε.

Equation (1.1) has recently been studied in more that one spatial dimen-sion (see [6,26–28,42]). In that case, the mathematical analysis is much moreinvolved. We conclude by mentioning that equation (1.1) has also been exam-ined from the numerical viewpoint (cf. [25]).

2. Functional setting and preliminary results

If W is a Sobolev-type space, then we set

W = {q ∈ W, m(q) = 0},

where

m(q) =1L

∫Ω

q(x) dx, Ω = (0, L).

Moreover, we set

q = q − m(q).

Let us define the linear unbounded operator

N = − d2

dx2: H2

per(Ω) → L2(Ω)

which is self-adjoint and nonnegative. If N is restricted to H2per(Ω), then it

turns out to be positive with compact inverse N−1. We consider the initial


and boundary value problem⎧⎪⎨⎪⎩

ρt + N(αNρ + f ′(ρ)) = 0,

ρ|t=0 = ρ0,

ρ is Ω − periodic,(2.1)

and its singular perturbation⎧⎪⎨⎪⎩

ερtt + ρt + N(αNρ + f ′(ρ)) = 0,

ρ|t=0 = ρ0, ρt|t=0 = ρ1,

ρ is Ω − periodic.(2.2)

We assume f ∈ C6(R) to be such that

lim inf|r|→+∞

f ′′(r) > −λ1, (2.3)

where λ1 > 0 is the first eigenvalue of N. Observe that condition (2.3) implies

f ′′(r) ≥ −C1, ∀r ∈ R, (2.4)

for some C1 ≥ 0. For instance, polynomials of degree 2p + 2 with strictlypositive leading coefficients (e.g., f(r) = (r2 − 1)p+1)) satisfy (2.3).

From now on, the same letter c (and sometimes cr, c′r, and ci, i ∈ N)

denotes positive constants that may change from line to line, but are alwaysindependent of ε. We denote by ‖.‖ and (., .) the usual norm and scalar productin L2(Ω).

We denote by W ′ the dual space of W . The space XY

denotes the clo-sure of a metric space X ⊂ Y in the topology of the complete metric space Y .For every r > 0, we endow (Hr

per(Ω))′ with the norm ‖q‖−r = (‖N−r/2q‖2 +|m(q)|2)1/2. We also note that ‖q‖r = (‖Nr/2q‖2 + |m(q)|2)1/2 is a norm onHr

per(Ω) which is equivalent to the usual Hr(Ω)−norm (see, e.g., [43]). Fur-thermore, there exist two positive constants C2, C3 such that

‖q‖−1 ≤ C2‖q‖ ≤ C3‖∇q‖, ∀q ∈ H1per(Ω).

We endow the Banach spaces H0ε = H1

per(Ω) × (H1per(Ω))′,H1

ε = H2per(Ω) ×

L2(Ω) and Hjε = Hj+1

per (Ω) × Hj−1per (Ω), j = 2, 3, . . ., with the norms

‖(p, q)‖H0ε

= (‖p‖21 + ε‖q‖2

−1)1/2,

‖(p, q)‖H1ε

= (‖p‖22 + ε‖q‖2)1/2,

‖(p, q)‖Hjε

= (‖p‖2j+1 + ε‖q‖2

j−1)1/2,

respectively. For ε=0, we agree that Hj0 =Hj+1

per (Ω) for any j ∈N. Then we set

Kδ = {u ∈ H1per(Ω), |m(u)| ≤ δ},

Kδ = {(u, v) ∈ H0ε , |m(u)| + ε0|m(v)| ≤ δ},

Kjδ = Kδ ∩ Hj+1

per (Ω),

Kjδ = Kδ ∩ Hj

ε ,

for some δ ≥ 0, where j ∈ N.


We start by recalling the following result (see [43]).

Theorem 2.1. We assume that (2.3) hold and that ρ0 ∈ H1per(Ω). Then problem

(2.1) possesses a unique solution ρ such that

ρ ∈ C([0, T ];H1per(Ω)) ∩ L2(0, T ;H3

per(Ω)), m(ρ(t)) = m(ρ0),

for any T > 0. Furthermore, if ρ0 ∈ H2per(Ω), then

ρ ∈ C([0, T ];H2per(Ω)) ∩ L2(0, T ;H4

per(Ω)).

Thanks to this result, we can define the semigroup

S(t) : H1per(Ω) → H1

per(Ω), ρ0 �→ ρ(t), t ≥ 0,

where ρ(t) is the solution to (2.1) at time t. This semigroup possesses theglobal attractor Aδ on Kδ which is bounded in K4

δ (cf., e.g., [38]).Let us now introduce the following weak formulation of problem (2.2).

Problem Pε. For any given T > 0, find ρ : [0, T ] → H2per(Ω) such that

ρ(0) = ρ0, ρt(0) = ρ1,

and, for almost every t ∈ [0, T ] and all q ∈ H3per(Ω),

d

dt[ε(ρt, q) + (ρ, q)] + α(ρx, (Nq)x) + ((f ′(ρ))x, qx) = 0. (2.5)

Observe that, taking q = 1 in (2.5), we obtain

εd2

dt2m(ρ) +

d

dtm(ρ) = 0,

so that

m(ρ(t)) = m(ρ0) + εm(ρ1)(1 − e−t/ε), ∀t ≥ 0, (2.6)

and

m(ρt(t)) = m(ρ1)e−t/ε, ∀t ≥ 0. (2.7)

Thus, when t goes to infinity, m(ρt(t)) → 0 and m(ρ(t)) → m(ρ0) + εm(ρ1).We now report a well-posedness result for Pε which is proven in [7] (see

also [22] and [46]).

Theorem 2.2. We assume that (2.3) holds and that (ρ0, ρ1) ∈ H0ε . Then Pε

possesses a unique solution ρ such that

(ρ, ρt) ∈ C([0, T ];H0ε )

and

ρtt ∈ L2(0, T ; (H3per(Ω))′),

for any T > 0. Moreover, if (ρ0, ρ1) ∈ H1ε , then

(ρ, ρt) ∈ C([0, T ];H1ε ), ρtt ∈ L2(0, T ; (H2

per(Ω))′).


Thanks to Theorem 2.2, we can define the semigroup

Sε(t) : H0ε → H0

ε , (ρ0, ρ1) �→ (ρ(t), ρt(t)), t ≥ 0,

where ρ(t) is the solution to Pε at time t. Moreover, arguing as in [22], forevery ε ∈ (0, ε0], we can prove the existence of the global attractor Aε,δ forSε(t) on Kδ that is bounded in K4

δ .In order to compare the dynamics of problems (2.1) and (2.2) (i.e., Pε),

we introduce the canonical lifting of the exponential attractor in the unper-turbed case (see [30]; cf. also [14,24]). Here the second component is recon-structed by means of the unperturbed equation, namely, we define a mappingL : H5

per(Ω) → H1per(Ω) by setting

Lρ = −N(αNρ + f ′(ρ)). (2.8)

Then we introduce the lifting of an exponential attractor Mδ for S(t) on K4δ ,

(Mδ)0 = {(ρ,Lρ) ∈ H2ε , ρ ∈ Mδ}.

More generally, if B is a bounded set in H5per(Ω), we indicate its lifting by

(B)0.We now recall that, following the proof of [22, Theorem 4.1], it is possible

to construct a bounded absorbing set for the semigroup Sε(t) on K4δ of the

form

B5,δ = {(u, v) ∈ K4δ , ‖(u, v)‖H4

ε≤ r},

with r independent of ε. Sometimes, B5,δ will be denoted by Bε5,δ. We assume

that

Bδ = {u ∈ K4δ , ‖u‖5 ≤ r}

is a bounded absorbing set for {S(t)}t≥0 in K4δ (see [38]). That is, there exists

t∗ > 0 such that Sε(t)B5,δ ⊂ B5,δ and S(t)Bδ ⊂ Bδ for all t ≥ t∗. From nowon, we set

B5,δ = Sε(t∗)B5,δ, Bδ = S(t∗)Bδ

and we will always assume that t∗ ≥ 1. Then we have

B5,δ ⊂ {(u, v) ∈ K4δ , ‖(u, v)‖H4

ε≤ r}

and

Bδ ⊂ {u ∈ K4δ , ‖u‖5 ≤ r}.

Note that B5,δ and Bδ are bounded attracting and absorbing sets for Sε(t)and S(t), respectively, as well. The Hausdorff semi-distance with respect tothe metric of E is defined as:

distE(A,B) = supa∈A

infb∈B

‖a − b‖E ,

whereas the symmetric Hausdorff distance between A and B reads

distsymE (A,B) = max{distE(A,B),distE(B,A)}.


A problem similar to Pε, but characterized by the following boundaryconditions

ρ(0, t) = ρ(L, t) = ρxx(0, t) = ρxx(L, t) = 0, ∀t ≥ 0,

was considered in [22]. Setting A = − d2

dx2 , with domain D(A) = H2(Ω)∩H10 (Ω),

it was defined, for every ε ≥ 0 and s ∈ R, the family of Hilbert spaces Hsε =

D(A(s+1)/2) × D(A(s−1)/2), with the norm

‖(p, q)‖Hsε

= (‖A(s+1)/2p‖2 + ε‖A(s−1)/2q‖2)1/2.

As we already mentioned, it was shown the existence of a closed boundedattracting set B5 for Sε(t) on H4

ε . This set is exponentially attracting in H0ε

as well (that is, any bounded set B ⊂ H0ε is attracted exponentially to B5

with respect to metric induced by the H0ε−norm), due to the transitivity of

exponential attraction property (see [22, Lemma 4.4]), first devised in [14,Theorem 5.1]. Then, it was constructed a robust family of exponential attrac-tors Mε on B5 with respect to the metric induced by the H0

ε−norm, whosecommon basin of attraction is the whole phase-space H0

ε , with a uniform upperbound on the fractal dimension and also a uniform rate of attraction of trajec-tories. Moreover, it was obtained an explicit estimate of the symmetric distancebetween Mε and the canonical lifting (M)0 of the corresponding unperturbedexponential attractor M, that is,

distsymH0

ε(Mε, (M)0) ≤ Cεσ, (2.9)

where C and 0 < σ ≤ 18 are independent of ε. More specifically, let a sequence

(uε, vε) ∈ Mε converge in the H0ε−norm to an element (u,Lu) ∈ (M)0 when

ε → 0. Then, ‖uε − u‖H1(Ω) → 0 and√

ε‖vε − Lu‖−1 → 0, when ε → 0, andtherefore, uε → u in H1

0 (Ω) and√

εvε → 0 in H−1(Ω), though we can-not say whether vε converges to Lu or not. In the present paper, as in [24], we

construct for Sε(t) a robust family of exponential attractors Mε,δ on B5,δ

H0ε

with respect to the metric induced by the H0ε -norm, whose common basin of

attraction is the whole phase-space Kδ, with a uniform upper bound on thefractal dimension as well as a uniform rate of attraction of trajectories. Here,inspired by [13], we show that the map ε �→ Mε,δ is Holder continuous in thefollowing sense

distsymH0

ε1(Mε1,δ,Mε2,δ) ≤ C ′|ε1 − ε2|σ, (2.10)

for all 0 < ε2 < ε1 ≤ ε0, where C ′ and 0 < σ ≤ 14 are independent of ε1 and

ε2. In addition, we also prove that

distH01(Mε,δ, (Mδ)0) ≤ C ′′εν , (2.11)

where C ′′ and 0 < ν ≤ 12 are independent of ε, and

limε→0

distH01((Mδ)0,Mε,δ) = 0. (2.12)

In other words, if (uε, vε) ∈ Mε,δ converges in the H01−norm to (u,Lu) ∈

(Mδ)0, then uε → u in H1per(Ω) and vε → Lu in (H1

per(Ω))′ as ε → 0.


In the next Sects. 3 and 4, we derive useful estimates which will serve inestablishing the main results of the paper (cf. Theorems 5.2, 5.3, 6.1 and 7.3)in Sects. 5 and 7.

3. Uniform estimates

In this section we prove some bounds which will be useful in Sect. 4.

Proposition 3.1. For any solution ρ given by Theorem 2.2 such that the tra-jectory (ρ(t), ρt(t))t≥0 lies in B5,δ, and any ε ∈ (0, ε0], we have

∫ t+1

t

‖ρt(s)‖23ds ≤ cr, ∀t ≥ 0. (3.1)

Proof. Let us take q = N3ρt(t) in (2.5). This gives

d

dt[ε‖ρt‖2

3] + α((N3/2ρ)x, N3ρt) + 2‖ρt‖23 − 2((N2f ′(ρ))x, N3/2ρt) = 0. (3.2)

We have

‖(N2f ′(ρ))x‖ ≤ cr,

since ‖ρ(t)‖H5(Ω) ≤ cr,∀t ≥ 0. We then deduce, recalling (2.7), that

d

dt[ε‖ρt‖2

3 + α‖ρ‖25] + c1‖ρt‖2

3 ≤ cr, (3.3)

whence (3.1). �

Proposition 3.2. For any solution ρ given by Theorem 2.2 such that thetrajectory (ρ(t), ρt(t))t≥0 lies in B5,δ, we have

‖(ρ(t), ρt(t))‖2H1

1≤ cr, ∀t ≥ 1, (3.4)

for any ε ∈ (0, ε0].

Proof. Taking q = Nρt(t) in (2.5) we obtain

d

dt[ε‖ρt‖2

1] + 2‖ρt‖21 + α((N2ρ)x, N1/2ρt) + 2((Nf ′(ρ))x, N1/2ρt) = 0. (3.5)

Therefore, we have

d

dt[ε‖ρt‖2

1] + ‖ρt‖21 ≤ cr, (3.6)

so thatd

dt[ε‖ρt‖2

1et/ε] ≤ cre

t/ε. (3.7)

We integrate (3.7) between s and t + 1, for any s ≤ t + 1. This yields

ε‖ρt(t + 1)‖21e

(t+1)/ε ≤ cε‖ρt(s)‖21e

s/ε + crε(e(t+1)/ε − es/ε

). (3.8)


Integrating now (3.8) between t and t + 1 with respect to s, we find

ε‖ρt(t + 1)‖21e

(t+1)/ε

≤ cε

∫ t+1

t

‖ρt(s)‖21e

s/εds + crε

∫ t+1

t

(e(t+1)/ε − es/ε

)ds. (3.9)

Therefore, we have

ε‖ρt(t + 1)‖21 ≤ ε

∫ t+1

t

‖ρt(s)‖21ds + crε. (3.10)

Thus, due to (3.1), we deduce

‖ρt(t)‖21 ≤ cr, ∀t ≥ 1, (3.11)

which gives

‖ρt(t)‖21 ≤ cr, ∀t ≥ 1, (3.12)

on account of (2.7). In other words, trajectories lying in B5,δ are not onlybounded in the H4

ε−norm but also in the H11−norm uniformly with respect

to ε. �Proposition 3.3. For any solution ρ given by Theorem 2.2 such that the tra-jectory (ρ(t), ρt(t))t≥0 lies in B5,δ, and any ε ∈ (0, ε0], we have∫ t

0

ε‖ρtt(s)‖21ds ≤ cr(1 + t), ∀t ≥ 0. (3.13)

Proof. We take q = Nρtt(t) in (2.5) and we findd

dt[‖ρt‖2

1 + 2((Nf ′(ρ))x, N1/2ρt) + 2α((N2ρ)x, N1/2ρt)] + 2ε‖ρtt‖21

= 2α((Nρt)x, N3/2ρt) + 2((f ′′(ρ)ρt)x, N3/2ρt). (3.14)

Therefore, we getd

dt[‖ρt‖2

1 + 2((Nf ′(ρ))x, N1/2ρt) + 2α((N2ρ)x, N1/2ρt)] + 2ε‖ρtt‖21

≤ cr‖ρt‖23. (3.15)

Integrating (3.15) between 0 and t, we find, on account of (3.4),∫ t

0

ε‖ρtt(s)‖21ds ≤ cr(1 + t), ∀t ≥ 0, (3.16)

and (3.13) follows, noting that∫ t

0m(ρtt(s))2ds ≤ c, ∀t ≥ 0. �

4. Estimates on the difference of two solutions

We now introduce the (nonlinear) manifold M = {(u, v) ∈ H0ε : v = Lu} and

we define the continuous semigroup S0(t) : M → M by setting

S0(t)(ρ0,Lρ0) = (S(t)ρ0,LS(t)ρ0), ∀ t ≥ 0.

Note that S0(t) has the global attractor (Aδ)0 on Kδ ∩ M.Let us first prove a useful bound on the solution to (2.1).


Proposition 4.1. Let ρ0 ∈ Bδ. Then the solution ρ given by Theorem 2.1 sat-isfies ∫ t

0

‖ρtt(s)‖2−1ds ≤ cr(1 + t), ∀t ≥ 0. (4.1)

Proof. First, we observe that ρ = ρt is the solution to the following linearizedproblem:

ρt + N(αNρ + f ′′(ρ)ρ) = 0, (4.2)ρ|t=0 = Lρ0, (4.3)

where ρ(t) = S(t)ρ0. We multiply (4.2) by N−1ρt and find

αd

dt‖ρ‖2

1 + 2‖ρt‖2−1 + 2

((f ′′(ρ)ρ)x, N−1/2ρt

)= 0. (4.4)

Since we have

‖(f ′′(ρ(t)))xρ(t)‖ ≤ cr‖ρt(t)‖1, ∀t ≥ 0,

then (4.4) yields

d

dt‖ρ‖2

1 + c‖ρt‖2−1 ≤ cr‖ρ‖2

1. (4.5)

If ρ0 ∈ Bδ, then we have ‖ρ(t)‖5 ≤ cr and ‖ρt(t)‖1 ≤ cr,∀t ≥ 0. Thus, (4.1)follows from (4.5). �

We now show the first basic estimate.

Proposition 4.2. Let (2.3) hold. Then, there exists t� > 0, independent of ε,such that

‖Sε(t)(ρ0, ρ1) − S0(t)(ρ0,Lρ0)‖2H0

1≤ εc(r, t), ∀t ≥ t�, (4.6)

for any (ρ0, ρ1) ∈ B5,δ and any ε ∈ (0, ε0].

Proof. We follow the strategy devised in [41]. For any solution ρε of (2.2), weset ρε = ρ1 + ρ2, where ρ1 and ρ2 solve the following problems

ερ1tt + ρ1

t + αN2ρ1 = 0, (4.7)ρ1|t=0 = 0, ρ1

t |t=0 = ρ1, (4.8)

and

ερ2tt + ρ2

t + N(αNρ2 + f ′(ρε)) = 0, (4.9)ρ2|t=0 = ρ0, ρ2

t |t=0 = 0, (4.10)

respectively. We then consider the difference of the solutions ρ and ρε toproblems (2.1) and (2.2) and we assume that the corresponding trajectories(ρε(t), ρε

t(t))t≥0 and (ρ(t))t≥0 lie in the absorbing sets B5,δ and Bδ, respec-tively. Thus we have

(ρε(t), ρεt(t)) = Sε(t)(ρ0, ρ1), (ρ(t), ρt(t)) = S0(t)(ρ0,Lρ0),


for (ρ0, ρ1) in B5,δ. Note that, due to (3.4), we have ‖ρ1‖H1(Ω) ≤ cr, andrecalling (2.8), we can see that

‖Lρ0‖1 ≤ cr.

We now set P = ρε − ρ and write P = ρ1 + v, where v = ρ2 − ρ satisfies thefollowing problem:

εvtt + vt + N(αNv + f ′(ρε) − f ′(ρ)) = −ερtt, (4.11)v|t=0 = 0, vt|t=0 = −Lρ0. (4.12)

Then we observe that

m(ρ1(t)) = εm(ρ1)(1 − e−t/ε), m(ρ1t (t)) = m(ρ1)e−t/ε, ∀t ≥ 0, (4.13)

which imply

m(ρ1(t))2 ≤ ε2m(ρ1)2,∫ t

0

m(ρ1t (s))

2ds ≤ εm(ρ1)2, ∀t ≥ 0. (4.14)

Let us take the L2-scalar product of (4.7) with ε−1N−1ρ1 and obtain

d

dt

[2(N−1/2ρ1

t , N−1/2ρ1) +

1ε‖ρ1‖2

−1

]+

2α

ε‖ρ1‖2

1 = 2‖ρ1t ‖2

−1. (4.15)

Similarly, we take the L2-scalar product of (4.7) with N−1ρ1t and obtain

d

dt

[ε‖ρ1

t ‖2−1 + α‖ρ1‖2

1

]+ 2‖ρ1

t ‖2−1 = 0. (4.16)

Adding together (4.15) and (4.16), with > 0 small, we get

dE1

dt+

c1

ε‖(ρ1, ρ1

t )‖2H0

ε≤ 0, (4.17)

where

E1(t) =

[2(N−1/2ρ1

t , N−1/2ρ1) +

1ε‖ρ1‖2

−1

]+[ε‖ρ1

t ‖2−1 + α‖ρ1‖2

1

]. (4.18)

Observe that

2(N−1/2ρ1t , N

−1/2ρ1) ≥ −2ε‖ρ1t ‖2

−1 − 12ε

‖ρ1‖2−1.

Clearly, there exists c1 independent of ε such that

E1(t) ≥ c1

(‖(ρ1(t), ρ1

t (t))‖2H0

ε+

1ε‖ρ1(t)‖2

−1

). (4.19)


Integrating (4.17) between 0 and t and noting that ρ1(0) = 0, we deduce∫ t

0

‖(ρ1(s), ρ1t (s))‖2

H0εds ≤ crε

2, ∀t ≥ 0. (4.20)

Therefore, on account of (4.14),∫ t

0

‖(ρ1(s), ρ1t (s))‖2

H0εds ≤ crε

2(1 + t), ∀t ≥ 0. (4.21)

On the other hand, noting that m(v(t)) = m(vt(t)) = 0,∀t ≥ 0, we take theL2-scalar product of (4.11) with ε−1N−1v and N−1vt, respectively, and obtain

d

dt

[2(N−1/2vt, N

−1/2v) +1ε‖v‖2

−1

]+

2α

ε‖v‖2

1 − 2ε(f ′(ρε) − f ′(ρ), v)

= 2‖vt‖2−1 − 2(N−1/2ρtt, N

−1/2v), (4.22)

andd

dt[ε‖vt‖2

−1 + α‖v‖21] + 2‖vt‖2

−1 + 2(f ′(ρε) − f ′(ρ), vt)

= −2ε(N−1/2ρtt, N−1/2vt). (4.23)

We now sum (4.22) and (4.23), for > 0 sufficiently small. It is easy torealize that

‖f ′(ρε) − f ′(ρ)‖H1(Ω) ≤ cr‖ρε − ρ‖1.

Moreover, we have, for any q ∈ (H1per(Ω))′,

(N−1/2ρtt, N−1/2q) ≤ c‖ρtt‖−1‖q‖−1.

We thus inferdE2

dt+

c1

ε‖(v, vt)‖2

H0ε

≤ c2E2 +c3

ε

(‖ρ1‖21 + ε2‖ρtt‖2

−1

), (4.24)

where

E2(t) = [2(N−1/2vt, N

−1/2v) + 1ε ‖v‖2

−1

]+ ε‖vt‖2

−1 + α‖v‖21, (4.25)

and

E2(t) ≥ c1

(‖(v(t), vt(t))‖2

H0ε

+1ε‖v(t)‖2

−1

). (4.26)

We deduce from (4.24), by applying the Gronwall lemma, that

E2(t) ≤[E2(0) +

c3

ε

∫ t

0

(‖ρ1(s)‖21 + ε2‖ρtt(s)‖2

−1

)ds

]ec2t ≤ crεe

c′rt,

on account of (4.1) and (4.21). Therefore,∫ t

0

‖(v(s), vt(s))‖2H0

εds ≤ crε

2ec′rt, ∀t ≥ 0, (4.27)

where the constants cr and c′r depend on r, but are independent of ε.

Let us multiply (4.16) by t, for any t > 0. This givesd

dt[εt‖ρ1

t ‖2−1 + αt‖ρ1‖2

1] + 2t‖ρ1t ‖2

−1 = ε‖ρ1t ‖2

−1 + α‖ρ1‖21. (4.28)


Integrating (4.28) between 0 an t, we find, on account of (4.20),

t‖(ρ1(t), ρ1t (t))‖2

H0ε

≤ crε2, ∀t ≥ 0. (4.29)

Let t� > 0, independent of ε, be such that e−(2t�)/ε ≤ ε. Any t� ≥ 12e realizes

this condition. Then, recalling (4.13), it follows that m(ρ1t (t))

2 ≤ εm(ρ1)2,∀t ≥t∗ (note that m(ρ1(t))2 ≤ ε2m(ρ1)2,∀t ≥ 0). Thus we have

t‖(ρ1(t), ρ1t (t))‖2

H0ε

≤ crε2, ∀t ≥ t∗. (4.30)

On the other hand, multiplying (4.23) by t, we find that

d

dt[εt‖vt‖2

−1 + αt‖v‖21] + t‖vt‖2

−1

≤ c(1 + t)‖(v, vt)‖2H0

ε+ tε2‖ρtt‖2

−1 + t‖ρ1‖21. (4.31)

We integrate (4.31) between 0 and t and obtain, due to (4.1), (4.21) and (4.27),

t‖(v(t), vt(t))‖2H0

ε≤ crε

2ec′rt, ∀t ≥ 0. (4.32)

Finally, it follows from (4.30) and (4.32) that

t‖(P (t), Pt(t))‖2H0

ε≤ crε

2ec′rt, ∀t ≥ t�, (4.33)

where cr and c′r depend on r, but are independent of ε.

Finally, estimate (4.6) follows from (4.33). �

We end this section by proving the second basic estimate.

Proposition 4.3. Let (2.3) hold and suppose ε1, ε2 ∈(0, ε0] are such that ε1 >ε2.Then, there exists t� > 0, independent of ε1 and ε2, such that∥∥∥∥Sε1(t) (ρ0, ρ1) − Sε2(t)

(ρ0,

√ε1√ε2

ρ1

)∥∥∥∥2

H01

≤ |ε1 − ε2|1/2c(r, t), ∀t ≥ t�, (4.34)

for any (ρ0, ρ1) ∈ Bε15,δ.

Proof. We first observe that (ρ0, (√

ε1/√

ε2)ρ1)∈Bε25,δ whenever (ρ0, ρ1)∈Bε1

5,δ.Let us consider two solutions ρε1 and ρε2 of (2.2) with ε = ε1 and ε = ε2, withinitial conditions (ρ0, ρ1) and (ρ0, (

√ε1/

√ε2)ρ1), respectively. We assume that

the trajectories (ρε1(t), ρε1t (t))t≥0 and (ρε2(t), ρε2

t (t))t≥0 lie in the absorbingsets Bε1

5,δ and Bε25,δ, respectively. Thus we have

(ρε1(t), ρε1t (t)) = Sε1(t) (ρ0, ρ1) , (ρε2(t), ρε2

t (t)) = Sε2(t)(

ρ0,

√ε1√ε2

ρ1

),

for (ρ0, ρ1) in Bε15,δ. Let us set P = ρε1 −ρε2 and ε = ε1 −ε2. Note that P solves

the following problem:

ε2Ptt + Pt + N(αNP + f ′(ρε1) − f ′(ρε2)) = −ερε1tt , (4.35)

P |t=0 = 0, Pt|t=0 =(

1 −√

ε1√ε2

)ρ1, (4.36)


and, for all t ≥ 0, we have

m(P (t)) =√

ε1(√

ε1 − √ε2)m(ρ1) + m(ρ1)

√ε1

(√ε2e

−t/ε2 − √ε1e

−t/ε1)

,

m(Pt(t)) = m(ρ1)√

ε1

(1√ε1

e−t/ε1 − 1√ε2

e−t/ε2

).

We observe that√

ε1e−t/ε1 − √

ε2e−t/ε2 = ε

(1

2√

ξ1

+t

ξ1

√ξ1

)e−t/ξ1 ,

1√ε1

e−t/ε1 − 1√ε2

e−t/ε2 = ε

(− 1

2ξ2

√ξ2

+t

ξ22

√ξ2

)e−t/ξ2 ,

where ξ1 = 1ε1 + (1 − 1)ε2 and ξ2 = 2ε1 + (1 − 2)ε2, for some 1,2 ∈[0, 1]. Let t� > 0, independent of ε1 and ε2, be such that t∗ ≥ 5

2e . Then, wehave e−t/ξ1 ≤ ξ1

√ξ1 and e−t/ξ2 ≤ ξ2

2

√ξ2,∀t ≥ t∗. Thus, we deduce

m(P (t))2 + m(Pt(t))2 ≤ cε(1 + t2), ∀t ≥ t∗, (4.37)

since (√

ε1 − √ε2)

2 ≤ ε. We take the L2-scalar product of (4.35) with N−1 ¯Pt.This yieldsd

dt[ε2‖ ¯Pt‖2

−1 + α‖ ¯P‖21] + 2‖ ¯Pt‖2

−1 + 2(f ′(ρε1) − f ′(ρε2), ¯Pt)=−2ε(ρε1tt , N

−1 ¯Pt).

(4.38)

Therefore, we getd

dt[ε2‖ ¯Pt‖2

−1 + α‖ ¯P‖21] + ‖ ¯Pt‖2

−1 ≤ cr‖P‖2H1(Ω) + ε2‖ρε1

tt ‖2−1. (4.39)

Applying the Gronwall lemma to (4.39), we obtain, on account of (3.13),

ε2‖ ¯Pt(t)‖2−1 + ‖ ¯P (t)‖2

1 ≤ cr[(√

ε1 − √ε2)2 + ε2ε−1

1 ]ec′rt

≤ crεec′

rt, ∀t ≥ 0, (4.40)

since εε−11 ≤ 1 and ‖ρ1‖ ≤ cr. Therefore, due to (4.37),

ε2‖Pt(t)‖2−1 + ‖P (t)‖2

1 ≤ crεec′

rt, ∀t ≥ t�, (4.41)

where cr and c′r depend on r, but not on ε1 and ε2. From the equality

√ε1Pt = (

√ε1 − √

ε2)Pt +√

ε2Pt,

we have

ε1‖Pt(t)‖2−1 − 2ε2‖Pt(t)‖2

−1 ≤ 2(√

ε1 − √ε2)

2‖Pt(t)‖2−1 ≤ crε, (4.42)

since ‖Pt(t)‖2−1 ≤ cr,∀t ≥ 0. Then, we infer from (4.41) and (4.42) that

ε1‖Pt(t)‖2−1 + ‖P (t)‖2

1 ≤ crεec′

rt, ∀t ≥ t�. (4.43)

We eventually deduce the estimate∥∥∥∥Sε1(t) (ρ0, ρ1) − Sε2(t)(

ρ0,

√ε1√ε2

ρ1

)∥∥∥∥2

H01

≤ εε−11 c(r, t), ∀t ≥ t�. (4.44)


To conclude, we note that, due to (4.6), there holds∥∥∥∥Sε1(t) (ρ0, ρ1) − Sε2(t)(

ρ0,

√ε1√ε2

ρ1

)∥∥∥∥2

H01

≤ ‖Sε1(t) (ρ0, ρ1) − S0(t)(ρ0,Lρ0)‖2H0

1

+∥∥∥∥Sε2(t)

(ρ0,

√ε1√ε2

ρ1

)− S0(t)(ρ0,Lρ0)

∥∥∥∥2

H01

≤ ε1c(r, t), ∀t ≥ t�. (4.45)

Hence, it follows from (4.44) and (4.45) that∥∥∥∥Sε1(t) (ρ0, ρ1) − Sε2(t)(

ρ0,

√ε1√ε2

ρ1

)∥∥∥∥2

H01

≤ min{ε1, εε−11 }c(r, t), ∀t ≥ t�.

(4.46)

Estimate (4.34) follows from (4.46) by noting that min{ε1, εε−11 } ≤ ε1/2. �

5. Exponential and global attractors

Let E be a metric space, X be a compact subset of E and {S(t)}t≥0 be acontinuous semigroup on E.

For the reader’s convenience, here below we recall the definition of anexponential attractor for the semigroup {S(t)}t≥0.

Definition 5.1. A compact set M is called an exponential attractor for S(t)for the topology of E if:(i) M is positively invariant under S(t), that is, S(t)M ⊂ M, ∀t ≥ 0;(ii) the fractal dimension of M is finite;(iii) there exists a constant c0 > 0 such that, for every bounded subset B ⊂

X, there exists a constant c1(B) > 0 such that

distE(S(t)B,M) ≤ c1e−c0t, ∀t ≥ 0.

We now give sufficient conditions ensuring the existence of uniform expo-nential attractors that are continuous with respect to ε. More precisely, wehave the

Theorem 5.1. Let E1, E2, V 1, V 2,W 1,W 2 be Banach spaces such that W i �V i � Ei, i = 1, 2. Set Eε = E1 ×E2, Vε = V 1 ×V 2,Wε = W 1 ×W 2 and endowthem with the following norms

‖(p, q)‖Eε= (‖p‖2

E1 + ε‖q‖2E2)1/2,

‖(p, q)‖Vε= (‖p‖2

V 1 + ε‖q‖2V 2)1/2,

‖(p, q)‖Wε= (‖p‖2

W 1 + ε‖q‖2W 2)1/2,

respectively, where ε ∈ [0, 1], with the convention that E0 = E1, V0 = V 1, andW0 = W 1. Let Bε(r) denote a closed ball in Wε of radius r > 0 and centeredat zero. Consider a one-parameter family of strongly continuous semigroups


{Sε(t)}ε acting on the phase-space Eε, for each ε ∈ [0, 1]. Then assume thatthere exist α, β, γ, ϑ ∈ (0, 1], κ ∈ (0, 1

2 ),Υj ≥ 0, and � > 0 (all independent ofε) such that, setting Bε = Bε(�), the following conditions hold:1. There exists a map L : B0 → V 2 which is Holder continuous of exponent

α. Here B0 is endowed with the metric topology of E1.2. There exists t� > 0, independent of ε, such that

Sε(t)Bε ⊂ Bε, ∀t ≥ t�,

and Bε is uniformly bounded (with respect to ε) in the E1−norm. Moreover,setting Sε(t�) = Sε, the map Sε satisfies, for every z1, z2 ∈ Bε,

Sεz1 − Sεz2 = Lε(z1, z2) + Kε(z1, z2),

where

‖Lεz1 − Lεz2‖Eε≤ κ‖z1 − z2‖Eε

, (5.1)‖Kεz1 − Kεz2‖Vε

≤ Υ1‖z1 − z2‖Eε. (5.2)

3. For any z1 = (η, ζ) ∈ Bε1 and any ε2 ∈ (0, ε1), there hold

‖Smε1 z1 − Sm

ε2 z2‖E1 ≤ Υm2 |ε1 − ε2|β , ∀m ∈ N, (5.3)

‖Sε1(t)z1 − Sε2(t)z2‖E1 ≤ Υ3|ε1 − ε2|γ , ∀t ∈ [t�, 2t�], (5.4)

where z2 = (η, (√

ε1/√

ε2)ζ) ∈ Bε2 . When ε1 > 0 and ε2 = 0, Smε2 z2 and Sε2(t)z2

are replaced in (5.3) and (5.4) by Lε1Sm0 Πε1z1 and Lε1S0(t)Πε1z1, respectively.

Here the “lifting” map Lε : B0 → Eε is defined by

Lεx ={

(x,Lx), if ε > 0,x, if ε = 0,

and Πε : Bε → B0 is the projection onto the first component when ε > 0, andthe identity map otherwise.

4. The map z �→ Sε(t)z is Lipschitz continuous on Bε endowed with themetric topology of Eε, with a Lipschitz constant independent of ε andt ∈ [t�, 2t�].

5. The map

(t, z) �→ Sε(t)z : [t�, 2t�] × Bε → Bε

is Holder continuous of exponent ϑ, where Bε is endowed with the metric topol-ogy of Eε.Then there exists a family of exponential attractors Mε on Bε = Bε

Eε with thefollowing properties:(i) Mε attracts Bε with an exponential rate which is uniform with respect to

ε, that is,

distEε(Sε(t)Bε,Mε) ≤ M1e

−ωt, ∀t ≥ 0,

for some M1 > 0 and some ω > 0.(ii) The fractal dimension of Mε is uniformly bounded with respect to ε, that

is,

dimEε[Mε] ≤ M2.


(iii) The family Mε is Holder continuous with respect to ε, that is, there exista positive constant M3 and τ ∈ (0, 1

2 ] such that

distsymEε1

(Mε1 ,Mε2) ≤ M3|ε1 − ε2|τ , (5.5)

for all 0 ≤ ε2 < ε1 ≤ 1, where M0 must be replaced by its lifting Lε1M0.In addition, there exist a positive constant M4 and σ ∈ (0, 1

2 ] such that

distE1(Mε,LεM0) ≤ M4εσ, (5.6)

for all 0 < ε ≤ 1, and

limε→0

distE1(LεM0,Mε) = 0. (5.7)

Here ω, τ, σ and Mj are independent of ε, ε1 and ε2, and they can becomputed explicitly.

Proof. Observe first that the existence of a family of exponential attractorsMε satisfying (i) and (ii) follows from [24]. In order to ensure that this familyfulfills (5.5), we follow a technique used in [34], based on an application of[13, Theorem 2.2]. We introduce the auxiliary operators Sε : E1 → E1, ε ∈(0, 1], Sε(t) = C−1

ε ◦ Sε(t) ◦ Cε and S0(t) = (S(t), 0),∀t ≥ 0, where Cε =(1 00 1√

ε

). We have Sε = Sε(t∗) : B1 → B1, for some t∗ > 0, and we set

Lε = C−1ε ◦ Lε ◦ Cε and Kε = C−1

ε ◦ Kε ◦ Cε. For any z = (u, v) ∈ B1, weset zi = (u, (1/

√εi)v), i = 1, 2, and we have zi ∈ Bεi

since Bε = CεB1. Thereholds ‖z‖E1 = ‖zi‖Eεi

, and we can easily deduce from (5.1) and (5.2) that

‖Lεz1 − Lεz2‖E1 ≤ κ‖z1 − z2‖E1 , (5.8)

‖Kεz1 − Kεz2‖V1 ≤ Υ1‖z1 − z2‖E1 . (5.9)

Also, we can deduce from (5.3) and (5.4) that

‖Smε1 z − Sm

ε2 z‖E1 ≤ Υm2 |ε1 − ε2|β , ∀m ∈ N, (5.10)

‖Sε1(t)z − Sε2(t)z‖E1 ≤ Υ3|ε1 − ε2|γ , ∀t ∈ [t�, 2t�]. (5.11)

It is thus clear that the maps Sε, Lε and Kε satisfy conditions 2. through 5.on B1 with respect to the metric induced by the E1−norm. Applying [34,Theorem 3.1], there exists a robust family of exponential attractors Nε for Sε

in B1E1 with respect to the metric induced by the E1−norm and such that

distE1(SεB1,Nε) ≤ M1e−ωt, (5.12)

dimE1 [Nε] ≤ M2, (5.13)distsym

E1(Nε1 ,Nε2) ≤ M3|ε1 − ε2|τ2 , (5.14)

where N0 is replaced by the lifting (M0, 0) of M0. Let us now set Mε = CεNε

for every ε ∈ (0, 1]. We note that, for any zi = (ηi, ζi) ∈ Mεi, and z0 =

(η0,Lη0) ∈ LεM0, we have zi = (ηi,√

εiζi) ∈ Nεiand z0 = (η0, 0) ∈ (M0, 0)

and there hold

‖z1 − z2‖Eε1≤ ‖z1 − z2‖E1 + M4(

√ε1 − √

ε2) (5.15)

‖z1 − z0‖Eε1≤ ‖z1 − z0‖E1 + M5

√ε1. (5.16)


For every ε ∈ (0, 1], we can check that Mε is an exponential attractor for theoperator Sε in Bε with respect to the metric induced by the Eε−norm, and(5.5) holds. In the case ε1 > 0 and ε2 = 0 we can say more. To achieve this goal,we follow a different construction devised in [24]. There a construction of expo-nential attractors Mε for Sε satisfying (i), (ii) and (5.5) with ε1 > 0 and ε2 = 0was given. On the other hand, a careful examination of the argument showsthat the upper semicontinuity estimate (5.6) holds as well. Indeed, thanks toAssumption 1, referring to the proof of [24], if En is a C( 1

2 +κ)n−net of Sn0 B0

in E0, then LεEn is not only a C ′( 12 + κ)αn−net of LεS

n0 B0 in Eε, but also

in E1, for some positive constants C and C ′. Thus, retracing the steps of theproof of the main theorem in [24] and taking advantage of (5.3), we can deduce(5.6). Finally, let us show the lower semicontinuity estimate (5.7). This is con-sequence of (5.5) and (5.6). Let us consider an element z0 = (u0,Lu0) ∈ LεM0.On account of (5.5), there exists a sequence (zεk

)k∈N in Mεk, with εk → 0, such

that zεk→ z0 in the Eεk

−norm, as k → +∞. But, due to (5.6) and the fact thatMε is uniformly bounded in V1, there also exist z0 = (u0,Lu0) ∈ LεM0 and asubsequence, still denoted by (zεk

)k∈N, such that zεk→ z0 in the E1−norm, as

k → +∞. Then, it is clear that z0 = z0 and the result follows. This completesthe proof of Theorem 5.1. �

Remark 5.1. A lower semicontinuity estimate analogous to (5.6) seems outof reach since one has to use the exponential attraction property of Mε thatholds in the Eε−norm only. We thus take this opportunity to correct estimate(78) in [2, Theorem 5.2] by replacing the symmetric Hausdorff distance withthe Hausdorff semidistance.

We now want to use this result, together with the transitivity of expo-nentially attraction property, to prove the existence of a family of robustexponential attractors Mε,δ for the semigroup Sε(t) on Kδ such that ε �→ Mε,δ

is continuous.Our main result is the following

Theorem 5.2. Let (2.3) hold. Then, for every ε ∈ (0, ε0], the semigroup Sε(t)

possesses an exponential attractor Mε,δ in B5,δ

H0ε

with respect to the metricinduced by the H0

ε−norm. Besides, the basin of attraction of each Mε,δ coin-cides with the whole phase space Kδ. Furthermore, the fractal dimension ofMε,δ has an upper bound which is independent of ε and the rate of attractionof trajectories with respect to the metric induced by the H0

ε -norm to Mε,δ isalso uniform in ε. The exponential attractors Mε,δ are Holder continuous withrespect to ε, i.e., for all 0 < ε2 < ε1 ≤ ε0,

distsymH0

ε1(Mε1,δ,Mε2,δ) ≤ M6|ε1 − ε2|ν , (5.17)

where M6 > 0 and 0 < ν ≤ 14 are independent of ε1 and ε2. In addition, we

have

distH01(Mε,δ, (Mδ)0) ≤ M7ε

σ, (5.18)


where the constant M7 > 0 and 0 < σ ≤ 12 are independent of ε, and

limε→0

distH01((Mδ)0,Mε,δ) = 0. (5.19)

Proof. On account of Theorem 5.1, we let

Eε = H0ε , Vε = H2

ε , Wε = H4ε ,

and we check all the assumptions 1–5.Observe first that there exists t∗ > 0 such that Sε(t)B5,δ ⊂ B5,δ for all

t ≥ t∗. Then, arguing as in [22], we can prove that B5,δ is an exponentiallyattracting set in Kδ. Thus we can apply Theorem 5.1 with Bε = B5,δ andwe can extend the basin of attraction to the whole phase-space by using thetransitivity of the exponential attraction.

Recalling (2.8), we see that L satisfies Assumption 1. Indeed, there existsa constant cr such that

‖Lρ1 − Lρ2‖−1 ≤ cr‖ρ1 − ρ2‖3 ≤ cr‖ρ1 − ρ2‖1/21 ,

for any ρ1 and ρ2 in Bδ. Observe that, for any q ∈ Bδ, we have ‖q‖3 ≤c‖q‖1/2

5 ‖q‖1/21 and ‖q‖5 ≤ cr. We now prove Assumption 2. Let us consider

two solutions ρ1 and ρ2 to (2.2) with initial conditions

ρi|t=0 = ρ0i, ρit|t=0 = ρ1i,

i = 1, 2, and set ρ = ρ1 − ρ2, ρ0 = ρ01 − ρ02 and ρ1 = ρ11 − ρ12. We assumethat both trajectories (ρi(t), ρit(t))t≥0, i = 1, 2, lie in B5,δ. We introduce thedecomposition ρ = ρ1 + ρ2, where ρ1 and ρ2 are the solutions to the followingproblems, respectively,

ερ1tt + ρ1

t + N(αNρ1 + γρ1) + C21ρ1 = 0, (5.20)

ρ1|t=0 = ρ0, ρ1t |t=0 = ρ1, (5.21)

and

ερ2tt + ρ2

t + N(αNρ2 + γρ2) + C21ρ2 = C2

1ρ, (5.22)ρ2|t=0 = 0, ρ2

t |t=0 = 0, (5.23)

where C21 is like in (2.4) and

γ(t) =∫ 1

0

f ′′(sρ1(t) + (1 − s)ρ2(t))ds.

We first observe that

m(ρ1(t)) = Ae− 1+√

1−4εC21

2ε t + Be− 1−√

1−4εC21

2ε t, ∀t ≥ 0,

m(ρ1t (t)) = Ce− 1+

√1−4εC2

12ε t + De− 1−

√1−4εC2

12ε t, ∀t ≥ 0,


where

A =12

(1 − (1 − 4εC2

1 )−1/2)

m(ρ0) − ε(1 − 4εC21 )−1/2m(ρ1),

B =12

(1 + (1 − 4εC2

1 )−1/2)

m(ρ0) + ε(1 − 4εC21 )−1/2m(ρ1),

C = C21 (1 − 4εC2

1 )−1/2m(ρ0) +12

(1 + (1 − 4εC2

1 )−1/2)

m(ρ1),

D = −C21 (1 − 4εC2

1 )−1/2m(ρ0) +12

(1 − (1 − 4εC2

1 )−1/2)

m(ρ1).

On the other hand, since 0 < ε ≤ ε0, there exist c0, c1 > 0, independent of ε,such that

εm(ρ1t (t))

2 + m(ρ1(t))2 ≤ c0e−c1t

(εm(ρ1)2 + m(ρ0)2

), ∀t ≥ 0.

Now, proceeding as in the proof of [22, Lemma 5.3], we can show that

‖(ρ1(t), ρ1t (t))‖2

H0ε

≤ cre−crt‖(ρ0, ρ1)‖2

H0ε, ∀t ≥ 0, (5.24)

and

‖(ρ2(t), ρ2t (t))‖2

H2ε

≤ c′re

c′rt‖(ρ0, ρ1)‖2

H0ε, ∀t ≥ 0, (5.25)

where cr and c′r depend on r, but not on ε. Estimates (5.24) and (5.25) entail

Assumption 2, provided we choose t∗ > 0 large enough in order to ensure thatcre

−crt∗< 1

2 . Note that t∗ can be also chosen in such a way that (4.6) and(4.34) hold so that Assumptions 3 follows.

Thus we are left to prove Assumptions 4 and 5. Indeed, defining

(ρ(t), ρt(t)) = Sε(t)(ρ01, ρ11), (ρ(t′), ρt(t′)) = Sε(t′)(ρ02, ρ12),

with (ρ01, ρ11), (ρ02, ρ12) ∈ B5,δ, and t, t′ ∈ [t∗, 2t∗], we obtain

‖Sε(t)(ρ01, ρ11) − Sε(t′)(ρ02, ρ12)‖H0ε

≤ ‖Sε(t)(ρ01, ρ11) − Sε(t′)(ρ01, ρ11)‖H0ε

+‖Sε(t′)(ρ01, ρ11) − Sε(t′)(ρ02, ρ12)‖H0ε. (5.26)

On the one hand, there holds

‖Sε(t)(ρ01, ρ11) − Sε(t′)(ρ01, ρ11)‖H0ε

≤ c(‖ρ(t) − ρ(t′)‖1 +

√ε‖ρt(t) − ρt(t′)‖−1

)

≤ c

∫ t′

t

‖ρt(s)‖1ds + c

∫ t′

t

√ε‖ρtt(s)‖−1ds

≤ c(r, t∗)|t′ − t|. (5.27)

On the other hand, it follows from (5.24)–(5.25) that

‖Sε(t′)(ρ01, ρ11) − Sε(t′)(ρ02, ρ12)‖H0ε

≤ c(r, t∗)‖(ρ01, ρ11) − (ρ02, ρ12)‖H0ε, ∀t′ > 0. (5.28)


Hence, we conclude with

‖Sε(t)(ρ01, ρ11) − Sε(t′)(ρ02, ρ12)‖H0ε

≤ c(r, t∗)(|t′ − t| + ‖(ρ01, ρ11) − (ρ02, ρ12)‖H0ε). (5.29)

This finishes the proof. �

We conclude this section by proving the upper semicontinuity of theglobal attractor Aε,δ at any ε2 > 0 in the H0

ε1-norm, for all 0 < ε2 < ε1 ≤ ε0,and at ε2 = 0 in the H0

1-norm.

Theorem 5.3. Let the assumptions of Theorem 5.2 hold. Then the global attrac-tors Aε,δ for the semigroup {Sε(t)}t≥0 are upper semicontinuous at any ε2 > 0with respect to the metric induced by the H0

ε1−norm and at ε2 = 0 with respectto the metric induced by the H0

1−norm, that is, for all 0 < ε2 < ε1 ≤ ε0,

limε1→ε2

distH0ε1

(Aε1,δ,Aε2,δ) = 0 (5.30)

and

limε→0

distH01(Aε,δ, (Aδ)0) = 0. (5.31)

Proof. In order to prove (5.30), we use the auxiliary operators Sε introducedin the proof of Theorem 5.1 once again. We denote by Aε,δ the global attrac-tor generated by this semigroup with respect to the metric induced by theH0

1-norm. We have Aε,δ = CεAε,δ, for any 0 < ε ≤ ε0, and

distH01(Sε(t)B1,Aε,δ) → 0, when t → +∞. (5.32)

Furthermore,

‖Sε1(t)(u, v) − Sε2(t)(u, v)‖2H0

1≤ c(r, t)|ε1 − ε2|1/2, ∀t ≥ t�, (5.33)

for any (u, v) ∈ B15,δ (cf. (4.34) and (5.11)). We are now ready to show that

limε1→ε2

distH01(Aε1,δ,Aε2,δ) = 0. (5.34)

To do so, let (u, v) ∈ Aε1,δ. Recalling that Aε1,δ ⊂ B15,δ, we have (cf. (5.33))

‖Sε1(t)(u, v) − Sε2(t)(u, v)‖2H0

1≤ c(r, t)|ε1 − ε2|1/2, ∀t ≥ t�. (5.35)

The definition of the global attractor Aε2,δ implies

distH01(Sε2(t)(u, v),Aε2,δ) → 0, when t → +∞. (5.36)

This shows that, if η > 0, then there exist (ϕ, φ) belonging to Aε2,δ and tη ≥ t�depending only on η such that

‖Sε2(tη)(u, v) − (ϕ, φ)‖2H0

1≤ η2

2. (5.37)

We now choose εη (which only depends on η) such that c(r, tη)|εη − ε2|1/2 ≤η2/2. For any ε2 < ε1 ≤ εη, we have, on account of (5.35),

‖Sε1(tη)(u, v) − Sε2(tη)(u, v)‖2H0

1≤ η2

2. (5.38)


Hence, we deduce from (5.37) and (5.38) that

‖Sε1(tη)(u, v) − (ϕ, φ)‖2H0

1≤ η2. (5.39)

This result is uniform with respect to (u, v), so that

distH0ε2

(Sε1(tη)Aε1,δ,Aε2,δ) ≤ η, (5.40)

whenever ε2 < ε1 ≤ εη. Finally, the invariance property of Aε1,δ entails thatSε1(tη)Aε1,δ = Aε1,δ. This completes the proof of (5.34). Recalling now (5.15),it is clear that

distH0ε1

(Aε1,δ,Aε2,δ) ≤ distH01(Aε1,δ,Aε2,δ) + M4(

√ε1 − √

ε2), (5.41)

and, passing to the limit ε1 → ε2, (5.30) follows. As far as the proof of (5.31)is concerned, we need not use the auxiliary operators Sε. A direct proof isavailable, as it was the case for (5.6). Indeed, let (ρ0, ρ1) ∈ Aε,δ. Recalling thatAε,δ ⊂ B5,δ, the following estimate holds (cf. (4.6)):

‖Sε(t)(ρ0, ρ1) − S0(t)(ρ0,Lρ0)‖2H0

1≤ c(r, t)ε, ∀t ≥ t�. (5.42)

Thus, owing to the definition of the global attractor (Aδ)0,

distH01(S0(t)(ρ0,Lρ0), (Aδ)0) → 0, when t → +∞. (5.43)

This shows that, if η > 0, then there exist (ϕ, φ) belonging to (Aδ)0 and tη ≥ t�depending only on η such that

‖S0(tη)(ρ0,Lρ0) − (ϕ, φ)‖2H0

1≤ η2

2. (5.44)

We now choose εη (which only depends on η) such that c(r, tη)εη ≤ η2/2. Forany 0 < ε ≤ εη, we have, on account of (5.42),

‖Sε(tη)(ρ0, ρ1) − S0(tη)(ρ0,Lρ0)‖2H0

1≤ η2

2. (5.45)

Hence, we deduce from (5.44) and (5.45) that

‖Sε(tη)(ρ0, ρ1) − (ϕ, φ)‖2H0

1≤ η2. (5.46)

This result is uniform with respect to (ρ0, ρ1), so that

distH01(Sε(tη)Aε,δ, (Aδ)0) ≤ η, (5.47)

whenever 0< ε≤ εη. Finally, the invariance property of Aε,δ entails that Sε(tη)Aε,δ = Aε,δ. This completes the proof of (5.31) (cf. also [2, Theorem 6.1]). �

6. Further results on the global attractors

This section is devoted to the study of the structure of the global attractorsand their stability properties with respect to ε under suitable assumptions onthe nonconstant equilibria.

We start by showing the following result.


Proposition 6.1. The semigroup {Sε(t)}t≥0 is a (strongly) continuous semi-group of class C1, for any ε ∈ (0, ε0]. Furthermore, Sε = Sε(t∗) defines aC1−mapping from H0

ε into H0ε . In addition to (4.34), there hold

‖S′ε(ρ0, ρ1)(s0, s1) − S′

0(ρ0,Lρ0)(s0, Ls0)‖H01

≤ c(r, t∗)ε1/2 (6.1)

and ∥∥∥∥S′ε1(ρ0, ρ1)(s0, s1) − S′

ε2

(ρ0,

√ε1√ε2

ρ1

)(s0,

√ε1√ε2

s1

)∥∥∥∥H0

1

≤ c(r, t∗)|ε1 − ε2|1/4, (6.2)

for any (ρ0, ρ1)∈Bε15 (Bε

5 for (6.1)), any (s0, s1)∈H0ε1 such that ‖(s0, s1)‖H0

1≤

cr, any 0 < ε ≤ ε0 and any 0 < ε2 < ε1 ≤ ε0.

Proof. Observe that

S′ε(t)(ρ0, ρ1)(s0, s1) = (s(t), st(t))

is defined by the problem

εstt + st + N(αNs + sf ′′(ρ)) = 0, (6.3)s|t=0 = s0, st|t=0 = s1, (6.4)

where (ρ, ρt) = Sε(t)(ρ0, ρ1). Similarly, S′(t)(ρ0)s0 = s(t) is the solution to theproblem

st + N(αNs + sf ′′(ρ)) = 0, (6.5)s|t=0 = s0, (6.6)

where ρ = S(t)ρ0. We now consider the difference of the solutions to (6.3) and(6.5), Q = sε − s, which satisfies the following equation:

εQtt + Qt + N(αNQ + sεf ′′(ρε) − sf ′′(ρ)) = −εstt, (6.7)

Q|t=0 = 0, Qt|t=0 = s1 − Ls0, (6.8)

where Ls0 is defined like L (see (2.8)), but via the linearized equation. We notethat

‖N1/2(sεf ′′(ρε) − sf ′′(ρ))‖ ≤ c(‖Q‖H1(Ω) + 1),

for any (ρ0, ρ1) ∈ B5. Proceeding as in Section 4, we find (6.1), where

S′0(t)(ρ0,Lρ0)(s0, Ls0) = (s(t), Ls(t)), ∀t ≥ 0.

We now consider the difference of two solutions to (6.3)

(sε1(t), sε1t (t)) = S′

ε1(t)(ρ0, ρ1)(s0, s1),

(sε2(t), sε2t (t)) = S′

ε2(t)(

ρ0,

√ε1√ε2

ρ1

)(s0,

√ε1√ε2

s1

). (6.9)

The function Q = sε1 − sε2 satisfies the following equation:

ε2Qtt + Qt + N(αNQ + sε1f ′′(ρε1) − sε2f ′′(ρε2)) = −εsε1tt , (6.10)

Q|t=0 = 0, Qt|t=0 =(

1 −√

ε1√ε2

)s1, (6.11)


where ε = ε1 − ε2. We can also show that∥∥∥∥S′ε1(t)(ρ0, ρ1)(s0, s1) − S′

ε2(t)(

ρ0,

√ε1√ε2

ρ1

)(s0,

√ε1√ε2

s1

)∥∥∥∥2

H01

≤ εε−11 c(r, t), ∀t ≥ t∗, (6.12)

for any (ρ0, ρ1) ∈ Bε15 and any (s0, s1) ∈ H0

ε1 such that ‖(s0, s1)‖H01

≤ cr. Weconclude as in the proof of Proposition 4.3, on account of (6.1) and (6.12).This completes the proof. �

Observe now that {S0(t)}t≥0 has a Lyapunov functional

J0(u, v) =∫

Ω

(α

2|∇u|2 + f(u)

)dx

on (Aδ)0. On the other hand, for every 0 < ε ≤ ε0, the semigroup {Sε(t)}t≥0

possesses a Lyapunov functional

Jε(u, v) =∫

Ω

(α

2|∇u|2 + f(u) +

ε

2|N−1/2v|2

)dx

on Aε,δ. Let us remark that, if (u, v) ∈ Aε,δ, there exists a complete orbit(ρ(t), ρt(t))t∈R in Aε,δ such that ρ(0) = u and ρt = v. We then have m(ρt(t)) =m(v)e−t/ε and, therefore, m(v) = m(ρt(t))et/ε, for all t ∈ R. Since m(ρt(t)) isbounded, we have m(v) = 0 by letting t → −∞. Therefore, m(ρ) is indepen-dent of t.

Setting

Kμδ = {u ∈ Kδ, m(u) = μ}, Kμ

δ = {(u, 0) ∈ Kδ, m(u) = μ},

Aμδ = Aδ ∩ Kμ

δ , Aμε,δ = Aε,δ ∩ Kμ

δ ,

it is clear that

Aδ =⋃

|μ|≤δ

Aμδ

and

Aε,δ =⋃

|μ|≤δ

Aμε,δ.

Moreover, Aμδ and Aμ

ε,δ are global attractors for the semigroups S(t) and Sε(t)restricted to Kμ

δ and Kμδ , respectively, which will be denoted by Sμ(t) and

Sμε (t). We consider the set of stationary solutions to (2.1) (with respect to S0)

subject to the additional condition m(ρ0) = μ

Z(Sμ0 ) = {(z, 0) ∈ H1

ε , Gz = N(αNz + f ′(z)) = 0, m(z) = μ}(cf. [40]) and we recall that an element (z, 0) of Z(Sμ

0 ) is hyperbolic if thedifferential operator G′(z) : H2

per(Ω) → L2(Ω) defined by G′(z)v = N(αNv +vf ′′(z)) is invertible. We also note that, if (z, 0) ∈ Z(Sμ

0 ) is hyperbolic, thenit is isolated. Also, note that Z(Sμ

0 ) contains the trivial solution ρ(t) = μ.A straightforward application of known results (see, e.g., [30, Sect. 3.8])

yields the following.


Theorem 6.1. Assume that μ ∈ [−δ, δ] and that all the nontrivial stationarysolutions to P0 subject to the condition m(ρ0) = μ are hyperbolic. Then theglobal attractors Aμ

ε,δ are lower semicontinuous at any ε ∈ [0, ε0]. Therefore,the global attractors Aμ

ε,δ are continuous at any ε. Moreover, there hold

distsymH0

1(Aμ

ε1,δ,Aμε2,δ) ≤ M8|ε1 − ε2|�, (6.13)

for all 0 < ε2 < ε1 ≤ ε0, and

distsymH0

1(Aμ

ε,δ, (Aμδ )0) ≤ M9ε

κ, (6.14)

for every 0 < ε ≤ ε0, where the constants M8,M9 > 0 and 0 < ≤ 14 , 0 <

κ ≤ 12 are independent of ε.

Proof. Since Z(Sμ0 ) ⊂ (Aδ)0 is compact in H1

ε , then Z(Sμ0 ) is a finite set con-

sisting of, say, m elements. Observe that, if (z, 0) is a hyperbolic stationarysolution to (2.1) with respect to Sμ

0 , then (z, 0) is a hyperbolic stationary solu-tion to (2.2). If all the stationary solutions are hyperbolic, it is well known thatthe global attractors (Aμ

δ )0 and Aμε,δ are the union of the unstable manifolds

with respect to Sμ0 and Sμ

ε of (zi, 0), i = 1, . . . ,m, respectively (see [29] fordetails). Moreover, for every ε ∈ (0, ε0], there exist local unstable manifoldsW(Sμ

ε |O(zi,0)) of Sμε at (zi, 0) and a corresponding local unstable manifold

W(Sμ0 |O(zi,0)) of Sμ

0 at (zi, 0) such that the global attractors are represented,respectively, as

(Aμδ )0 =

m⋃i=1

+∞⋃j=0

(Sμ0 )jW(Sμ

0 |O(zi,0)), Aμε,δ =

m⋃i=1

+∞⋃j=0

(Sμε )jW(Sμ

ε |O(zi,0)).

In addition, for any (zi, 0), i = 1, . . . , m, there exist a sufficiently small ε1 anda neighborhood O(zi,0), chosen independently of ε, 0 < ε ≤ ε1, such that thelocal unstable manifolds are graphs, that is,

W(Sμ0 |O(zi,0)) = {(u, v) ∈ H1

ε : (u, v)

= (u+, v+) + g0(u+, v+), (u+, v+) ∈ O(zi,0) ∩ E+(zi, 0)}and

W(Sμε |O(zi,0)) = {(u, v) ∈ H1

ε : (u, v)

= (u+, v+) + gε(u+, v+), (u+, v+) ∈ O(zi,0) ∩ E+(zi, 0)},

where g0, gε : O(zi,0) ∩ E+(zi, 0) → E−(zi, 0), g0(zi, 0) = 0 and gε(zi, 0) = 0.Note that E+(zi, 0) and E−(zi, 0) are the invariant linear subspaces corre-sponding to subsets of the spectrum of (Sμ

ε )′(zi, 0) which are in the domains{λ ∈ C : |λ| > 1} and {λ ∈ C : |λ| < 1}, respectively. These are inde-pendent of time and the dimension of E+ is finite. Moreover, we have E =E+(zi, 0) ⊕ E−(zi, 0) (cf. [2, Definition 6.4]). It follows from [44, Theorem 1]that the mappings g0, gε and their differentials g′

0, g′ε satisfy estimates of the


form (4.6)–(4.34) and (6.1)–(6.2), respectively. This result leads to the follow-ing comparison estimates:

distsymH1

1(W(Sμ

ε |O(zi,0)),W(Sμ0 |O(zi,0))) ≤ c0

√ε,

distsymH0

1(W(Sε1 |O(zi,0)),W(Sε2 |O(zi,0))) ≤ c0|ε1 − ε2|1/4,

where c0 is independent of ε, ε1 and ε2. Since there is a finite number of sta-tionary solutions (zi, 0), we can compare the whole global attractors (Aμ

δ )0and Aμ

ε,δ (compare with [41]). �

Remark 6.1. We can observe that the global attractors Aδ and Aε,δ have con-tinua of stationary solutions, due to the trivial solutions. Thus, we cannotexpect to prove the lower semicontinuity in that case. Once again, we takethis opportunity to correct estimate (99) in [2, Theorem 6.2] by replacing Aδ

and Aε,δ by Aμδ and Aμ

ε,δ, respectively.

7. Inertial manifolds

We first give the definition of an inertial manifold (see, e.g., [15,33,43]).

Definition 7.1. Let E be a metric space and {S(t)}t≥0 be a continuous semi-group on E. A set M is called an inertial manifold for S(t) if:(i) M is a finite dimensional Lipschitz manifold in E;(ii) M is smooth, that is, M is of class C1;(iii) M is positively invariant under the flow, that is, S(t)M ⊂ M, ∀t ≥ 0;(iv) M is exponentially attracting, that is, there exists a constant c0 such

that, for every u0 ∈ E, there exists a constant c1(u0) > 0 such that

distE(S(t)u0,M) ≤ c1e−c0t, ∀t ≥ 0.

We know that problem (2.1) possesses a bounded absorbing set Bδ inH5

per(Ω). Thus we can truncate the nonlinear term for large ‖ρ‖5. This allowsto construct an inertial manifold which is globally realized as a graph for thesemigroup generated by the modified equation and also by the original equa-tion (see [45, p. 875]). Then, since the resulting modified equation is identicalto the original one within the absorbing set, the intersection of this graphwith the absorbing set also defines an inertial manifold (note that it is also anexponential attractor) for the original equation (cf., e.g., [15,31]).

We now define a truncated function by setting

g(ρ) = θ

(‖ρ‖5

r

)f ′(ρ), (7.1)

where r is related with the absorbing set Bδ and is defined in Sect. 2 andθ : R

+ → [0, 1] is a C∞ function such that

θ(s) ={

1, if 0 ≤ s ≤ 1,0, if s > 2,

and

|θ′(s)| ≤ 2, ∀s ≥ 0.


We now consider an equation which is “equivalent” to the original equa-tion for t large, namely,

ρt + N(Nρ + g(ρ)) = 0. (7.2)

Then the following result holds.

Theorem 7.1. Let the assumptions of Theorem 5.2 hold. Then there exists aninertial manifold Mδ for the semigroup S(t) generated by equation (2.1) onKδ.

Proof. It is well known that there exists an inertial manifold Mδ for the semi-group S(t) generated by the modified equation (7.2) on Kδ (see [38,43]). Onthe other hand, arguing as in [45], one can show that Mδ is also an inertialmanifold for the semigroup S(t) generated by equation (2.1) on Kδ. �

We now consider the following system which is “equivalent” to the orig-inal equation for t large:

ερtt + ρt + N(Nρ + g(ρ)) = 0, (7.3)

where g is defined like g (see (7.1)) and where r is related with the absorbingset B5,δ. In this case, we can prove the

Theorem 7.2. Let the assumptions of Theorem 5.2 hold. Then, for any ε ∈(0, ε0], there exists an inertial manifold Mε,δ for the semigroup Sε(t) gener-ated by equation (2.2) on Kδ of the same dimension as Mδ (cf. Theorem 7.1).

Proof. This result was proven in detail in [3, Theorem 5.1] for a singularly per-turbed viscous Cahn–Hilliard–Gurtin equation in one and two space dimen-sion. We can thus follow the same steps with minor modifications.

First, we introduce the following change of variables: ρt = −(2ε)−1ρ +ε−1/2v and the auxiliary problem:

Ut + AU + G(U) = 0, (7.4)

where

U =(

ρv

), A =

(12ε − 1√

ε1√εN2 − 1

4ε√

εI 1

2ε

), G(U) =

(0

1√εNg(ρ)

).

Then, we show that, for any ε ∈ (0, ε0], there exists an inertial manifold Nε,δ

for the semigroup generated by (7.4) on Kδ of the same dimension as Mδ (cf.Theorem 7.1).

Now, for all ε ∈ (0, ε0], we introduce the auxiliary semigroups Tε(t) :H0

ε → H0ε defined by

Tε(t) (ρ0, ρ1) =(

ρ(t),√

ερt(t) +1

2√

ερ(t)

), ∀t ≥ 0,

where ρ is the solution to problem (7.3) with the initial conditions

ρ|t=0 = ρ0, ρt|t=0 = − 12ε

ρ0 +1√ερ1.


Introducing the matrix

Oε =

⎛⎝ 1 0

− 12ε

1√ε

⎞⎠ ,

we can also write

Sε(t) = Oε ◦ Tε(t) ◦ O−1ε , ∀ε > 0.

Clearly, Nε,δ is an inertial manifold for Tε(t), for every 0 < ε ≤ ε0, with respectto the metric induced by a norm ‖|.‖|H0

εequivalent to ‖.‖H0

ε. We set

Mε,δ = OεNε,δ, ∀ε ∈ (0, ε0].

For every ε ∈ (0, ε0],Mε,δ is an inertial manifold for the semigroup Sε(t)generated by (7.3) on Kδ with respect to the metric induced by the norm‖.‖H0

ε= ‖|C−1

ε .‖|H0ε. Indeed, we have, by definition,

distH0ε(Sε(t)Bε,Mε) = distH0

ε(TεB1,Nε), (7.5)

dimH0εMε = dimH0

εNε, (7.6)

and

Sε(t)Mε = OεTε(t)O−1ε Mε = OεTε(t)Nε ⊂ OεNε = Mε, ∀t ≥ 0. (7.7)

Finally, arguing as in [45], we deduce that Mε,δ is also an inertial manifoldfor the semigroup Sε(t) generated by the original equation (2.2) on Kδ. Thiscompletes the proof of the theorem. �

We observe that the inertial manifolds Mδ and Mε,δ, given by Theo-rems 7.1 and 7.2, respectively, are not only positively invariant, but also satisfyS(t)Mδ = Mδ and Sε(t)Mε,δ = Mε,δ,∀t ∈ R (see, e.g., [8,31]). We also notethat the lifting (Mδ)0 is an inertial manifold for the semigroups {S0(t)}t≥0

and {S0(t)}t≥0 on M. We now set

Mrε,δ = Mε,δ ∩ B5,δ, Mr

δ = Mδ ∩ Bδ.

As noted above, (Mrδ)0 and Mr

ε,δ are inertial manifolds for the semigroupsgenerated by the original equations and are only positively invariant underthe flow, that is, S0(t)(Mr

δ)0 ⊂ (Mrδ)0 and Sε(t)Mr

ε,δ ⊂ Mrε,δ,∀t ≥ 0.

Observe now that, if (u, s) ∈ Mrε,δ, then there exists a complete orbit

(ρ(t), w(t))t∈R in Mε,δ such that ρ(0) = u and w(0) = s. Hence we have

m(ρ(t)) = m(u) + εm(s)(1 − e−t/ε), ∀t ≥ 0,

m(w(t)) = m(s)e−t/ε, ∀t ≥ 0,

so that

m(s) = m(ρ(t))et/ε, ∀t ∈ R.

Since m(ρ(t)) and m(w(t)) are bounded for all t ≤ 0, it follows that m(s) = 0by letting t → −∞, and also m(ρ(t)) is independent of t.

We set

Mr,μδ = Mr

δ ∩ Kμδ , Mr,μ

ε,δ = Mrε,δ ∩ Kμ

δ .


It is clear that

Mrδ =

⋃|μ|≤δ

Mr,μδ

and

Mrε,δ =

⋃|μ|≤δ

Mr,με,δ .

Moreover, Mr,μδ and Mr,μ

ε,δ are inertial manifolds for the semigroups S(t) andSε(t) restricted to Kμ

δ and Kμδ , respectively.

We are now able to state the following result on the upper and lowersemicontinuity of the above inertial manifolds at ε = 0. These results werealso proven in detail in the viscous case (see [3, Propositions 6.2 and 6.3]).

Theorem 7.3. Let the assumptions of Theorem 5.2 hold and let μ ∈ [−δ, δ].Then the inertial manifolds Mr

ε,δ ∪ (Mδ)0 are upper semicontinuous at ε = 0and Mε,δ ∪ (Mr,μ

δ )0 are lower semicontinuous at ε = 0, with respect to themetric induced by the H0

1−norm, that is, for any η > 0, there exist tη, tη, εη

and εη such that

∀ε ≤ εη, distH01(Sε(tη)Mr

ε,δ, (Mδ)0) ≤ η (7.8)

and

∀ε ≤ εη, distH01((S(tη)Mr,μ

δ )0,Mε,δ) ≤ η. (7.9)

Remark 7.1. Using the same arguments, we can also prove that the familyof inertial manifolds Mr,μ

ε,δ are upper and lower semicontinuous at any ε2 > 0with respect to the metric induced by the H0

ε1−norm, that is, for all 0 < ε2 ≤ε1 ≤ ε0,

limε1→ε2

distsymH0

ε1(Mr,μ

ε1,δ,Mr,με2,δ) = 0. (7.10)

Remark 7.2. The existence and convergence, as ε goes to 0, of inertial manifoldsfor a singularly perturbed damped wave equation in one space dimension andunder stronger assumptions on the nonlinear term were obtained in [5,36,37].The method used in [36,37] works essentially for equations involving linearself-adjoint operators only. In [2], we successfully applied the latter methodto a singular perturbation of the standard viscous Cahn–Hilliard equation inone and two space dimensions. This method may also be applied to problem(2.2). However, our approach works for a larger class of nonlinear evolutiondynamical systems, even for equations that contain a non-self-adjoint operator(see [3]). Moreover, our results do not need stronger assumptions on g as in[5,36,37]. On the other hand, the results of [5] give the convergence of thefirst component of the inertial manifolds only. We somehow complete thoseresults by showing the strong convergence of the inertial manifolds for bothcomponents of the semigroup Sε(t).


Acknowledgments

The first author is grateful to the King Fahd University of Petroleum andMinerals for its support. The authors thank referee for his/her helpful andcareful reading of previous versions of this manuscript. His/her remarks andsuggestions greatly improved the quality of the present paper.

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A. BonfohDepartment of Mathematics and Statistics,King Fahd University of Petroleum and Minerals,P.O. Box 546, Dhahran 31261, Saudi Arabiae-mail: [email protected]

M. GrasselliDipartimento di Matematica “F. Brioschi”, Politecnico di Milano,Via E. Bonardi 9, 20133 Milano, Italye-mail: [email protected]

A. MiranvilleUniversite de Poitiers, Laboratoire de Mathematiques et Applications,UMR CNRS 6086. SP2MI, Boulevard Marie et Pierre Curie - Teleport 2,86962 Chasseneuil Futuroscope Cedex, Francee-mail: [email protected]

Received: 02 March 2009.

Accepted: 08 April 2010.

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Singularly perturbed 1D Cahn–Hilliard equation revisited

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