Phase-field systems with nonlinear coupling and dynamic boundary conditions

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Nonlinear Analysis 72 (2010) 2375–2399

Contents lists available at ScienceDirect

Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Phase-field systems with nonlinear coupling and dynamicboundary conditionsCecilia Cavaterra a, Ciprian G. Gal b, Maurizio Grasselli c,∗, Alain Miranville da Dipartimento di Matematica ‘‘F. Enriques’’, Università degli Studi di Milano, 20133 Milano, Italyb Department of Mathematics, University of Missouri, Columbia, MO 65211, USAc Dipartimento di Matematica ‘‘F. Brioschi’’, Politecnico di Milano, 20133 Milano, Italyd Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086, SP2MI, 86962 Chasseneuil Futuroscope Cedex, France

a r t i c l e i n f o

Article history:Received 10 June 2009Accepted 2 November 2009

MSC:35B4035B4135K5580A22

Keywords:Phase-field equationsDynamic boundary conditionsLaplace–Beltrami operatorGlobal attractorsExponential attractorsŁojasiewicz–Simon inequalityConvergence to equilibrium

a b s t r a c t

We consider phase-field systems of Caginalp type on a three-dimensional boundeddomain. The order parameter fulfills a dynamic boundary condition, while the (relative)temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann orRobin type. Moreover, the two equations are nonlinearly coupled through a quadraticgrowth function. Here we extend several results which have been proven by some ofthe authors for the linear coupling. More precisely, we demonstrate the existence anduniqueness of global solutions. Then we analyze the associated dynamical system andwe establish the existence of global as well as exponential attractors. We also discuss theconvergence of given solutions to a single equilibrium.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Awell-known evolutionary system of partial differential equationswhich describes the behavior of a two-phasematerialsubject to temperature variations, neglecting mechanical stresses, is the so-called Caginalp phase-field system (cf. [1,2], seealso [3–5])

δ∂tψ −∆ψ + f (ψ)− λ′ (ψ) θ = 0,∂t (εθ + λ (ψ))−∆θ = 0,

(1.1)

in Ω × (0,+∞), where Ω ⊂ R3 is a bounded domain occupied by the material for any time t ≥ 0. Here ψ is the orderparameter (or phase field), θ denotes the (relative) temperature, δ and ε are given positive constants, while λ is a functionrelated to the latent heat. Moreover, the function f : R→ R accounts for the presence of different phases (typically, f canbe the derivative of a double-well potential, i.e., f (y) = y(y2 − 1)).

∗ Corresponding author. Fax: +39 0223994561.E-mail addresses: [email protected] (C. Cavaterra), [email protected] (C.G. Gal), [email protected] (M. Grasselli),

[email protected] (A. Miranville).

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.11.002


2376 C. Cavaterra et al. / Nonlinear Analysis 72 (2010) 2375–2399

The mathematical literature regarding system (1.1) is rather vast when λ is assumed to be linear. For instance, werecall that global well-posedness results can be found in [6] (see also [7–9]). The analysis of the longtime behavior ofsolutions to equations like (1.1) was also carried out in several papers mainly devoted to establish the existence of globaland/or exponential attractors (see, for instance, [10–16]). Moreover, the asymptotic behavior of single solutions has beeninvestigated by means of suitable versions of the Łojasiewicz–Simon inequality (cf., e.g., [17–19]). All the mentioned resultsare essentially concerned with standard boundary conditions (that is, Dirichlet’s, Neumann’s or Robin’s). More recently, inorder to account for possible interactions with the walls, i.e., with the boundary Γ := ∂Ω , some physicists proposed adynamic boundary condition for ψ (cf., e.g., [20–22]). On account of this proposal, here we consider system (1.1) endowedwith the following boundary conditions:

∂tψ − α∆Γψ + ∂nψ + βψ + g (ψ) = 0,b∂nθ + cθ = 0,

(1.2)

on Γ × (0,+∞). Here α, β > 0 are given constants, n is the outward normal to Γ , ∆Γ denotes the Laplace–Beltramioperator on Γ and g : R→ R is a given nonlinear function. Furthermore, Γ is always assumed as smooth as is needed. Thehomogeneous boundary condition for θ subsumes the following cases: Dirichlet’s (b = 0 and c > 0), Neumann’s (b > 0and c = 0), Robin’s (b > 0 and c > 0).Eqs. (1.1)–(1.2) are also subject to the initial conditions

ψ|t=0 = ψ0, θ|t=0 = θ0. (1.3)

The initial and boundary value problems (1.1)–(1.3), always in the case of a linear λ, were first analyzed in the case(b, c) = (1, 0). More precisely, well-posedness and convergence to single stationary states were demonstrated in [23],taking g ≡ 0 and assuming f with a polynomially controlled growth of degree six. Existence of global and exponentialattractors was proven in [24], under rather weak assumptions on f and g (see also [25]). These results were furthergeneralized by assuming dynamic boundary conditions for θ in [26,27].Assuming λ linear is satisfactory for solid–liquid phase transitions. However, when one deals, for instance, with phase

transitions in ferromagnetic materials, where ψ represents the fraction of lattice sites at which the spins are pointing‘‘up’’, then a quadratic λ is a more appropriate choice (see [3] and references therein). In this case, as far as we know, theexisting contributions (see [18,28,29], cf. also [30], formemory effects, and [31, Sec. 7]) are all concernedwith homogeneousNeumann boundary condition forψ . Herewewant to extend the results contained in [26,27] to the case of aλwith quadraticgrowth, under sufficiently general assumptions on f and g , leaving the analysis of coupled dynamic boundary conditions(see [32]) and/or singular potentials (cf. [33,34]) to further investigations. The main novelties with respect to the previousresults on linearly coupled systems are the following:

(i) the proof of existence cannot be apparently carried out through a fixed-point argument as in [26], we thus employ aFaedo–Galerkin approach which is not standard due to the nature of the boundary conditions (see Section 2);

(ii) the approach used in [27] to show the convergence to a single equilibrium also needs to be modified because of thenonlinear coupling (see Section 5).

The plan of the paper goes as follows. In Section 2, we introduce the functional framework associated with (1.1)–(1.3).Then, we state and prove the well-posedness results for our problem, distinguishing between weak solutions (with growthrestrictions on f and g) and strong solutions (no growth restrictions on f and g). In Section 3, we first demonstrate theexistence of bounded absorbing sets and of compact absorbing sets for these weak and strong solutions. Hence, we deducethe existence of global attractors both for weak and strong solutions. Section 4 is devoted to the existence of exponentialattractors. Finally, in Section 5, we study the convergence of solutions to a single equilibrium.We conclude by observing that the results of this paper can be extended to the caseα = 0with appropriatemodifications

(see [27] for λ linear).

2. Well-posedness

Without loss of generality, from now on, we let δ = ε = 1.We denote by ‖·‖p and ‖·‖p,Γ the norms on Lp (Ω) and Lp (Γ ),respectively. In the case p = 2, 〈·, ·〉2 (or 〈·, ·〉2,Γ ) stands for the usual scalar product which induces the L2-norm (even forvector-valued functions). The norms onHs (Ω) andHs (Γ ) are indicated by ‖·‖Hs(Ω) and ‖·‖Hs(Γ ), respectively, for any s > 0.In order to account for all cases, we introduce the family of linear operators AK := −∆ on the Banach space L2 (Ω) withdomains

D(AK ) = H10 (Ω) ∩ H2(Ω), if K = D,

D(AK ) = u ∈ H2(Ω) : b∂nθ + cθ = 0, a.e. on Γ , if K ∈ N, R,

where K ∈ D,N, R and D,N, R stand for Dirichlet, Neumann, or Robin boundary conditions, respectively. We recall thatAK generates an analytic semigroup e−AK t on L2 (Ω). In addition, each AK is nonnegative and self-adjoint on L2 (Ω). We alsoset

Z1D = H10 (Ω) and Z1K = H

1(Ω), if K ∈ N, R ,


C. Cavaterra et al. / Nonlinear Analysis 72 (2010) 2375–2399 2377

endowed with the norms defined by

‖θ‖2Z1K=

‖∇θ‖22 , if K = D,

‖∇θ‖22 +cb

∥∥θ|Γ ∥∥22,Γ , if K = R,

‖∇θ‖22 + 〈θ〉2Ω , if K = N,

where we have defined

〈v〉Ω := |Ω|−1∫Ω

v (x) dx.

It is easy to check that, for each K ∈ D,N, R, the norm in Z1K is equivalent to the standard H1-norm. On account of the

dynamic boundary condition (see (1.2)), we also need to introduce the functional spacesVs = C s(Ω)‖·‖Vs

,where s ≥ 0 andthe norms ‖·‖Vs are given by

‖ψ‖Vs =(‖ψ‖2Hs(Ω) +

∥∥ψ|Γ ∥∥2Hs(Γ ))1/2 , ∀ψ ∈ C s (Ω) . (2.1)

Thus we have Vs = Hs (Ω)⊗ Hs (Γ ) , s ≥ 0. In general, any vectorΘ ∈ Vs will be of the form (θ1, θ2)with θ1 ∈ Hs (Ω, dx)and θ2 ∈ Hs (Γ , dS), and there need not be any connection between θ1 and θ2. We also set

Vs =Θ = (θ1, θ2) ∈ Vs : θ2 = θ1|Γ

, s > 1/2,

and V0 = V0, for the sake of notational simplicity. Furthermore, we notice that Vs is compactly embedded in Vs−1, for alls ≥ 1. We also recall that we have the following continuous embeddings: H1 (Ω) → L6 (Ω) ,H1/2 (Γ ) → L4 (Γ ) andH1 (Γ ) → Ls (Γ ), for any fixed s ∈ [1,+∞). For the latter two embeddings, note thatΩ is a smooth and compact three-dimensional Riemannianmanifoldwhose boundaryΓ is a two-dimensional Riemannianmanifold (see, e.g., [35, Chapter 2]).Let K ∈ D,N, R. We now restate problem (1.1)–(1.3) in weak and strong formulations. In the case K = N , we need to

define the enthalpy

IN (u, v) := 〈λ (u)+ v〉Ω .

This quantity is conserved in time for any given solution (see Eq. (2.7) below).The weak formulation of our problem reads

Problem PwK . For any given pair of initial data (ψ0, θ0) ∈ V1 × L2 (Ω), find

(ψ, θ) ∈ C([0,+∞);V1 × L2 (Ω)

), (2.2)

with

ψ ∈ L2 ([0,+∞);V2) ∩ H1 ([0,+∞);V0) , ∇θ ∈ L2([0,+∞); (L2(Ω))3

), (2.3)

such that

∂tψ −∆ψ + f (ψ)− λ′ (ψ) θ = 0, a.e. inΩ × (0,+∞), (2.4)

∂tψ − α∆Γψ + ∂nψ + βψ + g (ψ) = 0, a.e. on Γ × (0,+∞), (2.5)

〈∂t(θ + λ (ψ)), v〉2 + 〈∇θ,∇v〉2 + d〈θ, v〉2,Γ = 0, ∀v ∈ Z1K , a.e. in (0,+∞), (2.6)

which satisfies (1.3) and, if K = N ,

IN (ψ(t), θ(t)) = IN (ψ0, θ0) , ∀t ≥ 0. (2.7)

Here d = cb if K = R, d = 0 otherwise. The pair (ψ, θ)will be called weak solution.

The strong formulation is as follows.

Problem PsK . For any given pair of initial data (ψ0, θ0) ∈ V2 × Z1K , find

(ψ, θ) ∈ C([0,+∞);V2 × Z1K

), (2.8)

with

(∂tψ, ∂tθ) ∈ L2([0,+∞);V1 × L2 (Ω)

). (2.9)

AKθ ∈ L2 (Ω × [0,+∞)) . (2.10)

such that (ψ, θ) satisfies Eqs. (2.4), (2.5), (1.2) and (1.3) almost everywhere and, if K = N , (2.7). The pair (ψ, θ) will becalled strong solution.



Our goal is to prove that both PwK and PsK have a unique global solution which continuously depends on the initial data,

provided that the nonlinearities f , g andλ satisfy (possibly part of) the assumptions listed below. Of course, a strong solutionis also a weak solution.

(H1) f , g ∈ C1(R) satisfy

lim|y|→+∞

inf f ′ (y) > 0, lim|y|→+∞

inf g ′ (y) > 0. (2.11)

(H2) f and g satisfy the growth assumptions∣∣f ′ (y)∣∣ ≤ cf (1+ |y|2) , ∣∣g ′ (y)∣∣ ≤ cg (1+ |y|q) , ∀ y ∈ R, (2.12)

for some positive constants cf , cg , where q ∈ [1,+∞) is arbitrary.(H3) λ ∈ C2 (R) satisfies∣∣λ′′ (y)∣∣ ≤ cλ, ∀ y ∈ R, (2.13)

for some positive constant cλ.(H4) λ has the form

λ (y) = γ (y)− ay2, ∀y ∈ R,

where a > 0 and γ ∈ C2 (R) is such that γ ′ ∈ L∞ (R).(H5) f satisfies

f (y) y ≥ η1 |y|4 − η2, ∀ y ∈ R,

for some constants η1 > 0 and η2 ≥ 0.

Remark 2.1. Assumption (H4) is only needed to handle the case K = N , otherwise (H3) suffices. Note that (H4) is justifiedfrom a physical viewpoint (see, e.g., [3, Chap. 4]).

For the sake of completeness, we now state a known result which will be needed in the sequel (see [36, Lemma A.1]).

Lemma 2.2. Consider the following linear boundary value problem

−∆ψ = h1, inΩ,−α∆Γψ + ∂nψ + βψ = h2, on Γ .

Let (h1, h2) ∈ Vs, respectively, where s ≥ 0, s+ 12 6∈ N. Then the following estimate holds

‖ψ‖Hs+2(Ω) +∥∥ψ|Γ ∥∥Hs+2(Γ ) ≤ C (‖h1‖Hs(Ω) + ‖h2‖Hs(Γ )) ,

for some constant C > 0.

We now state and prove the first global existence result.

Theorem 2.3. Let f , g, λ satisfy assumptions (H1)–(H3). Then, for each K ∈ D,N, R, problem PwK admits a global weaksolution.

Proof. Let T > 0 be fixed. A unique local (and sufficiently smooth) solution to problem PwK on [0, T ] can be found by meansof a suitable Faedo–Galerkin approximation scheme (cf. [37] for a similar problem). To apply such an argument to a problemwith dynamic boundary conditions, we need to construct a convenient self-adjoint positive operator acting onV0 (see [38]).To this end, let us consider the operatorB0 given formally by

B0(ψ,ψ|Γ

)=(−∆ψ, (−∆ψ)|Γ

), (2.14)

for functions ψ ∈ C2(Ω)which satisfy the Wentzell boundary condition

(∆ψ)|Γ − α∆Γψ|Γ + ∂nψ + βψ|Γ = 0, on Γ . (2.15)

Here (∆ψ)|Γ stands for the trace of the function ∆ψ on the boundary Γ and should not be confused with the Laplace–Beltrami operator∆Γψ|Γ . The domain ofB0 is D (B0) = Ψ =

(ψ,ψ|Γ

): ψ ∈ C2

(Ω)and (2.15) holds. It is easy to show

that

〈B0Ψ ,Ξ〉V0 =

∫Ω

∇ψ · ∇ψdx+ α∫Γ

∇Γψ|Γ · ∇Γ ψ|Γ dS + β∫Γ

ψ|Γ ψ|Γ dS, (2.16)



for all Ψ ∈ D (B0) andΞ = (ψ, ψ|Γ ) ∈ V1. Thus,B0 is symmetric on V0. Let us now consider a function h ∈ C(Ω)and set

h = (h1, h2), with h1 := h|Ω and h2 := h|Γ . The equalityB0Ψ = h corresponds to the following boundary value problem:

−∆ψ = h1, inΩ, (2.17)−∆ψ = h2, on Γ . (2.18)

On account of (2.15), the boundary condition (2.18) becomes

− α∆Γψ|Γ + ∂nψ + βψ|Γ = h2, on Γ . (2.19)

We now define the ‘‘Wentzell version ofB0’’, B0, by setting B0Ψ = h on

D(B0) =Ψ ∈ V0 : Ψ = (ψ,ψ|Γ ) ∈ H2 (Ω)× H2 (Γ ) and (2.17)–(2.19) hold

.

Let B be the closure of B0. Then, using the techniques developed in [38], we can check that B is self-adjoint and positiveon V0, since β > 0 andB is the operator associated with the positive symmetric closed bilinear form defined by (2.16). Wenow recall the following characterization of the domain ofB (see [38, Thm. 3.2]):

D (B) =ψ ∈ H2 (Ω) : ψ|Γ ∈ H2 (Γ )

.

Thus, we have a complete system of eigenfunctions Ψii∈N for the operator B in V0 with Ψi ∈ D (B). Moreover, recallthat AK = −∆, K ∈ D,N, R, is a nonnegative and self-adjoint operator in L2 (Ω). Then, we also have a complete systemof eigenfunctions

ξKii∈N for the operator AK in L

2 (Ω) with ξKi ∈ D (AK ). According to the general spectral theory, theeigenvalues λi and λKi , K ∈ D,N, R, can be increasingly ordered and counted according to their multiplicities in orderto form a real divergent sequence. Moreover, the respective eigenvectors Ψi and ξKi turn out to form an orthogonal basisin V1, V0 and L2 (Ω) , Z1K , respectively. The eigenvectors Ψi and ξ

Ki may be assumed to be normalized in V0 and L2 (Ω),

respectively. We can now define the finite-dimensional spaces

Kn = span Ψ1,Ψ2, . . . ,Ψn , K∞ =∞⋃n=1

Kn,

P Kn = spanξK1 , ξ

K2 , . . . , ξ

Kn

, P K

∞=

∞⋃n=1

P Kn .

Clearly,K∞ and P K∞, K ∈ D,N, R, are dense subspaces of V1,V2,Z1K and D (AK ), respectively. For any n ∈ N, we look for

functions of the form

ψn(t) =n∑i=1

di (t)Ψi, θKn (t) =n∑i=1

ei (t) ξKi (2.20)

solving a suitable approximating problem. More precisely, for a fixed K ∈ D,N, R and for any given n ≥ 1, we look forC1-real functions di and ei, i = 1, . . . , n, only depending on time, which solve the approximating problem Pw,nK given by⟨

∂tψn,Ψ⟩V0+⟨Bψn,Ψ

⟩V0+ 〈f (ψn) ,Ψ 〉2 + 〈g (ψn) ,Ψ 〉2,Γ = 〈λ′ (ψn) θKn ,Ψ 〉2,⟨

∂tθKn ,Θ

⟩2 +

⟨AKθKn ,Θ

⟩2 = −〈λ

′ (ψn) ∂tψn,Θ〉2,(2.21)

with initial conditions⟨ψn (0) ,Ψ

⟩V0=⟨ψn0,Ψ

⟩V0,

⟨θKn (0) ,Θ

⟩2 =

⟨θn0,Θ

⟩2 , (2.22)

for all(Ψ ,Θ

)∈ Kn × P Kn . Here (ψn0, θn0) is the orthogonal projection of (ψ0, θ0) ontoKn × P Kn . Observe that

limn→∞

(ψn0, θn0) = (ψ0, θ0) , in V1 × L2 (Ω) . (2.23)

We aim to apply the standard Cauchy–Lipschitz theorem. For this purpose, we choose Ψ = Ψi and Θ = ξKi , 1 ≤ i ≤ n,and insert the representations (2.20) in place of the unknowns ψn and θKn in (2.21) and (2.22). After performing directcomputations (and using the above definitions), we can transform Pw,nK into a Cauchy problem for a system of nonlinearordinary differential equations in normal form for the vector-valued functions d = (d1, . . . , dn) and e = (e1, . . . , en), i.e.,

d′ (t) = U1 (d (t) , e (t)) ,e′ (t) = U2 (d (t) , e (t)) ,di(0) = 〈ψi0,Ψi〉V0 , ei(0) =

⟨θi0, ξ

Ki

⟩2 , i = 1, . . . , n,



where Ul : [0, Tn]× R2n → R2n, l = 1, 2, can be computed explicitly. Functions U1 and U2 are C1. These properties followfrom the smoothness of f , g and λ. We can thus apply the Cauchy–Lipschitz theorem and find a unique maximal solution(ψn, θ

Kn

)∈ C2([0, Tn);V2 × D(AK )) to (2.21)–(2.22), for some Tn ∈ (0, T ).

We will now deduce some a priori estimates on the approximating solutions(ψn, θ

Kn

). Such estimates, in particular,

ensure that the obtained solutions are defined on the whole interval [0, T ], for every n ∈ N. Throughout this proof, C willdenote a positive generic constant, depending at most on the physical parameters of the problem, but independent of n.Such a constant may vary even in the same line. Further dependencies of the constants will be pointed out if needed.Let us fix K ∈ D,N, R and consider our approximating solution

(ψn, θ

Kn

). For the sake of simplicity, we will drop the

superscript K from θKn . We take Ψ = ∂tψn(t) andΘ = θn(t) in (2.21). This gives

‖∂tψn (t)‖2V0 +12ddt

[α ‖∇Γψn (t)‖22,Γ + β ‖ψn (t)‖

22,Γ + ‖∇ψn (t)‖

22

]+

∫Ω

f (ψn (t)) ∂tψn (t) dx+∫Γ

g (ψn (t)) ∂tψn (t) dS

=⟨∂tψn (t) , λ′ (ψn (t)) θn (t)

⟩2 , (2.24)

12ddt‖θn (t)‖22 + ‖∇θn (t)‖

22 + d

∥∥θn|Γ (t)∥∥22,Γ = − ⟨λ′ (ψn (t)) ∂tψn (t) , θn (t)⟩2 . (2.25)

Adding together the above equations, then integrating with respect to time and exploiting (2.23), we get

‖ψn (t)‖2V1 +∫ t

0‖∂tψn (s)‖2V0 ds+ ‖θn (t)‖

22 +

∫ t

0

(‖∇θn(s)‖22 + d

∥∥θn|Γ (s)∥∥22,Γ ) ds+

∫Ω

F (ψn (t)) dx+∫Γ

G (ψn (t)) dS ≤ C,

where we have set

F (y) =∫ y

0f (r) dr, G (y) =

∫ y

0g (r) dr, ∀ y ∈ R. (2.26)

Recalling that, due to (2.11), F and G are bounded from below (independently of n), we infer

‖ψn (t)‖2V1 +∫ t

0‖∂tψn (s)‖2V0 ds+ ‖θn (t)‖

22 +

∫ t

0

(‖∇θn (s)‖22 + d

∥∥θn|Γ (s)∥∥22,Γ ) ds ≤ C, (2.27)

for all t ∈ [0, Tn). Observe that, in the case K = N (i.e., d = 0), we also need to control 〈θn(t)〉2Ω . Hence, on account of Eqs.(2.7), (2.13) and (2.27), we have

| 〈θn (t)〉Ω |2≤ I2N(ψ0, θ0)+ C(1+ ‖ψn(t)‖

42) ≤ C . (2.28)

In particular, bounds (2.27) and (2.28) imply that Tn = T , for every n ∈ N. In addition, we also learn that (ψn, θn) satisfiesthe following estimates:

‖ψn‖L∞([0,T ];V1)∩H1([0,T ];V0) ≤ C, (2.29)

‖θn‖L∞([0,T ];L2(Ω))∩L2([0,T ];Z1K

) ≤ C . (2.30)

Recalling (2.13), exploiting the Hölder inequality with exponents (5/2, 5/3) and the embedding H1 (Ω) ⊂ L6 (Ω), from(2.29) we also obtain∥∥λ′ (ψn) ∂tψn∥∥L6/5(Ω) ≤ C ‖∂tψn‖2 (1+ ‖ψn‖L3(Ω)) ≤ C ‖∂tψn‖2 . (2.31)

Observe now that

〈∂tθn, v〉2 =⟨∂tθn,Π

Kn v⟩2 = −

⟨AKθn + λ′ (ψn) ∂tψn,ΠKn v

⟩2 , (2.32)

whereΠKn : L2 (Ω)→ P Kn ⊂ V1, is the orthogonal projection with respect to the basis

ξKii∈N. Thus, recalling that L

6/5(Ω)

is continuously embedded in (Z1K )∗, from the uniform bounds (2.30)–(2.31), we deduce that

‖∂tθn‖L2([0,T ];(Z1K )∗) ≤ C . (2.33)



We are now ready to pass to the limit as n goes to+∞. On account of the above uniform inequalities, we can find ψ and θsuch that, up to subsequences,

ψn → ψ weakly ∗ in L∞ ([0, T ] ;V1) , ∂tψn → ∂tψ weakly in L2 ([0, T ] ;V0) , (2.34)

θn → θ weakly ∗ in L∞([0, T ] ; L2 (Ω)

)and weakly in L2

([0, T ] ;Z1K

), (2.35)

∂tθn → ∂tθ weakly in L2([0, T ] ;

(Z1K)∗)

. (2.36)

Due to (2.34)–(2.36) and classical compactness theorems, we also have

ψn → ψ strongly in C ([0, T ] ;V1−s) (2.37)

θn → θ strongly in L2([0, T ] ;H1−s(Ω)

), (2.38)

for all s ∈ (0, 1]. These strong convergences entail that, up to subsequences, ψn converges to ψ almost everywhere in Ω ,for any t ∈ [0, T ] and θn converges to θ almost everywhere inΩ × [0, T ]. Thanks to (2.29) and (2.30), we can easily controlthe nonlinear terms in (2.21). In particular, on account of (H2), we have

‖f (ψn)‖L2(Ω×[0,T ]) ≤ C, ‖g (ψn)‖Lq+1(Γ×[0,T ]) ≤ C . (2.39)

Then, using the continuity of f and g , (2.37) and (2.39), we easily deduce that, up to subsequences, f (ψn)weakly convergesto f (ψ) in L2 (Ω × [0, T ]), while g (ψn) weakly converges to g (ψ) in Lq+1 (Γ × [0, T ]), and hence in L2 (Γ × [0, T ]) sinceq ≥ 1. On the other hand, on account of (2.31), (2.34), (2.35) and (2.37), we obtain that, up to subsequences,

λ′ (ψn) θn → λ′ (ψ) θ weakly in L2 (Ω × [0, T ]) (2.40)

λ′ (ψn) ∂tψn → λ′ (ψ) ∂tψ weakly in L2([0, T ] ;

(Z1K)∗)

. (2.41)

For the sake of completeness, let us prove (2.41) ((2.40) is easier). Indeed, on account of (H3), it suffices to show thatψn∂tψnconverges weakly to ψ∂tψ in L2

([0, T ] ;

(Z1K)∗). For v ∈ L2 ([0, T ] ;Z1K ), we have

En =∫ T

0

∫Ω

(ψn∂tψn − ψ∂tψ) vdxdt

=

∫ T

0

∫Ω

(ψn − ψ) ∂tψnvdxdt +∫ T

0

∫Ω

(∂tψn − ∂tψ)ψvdxdt =: E1n + E2n . (2.42)

Since H1/2 (Ω) ⊂ L3 (Ω), it follows that∣∣E1n ∣∣ ≤ C ‖ψn − ψ‖L∞([0,T ];H1/2(Ω)) ‖∂tψnv‖L1([0,T ];L3/2(Ω))≤ C ‖ψn − ψ‖L∞([0,T ];H1/2(Ω)) ‖∂tψn‖L2(Ω×[0,T ]) ‖v‖L2

([0,T ];Z1K

) → 0,

as n→+∞, by (2.29) and (2.37). It remains to show that E2n → 0 as n→+∞. On account of the secondweak convergenceof (2.34), it turns out that we are finished if we deduce that ψv ∈ L2

([0, T ] ; L2 (Ω)

). But this is easy since

‖ψv‖L2(Ω×[0,T ]) ≤ C ‖ψ‖L∞([0,T ];L3(Ω)) ‖v‖L2([0,T ];Z1K

) ≤ C,by (2.37). It follows that En → 0 as n→+∞ so that (2.41) holds. In addition, we can also deduce that, up to subsequences,λ′ (ψn) ∂tψn converges to λ′ (ψ) ∂tψ weakly in L3/2(Ω × [0, T ]).By means of the above convergence properties (2.34)–(2.41), we can pass to the limit in the second equation of (2.21)

and show that θ solves (2.6) on (0, T ), for any T > 0. Moreover, by passing to the limit in the first equation of (2.21), wealso obtain that ψ (t) is a (weak) solution of the following boundary value problem:

−∆ψ (t) = h1 (t) , inΩ,−α∆Γψ (t)+ ∂nψ (t)+ βψ (t) = h2 (t) , on Γ , (2.43)

with h1 = −∂tψ − f (ψ) + λ′ (ψ) θ and h2 = −∂tψ − g (ψ). Since, from the above convergence properties, h1 ∈L2 (Ω × [0, T ]) and h2 ∈ L2 (Γ × [0, T ]), employing the regularity result of Lemma 2.2 (with s = 0), we deduce thatψ ∈ L2 ([0, T ] ;V2). Thus, the first equations of (1.1) and (1.2), respectively, are satisfied almost everywhere inΩ×(0, T ) andon Γ × (0, T ), respectively. Standard arguments imply that (ψ, θ) satisfies the initial conditions (1.3). The strong continuity(2.2) also follows from well-known results. Finally, the arbitrariness of T and estimate (2.27) allow us to deduce (2.3). Weomit the details.

We now state and prove the existence of a global strong solution.



Theorem 2.4. Let f , g, λ satisfy (H1) and (H3). Then, for each K ∈ D,N, R, problem PsK admits a global strong solution.

Proof. Let T > 0 be fixed. Recalling the proof of Theorem 2.3, we know that the approximating problem (2.21)–(2.22) hasa (unique) maximal solution

(ψn, θ

Kn

)∈ C2([0, Tn);V2 × D(AK )). We now show that we can obtain higher-order estimates

without requiring any growth assumptions on the nonlinearities f and g . We will do so by employing a two-step procedure.Step 1. First, for each ε > 0, we take a sequence of functions fε ∈ C1 (R) and gε ∈ C1 (R) such that they satisfy (2.11)

and (2.12) and fε → f , gε → g uniformly on compact subsets of R. In particular, such functions satisfy∣∣f ′ε (y)∣∣ ≤ cf ,ε (1+ |y|2) , ∣∣g ′ε (y)∣∣ ≤ cg,ε (1+ |y|q) , ∀ y ∈ R, (2.44)

for some positive constants cf ,ε, cg,ε that depend on ε (possibly behaving like O(ε−m

), for some m > 0). An example

of such approximating families ρεε>0 can be obtained by setting ρε (y) = ρ (y), for |y| ≤ ε−1, ρε (y) = ρ(ε−1

)+

ρ ′(ε−1

) (y− ε−1

), for y > ε−1 and ρε (y) = ρ

(−ε−1

)+ ρ ′

(−ε−1

) (y+ ε−1

)for y < −ε−1, where ρ can be either f or g .

Let us now replace the functions f and g in (2.5)–(2.6) by fε and gε , for a fixed ε > 0, respectively. Thus, according to theproof of Theorem 2.3, we obtain the existence of an approximating weak solution (ψε, θ ε) satisfying (2.5)–(2.6) and suchthat

‖ψε‖L∞([0,T ];V1)∩H1([0,T ];V0) + ‖θ

ε‖L∞([0,T ];L2(Ω))∩L2

([0,T ];Z1K

) ≤ C, (2.45)

where C is independent of ε. Indeed, (2.45) follows immediately from estimates (2.24)–(2.28) since Fε (y) ≥ −CF , Gε (y) ≥−CG, for all y ∈ R, where the positive constants CF , CG can be chosen independently of ε, because of assumption (2.11). Herewe have set, for y ∈ R,

Fε (y) =∫ y

0fε (r) dr, Gε (y) =

∫ y

0gε (r) dr.

Our next goal is to obtainmore regularity properties for the approximatingweak solution (ψε, θ ε) , for a fixed ε > 0.Wewill do sowith the aid of the Galerkin approximation scheme introduced in (2.21)–(2.22).We differentiate the first equationof (2.21) with respect to time and, in the resulting equation, we take Ψ = ∂tψε

n (t), while in the second equation of (2.21)we setΘ = ∂tθ εn (t). Adding the obtained relations, we have

12ddt

(∥∥∂tψεn (t)

∥∥22 +

∥∥∂tψεn (t)

∥∥22,Γ +

∥∥∇θ εn (t)∥∥22 + d ∥∥θ εn|Γ (t)∥∥22,Γ )+∥∥∂tψε

n (t)∥∥2

V1+∥∥∂tθ εn (t)∥∥22 + ∫

Ω

f ′ε(ψεn (t)

) (∂tψ

εn (t)

)2 dx+ ∫Γ

g ′ε(ψεn (t)

) (∂tψ

εn (t)

)2 dS=

⟨λ′′(ψεn (t)

) (∂tψ

εn (t)

)2, θ εn (t)

⟩2. (2.46)

Thanks to (2.11), there holds

−

∫Ω

f ′ε(ψεn (t)

) (∂tψ

εn (t)

)2 dx− ∫Γ

g ′ε(ψεn (t)

) (∂tψ

εn (t)

)2 dS ≤ C ∥∥∂tψεn (t)

∥∥2V0, (2.47)

where C is independent of ε. Observe that, due to assumption (2.11), the families fε and gε can be chosen in such a way that

f ′ε (y) ≥ −Cf , g ′ε (y) ≥ −Cg , (2.48)

independently of ε, for all y ∈ R. Moreover, using the interpolation inequality ‖u‖4 ≤ cΩ ‖u‖1/42 ‖u‖

3/4H1(Ω)

and then theYoung inequality with exponents (4/3, 4), we estimate the term on the right-hand side of (2.46) as follows (cf. (2.13)):∣∣∣⟨λ′′ (ψε

n

) (∂tψ

εn

)2, θ εn

⟩2

∣∣∣ ≤ C ∥∥∂tψεn

∥∥24

∥∥θ εn∥∥2≤ C

∥∥∂tψεn

∥∥1/22

∥∥∂tψεn

∥∥3/2H1(Ω)

∥∥θ εn∥∥2≤ η

(∥∥∇∂tψεn

∥∥22 +

∥∥∂tψεn

∥∥22

)+ Cη

∥∥∂tψεn

∥∥22

∥∥θ εn∥∥42 , (2.49)

for a sufficiently small η ∈ (0, 1) and a large positive constant Cη which are independent of ε. Then, on account of (2.47)–(2.49), an integration of (2.46) with respect to time yields

∥∥∂tψεn (t)

∥∥2V0+∥∥∇θ εn (t)∥∥22 + d ∥∥θ εn|Γ (t)∥∥22,Γ + ∫ t

0

(∥∥∂tψεn (s)

∥∥2V1+∥∥∂tθ εn (s)∥∥22) ds ≤ C, (2.50)



for all t ≥ 0, where C is independent of ε, t and n. Note that, owing to (2.50), we can use Eq. (2.32) to deduce∫ t

0

∥∥∆θ εn (s)∥∥22 ds ≤ C, ∀t ∈ [0, Tn). (2.51)

Indeed, on account of bounds (2.45) and (2.50), it is not difficult to check that λ′(ψεn

)∂tψ

εn is uniformly bounded (with

respect to n) in L2 (Ω × [0, T ]). The above estimates imply that Tn = T , for any n ∈ N. Moreover, on account of (2.45), (2.50)and (2.51), we deduce, up to subsequences, the following convergences:

∂tψεn → ∂tψ

ε weakly ∗ in L∞ ([0, T ] ;V0) , (2.52)

∂tψεn → ∂tψ

ε weakly in L2 ([0, T ] ;V1) , (2.53)

θ εn → θ ε weakly ∗ in L∞([0, T ] ;Z1K

), (2.54)

θ εn → θ ε weakly in L2 ([0, T ] ;D (AK )) , (2.55)

∂tθεn → ∂tθ

ε weakly in L2 (Ω × [0, T ]) . (2.56)

In addition, each of the convergence properties (2.40), (2.41) for(ψεn , θ

εn

)are now weak convergences in L2 (Ω × [0, T ]) at

least. Thus, it follows that (ψε, θ ε) satisfies Eqs. (1.1) and (1.2) almost everywhere, that is, (ψε, θ ε) is a strong solution to(2.5) and (2.6) on (0, T ), for any T > 0. On account of (2.50), we also have

‖∂tψε‖L∞([0,T ];V0)∩L2([0,T ];V1) + ‖θ

ε‖L∞([0,T ];Z1K

)∩H1([0,T ];L2(Ω))

≤ C, (2.57)

where C is independent of ε. To prove the last bound on ψε , we need to apply Lemma 2.2 (with s = 0) to the followingelliptic boundary value problem:

−∆ψε= hε1 := −∂tψ

ε+ λ′ (ψε) θ ε − fε (ψε) , a.e. inΩ × (0, T ) ,

−α∆Γψε+ ∂nψ

ε+ βψε

= hε2 := −∂tψε− gε (ψε) , a.e. on Γ × (0, T ) , (2.58)

which yields

‖ψε (t)‖2H2(Ω) +∥∥ψε|Γ (t)

∥∥2H2(Γ )

≤ Cε(∥∥hε1(t)∥∥22 + ∥∥hε2(t)∥∥22,Γ ) . (2.59)

Thus, owing to (2.44), (2.45) and (2.50) and the lower semicontinuity property of norms, it follows that

‖ψε (t)‖V2 ≤ Cε, (2.60)

where Cε > 0 depends on ε. Therefore, because of (2.60) and the continuous embeddings H2(Ω) ⊂ L∞(Ω),H2 (Γ ) ⊂L∞ (Γ ), we immediately deduce that ψε

∈ L∞ (Ω × [0, T ]) and ψε|Γ ∈ L

∞ (Γ × [0, T ]). Then fε (ψε) and gε (ψε) belong toL∞ (Ω × [0, T ]) and L∞ (Γ × [0, T ]), respectively.

Step 2. It remains to pass to the limit as ε → 0 in the above estimates for (ψε, θ ε) and find that the limit solution(ψ, θ) is also a strong solution of problem PsK . However, in order to prove this statement we need to deduce some additionalestimates for (ψε, θ ε)which are uniform with respect to ε. To this end, we consider the following problem:

−∆ψε+ fε (ψε) = jε1 := −∂tψ

ε+ λ′ (ψε) θ ε, a.e. inΩ × (0, T ) ,

−α∆Γψε+ ∂nψ

ε+ βψε

+ gε (ψε) = jε2 := −∂tψε, a.e. on Γ × (0, T ) , (2.61)

to which we apply the maximum principle from [36, Lemma A.2]. In order to do so, we first notice that, on account of theuniform estimates (in ε) (2.45) and (2.57), we have∥∥jε1(t)∥∥2 + ∥∥jε2(t)∥∥2,Γ ≤ C, (2.62)

where C is independent of ε. Thus, recalling again (2.48), the maximum principles for (2.61) states that

‖ψε (t)‖∞ + ‖ψε (t)‖∞,Γ ≤ Cf ,g + C

(∥∥jε1(t)∥∥2 + ∥∥jε2(t)∥∥2,Γ ) , (2.63)

where the constants Cf ,g , C are independent of ε. Thus, owing to (2.62) and (2.63), it follows that

‖ψε (t)‖∞ + ‖ψε (t)‖∞,Γ ≤ C . (2.64)

On the other hand, using the H2-regularity estimate of Lemma 2.2, and repeating the arguments of (2.58)–(2.60) once more,it is not difficult to realize, on account of (2.64), that

‖ψε (t)‖V2 ≤ C, (2.65)



where C is independent of ε.We are now ready to pass to the limit as ε → 0. Recalling (2.45), (2.57), (2.64), (2.65) and (2.51), we deduce, up to

subsequences, the following convergences:

∂tψε→ ∂tψ weakly ∗ in L∞ ([0, T ] ;V0) , (2.66)

∂tψε→ ∂tψ weakly in L2 ([0, T ] ;V1) ,

ψε→ ψ weakly ∗ in L∞ ([0, T ] ;V2) ,

θ ε → θ weakly ∗ in L∞([0, T ] ;Z1K

),

θ ε → θ weakly in L2 ([0, T ] ;D (AK )) ,∂tθ

ε→ ∂tθ weakly in L2 (Ω × [0, T ]) .

On the other hand, due to (2.66) and classical compactness theorems, we also have

ψε→ ψ strongly in C ([0, T ] ;V2−s) , (2.67)

θ ε → θ strongly in C([0, T ] ;H1−s(Ω)

), (2.68)

for all s ∈ (0, 1]. Furthermore, on account of (2.67), the fact that fε and gε are continuously differentiable and thatfε → f , gε → g uniformly on compact subsets of R, it is not difficult to show

fε (ψε)→ f (ψ) strongly in C([0, T ] ; L2 (Ω)

), (2.69)

gε (ψε)→ g (ψ) strongly in C([0, T ] ; L2 (Γ )

). (2.70)

Finally, the convergences λ′ (ψε) θ ε → λ′ (ψ) θ and λ′ (ψε) ∂tψε→ λ′ (ψ) ∂tψ are now strong in L2 (Ω × [0, T ]). We now

have all the ingredients to deduce that (ψ, θ) is a strong solution to (2.5) and (2.6) on (0, T ), for any T > 0. Indeed, wecan pass to the limit in the approximating problem for (ψε, θ ε) , ε > 0, and find that (ψ, θ) is a strong solution such that(ψ, θ) ∈ L∞

([0, T ];V2 × Z1K

). Moreover, due to the arbitrariness of T , from (2.50)–(2.51) we also deduce Eqs. (2.9) and

(2.10). Finally, we recover the continuity property Eq. (2.8) by means of [26, Thm. 2.3].

The uniqueness of solutions to both problems PwK and PsK is an immediate consequence of the following continuousdependence estimates.

Lemma 2.5. Let the assumptions of Theorem 2.3 hold and let (ψwi, θwi) be two solutions to PwK corresponding to the initial data(ψ0i, θ0i) ∈ V1 × L2 (Ω) , i = 1, 2. Then, for any t ≥ 0, the following estimate holds:

‖(ψw1 − ψw2) (t)‖2V1 + ‖(θw1 − θw2) (t)‖22 +

∫ t

0

[‖∂t (ψw1 − ψw2) (s)‖2V0 + ‖(θw1 − θw2) (s)‖

2Z1K

]ds

≤ CweLw t(‖ψ01 − ψ02‖

2V1 + ‖θ01 − θ02‖

22

), (2.71)

where Cw and Lw are positive constants depending on the norms of the initial data in V1 × L2 (Ω), onΩ and on the parametersof the problem, but are both independent of time.

Proof. Let us setψ := ψ1−ψ2 and θ := θ1− θ2 (the subscriptw is omitted here). Then, we easily realize that (ψ, θ) solvesthe system

〈∂tψ, u〉2 + 〈∇ψ,∇u〉2 + 〈Υ1, u〉2 + 〈∂tψ, u〉2,Γ+α〈∇Γψ,∇Γ u〉2,Γ + 〈βψ + Υ2, u〉2,Γ = 〈Υ3, u〉2, ∀u ∈ V1, a.e. in (0,∞), (2.72)

〈∂tθ, v〉2 + 〈∇θ,∇v〉 + d〈θ, v〉2,Γ = −〈Υ4, v〉2, ∀v ∈ Z1K , a.e. in (0,∞), (2.73)

where we have set

Υ1 := f (ψ1)− f (ψ2) , Υ2 := g (ψ1)− g (ψ2) , (2.74)

Υ3 := λ′(ψ1)θ1 − λ

′(ψ2)θ2, Υ4 := λ′(ψ1)∂tψ1 − λ

′(ψ2)∂tψ2. (2.75)

Taking u = ∂tψ(t) in (2.72), v = θ(t) in (2.73) and then adding the resulting equations, we deduce that

‖∂tψ (t)‖2V0 + ‖∇θ (t)‖22 + d

∥∥θ|Γ (t)∥∥22,Γ + 12 ddt [‖ψ (t)‖2V1 + ‖θ (t)‖22]= −〈Υ1 (t) , ∂tψ (t)〉2 − 〈Υ2 (t) , ∂tψ (t)〉2,Γ + 〈Υ3 (t) , ∂tψ (t)〉2 − 〈Υ4 (t) , θ (t)〉2 . (2.76)



We proceed as in [27, Proposition 3]. First, thanks to (2.27), we notice that

supt≥0

(‖ψi (t)‖2V1 + ‖θi (t)‖

22 +

∫ t

0‖∂tψi (s) ‖2V0ds

)≤ Cw, i = 1, 2, (2.77)

for some Cw > 0 depending only on the V1 × L2 (Ω)-norm of the initial data. We need to estimate all the remaining termson the right-hand side of (2.76). To this end, observe preliminarily that, owing to (2.12), we have

|f (y1)− f (y2)| ≤ cf(1+ |y1|2 + |y2|2

)|y1 − y2| ,

|g(y1)− g (y2)| ≤ cg(1+ |y1|q + |y2|q

)|y1 − y2| ,

(2.78)

where q ∈ [1,+∞) is fixed, but otherwise arbitrary.In order to estimate the first two terms on the right-hand side of (2.76), we use the Young and Hölder inequalities, (2.74)

and (2.78) as follows:∣∣〈Υ1 (t) , ∂tψ (t)〉2∣∣+ ∣∣〈Υ2 (t) , ∂tψ (t)〉2,Γ ∣∣≤(‖Υ1 (t)‖22 + ‖Υ2 (t)‖

22,Γ

)+14

[‖∂tψ (t)‖22 + ‖∂tψ (t)‖

22,Γ

]≤ Cc2f

(1+ ‖ψ1 (t)‖46 + ‖ψ2 (t)‖

46

)‖ψ (t)‖26

+ Cc2g(1+ ‖ψ1 (t)‖

2q3q,Γ + ‖ψ2 (t)‖

2q3q,Γ

)‖ψ (t)‖26,Γ +

14‖∂tψ (t)‖2V0 . (2.79)

Thanks to the continuous embeddings H1 (Ω) → L6 (Ω) and H1 (Γ ) → Ls (Γ ), for any s ∈ [1,+∞), we deduce from(2.79) that∣∣〈Υ1 (t) , ∂tψ (t)〉2∣∣+ ∣∣〈Υ2 (t) , ∂tψ (t)〉2,Γ ∣∣ ≤ Cw ‖ψ (t)‖2V1 + 14 ‖∂tψ (t)‖2V0 , (2.80)

where Cw > 0 only depends on the V1 × L2 (Ω)-norm of the initial data. Moreover, observe that

〈Υ3 (t) , ∂tψ (t)〉2 − 〈Υ4 (t) , θ (t)〉2 =⟨λ′ (ψ1(t))− λ′ (ψ2(t)) , ∂tψ (t) θ2 (t)

⟩2

−⟨λ′ (ψ1(t))− λ′ (ψ2(t)) , ∂tψ2 (t) θ (t)

⟩2 .

Then, on account of (2.13), we estimate the last two terms on the right-hand side of (2.76) as follows:∣∣〈Υ3 (t) , ∂tψ (t)〉2 − 〈Υ4 (t) , θ (t)〉∣∣≤ Cλ

(∣∣〈ψ (t) ∂tψ (t) , θ2 (t)〉2∣∣+ ∣∣〈ψ (t) , ∂tψ2 (t) θ (t)〉2∣∣)≤ Cλ

(‖ψ (t)‖4 ‖∂tψ (t)‖2 ‖θ2 (t)‖4 + ‖ψ (t)‖4 ‖θ (t)‖4 ‖∂tψ2 (t)‖2

)≤ C ‖ψ (t)‖2H1(Ω)

(‖θ2 (t)‖2Z1K

+ ‖∂tψ2 (t)‖22)+14‖∂tψ (t)‖22 +

12‖θ (t)‖2Z1K

. (2.81)

Thus, combining (2.76) with (2.80)–(2.81), we obtain, for K ∈ D, R,

12ddt

[‖ψ (t)‖2V1 + ‖θ (t)‖

22

]+12‖∂tψ (t)‖22 +

34‖∂tψ (t)‖22,Γ +

12‖θ (t)‖2Z1K

≤ GK (t) ‖ψ (t)‖2V1 , (2.82)

where

GK (t) := Cw(1+ ‖θ2 (t)‖2Z1K

+ ‖∂tψ2 (t)‖22).

It is not difficult to check, on account of (2.27) and (2.77), that GK ∈ L2(0, t), for all K ∈ D,N, R and any fixed t ≥ 0. In thecases K ∈ D, R, the proof of (2.71) follows from the Gronwall inequality. It is left to show (2.71) when K = N (we remindthat, in this case, d = 0 in (2.76)). Recall that every weak solution (ψi, θi) also satisfies (2.7). Then, it follows from (2.77) that

〈θ (t)〉2Ω ≤ C ‖θ (t)‖22 , ∀t ≥ 0. (2.83)

Adding relation (2.83) to (2.76), then recalling the definition of thenormofZ1N and arguing as above in (2.82),we immediatelyobtain estimate (2.71), for K = N .

A similar estimate holds for strong solutions.



Lemma 2.6. Let the assumptions of Theorem 2.4 hold and let (ψsi, θsi) be two global solutions to PsK corresponding to the initialdata (ψ0i, θ0i) ∈ V2 × Z1K , i = 1, 2. Then, for any t ≥ 0, the following estimate holds:

‖(ψs1 − ψs2) (t)‖2V1 + ‖(θs1 − θs2) (t)‖22 +

∫ t

0

[‖∂t (ψs1 − ψs2) (s)‖2V0 + ‖(θs1 − θs2) (s)‖

2Z1K

]ds

≤ CseLst(‖ψ01 − ψ02‖

2V1 + ‖θ01 − θ02‖

22

), (2.84)

where Cs and Ls are positive constants depending on the norms of the initial data in V2 × Z1K , onΩ and on the parameters of theproblem, but are both independent of time.

Remark 2.7. Note that (2.84) is similar to (2.71). However, they hold under different assumptions. Thus, the constants onthe right-hand side are different.

Proof. Here we use the same notation as in the proof of Lemma 2.5. Thanks to (2.64) and (2.78) we have

‖Υ1 (t)‖2 ≤ Cs ‖ψ (t)‖2 , ‖Υ2 (t)‖2,Γ ≤ Cs ‖ψ (t)‖2,Γ , ∀t ≥ 0,

so that∣∣〈Υ1 (t) , ∂tψ (t)〉2∣∣+ ∣∣〈Υ2 (t) , ∂tψ (t)〉2,Γ ∣∣ ≤ Cs [‖ψ (t)‖2 ‖∂tψ (t)‖2 + ‖ψ (t)‖2,Γ ‖∂tψ (t)‖2,Γ ]≤14‖∂tψ (t)‖2V0 + Cs ‖ψ (t)‖

2V0 . (2.85)

Furthermore, since estimates (2.81) and (2.83) also hold for strong solutions, combining (2.85)with (2.82)weobtain a similarinequality to (2.82), namely,

ddt

[‖ψ (t)‖2V1 + ‖θ (t)‖

22

]+12‖∂tψ (t)‖22 +

34‖∂tψ (t)‖22,Γ +

12‖θ (t)‖2Z1K

≤ Ls ‖ψ (t)‖2V1 , (2.86)

where Ls > 0 only depends on the norm of the initial data in V2 × Z1K . The proof follows again from the Gronwallinequality.

Straightforward consequences of the above results are the following.

Corollary 2.8. Let the assumptions of Theorem 2.3 hold. Then, for each K ∈ D,N, R, we can define a strongly continuoussemigroup

SwK (t) : V1 × L2 (Ω)→ V1 × L2 (Ω) ,

by setting, for all t ≥ 0,

SwK (t) (ψ0, θ0) = (ψw (t) , θw (t)) ,

where (ψw, θw) is the unique solution to PwK .

Corollary 2.9. Let the assumptions of Theorem 2.4 hold. Then, for each K ∈ D,N, R, we can define a semigroup

SsK (t) : V2 × Z1K → V2 × Z1K ,

by setting, for all t ≥ 0,

SsK (t) (ψ0, θ0) = (ψs (t) , θs (t)) ,

where (ψs, θs) is the unique solution to PsK. Moreover, SsK (t) is a closed semigroup in the sense of [39].

3. Existence of global attractors

Our aim in this section is to prove that SwK (t) and SsK (t) possess the global attractor. The preliminary step is to prove the

dissipativity of the semigroup, that is, the existence of a bounded absorbing set. In the case K = N , due to the enthalpyconservation Eq. (2.7), we need to put a constraint. More precisely, we set(

V1 × L2(Ω))M:=(u, v) ∈ V1 × L2(Ω) : |IN (u, v)| ≤ M

,(

V2 × Z1N)M:=(u, v) ∈ V2 × Z1N : |IN (u, v)| ≤ M

,



whereM ≥ 0 is fixed. Note that both spaces are complete metric spaces with respect to the metrics induced by the norms.Therefore, the phase space of SwK (t)will be

Y0,K =

V1 × L2(Ω), if K ∈ D, R ,(V1 × L2(Ω)

)M, if K = N,

(3.1)

while SsK (t)will act on

Y1,K =

V2 × Z1K , if K ∈ D, R ,(V2 × Z1N

)M, if K = N.

(3.2)

The main results of this section are

Theorem 3.1. Let f , g, λ satisfy assumptions (H1)–(H3)and (H5) . If K = N, assume in addition that λ fulfills (H4). Then, SwK (t)possesses the connected global attractor Aw

K ⊂ Y0,K which is bounded in V2 × H1 (Ω).

Remark 3.2. Actually, it can be proven thatAwK is bounded in V2 × H2 (Ω) arguing, e.g., as in [24].

Theorem 3.3. Let f , g, λ satisfy assumptions (H1), (H3)and (H5). If K = N, assume in addition that λ fulfills (H4). Then, SsK (t)possesses the connected global attractor AsK ⊂ Y1,K which is bounded in V3 × H3 (Ω).

The first result is concerned with the existence of a bounded absorbing set in Y0,K for both semigroups SwK (t) and SsK (t).

Lemma 3.4. Let f , g satisfy assumptions (H1) and (H5). Suppose that λ satisfies either (H3), if K ∈ D, R, or (H4), if K = N.Then, for any given initial data (ψ0, θ0) ∈ Y0,K , the following estimate holds:

‖(ψ (t) , θ (t))‖2Y0,K +∫ t+1

t

(‖∂tψ (s)‖2V0 + ‖θ (s)‖

2Z1K+ ‖ψ (s)‖4L4(Ω)

)ds

≤ CK(‖(ψ0, θ0)‖

2Y0,K + 〈F (ψ0) , 1〉2 + 〈G (ψ0) , 1〉2,Γ + 1

)e−ρt + C∗K , (3.3)

for each t ≥ 0, where F and G are defined as in (2.26). Here the constants ρ, CK , C∗K are independent of t and of the initial data.

Remark 3.5. Since we are dealing with gradient systems with a set of equilibria which is bounded in the phase space (seeSection 5), we could avoid to prove the existence of a bounded absorbing set and directly show the existence of the globalattractor (see, e.g., [40, Thm.A.3]). However, we prefer to prove the existence of a uniform dissipative estimate which canbe easily adapted to nonautonomous generalizations (see, e.g., [30,18,28] for standard boundary conditions).

Proof. Let us take u = 2ξψ(t), for some ξ > 0, in (2.5). Summing the resulting relationship with (2.24) and (2.25), we get

ddtE (t)+ κE (t) = Λ1 (t) , (3.4)

where κ > 0 and

E (t) := ‖ψ (t)‖2V1 + 2 〈F (ψ (t)) , 1〉2 + 2 〈G (ψ (t)) , 1〉2,Γ + ‖θ (t)‖22 + ξ ‖ψ (t)‖

2V0 + C .

Here the constant C > 0 is taken large enough in order to ensure that E (t) is nonnegative (recall that F and G are bothbounded from below). The functionΛ1 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 ‖∂tψ (t)‖2V0 − 2 ‖∇θ (t)‖

22 − 2d

∥∥θ|Γ (t)∥∥22,Γ+ ξκ ‖ψ (t)‖2V0 + κ ‖θ (t)‖

22 + κC + 2ξ

⟨λ′ (ψ (t)) ψ(t), θ(t)

⟩2 . (3.5)

Let us first discuss the case K ∈ D, R. Observe preliminarily that, owing to (H1), we have

|F (y)| ≤ 2f (y) y+ CF , |G (y)| ≤ 2g (y) y+ CG, (3.6)

F (y)− f (y) y ≤ C ′F |y|2+ C ′′F , G (y)− g (y) y ≤ C ′G |y|

2+ C ′′G , (3.7)

for any y ∈ R. All the positive constants in (3.6)–(3.7) depend on F andG, only. Moreover, recalling (H3) and using the Hölderand Young inequalities, we estimate the last term on the right-hand side of (3.5) as follows:

2ξ∣∣⟨λ′ (ψ)ψ, θ ⟩2∣∣ ≤ 2ξcλ 〈|ψ | , |θ |〉2 + 2ξcλ ⟨|ψ |2 , |θ |⟩2

≤ ‖θ‖2Z1K+ 2ξ 2c2λ

(‖ψ‖2V0 + ‖ψ‖

4L4(Ω)

). (3.8)



From (3.6)–(3.8) and assumptions H5, it follows that

Λ1 (t) ≤ −(2ξ − κ − 2κ

(C ′F + C

′

G

)− ξκ − ξ 2c

)‖ψ (t)‖2V1 − 2 ‖∂tψ (t)‖

2V0

−(ξ − κ)

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

− (1− κC0) ‖θ(t)‖2Z1K− [ξ (η1 − cξ)− κη1] ‖ψ (t)‖4L4(Ω) + C1,

where C0 > 0 and C1 > 0 depend on ξ , κ and C ′′F , C′′

G at most and c > 0 only depends on cλ. From now on, Ci stands for apositive constant which is independent of the initial data and of time.It is thus possible to pick ξ < 1 small enough and κ ∈ (0, ξ) (possibly even smaller than ξ 2) in order to have

ddtE (t)+ κE (t)+ C2

(‖ψ (t)‖2V1 + ‖∂tψ (t)‖

2V0 + ‖θ (t)‖

2Z1K+ ‖ψ (t)‖4L4(Ω)

)+ C3

(‖F (ψ (t))‖1 + ‖G (ψ (t))‖1,Γ

)≤ C1, (3.9)

from which we deduce that

ddtE (t)+ κE (t) ≤ C1. (3.10)

On the other hand, one can check that there exists a positive constant C4, independent of t and of the initial data, suchthat

‖(ψ (t) , θ (t))‖2Y0,K ≤ C4E (t) , ∀t ≥ 0. (3.11)

Then, applying the Gronwall lemma to (3.10), we infer that

E (t) ≤ C5E (0) e−κt + C6, ∀t ≥ 0. (3.12)

Integrating (3.9) over (t, t + 1) and employing estimate (3.12), we obtain

E (t)+∫ t+1

t

(‖ψ (s)‖2V1 + ‖∂tψ (s)‖

2V0 + ‖θ (s)‖

2Z1K

)ds

+

∫ t+1

t

(‖F (ψ (s))‖1 + ‖G (ψ (s))‖1,Γ + ‖ψ (s)‖

4L4(Ω)

)ds

≤ C7E (0) e−κt + C8, ∀t ≥ 0. (3.13)

Thus, on account of (3.11), we easily get the required estimate (3.3), for K ∈ D, R.To prove a similar uniform inequality when K = N , we need to use the enthalpy conservation (cf. (2.7)). Observe that

2ξ⟨λ′ (ψ)ψ, θ

⟩2 = 2ξ

⟨θ − 〈θ〉Ω , λ

′ (ψ)ψ⟩2 + 2ξ |Ω| IN(ψ0, θ0)

⟨λ′ (ψ)ψ

⟩Ω

− 2ξ |Ω| 〈λ (ψ)〉Ω⟨λ′ (ψ)ψ

⟩Ω. (3.14)

Employing the standard Poincaré inequality, we know that

‖v − 〈v〉Ω‖22 = ‖v‖

22 − |Ω| 〈v〉

2Ω ≤ CΩ ‖∇v‖

22 , (3.15)

for some CΩ > 0.Let us rewriteΛ1, defined by (3.5), taking (3.14) 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 ‖∂tψ (t)‖2V0 − 2 ‖∇θ (t)‖

22 + 2ξκ ‖ψ (t)‖

2V0 + κ ‖θ (t)‖

22

+ κC + 2ξ⟨θ(t)− 〈θ(t)〉Ω , λ

′ (ψ(t)) ψ(t)⟩2

+ 2ξ |Ω| IN(ψ0, θ0)⟨λ′ (ψ(t)) ψ(t)

⟩Ω− 2ξ |Ω| 〈λ (ψ(t))〉Ω

⟨λ′ (ψ(t)) ψ(t)

⟩Ω. (3.16)

It is worth mentioning that, from (2.7), we can also deduce the following estimate:

κ ‖θ‖22 = κ ‖(θ − 〈θ〉Ω)+ 〈θ〉Ω‖22

≤ 2κ ‖θ − 〈θ〉Ω‖22 + 2κ |Ω|

(IN(ψ0, θ0)− 〈λ (ψ)〉Ω

)2≤ 2κ ‖θ − 〈θ〉Ω‖

22 + κc ‖ψ‖

22 + κc ‖ψ‖

4L4(Ω) + CM , (3.17)



where c > 0 is independent of t, ξ , κ,M and the initial data, while CM > 0 depends onM, ξ , κ andΩ , but is independent oftime and the initial data. From now on, c will stand for a positive constant having these properties. There remain to estimateall the terms on the right-hand side of (3.14).We start by estimating the first term, arguing exactly as in (3.8). Recalling (H4),from (3.15) we obtain

2ξ∣∣⟨θ − 〈θ〉Ω , λ′ (ψ)ψ ⟩2∣∣ ≤ 2ξCγ 〈|θ − 〈θ〉Ω | , |ψ |〉2 + 4aξ ⟨|θ − 〈θ〉Ω | , ψ2⟩2

≤ ‖∇θ‖22 + cξ2‖ψ‖2V0 + cξ

2‖ψ‖4L4(Ω) . (3.18)

On the other hand,

2ξ |Ω| IN(ψ0, θ0)⟨λ′ (ψ)ψ

⟩Ω= 2ξ |Ω| IN(ψ0, θ0)

⟨γ ′ (ψ)ψ

⟩Ω− 4aξ |Ω| IN(ψ0, θ0)

⟨ψ2⟩Ω

≤ ξ 2 ‖ψ‖22 + ξ2‖ψ‖4L4(Ω) + CM , (3.19)

where the positive constant CM clearly depends onM . Finally, using the Hölder and Young inequalities, it is not difficult toshow that

− 2ξ |Ω| 〈λ (ψ)〉Ω⟨λ′ (ψ)ψ

⟩Ω= −4ξ |Ω| a2

⟨ψ2⟩2Ω− 2ξ |Ω| 〈γ (ψ)〉Ω

⟨γ ′ (ψ)ψ

⟩Ω

+ 4aξ |Ω| 〈γ (ψ)〉Ω⟨ψ2⟩Ω+ 2aξ |Ω|

⟨ψ2⟩Ω

⟨γ ′ (ψ)ψ

⟩Ω

≤ −4ξ |Ω| a2⟨ψ2⟩2Ω+(ξ 2 + ξ 4/3

)‖ψ‖44 + C9. (3.20)

Combining estimates (3.18)–(3.20), it easily follows that

2ξ⟨λ′ (ψ)ψ, θ

⟩2 ≤ −4ξ |Ω| a

2 ⟨ψ2⟩2Ω+ ‖∇θ‖22 + ξ

2 (c + 1) ‖ψ‖22 +[(1+ c) ξ 2 + ξ 4/3

]‖ψ‖4L4(Ω) + C

′

M , (3.21)

for a new positive constant C ′M . Using (3.17), (3.6), (3.7), (3.15) and (3.21), we infer from (3.16)

Λ1 (t) ≤ −[2ξ − κ (1+ c)− 2κ

(C ′F + C

′

G

)− 2ξκ − ξ 2 (c + 1)

]‖ψ (t)‖2V1

− 2 ‖∂tψ (t)‖2V0 −(ξ − κ)

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

− (1− 2κCΩ) ‖∇θ (t)‖22 − 4ξ |Ω| a2 ⟨ψ2⟩2

Ω

−(ξ[η1 − (1+ c) ξ + ξ 1/3

]− κ (η1 + c)

)‖ψ (t)‖4L4(Ω) + C10, (3.22)

where C10 > 0 depends onM, ξ , κ andΩ , but is independent of time and the initial data. It is thus possible to adjust ξ < 1small enough and κ ∈ (0, ξ) in order to deduce (cf. (3.4), (3.11) and (3.22))

ddtE (t)+ κE (t)+ κ ′

(‖ψ (t)‖2V1 + ‖ψ (t)‖

4L4(Ω)

)+ κ ′′

(‖∂tψ (t)‖2V0 + ‖∇θ (t)‖

22

)≤ C11. (3.23)

Applying a suitable version of the Gronwall inequality to (3.23) and taking (3.11) into account, we obtain the analogue ofestimate (3.3) in the case K = N , except for the L2

([t, t + 1] ;H1

)-norm of θ . More precisely, we have

E (t)+∫ t+1

t

(‖ψ (s)‖2V1 + ‖ψ (t)‖

4L4(Ω) + ‖∂tψ (s)‖

2V0 + ‖∇θ (s)‖

22

)ds

≤ C12E (0) e−κt + C13, ∀t ≥ 0. (3.24)

However, recalling (2.28), we easily obtain estimate (3.3) also for K = N . The proof is finished.

We now show the existence of a compact absorbing set, namely,

Lemma 3.6. Let the assumptions of Theorem 3.1 be satisfied. Then, there is a positive constant C independent of the initial dataand time and, for any R0 > 0, there exists t0 = t0(R0) > 0 such that∥∥SwK (t) (ψ0, θ0)∥∥V2×Z1K

≤ C, ∀t ≥ t0, (3.25)

for any (ψ0, θ0) ∈ B(R0) ⊂ Y0,K . HereB(R0) is a ball of radius R0, centered at 0.

Proof. We recall that all the following formal arguments can be justified by an approximation argument similar to the onedeveloped in Section 2. Recall first that (2.46) entails

ddt

(‖∂tψ (t)‖22 + ‖∂tψ (t)‖

22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ )



+ ‖∂tψ (t)‖22 + ‖∂tψ (t)‖22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ+ 2 ‖∂tψ (t)‖2V1 + 2 ‖∂tθ (t)‖

22 = Λ2 (t) , (3.26)

where

Λ2 (t) := −2⟨f ′ (ψ(t)) , ∂tψ2(t)

⟩2 − 2

⟨g ′ (ψ(t)) , ∂tψ2(t)

⟩2,Γ

+ ‖∂tψ (t)‖22 + ‖∂tψ (t)‖22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ + ⟨λ′′ (ψ (t)) ∂tψ2 (t) , θ (t)⟩2 .On account of (2.47)–(2.49), we have

ddt

(‖∂tψ (t)‖22 + ‖∂tψ (t)‖

22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ )+ ‖∂tψ (t)‖22 + ‖∂tψ (t)‖

22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ + 2 ‖∂tψ (t)‖2V1 + 2 ‖∂tθ (t)‖22≤ C ‖∂tψ (t)‖2V0 + ‖∂tψ (t)‖

22 + ‖∂tψ (t)‖

22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ+ η

(‖∇∂tψ‖

22 + ‖∂tψ‖

22

)+ Cη ‖∂tψ‖22 ‖θ‖

42 . (3.27)

Since η is small (cf. (2.49)), then we get

ddt

(‖∂tψ (t)‖22 + ‖∂tψ (t)‖

22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ )+ ‖∂tψ (t)‖22 + ‖∂tψ (t)‖

22,Γ + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ+C1

(‖∂tψ (t)‖2V1 + ‖∂tθ (t)‖

22

)≤ C2

(‖∂tψ (t)‖2V0 + ‖θ (t)‖

2Z1K+ ‖∂tψ(t)‖22 ‖θ(t)‖

42

)=: Λ3(t). (3.28)

Due to (3.3), we easily deduce that

supt≥0

∫ t+1

t|Λ3 (s) |ds ≤ C sup

t≥0

∫ t+1

t

(‖∂tψ(s)‖2V0 + ‖θ(s)‖

2Z1K

)ds+ C sup

t≥0‖θ(t)‖42 sup

t≥0

∫ t+1

t‖∂tψ(s)‖22 ds

≤ Q(‖(ψ0, θ0)‖

2Y0,K + 〈F (ψ0) , 1〉2 + 〈G (ψ0) , 1〉2,Γ + 1

)e−ρt + C . (3.29)

Here and in what follows C stands for a positive constant which is independent of t and of the initial data. This constant mayvary from line to line. Besides, Q denotes a positive nondecreasing monotone function which is independent of t . Using theso-called uniform Gronwall lemma (see, e.g., [41, Chap. III, Lemma 1.1]), on account of (3.29), from (3.28) and the hypothesisof the lemma (recall that f and g satisfy growth assumptions and so do F and G), we infer the existence of t#0 > 0 such that

‖∂tψ (t)‖22 + ‖∂tψ (t)‖22,Γ + ‖θ (t)‖

2Z1K+

∫ t+1

t

[‖∂tψ (s)‖2V1 + ‖∂tθ (s)‖

22

]ds

≤ Q(‖(ψ0, θ0)‖

2Y0,K

)e−ρt + C, ∀t ≥ t#0 . (3.30)

Applying now the maximum principle to the boundary value problem (2.58), as in the proof of Theorem 2.3, from (2.63),(3.3) and (3.30) we infer that

‖ψ (t)‖2∞+ ‖ψ (t)‖2

∞,Γ ≤ C(1+ ‖h1(t)‖22 + ‖h2(t)‖

22,Γ

)≤ Q

(‖(ψ0, θ0)‖

2Y0,K

)e−ρt + C, ∀t ≥ t#0 . (3.31)

In order to get the estimate for ψ (t) in V2,we rewrite problem Eq. (2.58) in the form−∆ψ = h3 := −f (ψ)+ λ′ (ψ) θ − ∂tψ, inΩ, a.e. in (t0,+∞),−α∆Γψ + ∂nψ + βψ = h4 := −∂tψ − g (ψ) , on Γ , a.e. in (t0,+∞),

(3.32)

and apply the result of Lemma 2.2. We obtain

‖ψ (t)‖2V2 ≤ C(‖h3 (t)‖22 + ‖h4 (t)‖

22,Γ

)≤ C

(‖f (ψ (t))‖22 + ‖g (ψ (t))‖

22,Γ + ‖θ (t)‖

22 + ‖ψ (t)‖

2V1 + ‖∂tψ (t)‖

2V0

),



which yields, on account of (3.3), (3.30) and (3.31), the following estimate:

‖ψ (t)‖V2 ≤ Q(‖(ψ0, θ0)‖

2Y0,K

)e−ρt + C, ∀t ≥ t#0 . (3.33)

By setting C = 2C + 1 and t0 = 1ρln (2Q (R0)) + t#0 , from (3.30) and (3.33) we obtain the desired claim. The proof is thus

complete.

Lemma 3.7. Let the assumptions of Theorem 3.3 be satisfied. Then, there is a positive nondecreasing monotone function Q and,for any R1 > 0, there exists t1 = t1(R1) > 0 such that∥∥SsK (t) (ψ0, θ0)∥∥V3×H3(Ω)

≤ Q (R1), ∀t ≥ t1, (3.34)

for any (ψ0, θ0) ∈ B(R1) ⊂ Y1,K . HereB(R1) is a ball of radius R1, centered at 0.

Proof. Observe first that, on account of Lemmas 3.4 and 3.6, it is easy to realize that SsK (t) has a bounded absorbing set inY1,K . Thus, without loss of generality, we assume that B(R1) is such a bounded absorbing set. As above, we need to obtainhigher-order estimates. Even in this case, we can proceed formally, on account of the approximation scheme obtained inthe proofs of Theorems 2.3 and 2.4. Let us differentiate (2.5) and (2.6) with respect to time. Then, we take u = ∂ttψ(t) andv = ∂tθ(t), respectively. Adding up the resulting relations, we obtain

12ddt

[‖∇∂tψ (t)‖22 + α ‖∇Γ ∂tψ (t)‖

22,Γ + β ‖∂tψ (t)‖

22,Γ + ‖∂tθ (t)‖

22

]+ ‖∂ttψ (t)‖22 + ‖∂ttψ (t)‖

22,Γ + ‖∇∂tθ (t)‖

22 + d

∥∥∂tθ|Γ (t)∥∥22,Γ= −

∫Ω

f ′ (ψ (t)) ∂tψ (t) ∂ttψ (t) dx−∫Γ

g ′ (ψ (t)) ∂tψ (t) ∂ttψ (t) dS

+⟨λ′′ (ψ (t)) ∂tψ (t) , ∂ttψ (t) θ (t)

⟩2 −

⟨λ′′ (ψ (t)) , ∂tψ2 (t) ∂tθ (t)

⟩2 . (3.35)

Using the Hölder and Young inequalities to estimate the last two terms on the right-hand side of equality (3.35), we get:∣∣⟨λ′′ (ψ) ∂tψ, ∂ttψθ ⟩2∣∣+ ∣∣⟨λ′′ (ψ) , ∂tψ2θt ⟩2∣∣ ≤ Cλ (‖∂ttψ‖2 ‖∂tψ‖4 ‖θ‖4 + ‖∂tψ‖24 ‖∂tθ‖2)≤12‖∂ttψ‖

22 + C

(‖∂tψ‖

2H1(Ω) ‖θ‖

2Z1K+ ‖∂tψ‖

2H1(Ω) ‖∂tθ‖2

).

Inserting this estimate on the right-hand side of (3.35), we deduce, for t ≥ 0,

ddt

Y (t)+ ‖∂ttψ (t)‖22 + 2 ‖∂ttψ (t)‖22,Γ + 2

(‖∇∂tθ (t)‖22 + d

∥∥∂tθ|Γ (t)∥∥22,Γ )≤ H1 (t)Y (t)+H2 (t) , (3.36)

where we have set

Y (t) := ‖∂tψ (t)‖2V1 + ‖∂tθ (t)‖22 ,

H1 (t) := C(‖θ (t)‖2Z1K

+ ‖∂tθ(t)‖2),

H2 (t) := C(∥∥f ′ (ψ (t)) ∂tψ (t)∥∥22 + ∥∥g ′ (ψ (t)) ∂tψ (t)∥∥22,Γ ) .

Owing to (3.30) and (3.31), we infer that

supt≥0

∫ t+1

t(H1 +H2) (s) ds ≤ Q (R1) . (3.37)

Then, using (3.37), we can apply to (3.36) the uniform Gronwall lemma once more and find t∗ > 0, depending on R1, suchthat

‖∂tψ (t)‖2V1 + ‖∂tθ (t)‖22 +

∫ t+1

t‖∂ttψ (s)‖2V0 ds ≤ Q (R1) , ∀t ≥ t∗. (3.38)

Let us consider again Eq. (2.6). Differentiating it with respect to t and taking v = ∂ttθ(t), we get

‖∂ttθ (t)‖22 +12ddt

(‖∇∂tθ (t)‖22 + d

∥∥∂tθ|Γ (t)∥∥22,Γ )= −

⟨(λ′′ (ψ) (∂tψ)

2) (t) , ∂ttθ (t)⟩2 − ⟨(λ′ (ψ) ∂ttψ) (t) , ∂ttθ (t)⟩2 ,



which yields, via standard Sobolev inequalities,

12‖∂ttθ (t)‖22 +

12ddt

(‖∇∂tθ (t)‖22 + d

∥∥∂tθ|Γ (t)∥∥22,Γ ) ≤ C ‖∂tψ (t)‖4H1(Ω) + Q (‖ψ (t)‖∞) ‖∂ttψ (t)‖22 . (3.39)

From (3.31), (3.36) and (3.38) we infer that

supt≥t1

∫ t+1

t

(‖∇θt (s)‖22 + d

∥∥∂tθ|Γ (s)∥∥22,Γ ) ds ≤ Q (R1),supt≥t1

∫ t+1

t

(‖∂tψ (s)‖4H1(Ω) + Q

(‖ψ (s)‖∞

)‖∂ttψ (s)‖22

)ds ≤ Q (R1).

We can now apply the uniform Gronwall lemma once again and infer from (3.39) the existence of t1 ≥ t∗ such that

‖∇∂tθ (t)‖22 + d∥∥∂tθ|Γ (t)∥∥22,Γ ≤ Q (R1) , ∀t ≥ t1. (3.40)

Therefore, (3.38) and (3.40) allow us to deduce, thanks to elliptic regularity results, the following bound:

‖θ (t)‖H3(Ω) ≤ Q (R1) , ∀t ≥ t1, (3.41)

for each K ∈ D,N, R.Furthermore, with the help of estimates (3.31), (3.33) and (3.38), we also obtain from Lemma 2.2

‖ψ (t)‖2V3 ≤ C(‖f (ψ (t))‖2H1(Ω) + ‖g (ψ(t))‖

2H1(Γ )

+Q(‖ψ (t)‖∞

)‖θ (t)‖2H1(Ω) + ‖ψ (t)‖

2V2 + ‖∂tψ (t)‖

2V1

), (3.42)

which yields, on account of the above estimates,

‖ψ (t)‖V3 ≤ Q (R1) , ∀t ≥ t1. (3.43)

Summing up, we conclude by observing that (3.34) follows from (3.41) and (3.43).

Proof of Theorem 3.1. On account of Corollary 2.8, Lemmas 3.4 and 3.6, the proof follows fromwell-known general results(see, e.g., [41]).

Proof of Theorem 3.3. Thanks to Lemma 3.7, the dynamical systems (Y1,K , SsK (t)) has an absorbing set in Y1,K which isbounded in V3 × H3 (Ω). Moreover, recalling Corollary 2.9, SsK (t) is also a closed semigroup. Thus, the proof follows from[39, Theorem 2 and Corollary 6].

4. Existence of exponential attractors

In this section, we establish the following results.

Theorem 4.1. Let f , g ∈ C2(R) and λ ∈ C3(R) satisfy the assumptions of Theorem 3.1. Then, SwK (t) possesses an exponentialattractor Mw

K , bounded in V2 × H2 (Ω), namely,

(I) MwK is compact and positively invariant with respect to S

wK (t), i.e.,

SwK (t)(MwK

)⊂Mw

K , ∀ t ≥ 0.

(II) The fractal dimension of MwK with respect to the Y0,K -metric is finite.

(III) There exist a positive nondecreasing monotone function Qw and a constant ρw > 0 such that

distY0,K(SwK (t) B,M

wK

)≤ Qw(‖B‖Y0,K )e

−ρw t , ∀t ≥ 0, (4.1)

where B is any bounded set of initial data in Y0,K . Here distY0,K denotes the non-symmetric Hausdorff distance in Y0,K and‖B‖Y0,K stands for the size of B in Y0,K .

Theorem 4.2. Let f , g ∈ C2(R) and λ ∈ C3(R) satisfy the assumptions of Theorem 3.3. Then, SsK (t) possesses an exponentialattractor Ms

K , bounded in V3 × H3 (Ω), namely,

(I) MsK is compact and positively invariant with respect to S

sK (t), i.e.,

SsK (t)(MsK

)⊂Ms

K , ∀t ≥ 0.

(II) The fractal dimension of MsK with respect to the Y1,K -metric is finite.



(III) There exist a positive nondecreasing monotone function Qs and a constant ρs > 0 such that

distY1,K(SsK (t) B,M

sK

)≤ Qs(‖B‖Y1,K )e

−ρst , ∀t ≥ 0, (4.2)

where B is any bounded set of initial data in Y1,K . Here distY1,K denotes the non-symmetric Hausdorff distance in Y1,K and‖B‖Y1,K stands for the size of B in Y1,K .

The proofs are based on a fundamental result on discrete semigroups (see [42]) which is reported here below for thereader’s convenience.

Theorem 4.3. Let V andW be two Banach spaces such that W is compactly embedded in V . Let X be a bounded subset of Wand consider a nonlinear mapΣ : X → X satisfying the smoothing property

‖Σ (x1)−Σ (x2)‖W ≤ C ‖x1 − x2‖V , (4.3)

for all x1, x2 ∈ X, where C > 0 depends on X. Then, the discrete dynamical system (X,Σn) possesses a discrete exponentialattractor M∗ ⊂ V , that is, a compact set with finite fractal dimension such that

Σ(M∗

)⊂M∗, (4.4)

distV(Σn (X) ,M∗

)≤ CXe−ρ∗n, n ∈ N, (4.5)

where CX and ρ∗ are positive constants which are independent of n, with the former depending on X.

Remark 4.4. Theorems 4.1 and 4.2 entail thatAwK andAsK have finite fractal dimension. Note that, if K = N, then Qw,Qs, ρs

and ρw depend onM .

The validity of the smoothing property, as well as the extension of the discrete case to the continuous one, areconsequences of the following lemmas.

Lemma 4.5. Let the assumptions of Theorem 4.1 be satisfied. Suppose that B(R0) is an absorbing ball of Y0,K such thatSwK (t) (B(R0)) ⊂ B(R0), for all t ≥ t0, where t0 > 0 is such that (3.25) holds. Let us set

(ψi(t), θi(t)) = SwK (t) (ψ0i, θ0i) ,

where (ψ0i, θ0i) ∈ B(R0), i = 1, 2. Then, the following estimate holds:

‖(ψ1 − ψ2) (t)‖2V2 + ‖(θ1 − θ2) (t)‖2Z1K≤ Q (R0) eLw t

(‖ψ01 − ψ02‖

2V1 + ‖θ01 − θ02‖

22

), ∀t ≥ t0 + 1. (4.6)

Here Q is a monotone increasing function depending only on R0,Ω,Γ and on the structural parameters of the problem.

Proof. Let us consider (2.72)–(2.73) and recall that there we have set ψ = ψ1 − ψ2 and θ = θ1 − θ2. We proceed asin [27] by differentiating (2.72) with respect to t , then taking u = ∂tψ (t) and adding the resulting relationship to (2.73)with v = ∂tθ (t). This procedure yields

ddt

[‖∂tψ (t)‖2V0 + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ ]+ 2 ‖∂tψ (t)‖2V1 + 2 ‖∂tθ (t)‖22= −2 〈∂t (Υ1 (t)) , ∂tψ (t)〉2 − 2 〈∂t (Υ2 (t)) , ∂tψ (t)〉2,Γ+ 2 〈∂t (Υ3 (t)) , ∂tψ (t)〉2 − 2 〈Υ4 (t) , ∂tθ (t)〉2 . (4.7)

We recall that Υ1,Υ2 and Υ3 are defined by (2.74)–(2.75). In addition, we have

∂tΥ1 = f ′ (ψ1) ∂tψ1 − f ′ (ψ2) ∂tψ2 = f ′ (ψ1) ∂tψ +[f ′ (ψ1)− f ′ (ψ2)

]∂tψ2,

where a similar formula holds for ∂tΥ2. Moreover, it is easy to check that

∂tΥ3 = λ′′ (ψ1) ∂tψθ1 + λ

′′ (ψ1) ∂tψ2θ +[λ′′ (ψ1)− λ

′′ (ψ2)]∂tψ2θ2

+ λ′ (ψ1) ∂tθ +[λ′ (ψ1)− λ

′ (ψ2)]∂tθ2. (4.8)

Consequently, from (3.25) and the (local) Lipschitz continuity of f ′ and g ′, it follows that

‖∂tΥ1 (t)‖2 + ‖∂tΥ2 (t)‖2,Γ ≤ Q (R0)(‖∂tψ (t)‖V0 + ‖∂tψ2 (t)‖V1 ‖ψ (t)‖V1

), (4.9)

for all t ≥ t0. Here we have also used the embedding H1(Ω) → L4(Ω). Moreover, owing to (3.25), the (local) Lipschitzcontinuity of λ′′ also implies that

2∣∣〈∂tΥ3 (t) , ∂tψ (t)〉2 − 〈Υ4 (t) , ∂tθ (t)〉2∣∣ ≤ Q (R0) ‖∂tψ2 (t)‖2H1(Ω) (‖ψ (t)‖2H1(Ω) + ‖θ (t)‖22)+Q (R0) ‖∂tψ (t)‖22 + C ‖∂tθ2(t)‖

22 ‖ψ (t)‖

2H1(Ω) + ‖∂tθ (t)‖

22 + ‖∂tψ (t)‖

2H1(Ω) , (4.10)



for all t ≥ t0. Consequently, owing to (4.8)–(4.10), from (4.7) we infer

ddt

[‖∂tψ (t)‖2V0 + ‖∇θ (t)‖

22 + d

∥∥θ|Γ (t)∥∥22,Γ ]+ ‖∂tψ (t)‖2V1 + ‖∂tθ (t)‖22≤ Q (R0)

(1+ ‖∂tψ2 (t)‖2H1(Ω) + ‖∂tθ2(t)‖

22

) (‖ψ (t)‖2H1(Ω) + ‖θ (t)‖

22 + ‖∂tψ (t)‖

22

), (4.11)

for all t ≥ t0. Then, on account of (3.30), the uniform Gronwall lemma entails that, for all t ≥ t0 + 1,

‖∂tψ (t)‖2V0 + ‖∇θ (t)‖22 + d

∥∥θ|Γ (t)∥∥22,Γ≤ Q (R0) sup

s∈[t0,t]

∫ s+1

s

(‖ψ (τ)‖2H1(Ω) + ‖θ (τ )‖

2Z1K+ ‖∂tψ (τ)‖

22

)dτ . (4.12)

Hence, thanks to inequality (2.71), we deduce

‖∂tψ (t)‖2V0 + ‖θ (t)‖2Z1K≤ Q (R0) eLw t

(‖ψ01 − ψ02‖

2V1 + ‖θ01 − θ02‖

22

), ∀t ≥ t0 + 1. (4.13)

Using now (4.13), we can argue exactly as in the proof of Lemma 3.6 (see (3.33)) to get the required estimate on theV2-normof ψ (t). The desired inequality (4.6) then follows. We leave the rigorous details to the reader.

The following lemma is a straightforward consequence of Lemma 4.5.

Lemma 4.6. Let the assumptions of Theorem 4.2 be satisfied. Suppose that B(R1) is an absorbing ball of Y1,K such thatSsK (t) (B(R1)) ⊂ B(R1), for all t ≥ t1 and some t1 > 0. Let us set

(ψi(t), θi(t)) = SsK (t) (ψ0i, θ0i) ,

where (ψ0i, θ0i) ∈ B(R1), i = 1, 2. Then, the function (ψ1 − ψ2, θ1 − θ2) satisfies the same estimate as in (4.6) for all t ≥ t1+1,with Lw replaced by Ls and Q depending on R1,Ω,Γ and on the structural parameters of the problem.

The following lemmas are concerned with the time regularity of SwK (t) and SsK (t), respectively.

Lemma 4.7. Let the assumptions of Lemma 3.6 hold. Then, there is a positive monotone increasing function Q such that∥∥SwK (t)(ψ0, θ0)− SwK (t)(ψ0, θ0)∥∥Y0,K≤ Q (R0)|t − t|1/2, (4.14)

for all t, t ∈ [t0,+∞) and any (ψ0, θ0) ∈ B(R0).

Proof. We claim that∫+∞

t0

∥∥∂sSwK (s)(ψ0, θ0)∥∥2V1×L2(Ω) ds ≤ Q (R0) . (4.15)

To this end, let us first recall that, due to (2.11), we have

‖ψ (t)‖2V1 + ‖θ (t)‖22 +

∫ t

0

(‖∇θ (s)‖22 + d

∥∥θ|Γ (s)∥∥22,Γ ) ds ≤ Q (R0) , (4.16)

for all t ≥ 0, where Q is independent of time. Furthermore, arguing as in (2.46)–(2.50), on account of (4.16), it is not difficultto deduce that∫ t

0

(‖∂tψ (s)‖2V1 + ‖∂tθ (s)‖

22

)ds ≤ Q (R0) , (4.17)

for all t ≥ 0. Finally, (4.15) is a straightforward consequence of (4.17).Observing that

SwK (t)(ψ0, θ0)− SwK (t)(ψ0, θ0) =

∫ t

t∂sSwK (s)(ψ0, θ0)ds, (4.18)

thanks to (4.15), we deduce (4.14).

Lemma 4.8. Let the assumptions of Lemma 3.7 hold. Then, there is a positive monotone increasing function Q such that∥∥SsK (t)(ψ0, θ0)− SsK (t)(ψ0, θ0)∥∥Y1,K≤ Q (R1)

(|t − t|1/6 + |t − t|1/4

), (4.19)

for all t, t ∈ [t1,+∞) and any (ψ0, θ0) ∈ B(R1).



Proof. Westill have (4.15).Moreover, (3.34) holds. Thus, using standard interpolation inequalities and theHölder inequality,from Eq. (4.18) we infer∥∥SsK (t)(ψ0, θ0)− SsK (t)(ψ0, θ0)∥∥2Y1,K = ∥∥ψ (t)− ψ (t)∥∥2V2 + ∥∥θ (t)− θ (t)∥∥2Z1K

≤ Q (R0)(∥∥ψ (t)− ψ (t)∥∥2/3V0

+∥∥θ (t)− θ (t)∥∥2)

≤ Q (R0)(|t − t|1/3 + |t − t|1/2

),

for all t > t1, that is, (4.19).

Proof of Theorem 4.1. Using Lemmas 3.4, 3.6 and 4.5, we can find a bounded subset X of Y0,K ∩ (V2×Z1K ) and t∗ > 0 such

that, settingΣK = SwK (t∗), themappingΣK : X → X enjoys the smoothing property (4.6). Therefore, Theorem 4.3 applies to

ΣK and there exists a compact setMw,∗K ⊂ X with finite fractal dimension (with respect to the Y0,K -metric) which satisfies

(4.4) and (4.5). Hence, setting

MwK =

⋃t∈[t∗,2t∗]

SwK (t)Mw,∗K ,

we deduce that (I) and (III) of Theorem 4.1 are fulfilled, while (II) is a consequence of (2.71) and (4.14).

Proof of Theorem 4.2. Set now V := V1 × L2 (Ω) andW := V2 × Z1K in Theorem 4.3. Using Lemmas 2.6, 3.7 and 4.6, wecan find a bounded subset X ofY1,K ∩ (V3×H3(Ω)) and t+ > 0 such that, definingΣK = SsK (t

+), the mappingΣK : X → Xenjoys the smoothing property (4.6). Therefore, Theorem 4.3 applies to ΣK and there exists a compact setM

s,∗K ∈ X with

finite fractal dimension (with respect to Y0,K -metric) which satisfies (4.4) and (4.5). Hence, setting, as above,

MsK =

⋃t∈[t+,2t+]

SsK (t)Ms,∗K ,

we deduce that (I), (II) and (III) of Theorem 4.2 are fulfilled, but with the metric of Y1,K replaced by that of Y0,K . Moreprecisely,Ms

K has finite fractal dimension (with respect to the Y0,K -metric), whereas only the following inequality is valid

distY0,K(SsK (t) B,M

sK

)≤ Qs(‖B‖Y1,K )e

−ρst , ∀t ≥ 0. (4.20)

In order to obtain the finite dimensionality and the required exponential convergence (4.2) ofMsK with respect to the Y1,K -

metric, there remains to recall that SsK (t) possesses the smoothing property (3.34) and to use the standard interpolationinequalities ‖·‖V2 ≤ C ‖·‖1/3V0 ‖·‖

2/3V3 , ‖·‖Z1K

≤ C ‖·‖1/22 ‖·‖1/2H2 , where the constant C is independent of t . The proof of

Theorem 4.2 is now complete.

5. Convergence to a single equilibrium

Here, for simplicity, we work with the strong solutions given by Theorem 2.4. However, thanks to Lemma 3.6, the resultsbelow can also be extended to weak solutions.Let us start by recalling the following straightforward proposition.

Proposition 5.1. Let the assumptions of Theorem 2.4 hold. For each K ∈ D,N, R, the semigroup SsK (t) has a (strict) Lyapunovfunctional defined by the free energy, namely,

LK (ψ0, θ0) :=12

[‖∇ψ0‖

22 + α ‖∇Γψ0‖

22,Γ + β ‖ψ0‖

22,Γ + ‖θ0‖

22

]+

∫Ω

F (ψ0) dx+∫Γ

G (ψ0) dS, (5.1)

where F and G are defined by Eq. (2.26). In particular, we have, for all t > 0,

ddt

LK (SsK (t)(ψ0, θ0)) = −‖∂tψ (t)‖22 − ‖∂tψ (t)‖

22,Γ − ‖∇θ (t)‖

22 − d

∥∥θ|Γ (t)∥∥22,Γ . (5.2)

Observe now that (ψ∞, θ∞) ∈ Y1,K is an equilibrium for PsKPsK if and only if it is a solution to the boundary value problem

−∆ψ∞ + f (ψ∞)− λ′(ψ∞)θ∞ = 0, inΩ,−α∆Γψ∞ + ∂nψ∞ + βψ∞ + g (ψ∞) = 0, on Γ ,−∆θ∞ = 0, inΩ,b∂nθ∞ + cθ∞ = 0, on Γ .

If K ∈ D, R, then θ∞ ≡ 0. Instead, when K = N we have

θ∞ = IN (ψ0, θ0)− 〈λ(ψ∞)〉Ω . (5.3)



Thus, the stationary problem reduces to the following:−∆ψ∞ + f (ψ∞)− λ′(ψ∞)θ∞ = 0, inΩ,−α∆Γψ∞ + ∂nψ∞ + βψ∞ + g (ψ∞) = 0, on Γ , (5.4)

where θ∞ = 0, if K = D, R, or θ∞ given by (5.3), if K = N . Note that, in the latter case, we are dealing with a nonlocalboundary value problem. However, it is not difficult to check that (5.4) possesses at least one solution (i.e., there exists atleast one stationary solution) by means of standard arguments (see, e.g., [43]).

Remark 5.2. Recalling Theorem 3.3, we note that AsK coincides with the unstable manifold of the set of equilibria (cf.,e.g., [41, Chap. 7, Sec. 4]).

We report here below some well-known properties of gradient systems with relatively compact orbits (see, e.g., [44]).

Lemma 5.3. For any (ψ0, θ0) ∈ Y1,K , the set ω(ψ0, θ0) is a nonempty compact connected subset of Y1,K . Furthermore, we have:

(i) ω(ψ0, θ0) is fully invariant for SsK (t);(ii) LK is constant on ω(ψ0, θ0);(iii) distY1,K

(SsK (t) (ψ0, θ0) , ω(ψ0, θ0)

)→ 0 as t →+∞;

(iv) ω(ψ0, θ0) consists of equilibria only.

Let us concentrate on the case K = N , since the other two cases are technically easier. It seems that we cannot proceed asin [27], arguing as in [23], but we need instead an adapted version of the Łojasiewicz–Simon inequality which can be provenas in [43, Lemma 4.1]. First, we introduce the functional

E(ψ) :=12

[‖∇ψ‖22 + α ‖∇Γψ‖

22,Γ + β ‖ψ‖

22,Γ +

∥∥m− 〈λ(ψ)〉Ω∥∥22]+

∫Ω

F (ψ) dx+∫Γ

G (ψ) dS, ∀ψ ∈ V2,

wherem ∈ R is fixed. It is not difficult to prove that the critical points of E are solutions to (cf. Eq. (5.4))−∆ψ + f (ψ)− λ′(ψ)

(m− 〈λ(ψ)〉Ω

)= 0, inΩ,

−α∆Γψ + ∂nψ + βψ + g (ψ) = 0, on Γ . (5.5)

Lemma 5.4. Let (H1), (H3)–(H5) hold and assume that F , G, λ are real analytic. Consider a solution ψ∞ ∈ V2 to (5.5). Then,there exist constants ξ ∈ (0, 1/2) and CL > 0, ζ > 0 depending on ψ∞ such that, for any ψ ∈ V2, if

‖ψ − ψ∞‖V2 ≤ ζ ,

then we have∥∥(−∆ψ + f (ψ)− λ′(ψ) (m− 〈λ(ψ)〉Ω) ,−α∆Γψ + ∂nψ + βψ + g (ψ))∥∥

V∗1

≥ CL|E (ψ)− E (ψ∞) |1−ξ . (5.6)

Remark 5.5. Note that ψ is bounded, so that the nonlinearities f (ψ), g(ψ) and λ (ψ), as well as their higher-orderderivatives, are bounded too.

Thanks to Lemma 5.4, we obtain the following extended version for the functionalLN whose proof goes as the one of [43,Lemma 4.6]. The constants are still denoted as in Lemma 5.4 for the sake of convenience.

Lemma 5.6. Let the assumptions of Lemma 5.4 hold. Consider a solution (ψ∞, θ∞) ∈ V2×Z1N to (5.4), with m = IN (ψ0, θ0) (cf.(5.3)). Then, there exist constants ξ ∈ (0, 1/2) and CL > 0, ζ > 0 depending on (ψ∞, θ∞) such that, for any (ψ, θ) ∈ V2 × Z1Nsatisfying IN (ψ, θ) = m, if

‖(ψ − ψ∞, θ − θ∞)‖V2×Z1N≤ ζ ,

then we have∥∥(−∆ψ + f (ψ)− λ′(ψ) (m− 〈λ(ψ)〉Ω) ,−α∆Γψ + ∂nψ + βψ + g (ψ))∥∥

V∗1+ ‖∇θ‖2

≥ CL|LN (ψ, θ)−LN (ψ∞, θ∞) |1−ξ . (5.7)

We can now state the main result of this section.



Theorem 5.7. Let K = N and let the assumptions of Theorem 3.3 hold. Suppose, in addition, that the nonlinearities F , G, λare real analytic. For any given initial datum (ψ0, θ0) ∈ Y1,N , the solution (ψ (t) , θ (t)) = SsN(t)(ψ0, θ0) converges to a singleequilibrium (ψ∞, θ∞) in the topology of V2 × Z1N , that is,

limt→+∞

(‖ψ(t)− ψ∞‖V2 + ‖θ(t)− θ∞‖Z1N

)= 0. (5.8)

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

‖ψ(t)− ψ∞‖V2 + ‖θ(t)− θ∞‖Z1N + ‖∂tψ (t) ‖V2 ≤ C(1+ t)−ξ/(1−2ξ), (5.9)

for all t ≥ 0.

Remark 5.8. If K ∈ D, R, we have θ∞ ≡ 0 and a similar result can be proven without requiring the real analyticity of λand replacing Eq. (5.6) with a simpler Łojasiewicz–Simon inequality for the functional

EK (ψ) :=12

[‖∇ψ‖22 + α ‖∇Γψ‖

22,Γ + β ‖ψ‖

22,Γ

]+

∫Ω

F (ψ) dx+∫Γ

G (ψ) dS.

Proof. In this proof, C will stand for a generic positive constant which is independent of time and depends at most on theinitial data, on the equilibrium (ψ∞, θ∞) and on the parameters of the problem. This constant may vary even within thesame line. Observe first that, if there exists t] ≥ 0 such thatLN

(ψ(t]), θ(t]))= L∞, then there is nothing to prove. Hence,

suppose that, for all t ≥ t0 ≥ 0, we have LN (ψ (t) , θ (t)) > L∞.We first observe that, by Lemma 5.6, the functional LNsatisfies the Łojasiewicz–Simon inequality (5.7) near every (ψ∞, θ∞) ∈ ω(ψ0, θ0). Since ω(ψ0, θ0) is compact in Y1,N , wecan cover it by the union of finitely many balls Bj with centers

(ψj∞, θ

j∞

)and radii rj, where each radius is such that (5.7)

holds in Bj. Since LN = L∞ on ω(ψ0, θ0), it follows from Lemma 5.6 that there exist uniform constants ξ ∈ (0, 1/2),CL > 0 (depending on (ψ∞, θ∞)) and a neighborhood U of ω(ψ0, θ0) such that (5.7) holds in U. Thus, recalling property(iii) of Lemma 5.3, we can find a time t1 > 0 such that (ψ (t) , θ (t)) belongs toU, for all t ≥ t1. Set now t2 := max t0, t1.On account of (5.2) and (5.7), we obtain, for every t ≥ t2,

−ddt(LN (ψ (t) , θ (t))−L∞)

ξ= ξ

(−ddt

LN (ψ (t) , θ (t)))(LN (ψ (t) , θ (t))−L∞)

ξ−1

≥ ξCL‖∂tψ‖

22 + ‖∂tψ‖

22,Γ + ‖∇θ‖

22∥∥(−∆ψ + f (ψ)− λ′(ψ) (m− 〈λ(ψ)〉Ω) ,−α∆Γψ + ∂nψ + βψ + g (ψ)

)∥∥V∗1+ ‖∇θ‖2

. (5.10)

Observe now that, recalling Eq. (2.5) and using the Poincaré–Wirtinger inequality, we have∥∥(−∆ψ + f (ψ)− λ′(ψ) (m− 〈λ(ψ)〉Ω) ,−α∆Γψ + ∂nψ + βψ + g (ψ))∥∥

V∗1

=∥∥(−∆ψ + f (ψ)− λ′(ψ)θ + λ′(ψ) (θ −m+ 〈λ(ψ)〉Ω) ,−α∆Γψ + ∂nψ + βψ + g (ψ)

)∥∥V∗1

≤ ‖∂tψ‖2 + ‖∂tψ‖2,Γ + C‖θ − 〈θ〉Ω ‖2≤ ‖∂tψ‖2 + ‖∂tψ‖2,Γ + C‖∇θ‖2. (5.11)

Inserting estimate (5.11) into estimate (5.10), we deduce that

−ddt(LN (ψ (t) , θ (t))−L∞)

ξ≥ C

(‖∂tψ (t)‖2 + ‖∂tψ (t)‖2,Γ + ‖∇θ (t)‖2

). (5.12)

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

∂tψ ∈ L1 ([t2,+∞) ;V0) , ∇θ ∈ L1([t2,+∞) ; (L2 (Ω))3

).

Consequently, we also have

∂tθ ∈ L1([t2,+∞) ;

(Z1N)∗)

.

We now recall that, due to Lemma 3.7, there exist an increasing unbounded sequence tk and an element (ψ∞, θ∞) ∈ω (ψ0, θ0) such that (ψ(tk), θ(tk)) → (ψ∞, θ∞) in V2 × Z1N as k goes to +∞. This fact, combined with the above L

1-integrability, implies that (ψ (t) , θ (t)) → (ψ∞, θ∞) in V0 ×

(Z1N)∗, as t goes to +∞ and in V2 × Z1N as well, appealing

once more to Lemma 3.7. Hence, ω (ψ0, θ0) = (ψ∞, θ∞) and (5.8) holds.There remains to prove (5.9). For t ≥ t2, it follows from (5.6), (5.11) and (5.12), that

ddt(LN (ψ (t) , θ (t))−L∞)

ξ+ C(LN (ψ (t) , θ (t))−L∞)

1−ξ≤ 0,



which yields

LN (ψ (t) , θ (t))−L∞ ≤ C(1+ t)−1/(1−2ξ), ∀t ≥ t2. (5.13)

Thus, integrating (5.12) over [t,+∞) and using (2.6), thanks to estimate (5.13), we get∫+∞

t

(‖∂tψ (s)‖2 + ‖∂tψ (s)‖2,Γ + ‖∇θ (s)‖2 + ‖∂tθ(s)‖(Z1N

)∗)ds ≤ C(1+ t)−ξ/(1−2ξ), ∀t ≥ t2,

and this estimate implies

‖ψ (t)− ψ∞‖2 + ‖ψ (t)− ψ∞‖2,Γ + ‖θ (t)− θ∞‖(Z1N)∗ ≤ C(1+ t)−ξ/(1−2ξ), ∀t ≥ t2.

Taking advantage of the above (lower-order) convergence estimates, we can prove the higher-order ones, arguing as in [27,Proof of Thm. 5.8] (see also [43, Sec. 5]).

Acknowledgement

The authors are grateful to the anonymous referee for the careful reading of the manuscript and for the appropriateremarks.

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Phase-field systems with nonlinear coupling and dynamic boundary conditions

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