Advances in Mathematical GakkotoshoSciences and Applications Tokyo, JapanVol.?, No.? (2008), pp.??-??

Nonisothermal Allen-Cahn equations

with coupled dynamic boundary conditions

Ciprian G. GalDepartment of Mathematical Sciences

University of MissouriColumbia, MO, 65211, USA(ciprian@math.missouri.edu)

Maurizio GrasselliDipartimento di Matematica “F. Brioschi”

Politecnico di MilanoI-20133 Milano, Italy

(maurizio.grasselli@polimi.it)

Alain MiranvilleLaboratoire de Mathematiques et Applications

Universite de PoitiersF-86962 Chasseneuil Futuroscope Cedex, France

(Alain.Miranville@math.univ-poitiers.fr)

Abstract. We consider a phase-field model of Caginalp type whose order parameter φand (relative) temperature θ are subject to coupled dynamic boundary conditions. Weshow that some previous authors’ results on the uncoupled case can be extended to thepresent situation. These results are essentially concerned with the existence of global andexponential attractors and the convergence of single solutions to single equilibria.

————————————————————-Communicated by Editors; Received April ??, ????.AMS Subject Classification 35B41, 35K55, 37L30, 80A22.

1 Introduction

A well-known mathematical model which describes the behavior of the phases in presenceof temperature variations is given by the Allen-Cahn equation suitably coupled with theheat equation (see [C, CL, F], cf. also [BS, M]):

ψt − α∆ψ + F ′ (ψ)− λ0θ = 0,

(εθ + λ0ψ)t − b∆θ = 0,(1.1)

in Ω×(0, +∞), where Ω is a bounded domain in R3, with smooth boundary Γ := ∂Ω. Hereψ is the order parameter, θ denotes the (relative) temperature, λ0, α, b, and ε are givenpositive constants. Moreover, F is a potential which accounts for the presence of differentphases. For instance, F can be a logarithmic potential which is usually approximated bya double-well potential, i.e., F (y) = (y2 − 1)

2.

System (1.1) has been studied extensively in the last decade by many people. In partic-ular, well-posedness results for (1.1), subject to standard boundary conditions (Dirichletor Neumann) can be found in [S, EZ] (see also [DKS, SA]). The analysis of such dissipativedynamical systems has been carried out in several papers (see, for instance, [BZ, BCH,BH, GP, GWZ, JB, K, Kp, L]), proving theorems on existence of global and/or exponen-tial attractors. Lately, the asymptotic behavior of single solutions has been investigatedby means of the ÃLojasiewicz-Simon inequality (see [CM1, CM2, GPS], where singularpotentials are considered, cf. also [AFI, CFP, Z]). System (1.1) can also be associatedwith a series of dynamic boundary conditions for the order parameter ψ and, possibly,for the temperature θ (see [GG1], cf. also [CFP, CGM, GG2, GM]). In particular, theseboundary conditions for (1.1) read, in Γ× (0, +∞):

ψt − β∆Γψ + α∂nψ + γψ + G′ (ψ) = 0,

aθt + b∂nθ + cθ = 0.(1.2)

Here β and γ are positive constants, G is a suitable nonlinear function and ∆Γ denotesthe Laplace-Beltrami operator on the surface Γ. Concerning the second condition of (1.2),we assume that a is positive and c is always nonnegative. System (1.1)-(1.2) is usuallyendowed with the initial conditions

ψ|t=0 = ψ0, θ|t=0 = θ0. (1.3)

Problem (1.1)-(1.3) was analyzed in [GG1] (see also [GG2]) showing that it defines adissipative dynamical system on a suitable phase-space. Also, it was established theexistence of the global attractor as well as an exponential attractor. The derivation ofsystem (1.1)-(1.3) can be illustrated by considering the free energy functional

EΩ (ψ, θ) :=

∫

Ω

[α

2|∇ψ|2 + F (ψ)− λ0ψθ − ε

2θ2

]dx,

so that we have∂ψEΩ = −α∆ψ + F ′ (ψ)− λ0θ,

and we define the enthalpy density (see [C]) by

e = −∂θEΩ = λ0ψ + εθ. (1.4)

Here ∂ψ and ∂θ stand for the variational derivatives.The governing equations for ψ and θ are thus derived from the relations

ψt = −∂ψEΩ, et +∇ · q = 0. (1.5)

Assuming the classical Fourier law q = −b∇θ, we deduce the standard Caginalp system(1.1). To introduce physically reasonable boundary conditions, we argue as in [GG1], butwe take into account a possible coupling between the two quantities due to their dynamiccharacter. More precisely, we consider the following boundary free energy functional(compare with EΩ (ψ, θ))

EΓ (ψ, θ) :=

∫

Γ

[β

2|∇Γψ|2 + g1 (θ)

(γ

2ψ2 + G (ψ)

)− g2 (θ) ψ − a

2θ2

]dS,

where g1 and g2 are suitably chosen functions. We remind that, in the original derivationit was taken G ≡ 1 and gi, i = 1, 2, were supposed to be constants (see [GG1]). The firstequation of (1.1) is subject to the dynamic boundary condition

ψt = −∂ψEΓ − α∂nψ, (1.6)

where the term α∂nψ in (1.6) is due to the contribution of the functional EΩ (ψ, θ). Thenwe obtain

ψt − β∆Γψ + α∂nψ + g1 (θ) (γψ + G′ (ψ))− g2 (θ) = 0. (1.7)

As far as possible boundary conditions for the second equation of (1.1) are concerned, fol-lowing [GG1] once more, we define the external enthalpy source on the boundary (comparewith (1.4)), namely,

eΓ = −∂θEΓ = aθ − g′1 (θ)(γ

2ψ2 + G (ψ)

)+ g′2 (θ) ψ. (1.8)

In this paper, since we assume to have a linear coupling in the bulk, we will simply takeg1 (θ) = 1, g2 (θ) = δθ, where δ is a given positive constant. From (1.8), we have

eΓ = aθ + δψ. (1.9)

Assume now that there is a source (or sink) on the boundary represented by a functionΦ (x, t, θ,∇θ) for t > 0 and x ∈ Γ. Then the first Law of Thermodynamics yields

∫

Ω

et dx +

∫

Γ

(eΓ)t dS = −∫

Ω

∇ · q dx +

∫

Γ

Φ (x, t, θ,∇θ) dS.

Then on account of the second equations of (1.1) and (1.5), we infer∫

Γ

[(aθ + δψ)t − Φ (x, t, θ,∇θ)] dS = 0, (1.10)

which certainly holds when

(aθ + δψ)t = Φ (x, t, θ,∇θ) , on Γ× (0, +∞). (1.11)

Following [GG1], we may assume that Φ only depends linearly on the temperature fluxb∂nθ across the boundary as well as on the temperature θ, by letting Φ (θ,∇θ) = −b∂nθ−cθ. Then the resulting boundary condition for the heat equation reads

(aθ + δψ)t + b∂nθ + cθ = 0, on Γ× (0, +∞). (1.12)

This is a reasonably general boundary condition which contains the usual (homogeneous)ones along with the so-called dynamic boundary condition when a > 0 and δ ≥ 0.

In this contribution, we consider a generalization of (1.1)-(1.3) by endowing the sameequations (1.1) with the new dynamic boundary conditions (holding on Γ× (0, +∞)):

ψt − β∆Γψ + α∂nψ + γψ + G′ (ψ)− δθ = 0,

(aθ + δψ)t + b∂nθ + cθ = 0.(1.13)

Note that we can recover (1.1)-(1.2) whenever δ = 0 (i.e., no temperature contributionon the first boundary condition in (1.13); also, the term δψt disappears from the secondboundary condition). Here we will consider only the case where the boundary conditionfor θ is also dynamic, i.e., we will suppose a > 0 and c ≥ 0.

Our assumptions on the nonlinearities F and G are the following: F, G ∈ C2(R) satisfythe conditions

lim|y|→+∞

inf F ′′ (y) > 0, lim|y|→+∞

inf G′′ (y) > 0. (1.14)

This will be enough in order to establish the existence of the global attractor, but only inthe case c > 0. When c = 0, we will also require that

F ′ (y) y ≥ ν1y2 − ν ′1, G′ (y) y ≥ ν2y

2 − ν ′2, (1.15)

for some νi > 0, ν ′i ≥ 0, i = 1, 2. In addition, to construct an exponential attractor, wewill need to assume the local Lipschitz continuity of G′′ and F ′′.

The plan of the paper is as follows. In Section 2, we state first a rigorous formulation ofthe problem that we want to study, by defining suitable energy spaces. Then we discussthe well-posedness and the existence of a bounded absorbing set. Hence, in the samesection, we show the existence of the global attractor and of an exponential attractor.The final Section 3 is devoted to the convergence of solutions to single equilibria.

The present results are proven by taking full advantage of the argument developed in[GG1]. Moreover, arguing as in [GG2], instead of taking a phase-space of type H2 ×H1,we can consider a larger one of type H1 × L2, provided that F and, if β = 0, G satisfysuitable growth restrictions (however, F can still be a double-well potential).

We conclude by observing that an interesting open problem is the mathematical anal-ysis of a nonlinear (possibly quadratic) coupling not only between the equations in thebulk (like, e.g., in [L], see also [GGP, GP, GPS, GWZ]), but also in the dynamic bound-ary conditions. Another interesting question concerns the possibility of handling singularpotentials. These issues will be the subject of a future investigation.

2 Well-posedness and attractors

To establish the well-posedness of problem (1.1), (1.3), (1.13), we need to introduce pre-liminary results related to a suitable functional setup. Without loss of generality, we letα = β = b = ε = 1 in what follows and, from now on, we denote by ‖·‖p and ‖·‖p,Γ thenorms on Lp (Ω) and Lp (Γ) , respectively. In the case p = 2, 〈·, ·〉2 stands for the usualscalar product. The norms on Hs (Ω) and Hs (Γ) are indicated by ‖·‖Hs(Ω) and ‖·‖Hs(Γ),respectively, for any s > 0. Furthermore, another appropriate space for our problem is

X2a = L2

(Ω, dx|Ω ⊕ adS|Γ

),

with norm

‖θ‖2X2

a=

∫

Ω

|θ (x)|2 dx +

∫

Γ

|θ (x)|2 adSx

. (2.1)

As usual (see, e.g., [GG1]), we identify X2a with L2 (Ω, dx) ⊕ L2 (Γ, adS). In order to

account for all cases, we introduce the family of linear operators AK := −∆ on theBanach space X2

a, where K ∈ W0,W1 and W0, W1 stand for the following boundaryconditions: dynamic with c = 0 and dynamic with c > 0, respectively. It is known thatAWj

generates a bounded analytic semigroup e−AWjt on X2

a (cf. [FGGR]) and each AK isnonnegative and self-adjoint on X2

a. The domain D (AWj

)cannot be described explicitly,

but is known to be contained in the set of functions θ ∈ H2loc (Ω) ∩ H1 (Ω) such that

∆θ ∈ X2a and a (∆θ)|Γ + ∂nθ + cθ = 0 on Γ, if a > 0, j = 0, 1. For the sake of convenience,

we also set ZK := H1(Ω), if K ∈ W0,W1 , and the norm in ZK is defined by

‖θ‖2ZK

=

‖∇θ‖2

2 + c∥∥θ|Γ

∥∥2

2,Γ, if K = W1,

‖∇θ‖22 + | 〈〈θ〉〉 |2, if K = W0,

where, from now on, we set

〈u〉Ω =1

|Ω|∫

Ω

u (x) dx, 〈〈u〉〉 =1

|Ω|+ a |Γ|

∫

Ω

u (x) dx +

∫

Γ

u (x) adSx

. (2.2)

It is easy to check that, for each K ∈ W0,W1, the norm in ZK is equivalent to thestandard H1 (Ω)-norm. We also introduce the additional functional spaces

Vs = Cs(Ω

)‖·‖Vs,

where s = 0, 1 and the norms ‖·‖Vsare given by

‖Π‖2V1

=

∫

Ω

|∇Π|2 dx +

∫

Γ

|∇ΓΠ|2 dS + γ

∫

Γ

∣∣Π|Γ∣∣2 dS

and

‖Π‖2V0

=

∫

Ω

|Π|2 dx +

∫

Γ

∣∣Π|Γ∣∣2 dS,

respectively. It is easy to see that the identification Vs = Hs (Ω)⊕Hs (Γ) holds. In general,any vector Π ∈ Vs will be of the form (Π1, Π2) with Π1 ∈ Hs (Ω) and Π2 ∈ Hs (Γ) andthere need not be any connection between Π1 and Π2.

We can now specify which class of problems we wish to solve. Let K ∈ W0,W1 .Then we formulate the

Problem PK. For any given pair of initial data (ψ0, θ0) ∈ V2 × ZK , find a pair

(ψ, θ) ∈ C ([0, +∞);V2 × ZK) , (ψt, θt) ∈ L2([0, +∞);V1 × X2

a

), (2.3)

which solves the system

ψt −∆ψ + F ′ (ψ)− λ0θ = 0, a.e. in Ω× (0, +∞) ,

(θ + λ0ψ)t −∆θ = 0, a.e. in Ω× (0, +∞) ,(2.4)

and fulfills almost everywhere on Γ× (0, +∞)

ψt −∆Γψ + ∂nψ + γψ + G′ (ψ)− δθ = 0,

(aθ + δψ)t + ∂nθ + cθ = 0,(2.5)

together with the initial conditions (1.3).Concerning the well-posedness of (2.4)-(2.5), the proof mimics that of [GG1, Theorem

3.2] and is based on a priori estimates which can be proven within a suitable approximationscheme. For the sake of completeness, we only give a few details here. We will show thatthe boundary contributions from the terms δθ and δψt in the first and second equationsof (2.5) do not produce additional technical difficulties and, in many cases, they do notcontribute to the basic differential (in)equalities when taking the inner product of theequations in (2.4) and (2.5) with suitable test functions, in L2 (Ω) and L2 (Γ) , respectively(see, for instance, Lemma 3 and its proof below). However, for the reader’s convenience,we report here the main steps of the construction of an appropriate approximation scheme.To this end, set

Y 1T := C0([0, T ];V2) ∩H2([0, T ];V0) ∩ C1([0, T ];V1),

Y 2T := C0([0, T ];D (AK)) ∩H1([0, T ]; ZK) ∩ C1([0, T ];X2

a).

Then consider two approximating sequences ψ0n ∈ V3 and θ0n ∈ D (AK) ∩ H2 (Ω) ,K ∈ W0,W1 , such that

‖ψ0n − ψ0‖V2 → 0, ‖θ0n − θ0‖ZK→ 0,

as n goes to +∞. Fix n ∈ N and set

Y nT :=(χ, η) ∈ Y 1

T × Y 2T :

χ (0, ·) = ψ0n, χt (0, ·) = ψ0n, (χt)|Γ (0, ·) = ψ0n|Γ, η (0, ·) = θ0n,

whereψ0n|Ω := ∆ψ0n − F ′ (ψ0n) + λ0θ0n,

ψ0n|Γ := ∆Γψ0n − ∂nψ0n − γψ0n −G′ (ψ0n) + δθ0n.

Observe that (ψ0n|Ω, ψ0n|Γ) ∈ V1.Let (χ, η) ∈ Y n

T and consider the systems

ψt −∆ψ = l1 := −F ′ (χ) + λ0η, in Ω× (0, T ),

∆Γψ − ∂nψ − γψ − ψt = l2 := G′ (χ) + δη, on Γ× (0, T ),

and θt −∆θ = l3 := −λ0χt, in Ω× (0, T ),

aθt + ∂nθ + cθ = l4 := −δχt, on Γ× (0, T ),

subject to the initial conditions

ψ (0, ·) = ψ0n, θ (0, ·) = θ0n.

Then on account of [GG1, Theorem 2.1 and Theorem 2.3], since (χ, η) ∈ Y nT , it is easy

to check that l1 ∈ H1 ([0, T ]; L2 (Ω)) , l2 ∈ H1 ([0, T ]; L2 (Γ)) and (l3, l4) ∈ L2 ([0, τ ];X2a) .

Thus, the first problem above has a unique solution

ψ ∈ C ([0, T ];V2) ∩ C1 ([0, T ];V0) ∩H1 ([0, T ];V1) ,

while the second problem admits a unique solution

θ ∈ C ([0, T ]; ZK) ∩ L2 ([0, T ];D (AK)) ∩H1([0, T ];X2

a

).

In order to produce the higher-order estimates in Y nT , we set

ψ = ψt, θ = θt

and observe that these functions (formally) solve the linear problems

ψt −∆ψ = l5 := −F ′′ (χ) χt + λ0ηt, in Ω× (0, T ),

∆Γψ − ∂nψ − γψ − ψt = l6 := G′′ (χ) χt + δηt, on Γ× (0, T ),

and θt −∆θ = l7 := −λ0χtt, in Ω× (0, T ),

aθt + ∂nθ + cθ = l8 := −δχtt, on Γ× (0, T ),

subject to the initial conditions

θ (0, ·) = θ0n|Ω, ψ (0, ·) = ψ0n|Ω,

where ψ0n|Ω = (ψ0n|Ω, ψ0n|Γ), θ0n|Ω = (θ0n|Ω, θ0n|Γ), and

θ0n|Ω := ∆θ0n − λ0ψ0n|Ω, θ0n|Γ := (∆θ0n)|Γ − λ0ψ0n|Γ.

Here (∆θ0n)|Γ stands for the restriction of the spatial Laplace operator ∆ to the bound-

ary Γ. Having made this identification, it is readily seen that θ0n|Ω ∈ X2a, since θ0n ∈

D (AK) ∩ H2 (Ω) and (ψ0n|Ω, ψ0n|Γ) ∈ V0. Arguing now exactly as in [GG1, Theo-rem 3.2, (3.14)-(3.15)], it is not difficult to see that we have l5 ∈ L2 ([0, T ] ; L2 (Ω)) ,l6 ∈ L2 ([0, T ] ; L2 (Γ)) and (l7, l8) ∈ L2 ([0, T ] ;X2

a) . This allows us to define a mappingS : Y n

T → Y nT , S (χ (t) , η (t)) = (ψ (t) , θ (t)). By using the contraction mapping principle

and by choosing a sufficiently small time T > 0, one can show that the mapping S is acontraction of a bounded closed subset of Y n

T into itself, with respect to an appropriatemetric (see [GG1, (3.22)]). This immediately implies the existence and uniqueness of alocal approximating solution (ψn, θn) ∈ Y n

T , for some T > 0 small enough and dependingon n. Then taking advantage of its smoothness, we can find some a priori estimates whichare independent of n, proving that the solution is in fact global (cf. Lemma 3 below).Finally, passing to the limit as n → +∞, we recover a solution which satisfies our originalassumptions. Its uniqueness follows from Theorem 5 below.

Consequently, we have the following.

Theorem 1 For each K ∈ W0,W1 , problem PK admits a unique solution such that

θ ∈ L2loc ([0, +∞);D (AK)) .

Moreover, setting

IW0 (u, v) :=1

(|Ω|+ a |Γ|)

∫

Ω

(λ0u (x) + v (x)) dx +

∫

Γ

(δu (x) + av (x)) dSx

, (2.6)

there holds, for all t ≥ 0,

IW0 (ψ(t), θ(t)) = IW0 (ψ0, θ0) , if K = W0. (2.7)

Remark 2 Observe that, if (ψ, θ) solves (PK) , with K = W0, then the above quantityIW0 (ψ(t), θ(t)) is conserved because of the boundary conditions.

In order to prove the existence of global and exponential attractors for problem PK,following [GG1, Section 4], we define the phase space in the following way:

YK :=

V2 ×H1 (Ω) , if K = W1,

(V2 ×H1 (Ω)) ∩ |IW0 (ψ0, θ0)| ≤ M , if K = W0,(2.8)

where M ≥ 0 is fixed. Note that YK is a complete metric space with respect to the metricinduced by the norm of V2 ×H1 (Ω) .

The first result is concerned with uniform a priori estimates for the solutions to PK.

Lemma 3 For any given pair of initial data (ψ0, θ0) ∈ YK, the following estimate holds:

‖(ψ (t) , θ (t))‖2YK

+ ‖ψt (t)‖2V0

+

t+1∫

t

(‖ψt (s)‖2

V1+ ‖θt (s)‖2

X2a

)ds ≤ Q

(‖(ψ0, θ0)‖2YK

)e−ρt + C, (2.9)

for each t ≥ 0, where Q is a positive monotone nondecreasing function and the constantsρ, C are independent of t and of the initial data.

Remark 4 It is worth mentioning that, when K = W0, both the function Q and theconstant C in estimate (2.9) depend on M.

Proof. We take the inner product in L2 (Ω) of the first equation of (2.4) with ψt and2ξψ, respectively (for some ξ > 0), then take the inner product in L2 (Ω) of the secondequation of (2.4) with θ. Integrating by parts, using the boundary conditions of (2.5) andthen adding the resulting relationships, we obtain

Et (t) + κE (t) = Λ1 (t) , (2.10)

where κ > 0 and

E (t) := ‖ψ (t)‖2V1

+ 2 〈F (ψ (t)) , 1〉2 + 2 〈G (ψ (t)) , 1〉2,Γ + ‖θ (t)‖2X2

a+ ξ ‖ψ (t)‖2

V0+ C.

Here the constant C > 0 is taken large enough in order to ensure that E is nonnegative(recall that F and G are both bounded from below). The function Λ1 is given by

Λ1 (t) := 2κ[〈F ′ (ψ (t))− F ′ (ψ (t)) ψ (t) , 1〉2 + 〈G′ (ψ (t))−G′ (ψ (t)) ψ (t) , 1〉2,Γ

]

− (2ξ − κ) ‖ψ (t)‖2V1− 2 (ξ − κ)

[〈F ′ (ψ (t)) , ψ (t)〉2 + 〈G′ (ψ (t)) , ψ (t)〉2,Γ

]

−2 ‖ψt (t)‖2V0− 2 ‖∇θ (t)‖2

2 − 2c∥∥θ|Γ (t)

∥∥2

2,Γ

+ξκ ‖ψ (t)‖2V0

+ κ ‖θ (t)‖2X2

a+ 2ξλ0 〈θ(t), ψ(t)〉2 + 2ξδ 〈θ(t), ψ(t)〉2,Γ + κC. (2.11)

We first discuss the case when K = W1. Observe preliminarily that, owing to (1.14),we have

C∗ |F ′ (y)| (1 + |y|) ≤ 2F ′ (y) y + CF , C∗ |G′ (y)| (1 + |y|) ≤ 2G′ (y) y + CG, (2.12)

F (y)− F ′ (y) y ≤ C ′F |y|2 + C

′′F , G (y)−G′ (y) y ≤ C ′

G |y|2 + C′′G, (2.13)

for any y ∈ R. Here all the constants involved in (2.12)-(2.13) are positive and sufficientlylarge constants which depend on F and G, only. From (2.11) and the fact that the traceoperator from ZK to L2 (Γ) , is continuous, it follows that

Λ1 (t) ≤ − (2ξ − κ− 2κ (C ′

F ′ + C ′G′)− ξκ− ξ2λ2

0 − ξ2δ2) ‖ψ (t)‖2

V1− 2 ‖ψt (t)‖2

V0

−C∗ (ξ − κ)(〈|F ′ (ψ (t))| , 1 + |ψ (t)|〉2 + 〈|G′ (ψ (t))| , 1 + |ψ (t)|〉2,Γ

)

− (1− κC0) ‖θ(t)‖2ZK

+ C1,

where C0 > 0 and C1 > 0 depend on ξ, κ and CF , CG at most. From now on, Ci standsfor a positive constant which is independent of the initial data and of time.

It is thus possible to adjust κ and ξ in order to have

Et (t) + κE (t) + C2

(‖ψ (t)‖2V1

+ ‖ψt (t)‖2V0

+ ‖θ (t)‖2ZK

)

+C3

(〈|F ′ (ψ (t))| , 1 + |ψ (t)|〉2 + 〈|G′ (ψ (t))| , 1 + |ψ (t)|〉2,Γ

)≤ C1.

Then, applying a suitable version of Gronwall’s inequality (see, e.g., [GGP, Lemma 2.5]),we deduce that

E (t) +

t+1∫

t

(‖ψ (s)‖2V1

+ ‖ψt (s)‖2V0

)ds +

t+1∫

t

〈|F ′ (ψ (s))| , 1 + |ψ (s)|〉2 ds

+

t+1∫

t

〈|G′ (ψ (s))| , 1 + |ψ (s)|〉2,Γ ds +

t+1∫

t

‖θ (s)‖2ZK

ds

≤ C4E (0) e−κt + C6, ∀ t ≥ 0. (2.14)

To prove a similar uniform inequality when K = W0, we need to use the enthalpy conser-vation (2.7). Observe that the last terms on the right-hand side of (2.11) can be rewrittenas follows:

2ξλ0 〈θ, ψ〉2 + 2ξδ 〈θ, ψ〉2,Γ = 2ξλ0 〈θ − 〈〈θ〉〉 , ψ〉2 + 2ξδ 〈θ − 〈〈θ〉〉 , ψ〉2,Γ

+2ξ (|Ω|+ a |Γ|) IW0(ψ0, θ0) 〈〈θ〉〉 − 2ξ (|Ω|+ a |Γ|) 〈〈θ〉〉2 . (2.15)

By using a generalized version of the Friedrich’s inequality, we can find C > 0, independentof v, such that, for all v ∈ H1(Ω),

‖v − 〈〈v〉〉‖2X2

a= ‖v − 〈〈v〉〉‖2

2 + a ‖v − 〈〈v〉〉‖22,Γ

= ‖v‖2X2

a− (|Ω|+ a |Γ|) 〈〈v〉〉2 ≤ C ‖∇v‖2

2 . (2.16)

We rewrite Λ1, defined by (2.11), by taking (2.15) into account. We obtain

Λ1 (t) := 2κ[〈F ′ (ψ (t))− F ′ (ψ (t)) ψ (t) , 1〉2 + 〈G′ (ψ (t))−G′ (ψ (t)) ψ (t) , 1〉2,Γ

]

− (2ξ − κ) ‖ψ (t)‖2V1− 2 (ξ − κ)

[〈F ′ (ψ (t)) , ψ (t)〉2 + 〈G′ (ψ (t)) , ψ (t)〉2,Γ

]

−2 ‖ψt (t)‖2V0

+ 2ξκ ‖ψ (t)‖2V0− 2 (ξ − κ) (|Ω|+ a |Γ|) 〈〈θ(t)〉〉2

+2ξ (|Ω|+ a |Γ|) IW0(ψ0, θ0) 〈〈θ(t)〉〉+ κ ‖θ (t)‖2X2

a− 2ξ (|Ω|+ a |Γ|) 〈〈θ(t)〉〉2

−2 ‖∇θ (t)‖22 + 2ξλ0 〈θ(t)− 〈〈θ(t)〉〉 , ψ〉2 + 2ξδ 〈θ(t)− 〈〈θ(t)〉〉 , ψ〉2,Γ + κC. (2.17)

Using (2.16), we infer from (2.17)

Λ1 (t) ≤ − (2ξ − κ− 2κ (C ′F ′ + C ′

G′)− ξκ) ‖ψ (t)‖2V1− 2 ‖ψt (t)‖2

V0

−2 (ξ − κ)[〈F ′ (ψ (t)) , ψ (t)〉2 + 〈G′ (ψ (t)) , ψ (t)〉2,Γ

]

+2ξ (|Ω|+ a |Γ|) IW0(ψ0, θ0) 〈〈θ(t)〉〉 − 2 (ξ − κ) (|Ω|+ a |Γ|) 〈〈θ(t)〉〉2

−(1− 2κC

)‖∇θ (t)‖2

2 + ξ2C(λ2

0 + δ2) ‖ψ (t)‖2

V0+ C7. (2.18)

It is now left to estimate 〈〈θ〉〉 . We integrate all equations of (2.4) over Ω and (2.5) overΓ. Then adding the resulting equations, we obtain

d

dt〈〈θ〉〉+ λ2

0 〈〈θ〉〉 =(λ0 − δ) |Γ||Ω|+ a |Γ| 〈ψt〉Γ +

λ0γ |Γ||Ω|+ a |Γ| 〈ψ〉Γ

+λ0 |Γ|

|Ω|+ a |Γ| 〈G′ (ψ)〉Γ +

λ0 |Ω||Ω|+ a |Γ| 〈F

′ (ψ)〉Ω +

(λ2

0a− λ0δ) |Γ|

|Ω|+ a |Γ| 〈θ〉Γ , (2.19)

where we have set

〈u〉Γ =1

|Γ|∫

Γ

u (x) dSx.

We now multiply (2.19) by

ζ :=2ξ (|Ω|+ a |Γ|) IW0 (ψ0, θ0)

λ20 − κ

,

provided that κ < λ20, and we add the resulting relation to (2.10). Then on account of

(2.18), we inferd

dt[E (t) + ζ 〈〈θ (t)〉〉] + κ [E (t) + ζ 〈〈θ (t)〉〉]

≤ −(2ξ − κ− 2κ (C ′

F ′ + C ′G′)− ξκ− ξ2

(λ2

0 + δ2)C

)‖ψ (t)‖2

V1− 2 ‖ψt (t)‖2

V0

−2 (ξ − κ)[〈F ′ (ψ (t)) , ψ (t)〉2 + 〈G′ (ψ (t)) , ψ (t)〉2,Γ

]− 2 (ξ − κ) (|Ω|+ a |Γ|) 〈〈θ(t)〉〉2

−(1− 2κC

)‖∇θ (t)‖2

2 +(λ0 − δ) |Γ| ζ|Ω|+ a |Γ| 〈ψt(t)〉Γ +

λ0γ |Γ| ζ|Ω|+ a

b|Γ| 〈ψ(t)〉Γ

+λ0 |Γ| ζ|Ω|+ a |Γ| 〈G

′ (ψ(t))〉Γ +λ0 |Ω| ζ|Ω|+ a |Γ| 〈F

′ (ψ(t))〉Ω

+ζ

(λ2

0a− λ0δ) |Γ|

|Ω|+ a |Γ| 〈θ(t)〉Γ + C8. (2.20)

Due to assumption (1.15), it follows that, for any positive constants η, η′, there existpositive constants Cη, Cη′ such that

|〈F ′ (ψ(t))〉Ω| ≤ 〈|F ′ (ψ(t))|〉Ω ≤ η 〈F ′ (ψ (t)) , ψ (t)〉2 + Cη, (2.21)

|〈G′ (ψ (t))〉Γ| ≤ 〈|G′ (ψ (t))|〉Γ ≤ η′ 〈G′ (ψ (t)) , ψ (t)〉2,Γ + Cη′ . (2.22)

Using (2.21)-(2.22) with η and η′ small enough, it is easy to see that

λ0 |Ω| ζ|Ω|+ a |Γ| 〈F

′ (ψ(t))〉Ω − 2 (ξ − κ) 〈F ′ (ψ (t)) , ψ (t)〉2 ≤ C9,

λ0 |Γ| ζ|Ω|+ a |Γ| 〈G

′ (ψ(t))〉Γ − 2 (ξ − κ) 〈G′ (ψ (t)) , ψ (t)〉2,Γ ≤ C10.

Also, observe that

(λ0 − δ) |Γ| ζ|Ω|+ a |Γ| 〈ψt〉Γ ≤ ‖ψt‖2

V0+ C11,

ζ(λ2

0a− λ0δ) |Γ|

|Ω|+ a |Γ| 〈θ〉Γ ≤ κ ‖θ‖2X2

a+ C12.

Thus, thanks to the above inequalities, we deduce from (2.16) and (2.20) that

d

dt[E (t) + ζ 〈〈θ (t)〉〉] + κ [E (t) + ζ 〈〈θ (t)〉〉]

+κ′(‖ψ (t)‖2

V1+ ‖ψt (t)‖2

V0+ ‖∇θ (t)‖2

2

) ≤ C13. (2.23)

On the other hand, one can check that there exists a positive constant CM , independentof t and of the initial data, such that

Υ(t) ≤ CM (E (t) + ζ 〈〈θ (t)〉〉) , (2.24)

whereΥ(t) := ‖ψ (t)‖2

V1+ ‖θ (t)‖2

X2a.

The positive constant C is suitably chosen so that the sign of Υ(t) is nonnegative.Applying Gronwall’s inequality to (2.23) and taking (2.24) into account, we obtain the

analogue of estimate (2.14), that is,

Υ(t) +

t+1∫

t

(‖ψ (s)‖2V1

+ ‖ψt (s)‖2V0

+ ‖∇θ (s)‖22

)ds

≤ Q(‖(ψ (0) , θ (0))‖2

YK

)e−κt + C14, ∀ t ≥ 0. (2.25)

In order to prove (2.9) in all cases K ∈ W0,W1 , we need to differentiate all equationsin (2.4)-(2.5) with respect to time. This can be justified by means of the approximationprocedure devised in [GG1]. Thus, repeating the proof in [GG1, (4.20)-(4.23)] word byword, we obtain (2.9). We leave its rigorous proof to the reader.

Adapting the proofs of [GG1, Lemma 3.3], the following continuous dependence resultfor the solutions to problem PK is rather standard.

Lemma 5 Let (ψi, θi) be the solution to PK corresponding to the initial data (ψ0i, θ0i) ∈YK , i = 1, 2. Then for any t ≥ 0, the following estimate holds:

‖(ψ1 − ψ2) (t)‖2V1

+ ‖(θ1 − θ2) (t)‖2X2

a

+

t∫

0

[‖(ψ1 − ψ2)t (s)‖2

V0+ ‖∇ (θ1 − θ2) (s)‖2

2 + c∥∥(

θ1|Γ − θ2|Γ)(s)

∥∥2

2,Γ

]ds

≤ CeLt(‖ψ01 − ψ02‖2

V1+ ‖θ01 − θ02‖2

X2a

), (2.26)

where C and L are positive constants which depend on the norms of the initial data, onΩ, Γ and on the parameters of the problem, but are both independent of time.

The following result is a straightforward consequence of Theorem 1 and Lemma 3.

Theorem 6 For each K ∈ W0,W1 , PK defines a semigroup SK (t) : YK → YK bysetting, for all t ≥ 0,

SK (t) (ψ0, θ0) = (ψ (t) , θ (t)) ,

where (ψ, θ) is the unique solution to PK. Moreover, SK (t) has a bounded absorbing setin YK .

Remark 7 Assume that (ψ0n, θ0n) → (ψ0, θ0) in YK and SK (t) (ψ0n, θ0n) → (ψ, θ) as ngoes to +∞. Then thanks to Lemma 5, we can easily conclude that SK (t) (ψ0, θ0) = (ψ, θ).Therefore, we have that SK (t) is a closed semigroup in the sense of [PZ]. This is latterused to prove (see Theorem 9) that there exists a connected global attractor AK⊂ YK forthe semigroup SK (t) .

The next lemma shows the existence of a compact absorbing set and follows from[GG1, Lemma 4.4].

Lemma 8 Let K ∈ W0, W1 be fixed. Then there is a positive monotone nondecreasingfunction Q and, for any R0 > 0, there exists t0 = t0(R0) > 0 such that

‖(ψ (t) , θ (t))‖V3×H2(Ω) ≤ Q(R0), ∀ t ≥ t0, (2.27)

for any (ψ0, θ0) ∈ B(R0) ⊂ YK , where B(R0) is the ball of radius R0 centered at 0.

Proof. The following estimates will also be deduced by a formal argument. We differen-tiate (2.4) and (2.5) with respect to time and we obtain

ψtt −∆ψt + F ′′ (ψ) ψt − λ0θt = 0, a.e. in Ω× (0, +∞) ,

(θ + λ0ψ)tt −∆θt = 0, a.e. in Ω× (0, +∞) ,(2.28)

subject to boundary conditions

ψtt −∆Γψt + ∂nψt + γψt + G′′ (ψ) ψt − δθt = 0,

(aθ + δψ)tt + ∂nθt + cθt = 0.(2.29)

Then we multiply the equations in (2.28) by ψtt(t) and by θt(t), respectively. Integratingover Ω, using the boundary conditions (2.29) and adding the resulting relations, we obtain

1

2

d

dt

[‖ψt (t)‖2

V1+ ‖θt (t)‖2

X2a

]+ ‖ψtt (t)‖2

V0+ ‖∇θt (t)‖2

2 + c∥∥(θ|Γ)t (t)

∥∥2

2,Γ

= −∫

Ω

F ′′1 (ψ (t)) ψt (t) ψtt (t) dx−

∫

Γ

F ′′2 (ψ (t)) ψt (t) ψtt (t) dS. (2.30)

Using Holder’s and Young’s inequalities, we have

d

dt

(‖ψt (t)‖2

V1+ ‖θt (t)‖2

X2a

)+ ‖ψtt (t)‖2

V0+ 2

(‖∇θt (t)‖2

2 + c∥∥(θ|Γ)t (t)

∥∥2

2,Γ

)

≤ C(‖F ′′

1 (ψ (t)) ψt (t)‖22 + ‖F ′′

2 (ψ (t)) ψt (t)‖22,Γ

). (2.31)

Here and in the sequel, C stands for a positive constant which is independent of t and ofthe initial data. This constant may vary from line to line. Besides, Q denotes a positivemonotone nondecreasing function which is independent of t.

Thanks to the embedding H2(Ω) → C(Ω), we infer from (2.31)

d

dt

(‖ψt (t)‖2

V1+ ‖θt (t)‖2

X2a

)+ ‖ψtt (t)‖2

V0+ 2

(‖∇θt (t)‖2

2 + c∥∥(θ|Γ)t (t)

∥∥2

2,Γ

)

≤ Q(‖ψ (t)‖2

V2

) ‖ψt (t)‖2V0

, (2.32)

which yields, owing to (2.9),

d

dt

(‖ψt (t)‖2

V1+ ‖θt (t)‖2

X2a

)≤ Q (R0) ‖ψt (t)‖2

V0. (2.33)

Then recalling (2.9) again, we can apply to (2.33) the so-called uniform Gronwall’s lemma(see, e.g., [T]) and find t1 ≥ 1, depending on R0, such that

‖ψt (t)‖2V1

+ ‖θt (t)‖2X2

a≤ Q (R0) , ∀ t ≥ t1. (2.34)

Next, we multiply the second equations of (2.28) and (2.29) by θtt and (θtt)|Γ , respectively,then integrate over Ω and Γ. We have

‖θtt (t)‖22 + a ‖θtt (t)‖2

2,Γ +1

2

d

dt

(‖∇θt (t)‖2

2 + c∥∥θt|Γ (t)

∥∥2

2,Γ

)

= −λ0 (ψtt (t) , θtt (t))2 − δ (ψtt (t) , θtt (t))2,Γ

≤ λ0 ‖ψtt (t)‖2 ‖θtt (t)‖2 + δ ‖ψtt (t)‖2,Γ ‖θtt (t)‖2,Γ ,

which yields, owing to Young’s inequality,

1

2

(‖θtt (t)‖2

2 + a ‖θtt (t)‖22,Γ

)+

1

2

d

dt

(‖∇θt (t)‖2

2 + c∥∥θt|Γ (t)

∥∥2

2,Γ

)

≤ λ20

2‖ψtt (t)‖2

2 +δ2

2a‖ψtt (t)‖2

2,Γ .

Thus, we have, in particular,

d

dt

(‖∇θt (t)‖2

2 + c∥∥θt|Γ (t)

∥∥2

2,Γ

)≤ C ‖ψtt (t)‖2

V0. (2.35)

We infer from (2.32) and (2.34) that

supt≥t1

t+1∫

t

[‖ψtt (s)‖2

V0+ 2

(‖∇θt (s)‖2

2 + c∥∥(θ|Γ)t (s)

∥∥2

2,Γ

)]ds ≤ C(R0). (2.36)

We can now apply the uniform Gronwall’s lemma once more and infer from (2.35) and(2.36) the existence of t2 ≥ t1 such that

‖∇θt (t)‖22 + c

∥∥θt|Γ (t)∥∥2

2,Γ≤ C (R0) , ∀ t ≥ t2. (2.37)

Therefore, (2.34) and (2.37) allow us to deduce from the second equations of (2.4) and(2.5), via standard elliptic regularity results, the following estimate:

‖θ (t)‖2H2(Ω) ≤ C

(‖θt (t)‖2

H1(Ω) + ‖ψt (t)‖2H1(Ω)

)≤ C (R0) , ∀ t ≥ t2, (2.38)

for each K ∈ W0,W1. Furthermore, using a standard H3-regularity result from [GG1,Lemma 2.2], we obtain

‖ψ (t)‖2H3(Ω) + ‖ψ (t)‖2

H3(Γ)

≤ C(‖F ′

1 (ψ (t))‖2H1(Ω) + ‖F ′

2 (ψ (t))‖2H1(Γ) + ‖θ (t)‖2

H1(Ω) + ‖ψt (t)‖2V1

),

which gives, on account of the above estimates,

‖ψ (t)‖2V3≤ C (R0) , ∀ t ≥ t1. (2.39)

We finally conclude by observing that (2.27) follows from (2.38) and (2.39).Thanks to Lemmas 3 and 8 and recalling Remark 7, we can apply [PZ, Corollary 6]

to prove the following result.

Theorem 9 Let K ∈ W0,W1 and let F, G ∈ C2(R) satisfy (1.14). In addition, assumethat (1.15) holds if K = W0. Then there exists a connected global attractor AK⊂ YK forthe semigroup SK (t) . Moreover, AK is bounded in V3 ×H2 (Ω) .

The next lemma is concerned with the smoothing property for the difference of anytwo trajectories to problem PK and the Lipschitz continuity of the solutions in the metricof V2 × ZK . The proof is analogous to the one in [GG1, Lemma 4.6] and uses the sameideas as in the proof of Lemma 8. The required changes are minor. Therefore, we leavethe rigorous details to the reader.

Lemma 10 Let K ∈ W0,W1 and let F, G ∈ C2,1(R) satisfy the assumptions of The-orem 9. Denote by (ψi, θi) the solution to PK with initial data (ψ0i, θ0i) ∈ YK , i = 1, 2.Then the following estimate holds:

‖(ψ1 − ψ2) (t)‖2V2

+ ‖(θ1 − θ2) (t)‖2ZK

+

t∫

0

[‖(ψ1 − ψ2)t (s)‖2

V1+ ‖(θ1 − θ2)t (s)‖2

X2a

]ds

≤ CeLt(‖ψ01 − ψ02‖2

V2+ ‖θ01 − θ02‖2

ZK

), ∀ t ≥ 0. (2.40)

Furthermore, we have the following smoothing estimate:

‖(ψ1 − ψ2) (t)‖2V3

+ ‖(θ1 − θ2) (t)‖2H2(Ω)

≤ C

(t2 + t + 1

t2

)eLt

(‖ψ01 − ψ02‖2V2

+ ‖θ01 − θ02‖2ZK

), ∀ t > 0. (2.41)

Here C and L are positive constants which only depend on the norms of the initial data,on Ω, Γ and on the structural parameters of the problem.

Finally, we state the main result of this section, namely,

Theorem 11 Let K ∈ W0,W1 and let F, G ∈ C2,1(R) satisfy the assumptions ofTheorem 9. Then SK(t) possesses an exponential attractor MK ⊂ YK , namely,

(i) MK is compact and semi-invariant with respect SK (t) , i.e.,

SK (t) (MK) ⊂MK , ∀ t ≥ 0.

(ii) The fractal dimension dimF (MK ,YK) of MK is finite.(iii) MK attracts exponentially fast any bounded subset B of YK , that is, there exist

a positive monotone nondecreasing function Q and a constant ρ > 0 such that

distYK(SK (t) B,MK) ≤ Q(‖B‖YK

)e−ρt, ∀ t ≥ 0.

Here distYKdenotes the non-symmetric Hausdorff distance between sets in YK and ‖B‖YK

stands for the size of B in YK .

The proof of Theorem 11 follows from [GG1, Theorem 4.5], employing Lemma 10and the fact that SK (t) is Holder continuous on (0, T ] × YK , a fact which can be easilychecked (see, e.g., [GG1, Lemma 4.7]). We remind that this argument is based on thegeneral theorem on the existence of an exponential attractor proven in [EMZ]. The detailsare left to the reader.

Remark 12 Theorem 11 entails that AK has finite fractal dimension.

3 Convergence to equilibria

Here we work within the framework used in [GG2]. The following proposition can beeasily proven.

Proposition 13 Let the hypotheses of Theorem 9 hold. For each K ∈ W0,W1 , thesemigroup SK (t) has a (strict) Lyapunov functional defined by the free energy, namely,

LK(ψ0, θ0) =1

2

[‖∇ψ0‖2

2 + ‖∇Γψ0‖22,Γ + γ ‖ψ0‖2

2,Γ + ‖θ0‖22 + a ‖θ0‖2

2,Γ

]

+

∫

Ω

F (ψ0) dx +

∫

Γ

G (ψ0) dS. (3.1)

In particular, we have, for all t > 0,

d

dtLK(SK (t) (ψ0, θ0)) = −‖ψt (t)‖2

2 − ‖ψt (t)‖22,Γ − ‖∇θ (t)‖2

2 − c∥∥θ|Γ (t)

∥∥2

2,Γ. (3.2)

We now examine more closely the set of equilibria. We first observe that (ψ∞, θ∞) ∈YK is an equilibrium for PK if and only if it is a solution to the boundary value problem

−∆ψ∞ + F ′ (ψ∞)− λ0θ∞ = 0, in Ω,

−∆Γψ∞ + ∂nψ∞ + γψ∞ + G′ (ψ∞)− δθ∞ = 0, on Γ,

−∆θ∞ = 0, in Ω,

∂nθ∞ + cθ∞ = 0, on Γ.

Then it is not difficult to realize that, when K = W1 , we obtain θ∞ ≡ 0. Otherwise, whenK = W0, θ∞ must equal to a constant which is uniquely determined from (2.7). Thus,the above stationary problem reduces to the following:

−∆ψ∞ + F ′ (ψ∞)− λ0θ∞ = 0, in Ω,−∆Γψ∞ + ∂nψ∞ + γψ∞ + G′ (ψ∞)− δθ∞ = 0, on Γ,

(3.3)

where

θ∞ =

0, for K = W1,

IK (ψ0, θ0)−(

λ0|Ω|〈ψ∞〉Ω|Ω|+a|Γ| +

δ|Γ|〈ψ∞〉Γ|Ω|+a|Γ|

), for K = W0,

(3.4)

where we recall that 〈·〉Ω , 〈·〉Γ stand for the spatial averages over Ω and Γ, respectively.It is not difficult to check that (3.3) possesses at least one solution (i.e., there exists atleast one stationary solution) by means of standard arguments (see, e.g., [CFP], [CM2],[WZ]).

Remark 14 Since we are dealing with a gradient system, then the global attractor AK

coincides with the unstable manifold of the set PK of the stationary points (cf., e.g., [T]).

Therefore, on account of the above results, we report some standard implications (cf.,e.g., [CH, Chap. 9]).

Lemma 15 For any (ψ0, θ0) ∈ YK, the set ω(ψ0, θ0) is a nonempty compact connectedsubset of YK. Furthermore, we have(i) ω(ψ0, θ0) is fully invariant for SK (t);(ii) LK is constant on ω(ψ0, θ0);(iii) distYK

(SK (t) (ψ0, θ0) , ω(ψ0, θ0)) → 0 as t → +∞;(iv) ω(ψ0, θ0) only consists of equilibria.

Next, arguing as in [GG2] (see also [CFP]), we assume, without loss of generality, that(ψ0, θ0) satisfies the condition IK (ψ0, θ0) = 0, whenever K = W0. Indeed, it suffices toreplace the solution (ψ, θ) corresponding to (ψ0, θ0) by (ψ − c, θ) with

c = (|Ω|+ a |Γ|) IK (ψ, θ) / (|Ω|λ0 + δ |Γ|)

and to note that (ψ − c, θ) satisfies the phase-field equations (2.4)-(2.5) with F ′ (y) andG′ (y) replaced by F ′ (y + c) and G′ (y + c) , respectively. The convergence of (ψ(t), θ(t))as t → +∞ is not affected by this normalization. Thus, recalling (2.8), we are now dealingwith a Hilbert space YK and we will indicate by Y∗K its dual. Moreover, if K = W0, wewill replace, in the functional LK : YK → R, F (y) and G (y) by F (y + c) and G (y + c) ,respectively.

The version of the ÃLojasiewicz-Simon inequality that we need is given by the

Lemma 16 Let (ψ∞, θ∞) ∈ YK satisfy (3.3)-(3.4), that is, (ψ∞, θ∞) is a critical pointof LK. Assume that F and G are real analytic. Then there exist constants ξ ∈ (0, 1/2),CL, and ζ > 0 depending on (ψ∞, θ∞) such that, for any (ψ, θ) ∈ YK satisfying ‖ (ψ, θ)−(ψ∞, θ∞) ‖V2×H1(Ω) ≤ ζ, we have

CL ‖L′K (ψ, θ)‖Y∗K ≥ |LK (ψ, θ)− LK (ψ∞, θ∞) |1−ξ, (3.5)

where L′K denotes the Frechet derivative of LK .

Remark 17 The proof of Lemma 16 is based on the same arguments as those used in[CFP] (see also [GG2]). Note that ψ∞ and ψ are bounded functions.

We can now state the main result of this section.

Theorem 18 Let the assumptions of Theorem 9 hold and fix K ∈ W0,W1 . Assume,in addition, that the nonlinearities F and G are real analytic. Then for any given initialdatum (ψ0, θ0) ∈ YK, the solution t 7→ (ψ (t) , θ (t)) = SK(t)(ψ0, θ0) to PK converges to asingle equilibrium (ψ∞, θ∞) in the topology of YK, that is,

limt→+∞

(‖ψ(t)− ψ∞‖V2 + ‖θ(t)− θ∞‖H1(Ω)

)= 0. (3.6)

Moreover, there exist C > 0 and ξ ∈ (0, 1/2) depending on (ψ∞, θ∞) such that

‖ψ(t)− ψ∞‖V2 + ‖θ(t)− θ∞‖H1(Ω) + ‖ψt (t) ‖V0 ≤ C(1 + t)−ξ/(1−2ξ), (3.7)

for all t ≥ 0.

Proof. We observe that, if there exists t‖ ≥ 0 such that LK

(ψ

(t‖

), θ

(t‖

))= L∞, then,

for all t ≥ t‖, LK (ψ (t) , θ (t)) = L∞, that is,

ψ (t) = ψ∞, θ (t) = θ∞, for all t ≥ t‖.

In that case, there is nothing to prove.Without loss of generality, assume now that we have LK (ψ (t) , θ (t)) > L∞, for all

t ≥ t0. We first observe that, by Lemma 15 and Lemma 16, the functional LK satisfiesthe ÃLojasiewicz-Simon inequality (3.5) near every (ψ∞, θ∞) ∈ ω(ψ0, θ0). Since ω(ψ0, θ0) iscompact in YK , we can cover this set it by the union of finitely many balls Bj with centers(ψj∞, θj

∞)

and radii rj, where each radius rj is such that (3.5) holds in Bj. Since LK = L∞on ω(ψ0, θ0), it follows from Lemma 16 that there exist uniform constants ξ ∈ (0, 1/2),CL > 0 and a neighborhood U of ω(ψ0, θ0) such that

CL ‖L′K (ψ, θ)‖Y∗K ≥ |LK (ψ, θ)− L∞|1−ξ, ∀ (ψ, θ) ∈ U . (3.8)

Recalling property (iii) of Lemma 15, we can find a time t1 > 0 such that (ψ (t) , θ (t))belongs to U , for all t ≥ t1. Set now t2 := max t0, t1. Recalling (3.2), we obtain, forevery t ≥ t2,

− d

dt(LK (ψ (t) , θ (t))− L∞)ξ = ξ

(− d

dtLK (ψ (t) , θ (t))

)(LK (ψ (t) , θ (t))− L∞)ξ−1

≥ ξ

CL

‖ψt (t)‖22 + ‖ψt (t)‖2

2,Γ + ‖∇θ (t)‖22 + c

∥∥θ|Γ (t)∥∥2

2,Γ

‖L′K (ψ (t) , θ (t))‖Y∗K. (3.9)

Assume first that K = W1. Using Green’s formula on Ω, we obtain

〈L′K (ψ, θ) , (h, k)〉Y∗K ,YK= −

∫

Ω

∆ψhdx +

∫

Γ

∂nψhdS +

∫

Ω

F ′ (ψ) hdx

+

∫

Γ

(−∆Γψ + γψ) hdS +

∫

Γ

G′ (ψ) hdS +

∫

Ω

θkdx +

∫

Γ

aθkdS

=

∫

Ω

(−∆ψ + F ′ (ψ)− λ0θ) hdx +

∫

Γ

(−∆Γψ + ∂nψ + γψ + G′ (ψ)− δθ) hdS

+

∫

Ω

θ (k + λ0h) dx +

∫

Γ

θ (ka + δh) dS. (3.10)

Hence, by using the Cauchy-Schwarz inequality and the fact that H1 (Ω) ⊂ X2a, we obtain

‖L′K (ψ, θ)‖Y∗K = sup‖(ψ,θ)‖YK

≤1

〈L′K (ψ, θ) , (h, k)〉Y∗K ,YK≤ C(‖−∆ψ + F ′ (ψ)− λ0θ‖2

+ ‖−∆Γψ + ∂nψ + γψ + G′ (ψ)− δθ‖2,Γ + ‖θ‖H1(Ω)), (3.11)

for K = W1. In order to prove an estimate similar to (3.11) for K = W0, we proceed asin [GG2]. Since (h, k) ∈ YK and IW0 (h, k) = 0, we easily deduce that

〈L′K (ψ, θ) , (h, k)〉Y∗α,a,Yα,a= 〈−∆ψ + F ′ (ψ)− λ0θ, h〉2

+ 〈−∆Γψ + ∂nψ + γψ + G′ (ψ)− δθ, h〉2,Γ

+ 〈θ − 〈〈θ〉〉 , k + λ0h〉2 + 〈θ − 〈〈θ〉〉 , ak + δh〉2,Γ ,

where

〈〈u〉〉 =1

(|Ω|+ a |Γ|)

∫

Ω

u (x) dx +

∫

Γ

u (x) adSx

.

An inequality similar to (3.11) follows by arguing as above and by using a generalizedPoincare’s inequality (cf., e.g., [GG1]):

‖v − 〈〈v〉〉‖2X2

a= ‖v‖2

X2a− (|Ω|+ a |Γ|) 〈〈v〉〉2 ≤ C∗ ‖∇v‖2

2 . (3.12)

From (3.11), we obtain, for all t ≥ t2,

C ‖L′K (ψ(t), θ(t))‖Y∗K ≤ ‖ψt (t)‖2 + ‖ψt (t)‖2,Γ + ‖∇θ (t)‖2 + c∥∥θ|Γ (t)

∥∥2,Γ

, (3.13)

where c = 0, if K = W0.

Inserting estimate (3.13) into estimate (3.9), we deduce that

− d

dt(LK (ψ (t) , θ (t))− L∞)ξ

≥ C(‖ψt (t)‖2 + ‖ψt (t)‖2,Γ + ‖∇θ (t)‖2 + c

∥∥θ|Γ (t)∥∥

2,Γ

). (3.14)

By integrating this inequality over [t2, +∞) and using the fact that LK (ψ (t) , θ (t)) → L∞as t goes to +∞, we infer

ψt ∈ L1 ([t2, +∞) ;V0) ,

θ ∈ L1([t2, +∞) ; H1 (Ω)

), if K = W1,

∇θ ∈ L1([t2, +∞) ; (L2 (Ω))3

), if K = W0.

Consequently, we also have θt ∈ L1([t2, +∞) ; (H1 (Ω))

∗). We now recall that, due to

Lemma 15, there exists an increasing unbounded sequence tn and an element (ψ∞, θ∞) ∈ω (ψ0, θ0) such that (ψ(tn), θ(tn)) → (ψ∞, θ∞) in V2×H1 (Ω) as n goes to +∞. This fact,combined with the above L1-integrability, implies that (ψ (t) , θ (t)) → (ψ∞, θ∞) as t goesto +∞ in V0× (H1 (Ω))

∗and in V2×H1 (Ω) , appealing once more to Lemma 15. Hence,

ω (ψ0, θ0) = (ψ∞, θ∞) and (3.6) holds. Estimate (3.7) follows by means of the samearguments as those used in [GG2, Theorem 21]. Therefore, we omit its proof.

Remark 19 It is also worth noting that, when both nonlinearities F and G are realanalytic, we can in fact show, by using the smoothing property of the solutions (ψ, θ) , theconvergence result (3.6) and the rate estimate (3.7) in higher order norms, provided thatΓ is smooth accordingly.

Acknowledgments. The authors wish to thank Professors Umberto Mosco and JurgenSprekels for pointing out to them possible coupling effects in the boundary conditions.The second author has been partially supported by the Italian PRIN Research Project2006 Problemi a frontiera libera, transizioni di fase e modelli di isteresi.

4 References

[AFI] S. Aizicovici, E. Feireisl, F. Issard-Roch, Franoise, Long-time convergence of solu-tions to a phase-field system, Math. Methods Appl. Sci. 24 (2001), 277-287.

[BZ] P.W. Bates, S. Zheng, Inertial manifolds and inertial sets for the phase-field equa-tions, J. Dynamics Differential Equations 4 (1992), 375-397.

[BCH] D. Brochet, X. Chen, D. Hilhorst, Finite dimensional exponential attractor forthe phase-field model, Appl. Anal. 49 (1993), 197-212.

[BH] D. Brochet, D. Hilhorst, Universal attractor and inertial sets for the phase-fieldmodel, Appl. Math. Lett. 4 (1991), 59-62.

[BS] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, New York,1996.

[C] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration.Mech. Anal. 92 (1986), 205-245.

[CH] T. Cazenave, A. Haraux, An introduction to semilinear evolution equations, OxfordUniversity Press, New York, 1998.

[CFP ] R. Chill, E. Fasangova, J. Pruss: Convergence to steady states of solutions of theCahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math.Nachr. 13 (2006), 1448-1462.

[CGM ] L. Cherfils, S. Gatti, A. Miranville, Existence of global solutions to the Caginalpphase-field system with dynamic boundary conditions and singular potentials, J.Math. Anal. Appl. 343 (2008), 557-566.

[CM1] L. Cherfils, A. Miranville, Some remarks on the asymptotic behavior of the Cagi-nalp system with singular potentials, Adv. Math. Sci. Appl. 16 (2007), 107-129.

[CM2] L. Cherfils, A. Miranville, On the Caginalp system with dynamic boundary con-ditions and singular potentials, Appl. Math., to appear.

[CL] J.B. Collins, H. Levine, Diffuse interface models of diffusion limited crystal growth,Phys. Rev. B 31 (1985), 6119-6222 [Errata, Phys. Rev. B 33 (1985), 2020].

[DKS] A. Damlamian, N. Kenmochi, N. Sato, Subdifferential operator approach to aclass of nonlinear systems for Stefan problems with phase relaxation, NonlinearAnal. 23 (1994), 115-142.

[EMZ] M. Efendiev, A. Miranville, S. Zelik, Exponential attractors for a nonlinearreaction-diffusion system in R3, C. R. Math. Acad. Sci. Paris 330 (2000), 713-718.

[EZ] C.M. Elliott, S. Zheng, Global existence and stability of solutions to the phase-fieldequations, in “Free boundary problems”, Internat. Ser. Numer. Math. 95, 46-58,Birkhauser Verlag, Basel, 1990.

[FGGR] A. Favini, G. Ruiz Goldstein, J.A. Goldstein, S. Romanelli, The heat equationwith general Wentzell boundary conditions, J. Evol. Equ. 2 (2002), 1-19.

[F ] G.J. Fix, Phase field models for free boundary problems, in “Free boundary problems:theory and applications, Vol. II ”(A. Fasano, M. Primicerio, Eds.), Pitman Res.Notes Math. Ser. 79, 580-589, Longman, London, 1983.

[GG1] C.G. Gal, M. Grasselli, The non-isothermal Allen-Cahn equation with dynamicboundary conditions, Discrete Contin. Dyn. Syst., to appear.

[GG2] C.G. Gal, M. Grasselli, On the asymptotic behavior of the Caginalp system withdynamic boundary conditions, Commun. Pure Appl. Anal., to appear.

[GM ] S. Gatti, A. Miranville, Asymptotic behavior of a phase-field system with dynamicboundary conditions, in “Differential Equations: Inverse and Direct Problems”(A.Favini, A. Lorenzi, Eds.), Ser. Lect. Notes Pure Appl. Math. 251, 149-170,Chapman & Hall/CRC, Boca Raton, 2006.

[GGP ] C. Giorgi, M. Grasselli, V. Pata, Uniform attractors for a phase-field model withmemory and quadratic nonlinearity, Indiana Univ. Math. J. 48 (1999), 1395-1445.

[GP ] M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Com-mun. Pure Appl. Anal. 3 (2004), 849-881.

[GPS] M. Grasselli, H. Petzeltova, G. Schimperna, Long time behavior of solutions to theCaginalp system with singular potential, Z. Anal. Anwendungen 25 (2006), 51-72.

[GWZ] M. Grasselli, H. Wu, S. Zheng, Asymptotic behavior of a non-isothermal Ginz-burg-Landau model, Quart. Appl. Math., to appear.

[JB] A. Jimenez-Casas, A. Rodrıguez-Bernal, Asymptotic behaviour for a phase fieldmodel in higher order Sobolev spaces, Rev. Mat. Complut. 15 (2002), 213-248.

[K] V.K. Kalantarov, On the minimal global attractor of a system of phase field equations(Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 188(1991), Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsii. 22, 70-86, 186[translation in J. Math. Sci. 70 (1994), 1767-1777].

[Kp] O.V. Kapustyan, An attractor of a semiflow generated by a system of phase-fieldequations without uniqueness of the solution (Ukrainian), Ukraın. Mat. Zh. 51(1999), 1006-1009 [Translation in Ukrainian Math. J. 51 (1999), 1135-1139 (2000)].

[L] Ph. Laurencot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc.Edinburgh Sect. A 126 (1996), 167-185.

[M ] G.B. McFadden, Phase-field models of solidification, Contemp. Math. 306 (2002),107-145.

[PZ] V. Pata, S. Zelik, A result on the existence of global attractors for semigroups ofclosed operators, Commun. Pure Appl. Anal. 6 (2007), 481-486.

[SA] N. Sato, T. Aiki, Phase field equations with constraints under nonlinear dynamicboundary conditions, Commun. Appl. Anal. 5 (2001), 215-234.

[S] G. Schimperna, Abstract approach to evolution equations of phase field type andapplications, J. Differential Equations 164 (2000), 395-430.

[T ] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.

[WZ] H. Wu, S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation withdynamic boundary conditions, J. Differential Equations 204 (2004), 511-531.

[Z] Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neu-mann boundary conditions, Commun. Pure Appl. Anal. 4 (2005), 683-693.