Advances in Differential Equations Volume 12, Number 11 (2007), 1241–1274

GLOBAL WELL-POSEDNESS FORTHE NON-ISOTHERMAL CAHN–HILLIARD

EQUATION WITH DYNAMIC BOUNDARY CONDITIONS

Ciprian G. Gal

Department of Mathematics, University of Missouri, ColumbiaColumbia, MO 65201

(Submitted by: J.A. Goldstein)

Abstract. We consider a model of non-isothermal phase transitiontaking place in a confined container. The order parameter φ is governedby a Cahn–Hilliard-type equation which is coupled with a heat equationfor the temperature θ. The former is subject to a nonlinear dynamicboundary condition recently proposed by some physicists to account forinteractions with the walls. The latter is endowed with a boundarycondition which can be a standard one (Dirichlet, Neumann or Robin)or even dynamic. We thus formulate a class of initial- and boundary-value problems whose local existence and uniqueness is proven by meansof the contraction mapping principle. The local solution becomes globalowing to suitable a priori estimates.

1. Introduction

The Cahn–Hilliard equation describes spinodal decomposition of binarymixtures that appears, for example, in cooling processes of alloys, glasses orpolymer mixtures, in the absence of temperature variations and mechanicalstresses (see [8], [33, 34], [29] and the references cited therein). Consider ahomogeneous substance, initially homogeneous, with controllable thermody-namic properties. Those properties are adjusted so that the substance entersan extremely unstable state, such that the unfolding of its instabilities en-tails phase separation into a fine-scale heterogenous mixture of phases. Thetypical scenario involves lowering the temperature of the binary alloy to alevel where the alloy can no longer exist in equilibrium in its homogeneousstate. The most basic properties of this important process is described by

Accepted for publication: August 2007.AMS Subject Classifications: 35K55, 74N20, 35B40, 35B45, 37L30.

1241

1242 Ciprian G. Gal

the equation:

ψt −∆(−∆ψ + F

′1 (ψ)

)= 0 in Ω× (0,∞) , (1.1)

whereµ = −∆ψ + F

′1 (ψ) in Ω× (0,∞) , (1.2)

is referred to as the chemical potential which is equal to the Frechet derivativeof the functional EΩ (ψ) (see (1.6) below) with respect to the concentrationvariable ψ, Ω is a bounded domain in Rn, n = 1, 2, 3 with smooth boundaryΓ = ∂Ω, ψ is the relative concentration difference of the mixture componentsand F1 is a general potential function. For example, F1 can be a logarithmicpotential, approximated by the double well potential F1 (ψ) = 1

4

(ψ2 − 1

)2.In literature, the usual boundary conditions considered are the following:

∂nµ = 0 on Γ (1.3)

and∂nψ = 0 on Γ. (1.4)

Equation (1.1) is supplemented by the initial condition (1.5) ψ (0, x) = ψ0 (x)in Ω. In the above (1.3)–(1.4) , ∂n denotes the exterior normal derivative atΓ = ∂Ω and n = n (x) is the exterior normal at x ∈ Γ. The boundarycondition (1.3) means that there cannot be any exchange of the mixturesconstituents through the boundary Γ; (1.4) simply describes the fact thatthe boundary Γ acts as an impenetrable wall. It easy to see from (1.1)and (1.3) that the total mass 〈ψ〉Ω := (1/ |Ω|)

∫Ω ψ dx is conserved for all

time t. The boundary condition (1.4) is usually referred to as the variationalboundary condition, which together with (1.3) results in the fact that thefollowing energy functional is decreasing:

EΩ (ψ) :=∫

Ω

[12 |∇ψ|

2 + F1 (ψ)]dx. (1.6)

In recent years, physicists proposed that the following energy functionalshould be added to (1.6):

EΓ (ψ) :=∫

Γ

[α2|∇Γψ|2 +

βψ2

2

]dS (1.7)

to form a total energy functional

E (ψ) := EΩ (ψ) + EΓ (ψ) . (1.8)

This is considered when the effective interaction between the wall (i.e., Γ)and both mixture components is short-ranged (see, e.g., [19, 20] and their

Global well-posedness 1243

references). It turns out that instead of the boundary condition (1.4), thefollowing dynamic boundary condition

1dψt = α∆Γψ − ∂nψ − βψ on Γ (1.9)

is proposed when one has to account for the interaction of the binary compo-nents with the walls or when one of the chemical components is preferentiallyattracted to the walls (that is, Γ = ∂Ω). Here the constants α > 0, d > 0,and β > 0, and ∆Γ denotes the tangential Laplace operator on the surfaceΓ of Ω. We recall that, phenomenologically speaking, boundary condition(1.9) means that the density at the surface relaxes towards equilibrium witha rate proportional to the driving force given by the Frechet derivative ofthe free energy functional EΓ (ψ). The term ∂nψ is due to the contributioncoming from the variation of the free energy functional EΩ (ψ) .

There is a fairly large literature on such Cahn–Hilliard models. TheCauchy problem (1.1) , (1.3) , (1.4) originated many mathematical papersdevoted to issues like global existence, uniqueness and large-time behav-ior of solutions (see [17, 18], [45, 46], [42], [30]). The problem (1.1) withboundary conditions (1.3) , (1.9) has recently been studied in [11], [32], [37],[39], [43]. For instance, R. Racke and S. Zheng [39] show the existence anduniqueness of a global solution when the potential F1 has a polynomiallycontrolled growth of degree 6, and later J. Pruss, R. Racke and S. Zheng[37] study the problem of maximal Lp-regularity and asymptotic behaviorof the solution and prove the existence of a global attractor to the Cahn–Hilliard equation (1.1) with boundary conditions (1.3), (1.9). The limitingcase d = ∞ is also discussed in [39]. Besides, the problem has also beenanalyzed as a dissipative dynamical system in [32], when the nonlinearityF1 is quite general. There, the main result is the construction of a family ofexponential attractors for the viscous Cahn–Hilliard equation (obtained byadding δ0∆ψt in (1.1)) which is robust with respect to the viscosity coeffi-cient δ0. However, another well-known mathematical model which describesthe behavior of the phases, but in the presence of temperature variations, isgiven by the Cahn–Hilliard equation suitably coupled with the heat equation(see (1.10), (1.11) below). The resulting system governs the order param-eter (or phase-field ψ) and the temperature θ. Linearizing with respect toa suitable critical temperature at which the two phases coexist, one obtainsthe phase-field system (see [10] and [27])

δψt −∆(−∆ψ + F

′1(ψ)− λ0θ

)= 0, (1.10)

(θ + λ0ψ)t −∆θ = 0, (1.11)

1244 Ciprian G. Gal

in Ω × (0,+∞). Let us illustrate this by means of a sufficiently generalphenomenological argument. Let us consider first the free-energy functional

EΩ (ψ, θ) :=∫

Ω

[12|∇ψ|2 + F1 (ψ)− λ0 (ψ − ψc) θ −

12θ2]dx, (1.12)

where we assume that the temperature θ varies in time inside the medium,λ0 > 0 is the latent heat and ψc is a positive constant. Then one obtainsthe expression for the chemical potential µ and internal energy e as follows:

µ =∂EΩ

∂ψ= −∆ψ + F

′1 (ψ)− λ0θ,

e = −∂EΩ

∂θ= λ0 (ψ − ψc) + θ.

In order to obtain the governing equations, we assume that µ and e satisfyconservation laws. They read as follows:

ψt + divQ = 0, (1.13)

et + divq = 0, (1.14)where Q is the phase flux of the order parameter ψ and is assumed tobe according to Fick’s law to be Q = −∇µ and q is the heat flux of theinternal temperature which is assumed to satisfy the Fourier law, that is,q = −∇θ. We recall that equation (1.13) is a mass conservation equation,while equation (1.14) is an energy balance equation. Thus, from (1.13),(1.14), we obtain

ψt = ∆µ, µ = −∆ψ + F′1 (ψ)− λ0θ for t > 0, x ∈ Ω, (1.15)

θt −∆θ = −λ0ψt for t > 0, x ∈ Ω. (1.16)Equation (1.15) is of fourth order, so we need two boundary conditions andfor (1.16), we need to impose one boundary condition only. In order to intro-duce physically reasonable boundary conditions, we consider the followingboundary energy functional

EΓ (ψ, θ) :=∫

Γ

[α2|∇Γψ|2 +

βψ2

2+ F2 (ψ)− a

2θ2]dS,

where F2 is a nonlinear potential function. Then, equation (1.15) is subjectto the nonlinear dynamic boundary condition (compare with (1.9))

1dψt = α∆Γψ − ∂nψ − βψ − F

′2 (ψ) , on Γ× (0,+∞).

In the traditional approach, equation (1.16) is assumed to hold everywherein the region Ω and then the boundary conditions are appended later, like

Global well-posedness 1245

Dirichlet, Neumann and Robin boundary conditions. We will derive all theboundary conditions (including dynamic) for (1.16) as part of an energybalance law. Our methods are based on the derivation of dynamic boundaryconditions in the context of heat and wave equations discussed in [26]. Weobserve that (1.14) and (1.16) do not show how to model a heat or coolingsource that is located on the boundary Γ. Following [26], we define theexternal energy source at the boundary as

eΓ = −∂EΓ

∂θ= aθ. (1.17)

Now, suppose that there is a source (or sink) on the boundary representedby the function Φ (x, t, θ,∇θ) for t > 0 and x ∈ Γ. The first law of thermo-dynamics (1.14) takes the form∫

Ωetdx+

∫Γ

(eΓ)t dS = −∫

Ωdivq +

∫Γ

Φ (x, t, θ,∇θ) dS, (1.18)

where, in this case, the rate of change of heat flow in the region must containa term like

∫Γ (eΓ)t dS =

∫Γ aθ dS (see (1.17)) to represent the heat (or

cooling) generated on the boundary Γ due to the source Φ. Equating theterms that hold in Ω and the usual argument lead to the same equation(1.16) that holds in Ω and for t > 0. Then, we see that (1.18) reduces to∫

Γ[(eΓ)t − Φ (x, t, θ,∇θ)] dS =

∫Γ

[aθt − Φ (x, t, θ,∇θ)] dS = 0. (1.19)

A sufficient condition for (1.19) to hold is given by

aθt = Φ (x, t, θ,∇θ) on Γ and t > 0. (1.20)

Here we assume that a > 0. Next, we show how the all the standard boundaryconditions for equation (1.16) can be written for various choices of Φ.

First, we choose Φ ≡ 0. Then, we get θt ≡ 0 on Γ, therefore θ (x, t) =θ0 (x) , for x ∈ Γ, t ≥ 0, where θ0 is the initial condition correspondingto the equation (1.16) . Thus, we obtain a Dirichlet boundary condition forour non-constant temperature θ. Next, we move to derive an inhomogeneousNeumann boundary condition. For this, suppose that Φ = f (t) , where fis not identically zero. Then, if θ (x, t) and Γ are sufficiently regular, wehave aθt = f (t) on Γ; hence, (∇θ)t = ∇ (θt) = 0 on Γ. It follows that∇θ (x, t) = g (x) holds on Γ, so that

∂nθ (x, t) = g (x) on Γ and t > 0.

1246 Ciprian G. Gal

To obtain a Robin boundary condition, we consider the choice of

Φ (x, t, θ,∇θ) = aeCrf (t) ,

for C ∈ R and r is defined as the parameter describing the line L whichpasses through x and contains the outward unit vector −→n such that r > 0 atall points on L ∩Ω which are close to x. We have θt (x, t) = eCrf (t); hence,

∂θt∂n

(x, t) = (∂nθ)t (x, t) = CeCrf (t) on Γ, t > 0.

It follows that(∂nθ)t − Cθt = 0 on Γ,

which implies∂nθ − Cθ = g (x) on Γ.

In order to model a heat (or cooling) source acting on the boundary of theregion Ω, we assume that the source Φ depends linearly on the temperatureflux ∂nθ across the boundary as well as the temperature θ. That is, we let

Φ (x, t, θ,∇θ) = −b∂nθ − cθ;then our boundary condition for the governing equation (1.16) becomes

aθt + b∂nθ + cθ = 0 on Γ. (1.21)

Regarding (1.21), we shall suppose that a, b, and c are always nonnegativesince we want to analyze a dissipative phenomenon. The condition (1.21)is a reasonably general condition which contains the usual (homogeneous)ones along with the so-called Wentzell boundary condition when a > 0 (see,e.g., [21] and [22] and their references). More precisely, we will considerthe following cases: Wentzell boundary condition (a > 0, b > 0, c ≥ 0),Neumann-Robin boundary condition (a = 0, b > 0, c ≥ 0), and Dirichletboundary condition (a = b = 0, c > 0). We recall that equation (1.15) isendowed with the boundary conditions

∂nµ = 0 on Γ, (1.22)1dψt = α∆Γψ − ∂nψ − βψ − F

′2 (ψ) on Γ, (1.23)

where we assume that α, d > 0, β > 0. Notice that equations (1.15) , (1.22)entail the mass conservation.

Finally, we will also mention the following papers in [1], [13], [14], [7], [10],[27]. In all these articles, the differential model describing non-isothermalphase separation is given by the system (1.15), (1.16), when both θ andψ satisfy Dirichlet or Neumann boundary conditions. For the analysis on

Global well-posedness 1247

such systems or related problems (when memory effects may also be incor-porated in the model (1.15), (1.16)), we refer the reader to [27] where plentyof references are properly quoted. Well-posedness and maximal regularity,as well as asymptotic behavior for the system (1.15), (1.16) (when F1 has apolynomially controlled growth of degree 6), equipped with linear dynamicboundary conditions for ψ (by taking F2 ≡ 0 in (1.23)) and Robin and Neu-mann boundary conditions for the temperature function θ (that is, the casea = 0, b > 0, c ≥ 0 in (1.21)) was studied in [38]. There, the authors alsoprove convergence of solutions to steady states as t→ +∞. However, as faras we know, the system (1.15), (1.16), (1.21)–(1.23) has not yet been ana-lyzed when F1 and F2 are quite general potential functions and θ satisfiesa dynamic boundary condition. Our goal in this paper is to deal with theissue of well posedness for such problems. Besides, the problem has also beenanalyzed as a dissipative dynamical system in [23], proving the existence ofa global attractor A as well as an exponential attractor M. This fact, inparticular, entails that A has finite fractal dimension. In a different orderof ideas, we will also mention [24, 25], where we have considered a Cahn–Hilliard model (1.1) for a phase separation in a binary fluid mixture whichis contained within a medium with porous walls. This course of phase sep-aration is completely changed from the one described by the Cahn–Hilliardequations with classical (traditional) boundary conditions considered above(see (1.22), (1.23)). In that case [24, 25], the wall Γ in the medium is porousor (semi)permeable. The derivation of the mass conservation law (1.13) thatincludes a mass (density) source on the boundary will give rise to Wentzell(dynamical) boundary conditions for the chemical potential µ. This is verysimilar to the derivation of (1.21) from the energy conservation law (1.18) .We could also consider such a system here, but the kinetic equations aresignificantly different, so we will investigate this problem in a forthcomingarticle.

This paper is concerned with the global existence and uniqueness of solu-tions to the system of initial-value problems given by

ψt = ∆µ in Ω× (0,∞) , (1.24)

µ = −∆ψ + F′1 (ψ)− λ0θ, in Ω× (0,∞) , (1.25)

θt −∆θ = −λ0ψt, in Ω× (0,∞) , (1.26)

with the boundary conditions

∂nµ = 0, on Γ× (0,∞) (1.27)

1248 Ciprian G. Gal

1dψt = α∆Γψ − ∂nψ − βψ − F

′2 (ψ) , on Γ× (0,∞) , (1.28)

andaθt + b∂nθ + cθ = 0, on Γ× (0,∞) , (1.29)

ψ|t=0 = ψ0, θ|t=0 = θ0, (1.30)where we recall again the following cases: if a > 0, b > 0, and c ≥ 0, then(1.29) is a dynamical (Wentzell) boundary condition; if a = 0, b > 0, c ≥ 0then (1.29) includes Neumann and Robin boundary conditions; if a = b = 0and c > 0, then (1.29) is a Dirichlet boundary condition. We could alsoconsider external forces acting on the system of equations (1.24)–(1.30) , butfor the sake of simplicity, we confine our attention to the case above.

We will employ classical methods, that is, fixed-point theorems and stan-dard energy methods. In Section 2, we discuss the auxiliary linear problemsassociated with our equations, then in Section 3, we formulate an approx-imate problem (Pε) (ε 1) and give existence results in the case whenΩ ⊂ Rn, n = 2, 3. Finally, in Section 4, the original system (1.24)–(1.30) isobtained from the problem Pε in the limit as ε→ 0+. The one-dimensionalcase is discussed in Section 5.

Throughout this article, we assume that the functions F1, F2 : R→ R aregiven C2-functions satisfying

lim|s|→∞

inf F′′i (s) > 0, for all i = 1, 2. (1.31)

Note that the typical double-well potential F1(s) = (s2−1)2 satisfies (1.31) .

2. Preliminary results

To establish the well-posedness of problem (1.24)–(1.30), we need to intro-duce some preliminary results related to auxiliary problems. We shall showthat, using sufficiently strong a priori estimates, we will be able to prove thewell-posedness of our problem in a suitable Sobolev setup. From now on,we denote by ‖·‖p and ‖·‖p,Γ , the norms on Lp (Ω) and Lp (Γ) , respectively.In the case p = 2, 〈·, ·〉2 stands for the usual scalar product. The normson Hs (Ω) and Hs (Γ) are indicated by ‖·‖Hs and ‖·‖Hs(Γ), respectively, forany s > 0. The initial conditions θ0 and φ0 are identified with the pairs(θ0, θ0|Γ

)∈ Ω× Γ, if a > 0, and

(φ0, φ0|Γ

)∈ Ω× Γ, respectively.

We begin to consider the following linear problem:

θt −∆θ = h1, in Ω× (0,+∞), (2.1)

aθt + b∂nθ + cθ = 0, on Γ× (0,+∞), (2.2)

Global well-posedness 1249

θ|t=0 = θ0. (2.3)As we mentioned in the introduction, we distinguish between three possiblecases, namely Wentzell, Neumann-Robin, and Dirichlet boundary conditions.The first case is less standard; however, it has recently been considered bya number of authors (see, e.g., [21, 22, 24] and references therein; cf. also[22]). In this case, the appropriate space for problem (2.1)–(2.3) is

Xpa = Lp

(Ω, dx|Ω ⊕

adS

b |Γ

), 1 ≤ p <∞,

with norm

‖θ‖Xpa =(∫

Ω|θ (x)|p dx+

∫Γ|θ (x)|p adSx

b

) 1p. (2.4)

Here we identify Xpa with Lp (Ω, dx) ⊕ Lp

(Γ, adSb

)and, if θ ∈ C

(Ω), we

identify θ with the vector Θ =(θ|Ω, θ|Γ

)∈ C (Ω) × C (Γ) . Then we define

Xpa as the completion of C(Ω) with respect to the norm (2.4) (see [21] for

details). In the remaining cases, i.e., when a = 0, we set Xp0 := Lp (Ω, dx) .

Other authors have worked in a similar framework if a > 0. For instance, [3]deals with a general class of inhomogeneous parabolic initial- and boundary-value problems with dynamical boundary conditions. A simpler and moregeneral approach can be found in [21] (see also [22]), while the abstractapproach used in [35] allows consideration of dynamic boundary conditionsinvolving surface diffusion represented by the Laplace–Beltrami operator ∆Γ

(see also below).In order to account for all the cases, we introduce the family of linear oper-

ators AK := −∆ on the Banach space Xpa, where K ∈ D,N,R,N,W0,W1

and D, N, R, W0, W1 stand for the following boundary conditions: Dirichlet,Neumann, Robin, Wentzell with c = 0, Wentzell with c > 0, respectively.Correspondingly, we let

D (AD) =θ ∈W 2,p (Ω) : θ = 0 on Γ

,

D (AN ) =θ ∈W 2,p (Ω) : ∂nθ = 0 on Γ

,

D (AR) =θ ∈W 2,p (Ω) : b∂nθ + cθ = 0 on Γ

, if b > 0, c ≥ 0,

D(AWj

)⊂θ ∈W 2,p

loc (Ω) ∆θ ∈ Xpa; ∂nθ, θ are well-defined

members of L2 (Γ) in the trace sense, and

a (∆θ)|Γ + b∂nθ + cθ = 0 on Γ

if a > 0, b > 0, j = 0, 1.

Each domain is endowed with the graph norm.

1250 Ciprian G. Gal

It is known that AWj generates a bounded analytic semigroup e−AWj t

on Xpa = Lp (Ω, dx) ⊕ Lp

(Γ, adSb

), for 1 ≤ p ≤ ∞, j = 0, 1 (cf. [21]). On

the other hand, it is standard that AK , when K ∈ D,N,R , generates abounded analytic semigroup e−AKt on Xp

0 = Lp (Ω, dx) and D (AK) is denselycontained in W 2,p (Ω) . In addition, each AK is nonnegative and self-adjointon X2

a. For our convenience, we also set

ZD = H10 (Ω), ZK = H1(Ω), if K ∈ N,R,W0,W1 .

We can now state a well-posedness theorem which can be easily proven viaarguments based on semigroup theory and energy estimates.

Theorem 1. Let θ0 ∈ ZK , K ∈ D,N,R,W0,W1 and h1 ∈ L2([0, T ] ; X2

a

),

for any fixed T > 0. Then problem (2.1)–(2.3) admits a unique solution θsuch that

θ ∈ C ([0, T ] ;ZK) ∩ L2 ([0, T ] ;D (AK)) ,θt ∈ C ([0, T ] ;Z∗K) ∩ L2

([0, T ] ; X2

a

).

Moreover, the following estimate holds:

‖θ (t)‖2ZK +∫ t

0

(‖θt (s)‖2X2

a+ ‖θ (s)‖2D(AK)

)ds

≤ C(‖θ0‖2ZK +

∫ t

0‖h1 (s)‖2X2

ads)

+ CK

(∫ t

0‖h1 (s)‖X2

ads)2, (2.5)

for all 0 ≤ t ≤ T . Here C is a positive constant that depends on Ω andCK = 0 if K ∈ D,R,W1 .

We now consider the linear problem of finding a function φ such that

εφt −∆φ = q1, in Ω× (0,+∞), (2.6)

α∆Γφ− ∂nφ− βφ−φtd

= q2, on Γ× (0,+∞), (2.7)

φ|t=0 = φ0. (2.8)Problems of this sort have been studied, e.g., in [24, 32, 35, 39] (cf. also [2, 3]for related problems).

Let us introduce the functional spaces

Vs = Cs(Ω)‖·‖Vs ,

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

‖Π‖2V1=∫

Ω|∇Π|2 dx+ α

∫Γ|∇ΓΠ|2 dS + β

∫Γ

∣∣Π|Γ∣∣2 dS

Global well-posedness 1251

and‖Π‖2V0

=∫

Ωε |Π|2 dx+

1d

∫Γ

∣∣Π|Γ∣∣2 dS,respectively. It easy to see that the following identification holds: Vs =Hs (Ω)⊕Hs (Γ). Note that when ε = 0, the space V0 defined above becomesL2 (Γ) endowed with the natural inner product 〈·, ·〉2,Γ . Then we recall anelliptic regularity lemma which was proven in [32].

Lemma 2. Consider the following linear problem:

−∆φ = j1, in Ω ⊂ Rn, n = 2, 3, (2.9)

−α∆Γφ+ ∂nφ+ βφ = j2, on Γ. (2.10)Let (j1, j2) ∈ Hs (Ω) × Hs (Γ), respectively, where s ≥ 0, s + 1

2 /∈ N. Thenthe following estimate holds:

‖φ‖Hs+2 + ‖φ‖Hs+2(Γ) ≤ C(‖j1‖Hs + ‖j2‖Hs(Γ)

), (2.11)

for some constant C > 0.

Observe that (2.9)–(2.10) is an elliptic boundary-value problem in thesense specified in [28] (see also [36, 44]).

The existence and uniqueness of the solution to problem (2.6)–(2.8) canbe derived by means of energy estimates and Lemma 2 . There holds

Theorem 3. Let q1 ∈ H1([0, T ] ;H1 (Ω)

), q2 ∈ H1

([0, T ] ;H1 (Γ)

)and

φ0 ∈ V3 = H3 (Ω)⊕H3 (Γ). Set

φ1 :=∆φ0 + q1 (0, ·)

ε∈ H1 (Ω) (2.12)

andφ1|Γ := d (∆Γφ0 − ∂nφ0 − βφ0 − q2 (0, ·)) ∈ H1 (Γ) . (2.13)

Then problem (2.6)–(2.8) has a unique solution φ such that

φ ∈ C ([0, T ] ; V3) ∩ C1 ([0, T ] ; V1) ,

φt ∈ C ([0, T ] ; V1) ∩H1 ([0, T ] ; V0) .Moreover, the following estimates hold:

‖φ (t)‖2V1+∫ t

0

‖φt (τ)‖2V0dτ ≤ C

(‖φ0‖2V1

+∫ t

0

[‖q1 (s)‖22 + ‖q2 (s)‖22,Γ

]ds), (2.14)

‖φt (t)‖2V1+∫ t

0

‖φtt (τ)‖2V0dτ ≤ C

(‖φ1‖2V1

+∫ t

0

[‖q1t (s)‖22 + ‖q2t (s)‖22,Γ

]ds),

(2.15)

1252 Ciprian G. Gal

‖φ (t)‖2Vs≤ C

(‖φ1‖2Vs−2

+ ‖q1 (t)‖2Hs−2 + ‖q2 (t)‖2Hs−2(Γ)

+∫ t

0

[‖q1t (s)‖22 + ‖q2t (s)‖22,Γ

]ds), (2.16)

for all 0 ≤ t ≤ T , s = 2, 3, where C > 0 depends at most on Ω, α, β, d, andε, but it is independent of φ0, q1, q2, T, and t.

Remark 4. The result in Theorem 3 is valid in bounded domains whenn = 2, 3. When the dimension is n = 1, we note that ∆Γ is not present inthe boundary condition (2.7) . This case will be discussed in Section 5.

3. Well-posedness of the approximate problem

Here we prove that an approximate problem of (1.24)–(1.30) has a uniqueglobal solution which continuously depends on the initial data. However,we first need to specify more rigorously which class of problems we want tosolve. Let K ∈ D,N,R,W0,W1; then we formulate

Problem (PKε ). For any ε > 0 and any given pair of initial data

(φ0, u0, θ0) ∈ V3 ×H1 (Ω)× ZK , (3.1)

find (φ, u, θ) ∈ C([0,+∞),V3×H1(Ω)×ZK)∩H1loc([0,∞),V1×L2(Ω)×X2

a)solving the system

εut −∆u = −φt, (3.2)

εφt −∆φ = u− F ′1 (φ) + λ0θ, (3.3)θt −∆θ = −λ0φt, (3.4)

almost everywhere on Ω × (0,∞), subject to boundary conditions (1.27),(1.28), and (1.29), and fulfilling initial conditions (1.30).

In order to show the global existence of the problem PKε , we shall needestimates on the nonlinearities F1 and F2. Thus, owing to the embeddingH2 ⊂ C and recalling the assumptions on Fi, i = 1, 2, it is not difficult torealize the validity of the following:

Lemma 5. Let Φ ∈ C ([0, T ] ,V3) and Ψ ∈ C ([0, T ] ,V0). Set Φ0 := Φ (0, ·) .Then, the following estimates hold:∫

Ω

∣∣∣F (s)1 (Φ (t, x)) Ψ (t, x)

∣∣∣2 dx+∫

Γ

∣∣∣F (s)

2 (Φ (t, x)) Ψ (t, x)∣∣∣2 dS ≤

≤ C[Q1

(‖Ψ (t, ·)‖2V0

)+Q1

(‖Φ (t, ·)‖2V2

)], s = 1, 2, (3.5)∥∥∥F ′1 (Φ (t, ·))

∥∥∥2

2+∥∥∥F ′2 (Φ (t, ·))

∥∥∥2

2,Γ≤ C

(Q1

(‖Φ0‖2V2

)+

Global well-posedness 1253

+ t

∫ t

0Q1

(‖Φ (τ, ·)‖2V2

)dτ + t

∫ t

0Q1

(‖Φt (τ, ·)‖2V1

)dτ), (3.6)

for all t ∈ [0, T ]. Here C > 0 is independent of Φ, Ψ, and t, while Q1 is amonotone-nondecreasing function that depends at most on the size of F1 andF2.

Note that (3.6) follows from (3.5) with s = 2, thanks to the fundamentaltheorem of calculus. In order to prove existence and uniqueness, we shalluse first the contraction-mapping principle to find the unique local solution.Then, thanks to suitable a priori estimates, the local solution will becomea global one. From now on, we assume that the following condition (see(3.7) below) on the initial data φ0, θ0, and u0 = µ0 + εµ1, with µ0|Ω =−∆φ0 + F

′1 (φ0)− λ0θ0, is satisfied:

φ0 ∈ V3, θ0 ∈ ZK , K ∈ D,R,N,W0,W1 , µ1 ∈ H1 (Ω) . (3.7)

Clearly, this implies that u0 ∈ H1 (Ω). Moreover, due to the embeddingH2 ⊂ C, it follows that

φ1|Ω =∆φ0 + u0 − F

′1 (φ0) + λ0θ0

ε= µ1|Ω ∈ H1 (Ω) , (3.8)

φ1|Γ = d(α∆Γφ0 − ∂nφ0 − βφ0 − F

′2 (φ0)

)∈ H1 (Γ) .

Then, we have the following result on the global existence and uniquenessof the problem

(PKε).

Theorem 6. Suppose that φ0, u0, and θ0 satisfy (3.7). For each K ∈D,N,R,W0,W1, problem (PKε ) admits a unique solution. Moreover, thereholds

φ ∈ C1 ([0,+∞); V1) ∩H2loc ([0,+∞); V0) ,

θ ∈ L2loc ([0,+∞);D (AK)) , u ∈ L2

loc

([0,+∞);H2 (Ω)

).

Moreover, setting

〈φ〉Ω =1|Ω|

∫Ωφ (x) dx, 〈〈θ〉〉 =

1(|Ω|+ a

b |Γ|)( ∫

Ωθ (x) dx+

∫Γθ (x)

adS

b

),

there holds for all t ≥ 0(i) 〈εu (t) + φ (t)〉Ω = 〈εu0 + φ0〉Ω and, in addition, 〈θ (t) + λ0φ (t)〉Ω =

〈θ0 + λ0φ0〉Ω , if K = N and(ii) 〈εu (t) + φ (t)〉Ω = 〈εu0 + φ0〉Ω and, in addition, 〈〈θ(t)〉〉+λ0〈φ(t)〉Ω =

〈〈θ0〉〉+ λ0 〈φ0〉Ω , if K = W0.

1254 Ciprian G. Gal

Proof. We begin to observe that if (φ, u, θ) solves(PKε)

with K ∈ N,W0 ,then the quantities above are conserved because of the boundary conditions.Let us now consider δ ∈ (0, T ] and define

Y1 := C ([0, δ] ; V3) ∩ C1 ([0, δ] ; V1) ∩H2 ([0, δ] ; V0) ,

Y2 := C([0, δ] ;H1 (Ω)

)∩ L2

([0, δ] ;H2 (Ω)

)∩H1

([0, δ] ;L2 (Ω)

),

Y3 := C ([0, δ] ;ZK) ∩ L2 ([0, δ] ;D (AK)) ∩H1([0, δ] ; X2

a

),

andYδ := Y1 × Y2 × Y3.

For (χ, v, η) ∈ Yδ, we define (φ, u, θ) to be the solution ofεut −∆u = −χt := l1 in Ω,

∂nu = 0 on Γ, (3.9)εφt −∆φ = v − F ′1 (χ) + λ0η := l2 in Ω,

α∆Γφ− ∂nφ− βφ− φtd = F

′2 (χ) := l3 on Γ,

(3.10)

and θt −∆θ = −λ0χt := l4 in Ω,aθt + b∂nθ + cθ = 0 on Γ. (3.11)

For (χ, v, η) ∈ Yδ, we have l1 ∈ H1([0, δ] ;L2 (Ω)

), and thus by known ex-

istence results, we have a unique solution u ∈ Y2 to the problem (3.9) . Onthe other hand, by Lemma 5, the right-hand sides l2 and l3 in (3.10) sat-isfy the assumptions of Theorem 3, that is, l2 ∈ C

([0, δ] ;H1 (Ω)

), l2t ∈

L2([0, δ] ;H1 (Ω)

)and l3 ∈ C

([0, δ] ;H1 (Γ)

), l3t ∈ L2

([0, δ] ;H1 (Γ)

). Thus,

from the conditions on the initial data, the problem (3.10) has a unique solu-tion φ ∈ Y1. Finally, since l4 ∈ L2

([0, δ] ; X2

a

), it follows from Theorem 1 that

there is a unique solution θ ∈ Y3 to problem (3.11) . Therefore, the mappingS : (χ, v, η)→ (φ, u, θ) is well defined as a map from Yδ to Yδ.

Next, we define a new set Pδ ⊂ Yδ to be the set that consists of functions(φ, u, θ) ∈ Yδ, with φ|t=0 = φ0, u|t=0 = u0, φt|t=0 = φ1, θ|t=0 = θ0 that satisfythe following:

max0≤t≤δ

‖u (t)‖2H1 +∫ δ

0

(‖ut (τ)‖22 + ‖u (τ)‖2H2

)dτ ≤M1,

max0≤t≤δ

‖φ (t)‖2V2≤M2,

max0≤t≤δ

(‖φt (t)‖2V1

+ ‖φ (t)‖2V3

)+∫ t

0‖φtt (τ)‖2V0

dτ ≤ 2M3,

Global well-posedness 1255

max0≤t≤δ

‖θ (t)‖2ZK +∫ δ

0

(‖θt (τ)‖2X2

a+ ‖θ (τ)‖2D(AK)

)dτ ≤M4,

where the positive constants Mi (i = 1, 2, 3, 4) will be specified below.

We shall prove that S : Pδ → Pδ is a contraction in a suitable norm on Pδif δ > 0 is sufficiently small. Let (χ, v, η) ∈ Pδ and (φ, u, θ) = S ((χ, v, η)) .Then, we conclude from (3.9), standard energy arguments (similar to (2.5))and the obvious inclusion V1 ⊆ L2 (Ω) , for t ∈ [0, δ] that

‖u (t)‖2H1 +∫ t

0

(‖ut (τ)‖22 + ‖u (τ)‖2H2

)dτ ≤

≤ C1

(‖u0‖2H1 +

∫ δ

0‖χt (τ)‖22 dτ +

√δ(∫ δ

0‖χt (τ)‖22 dτ

)1/2)≤ C1

(‖u0‖2H1 + 2δM3 + 2

√δ√M3

), (3.12)

where C1, C2, . . . are positive constants that depend at most on ε. We choose

M1 ≥ 2C1 ‖u0‖2H1 . (3.13)

On the other hand, the estimate (2.16) from Theorem 3 and the estimate(3.6) from Lemma 5, yields for t ∈ [0, δ] that

‖φ (t)‖2V2≤ C2

(‖φ1‖2V0

+‖v (τ)‖22 +∥∥∥F ′1 (χ (τ))

∥∥∥2

2+∥∥∥F ′2 (χ (τ))

∥∥∥2

2,Γ+‖η (τ)‖22

+∫ δ

0

[‖vt (τ)‖22+

∥∥∥F ′′1 (χ (τ))χt (τ)∥∥∥2

2+‖ηt (τ)‖22+

∥∥∥F ′′2 (χ (τ))χt (τ)∥∥∥2

2,Γ

]dτ)

≤ C2

(‖φ1‖2V0

+Q1

(‖φ0‖2V2

)+M1 +M4+

+ (δ + 1)∫ δ

0

[Q1

(‖χ (τ)‖2V2

)+Q1

(‖χt (τ)‖2V1

)]dτ

)≤ C2

(‖φ1‖2V0

+Q1

(‖φ0‖2V2

)+M1 +M4 +

(δ2 + δ

)Q1 (M3)

). (3.14)

We choose

M2 ≥ 2C2

(‖φ1‖2V2

+Q1

(‖φ0‖2V2

)+M1 +M4

). (3.15)

The estimate (2.5) from Theorem 1 yields

max0≤t≤δ

‖θ (t)‖2ZK +∫ δ

0

(‖θt (τ)‖2X2

a+ ‖θ (τ)‖2D(AK)

)dτ

≤ C3

(‖θ0‖2ZK +

∫ t

0‖χt (τ)‖2X2

adτ)

+ CK

(∫ t

0‖χt (τ)‖X2

adτ)2

1256 Ciprian G. Gal

≤ C3

(‖θ0‖2ZK + δM3

)+ CK

√δ√M3. (3.16)

We recall that CK = 0 when K ∈ D,R,W1 . Moreover, we have also usedthe fact that V0 = X2

a, up to an equivalent norm if a > 0 and V0 ⊂ X20 =

L2 (Ω) . We chooseM4 ≥ 2C3 ‖θ0‖2ZK . (3.17)

Recalling (2.15) of Theorem 3 and (3.5) of Lemma 5 yields for t ∈ [0, δ] that

‖φt (t)‖2V1+∫ t

0‖φtt (τ)‖2V0

dτ

≤ C4

(‖φ1‖2V1

+∫ δ

0

[‖vt (τ)‖22 +

∥∥∥F ′′1 (χ (τ))χt (τ)∥∥∥2

2+ ‖ηt (τ)‖22

]dτ

+∫ δ

0

∥∥∥F ′′2 (χ (τ))χt (τ)∥∥∥2

2,Γdτ

)≤ C4

(‖φ1‖2V1

+M4 +M1 + δQ2 (M3)), (3.18)

where Q2 is a monotone-nondecreasing function that depends on Q1. Fur-thermore, using estimate (2.16) of Theorem 3, the Sobolev embeddingsH2 (Ω) → C (Ω) , H2 (Γ) → C (Γ) and the estimates (3.5) , (3.6) of Lemma 5,we deduce

‖φ (t)‖2V3≤ C5

(‖φ1‖2V1

+ ‖v (t)‖2H1 + ‖η (t)‖2ZK

+∥∥∥F ′1 (χ (t))

∥∥∥2

H1+∥∥∥F ′2 (χ (t))

∥∥∥2

H1(Γ)

+∫ t

0

[∥∥∥F ′′1 (χ (τ))χt (τ)∥∥∥2

2+∥∥∥F ′′2 (χ (τ))χt (τ)

∥∥∥2

2,Γ

]dτ

)≤ C5

(‖φ1‖2V1

+M1 +M4 +Q3 (M2) +Q3

(‖φ0‖2V3

)+ (δ + 1)

∫ δ

0

(Q3

(‖χ (τ)‖2V2

)+Q3

(‖χt (τ)‖2V1

))dτ (3.19)

≤ C6

[‖φ1‖2V1

+M1 +Q3 (M2) +M4 +Q3(‖φ0‖2V3) +

(δ2 + δ

)Q3 (M3)

],

for some monotonic function Q3 that depends on Q1. Correspondingly, weselect

M3 ≥ max

2C4(‖φ1‖2V1+M4 +M1), 2C6(‖φ1‖2V1

+M1 +Q3(M2) +M4 +Q1(‖φ0‖2V3)). (3.20)

Global well-posedness 1257

We now choose δ > 0 small enough such that the following inequalities hold:

4(δM3 +

√δ√M3

)≤M1, (3.21)

2(δ2 + δ

)Q1 (M3) ≤M2, (3.22)

2δM3 + 2CK√δ√M3 ≤M4, (3.23)

max 2δ (δ + 1)Q3 (M3) , 2δQ2 (M3) ≤M3. (3.24)This choice combined with (3.13), (3.15), (3.17) and (3.20), on account of(3.12), (3.14), (3.16), (3.18), and (3.19) entails that S maps Pδ into itself. Letus now show that, possibly choosing δ smaller, S : Pδ → Pδ is a contractionmapping with respect to the metric induced by the norm

‖(χ, v, η)‖2Yδ := max0≤t≤δ

‖χ (t)‖2V1+∫ δ

0

(ε ‖χt (s)‖22 + ‖χt (s)‖22,Γ

)ds (3.25)

+ε max0≤t≤δ

‖v (t)‖22 + max0≤t≤δ

‖η (t)‖2X2a.

Clearly, Pδ is a closed metric space with respect to the metric defined above.Let (χj , vj , ηj) ∈ Pδ for j = 1, 2 and (φj , uj , θj) = S ((χj , vj , ηj)) . Then, letφ := φ1 − φ2, u := u1 − u2, θ := θ1 − θ2, χ := χ1 − χ2, v := v1 − v2 andη := η1 − η2 satisfy εut −∆u = −χt := l1 in Ω,

∂nu = 0 on Γ,u (0, ·) = u0,

(3.26)

εφt −∆φ = v + F

′1 (χ1)− F ′1 (χ2) + λ0η := l2 in Ω,

α∆Γφ− ∂nφ− βφ− φtd = F

′2 (χ1)− F ′2 (χ2) := l3 on Γ,

φ (0, ·) = φ0,

(3.27)

θt −∆θ = −λ0χt := l4, in Ω,aθt + b∂nθ + cθ = 0, on Γ.

θ (0, ·) = θ0.(3.28)

Let us take the scalar product in L2(Ω) of the first equation of (3.27) timesφt(t). Then, on account of the second equation of (3.27), we find12d

dt‖φ(t)‖2V1

+ε ‖φt (t)‖22+1d‖φt (t)‖22,Γ = −〈l2 (t) , φt (t)〉2−〈l3 (t) , φt (t)〉2,Γ .

(3.29)Then, it is easy to deduce from (3.29) the inequality

d

dt‖φ (t)‖2V1

+ ‖φt (t)‖2V0≤ Q4(M3)

(‖χ(t)‖2V0

+ ‖v(t)‖22 + ‖η(t)‖2X2a

)

1258 Ciprian G. Gal

(note that Q4 depends at most on ε), which entails, for all t ∈ [0, δ],

‖φ (t)‖2V1+∫ t

0‖φt (s)‖2V0

ds ≤ δQ4(M3) ‖(χ, v, η)‖2Yδ . (3.30)

Consider now (3.28) and take the scalar product in L2(Ω) of the first equationwith θ(t). We obtain

12d

dt‖θ (t)‖2X2

a+ ‖∇θ (t)‖22 +

c

b

∥∥θ|Γ (t)∥∥2

2,Γ= −〈λ0χt(t), θ(t)〉2 ,

which givesd

dt‖θ (t)‖2X2

a≤ C7‖χt(t)‖V0‖θ(t)‖X2

a. (3.31)

Using now a suitable version of Gronwall’s lemma, from (3.31), we deduce

‖θ (t)‖2X2a≤ C8

(∫ t

0‖χt(s)‖V0ds

)2,

so that‖θ (t)‖2X2

a≤ δC9 ‖(χ, v, η)‖2Yδ , (3.32)

for all t ∈ [0, δ]. Similarly, considering (3.26) and taking the scalar productin L2 (Ω) of the first equation of (3.26) with u (t) , we obtain

ε

2d

dt‖u (t)‖22 + ‖∇u (t)‖22 = −〈χt(t), u(t)〉2 . (3.33)

Arguing now as in (3.31)–(3.32) , we deduce that

ε ‖u (t)‖22 ≤ δC10 ‖(χ, v, η)‖2Yδ , (3.34)

where C10 depends at most on ε. Hence, on account of (3.30), (3.32) and(3.34), we can say that S is a contraction of the complete metric space Pδ initself (cf. (3.25)), provided that δ > 0 is possibly smaller than the one whichfulfills (3.21)–(3.24). Therefore, owing to the contraction-mapping principle,we conclude that problem

(PKε)

has a unique local solution (φ, u, θ) ∈ Pδ.We now show that this solution is indeed a global one. In order to achieve

that, we shall prove a suitable set of a priori estimates. Let us fix K ∈D,N,R,W0,W1 and consider our local solution (φ, u, θ). This solution cancertainly be (uniquely) extended on a right-maximal time interval [0, Tmax),where Tmax > δ. In order to obtain higher-order estimates for our solutionand to make our arguments more rigorously, we need our local solution tobe more smooth, so we can differentiate equations (3.9) , (3.10). For thispurpose, consider (φ, u, θ) ∈ Yδ = Y 1

δ × Y 2δ × Y 3

δ , where

Y 1δ := C1 ([0, δ] ; V2) ∩ C2 ([0, δ] ; V0) ∩H2 ([0, δ] ; V1) ,

Global well-posedness 1259

Y 2δ := C1

([0, δ] ;H2 (Ω)

)∩ C2

([0, δ] ;L2 (Ω)

)∩H2

([0, δ] ;L2 (Ω)

),

Y 3δ := C ([0, δ] ;ZK) ∩ L2 ([0, δ] ;D (AK)) ∩H1

([0, δ] ; X2

a

),

such thatφ0 (0, ·) = φ0n, u0 (0, ·) = u0n, θ0 (0, ·) = θ0n,

where φ0n ⊂ V4, u0n ⊂ H4 (Ω) and θ0n ⊂ ZK are such that

‖φ0n − φ0‖V3 → 0, ‖u0n − u0‖H1 → 0, ‖θ0n − θ0‖ZK → 0,

as n goes to +∞. For a fixed n ∈ N, we consider the systems εut −∆u = −φt in Ω× (0, δ) ,∂nu = 0 on Γ× (0, δ) ,

u (0, ·) = u0n,(3.35)

εφt −∆φ = u− F ′1 (φ) + λ0θ in Ω× (0, δ) ,

α∆Γφ− ∂nφ− βφ− φtd = F

′2 (φ) on Γ× (0, δ) ,

φ (0, ·) = φ0n,

(3.36)

and θt −∆θ = −λ0φt, in Ω× (0, δ) ,aθt + b∂nθ + cθ = 0, on Γ× (0, δ) ,

θ (0, ·) = θ0n.(3.37)

Now, arguing exactly as in the proof of the local existence (see Theorem 6), itis now easy to check that problem (3.36) has a unique solution φ ∈ Y 1

δ , whileproblems (3.35) and (3.37) , respectively, admit a unique solution u ∈ Y 1

δ andθ ∈ Y 3

δ , respectively. Note that the functions (φn, un, θn) ∈ Yδ depend on n.By performing rigorously our estimates on (φn, un, θn) , the idea is to showthat the solution exists on any bounded time interval (hence on [0,+∞))and then passing to the limit as n→∞ in all these estimates, to recover thesolution corresponding to our original assumptions. In what follows, for thesake of simplicity, we drop the n-dependence from the solution (φn, un, θn) .

We begin by multiplying the equations (3.35) , (3.36) , and (3.37) by u, φt,and θ respectively, and then integrating with respect to x, we deduce that

ε

2d

dt‖u (t)‖22 + ‖∇u (t)‖22 = −〈φt, u〉2 , (3.38)

ε ‖φt (t)‖22 +1d‖φt (t)‖22,Γ +

12d

dt

[‖α∇Γφ (t)‖22,Γ + ‖βφ (t)‖22,Γ + ‖∇φ (t)‖22

]+∫

ΩF′1 (φ (t))φt (t) dx+

∫ΓF′2 (φ (t))φt (t) dS = 〈φt, u〉2 + λ0 〈φt, θ〉2 ,

(3.39)

1260 Ciprian G. Gal

and12d

dt‖θ (t)‖2X2

a+ ‖∇θ (t)‖22 +

c

b

∥∥θ|Γ (t)∥∥2

2,Γ= −λ0 〈φt, θ〉2 . (3.40)

Inserting now the scalar products from the right-hand side of (3.39) fromboth (3.38) and (3.40) , and then integrating the resulting relation with re-spect to t, we deduce for each 0 < ε ≤ 1 that

ε ‖u (t)‖22 + ‖φ (t)‖2V1+∫ t

0

(‖∇u (τ)‖22 +

1d‖φt (t)‖22,Γ + ε ‖φt (t)‖22

)dτ

+ ‖θ (t)‖2X2a

+∫ t

0

(‖∇θ (τ)‖22 +

c

b

∥∥θ|Γ (τ)∥∥2

2,Γ

)dτ

+∫

ΩF1 (φ (t)) dx+

∫ΓF2 (φ (t)) dS ≤ C, (3.41)

where C > 0 depends at most on ‖u0n‖H1 , ‖φ0n‖V1, and ‖θ0n‖X2

a, but is

independent of n, t, ε, and δ. From now on, C stands for a positive constantthat depends on ‖u0n‖H1 , ‖φ0n‖V3

, ‖φ1n‖V1, and ‖θ0n‖ZK , but is indepen-

dent of n, ε, t, and δ, and is possibly taking different values even in the sameline. We have

‖θ (t)‖2X2a≤ C, ε ‖u (t)‖22 ≤ C, ‖φ (t)‖2V1

≤ C, (3.42)∫ t

0

(‖∇θ (τ)‖22 +

c

b

∥∥θ|Γ (τ)∥∥2

2,Γ

)dτ ≤ C (3.1)∫

ΩF1 (φ) dx ≤ C,

∫ t

0‖∇u (τ)‖22 dτ ≤ C, (3.43)∫

ΓF2 (φ) dS ≤ C, ε

∫ t

0‖φt (τ)‖22 dτ ≤ C, (3.2)∫ t

0

1d‖φt (τ)‖22,Γ dτ ≤ C. (3.44)

Next, we differentiate (3.36) with respect to t (note that we can do so,since φ ∈ C1 ([0, Tmax] ; V2) ∩ C2 ([0, Tmax] ; V0)), take the inner product inL2 (Ω) of the resulting relation with φt, and then use the second equation.Combining this last relation with the relations that we obtain by taking theinner products in L2 (Ω) of (3.35) , (3.37) with ut and θt respectively, wededuce

12d

dt

(ε ‖φt (t)‖22 +

1d‖φt (t)‖22,Γ + ‖∇u (t)‖22 + ‖∇θ (t)‖22 +

c

b

∥∥θ|Γ (t)∥∥2

2,Γ

)

Global well-posedness 1261

+ ‖φt (t)‖2V1+ ε ‖ut (t)‖22 +

∫ΩF′′1 (φ (t))φ2

t (t) dx

+∫

ΓF′′2 (φ (t))φ2

t (t) dS + ‖θt (t)‖2X2a

= 0. (3.45)

We first estimate ‖φt (t)‖22 , by multiplying (3.35) by φt and integrating byparts, leading us to

‖φt (t)‖22 ≤ ε |〈ut, φt〉2|+ |〈∇u,∇φt〉2|

≤ ε ‖ut‖2 ‖φt‖2 + ‖∇u‖2 ‖φt‖H1 . (3.46)

By the inequality ab ≤ 12

(a2 + b2

)and obvious inclusion V1 ⊂ H1 (Ω) ,

(3.46) yields‖φt (t)‖22 ≤ ε

2 ‖ut‖22 + 2C ‖∇u‖2 ‖φt‖V1. (3.47)

Furthermore, integrating with respect to t in (3.47) , we deduce the followingestimate from (3.43):∫ t

0‖φt(τ)‖22dτ ≤ ε2

∫ t

0‖ut‖22 + 2C

(∫ t

0‖∇u(τ)‖22

)1/2(∫ t

0‖φt(τ)‖2V1

dτ)1/2

≤ ε2

∫ t

0‖ut (τ)‖22 dτ + C

(∫ t

0‖φt (τ)‖2V1

dτ)1/2

. (3.48)

Moreover, due to the assumptions (1.31) on Fi and Gin, we have that−G′in (s) ≥ Ni, ∀s ∈ R, for each i, and some Ni > 0, uniformly in n. Weestimate the integral terms in (3.45) involving such nonlinearities as follows:

−∫

ΩF′′1 (φ (t))φ2

t (t) dx−∫

ΓF′′2 (φ (t))φ2

t (t) dS ≤ max N1, N2 ‖φt (t)‖2V0.

(3.49)Collecting now (3.48) and (3.49) , and using (3.44), one deduces that

−∫ t

0

(∫ΩF′′1 (φ (t))φ2

t (t) dx+∫

ΓF′′2 (φ (t))φ2

t (t) dS)dτ

≤ ε2

∫ t

0‖ut (τ)‖22 dτ + C

(∫ t

0‖φt (τ)‖2V1

dτ)1/2

+ C. (3.50)

Applying now a suitable version of Gronwall’s inequality to (3.45) and usingestimate (3.50), we obtain the following estimates for each 0 < ε ≤ 1/2:

ε‖φt(t)‖22 ≤ C,1d‖φt(t)‖22,Γ ≤ C, ‖∇u(t)‖22 ≤ C,

‖∇θ(t)‖22 +c

b‖θ|Γ(t)‖22,Γ ≤ C, (3.51)

1262 Ciprian G. Gal∫ t

0‖θt (τ)‖2X2

adτ ≤ C,

∫ t

0‖φt (τ)‖2V1

dτ ≤ C, ε2

∫ t

0‖ut (τ)‖22 dτ ≤ C. (3.52)

Collecting now (3.51) and (3.42) , we easily notice that

‖θ (t)‖2ZK ≤ C. (3.53)

Now, we integrate (3.36) over Ω and get∫Ω

(εφt −∆φ) dx =∫

Ω

(u− F ′1 (φ)− λ0θ

)dx,∫

Ωεφt dx−

∫Γ∂nφdS =

∫Ω

(u− F ′1 (φ)− λ0θ

)dx.

From the boundary condition of (3.36), we conclude that∫Γ∂nφdS =

∫Γα∆ΓφdS −

∫Γβφ dS −

∫ΓF′2 (φ) dS −

∫Γ

φtddS

= −∫

ΓβφdS −

∫ΓF′2 (φ) dS −

∫Γ

φtddS.

Thus, we deduce that∫Ωεφt dx+

∫Γ

φtddS +

∫Γβφ dS +

∫ΩF′1 (φ) dx

+∫

Ωλ0θdx+

∫ΓF′2 (φ) dS =

∫Ωu dx. (3.54)

On the other hand, using the estimates of (3.42)–(3.44) and (3.51)–(3.53) ,we obtain ∣∣∣ ∫

Ωεφt dx

∣∣∣+∣∣∣ ∫

Γ

φtddS∣∣∣ ≤ C, ∣∣∣ ∫

Γβφ dS

∣∣∣ ≤ C,∣∣∣ ∫ΩF′1(φ)dx

∣∣∣ ≤ C, ∣∣∣ ∫ΓF′2 (φ) dS

∣∣∣ ≤ C, ∣∣∣ ∫Ωλ0θ dx

∣∣∣ ≤ C.Therefore, we obtain ∣∣∣ ∫

Ωu dx

∣∣∣ ≤ C,which combined with (3.51) , yields the estimate

‖u‖H1 ≤ C. (3.55)

Note that, owing to (3.42)–(3.44) and (3.51)–(3.53), from (3.35) and (3.37)we also deduce

supt≥0

∫ t

0‖θ (s)‖2D(AK) ds ≤ C, sup

t≥0

∫ t

0‖∆u (s)‖22 ds ≤ C.

Global well-posedness 1263

To prove the last bound on φ, we first need to apply the maximum principleto the following system (see [32], Lemma A.2)

−∆φ+ F′1 (φ) = g1 := −εφt + u+ λ0θ, in Ω,

−α∆Γφ+ ∂nφ+ βφ+ F′2 (φ) = g2 := 1

dψt, on Γ,

which yields for each 0 < ε ≤ 1

‖φ (t)‖2∞ + ‖φ (t)‖2∞,Γ ≤ C(

1 + ‖g1(t)‖22 + ‖g2(t)‖22,Γ),

and, owing to (3.51), (3.53) and (3.55), it follows that

‖φ (t)‖2∞ + ‖φ (t)‖2∞,Γ ≤ C. (3.56)

On the other hand, recalling (2.11), we have

‖φ (t)‖V2≤ C

(‖j1(t)‖2 + ‖j2(t)‖2,Γ

), (3.57)

where

j1 := εφt − u+ F′1 (φ)− λ0θ, j2 :=

φtd

+ F′2 (φ) .

Thus, thanks to (3.51), (3.53), (3.55) and (3.56), from (3.36) we infer

‖φ (t)‖V2≤ C. (3.58)

It remains to prove the estimate for φ in V3. Next, we differentiate (3.35)and (3.36) with respect to t (noting that u ∈ C2

([0, Tmax] ;L2 (Ω)

)), and

multiply by φtt and utt respectively. Integrating by parts, we obtain thefollowing relations:

12d

dt‖φt (t)‖2V1

+ ε ‖φtt (t)‖22 +1d‖φtt (t)‖22,Γ

+∫

ΩF′′1 (φ (t))φt (t)φtt (t) dx+

∫ΓF′′2 (φ (t))φt (t)φtt (t) dS

= (ut, φtt)2 + (λ0θt, φtt)2 , (3.59)ε

2d

dt‖ut (t)‖22 + ‖∇ut‖22 = − (ut, φtt)2 .

Combining the above two relations, we obtain12d

dt‖φt (t)‖2V1

+ ε ‖φtt (t)‖22 +ε

2d

dt‖ut (t)‖22 + ‖∇ut‖22 +

1d‖φtt (t)‖22,Γ

= −∫

ΩF′′1 (φ (t))φt (t)φtt (t) dx−

∫ΓF′′2 (φ (t))φt (t)φtt (t) dS + 〈λ0θt, φtt〉2 .

(3.60)

1264 Ciprian G. Gal

We estimate the last term on the right-hand side of (3.60) as follows:

|λ0 〈θt, φtt〉2| ≤ C ‖θt‖2 ‖φtt‖2 ≤ε

2‖φtt‖22 +

C

ε‖θt‖22 . (3.62)

Thus, we obtain from (3.52)∫ t

0|λ0 〈θt, φtt〉2| ≤

ε

2

∫ t

0‖φtt‖22 +

C

ε. (3.63)

Moreover, using estimate (3.52) again, but now in (3.48) , we easily deduce∫ t

0‖φt‖22 dτ ≤ C. (3.64)

It remains to estimate the nonlinear terms in (3.60). Nevertheless, in (3.56) ,we have obtained uniform estimates (with respect to ε) for the L∞ (Ω) andL∞ (Γ) norms of the function φ. It follows that

−∫

ΩF′′1 (φ (t))φt (t)φtt (t) dx−

∫ΓF′′2 (φ (t))φt (t)φtt (t) dS

≤ Q5 (‖φ‖∞) ‖φtt‖2 ‖φt‖2 +Q5

(‖φ‖∞,Γ

)‖φtt‖2,Γ ‖φt‖2,Γ (3.65)

≤ Q5(‖φ‖∞)(ε

4‖φtt‖22 +

C

ε‖φt‖22

)+Q5(‖φ‖∞,Γ)

(η ‖φtt‖22,Γ + Cη ‖φt‖22,Γ

),

for a sufficiently small constant η > 0 and large positive constant Cη. Herethe monotonic function Q5 depends only on the size of the nonlinearities F1

and F2 and is independent of ε, δ, and t. Integrating now (3.60) with respectto t, and using estimates (3.56) and (3.62)–(3.65) , we deduce

‖φt (t)‖2V1+ ε ‖ut (t)‖22 +

∫ t

0

(ε ‖φtt‖22 + ‖∇ut‖22 +

1d‖φtt‖22,Γ

)dτ

≤ C

ε

∫ t

0‖φt‖22 dτ +

∫ t

0‖φt‖22,Γ dτ +

C

ε

∫ t

0‖θt‖22 dτ ≤ C +

C

ε, (3.66)

by (3.64) and (3.52) . Consequently, (3.66) yields

ε ‖φt (t)‖2V1≤ C, ε2

∫ t

0‖φtt (t)‖22 dτ ≤ C,

∫ t

0

ε

d‖φtt (t)‖22,Γ ≤ C, (3.67)

ε2 ‖ut (t)‖22 ≤ C, ε∫ t

0‖∇ut‖22 ≤ C.

Finally, we have the elliptic estimate (see (2.11) of Lemma 2)

‖φ (t)‖V3≤ C

(‖j1‖H1 + ‖j2‖H1(Γ)

), (3.68)

Global well-posedness 1265

where j1 := −u + εφt + F′1 (φ) − λ0θ and j2 := βφ + φt

d + F′2 (φ) . Since by

(3.53) , (3.58) , (3.55) and (3.67) , ‖j1‖H1 ≤ C and ‖j2‖H1(Γ) ≤Cε , we easily

deduce the required estimate,

‖φ (t)‖V3≤ C

ε. (3.69)

Collecting (3.51), (3.53), (3.67) and (3.69), we conclude that the local solu-tion (φn, un, θn) is global. Finally, it remains to pass to the limit as n→∞,in all the estimates obtained above in the appropriate topologies (see above(3.35)), recalling that all the constants involved in the proof are independentof n. This completes the proof of Theorem 6.

We conclude by stating the following continuous dependence result.

Lemma 7. Let (φi, ui, θi) be the solution to(PKε)

corresponding to the initialdata (φ0i, u0i,θ0i) , i = 1, 2. Then, for any t ≥ 0, the following estimate holds:

‖φ1 (t)− φ2 (t)‖2V1+ ‖θ1 (t)− θ2 (t)‖2X2

a+ ε ‖u1 (t)− u2 (t)‖22

+∫ t

0

[‖∇u (s)‖22 + ‖φt (s)‖2V0

+ ‖∇θ (s)‖22 +c

b

∥∥θ|Γ (s)∥∥2

2,Γ

]ds

≤ CeLt(‖φ01 − φ02‖2V1

+ ‖θ01 − θ02‖2X2a

+ ε ‖u01 − u02‖22), (3.70)

where C and L are positive constants depending on the norms of the initialdata in V3 ×H1 (Ω) × ZK , on Ω, Γ and on the parameters of the problem,but are both independent of time.

Estimate (3.70) can be easily derived by writing down problem(PKε)

forφ = φ1−φ2, u = u1−u2 and θ = θ1−θ2 (see (3.26)–(3.28)) and then arguingexactly as in (3.29)–(3.34) . Of course, in order to estimate the differences ofthe nonlinear functions, one must use (3.56).

4. Global existence for the original system

We turn now to the original problem (1.24)–(1.30), that is,

ψt = ∆µ in Ω× (0,∞) , (4.1)

µ = −∆ψ + F′1 (ψ)− λ0θ, in Ω× (0,∞) , (4.2)

θt −∆θ = −λ0ψt, in Ω× (0,∞) , (4.3)with the boundary conditions

∂nµ = 0, on Γ× (0,∞) , (4.4)

1266 Ciprian G. Gal

1dψt = α∆Γψ − ∂nψ − βψ − F

′2 (ψ) , on Γ× (0,∞) , (4.5)

andaθt + b∂nθ + cθ = 0, on Γ× (0,∞) , (4.6)

ψ|t=0 = ψ0, θ|t=0 = θ0. (4.7)

Recall that the nonlinear terms Fi are assumed to be C2 (R,R) and satisfythe following condition

lim|s|→∞

inf F′′i (s) > 0, i = 1, 2.

To prove our main result, we need the following lemma:

Lemma 8. Let X0⊂ E ⊂ X1 be Banach spaces, the imbedding X0→ E beingcompact. Then the following embeddings are compact:

(i) (Aubin): L2 ([0, T ] ; X0) ∩ϕ : ∂ϕ∂t ∈ L

2 ([0, T ] ; X1)→ L2 ([0, T ] ; E)

and(ii) (Simon): L∞ ([0, T ] ; X0) ∩

ϕ : ∂ϕ∂t ∈ L

r ([0, T ] ; X1)→ C ([0, T ] ; E)

if 1 < r ≤ ∞.

We have the following result:

Theorem 9. Suppose ψ0 ∈ V3, θ0 ∈ ZK , K ∈ D,N,R,W0,W1 . Then theinitial-value problem (4.1)–(4.7) admits a unique global solution (ψ, θ) suchthat for any ω > 0, we have

ψ ∈ C ([0,+∞) ; V2−ω) ,

ψt ∈ L2loc ([0,+∞) ; V1) , ψ ∈ L2

loc ([0,+∞) ; V3−ω) ,

µ ∈ L2loc

([0,+∞) ;H3 (Ω)

)∩ L∞loc

([0,+∞) ;H1 (Ω)

),

θ ∈ C([0,+∞) ;H1−ω (Ω)

)∩ L2

loc ([0,+∞) ;D (AK)) ,

θt ∈ L2loc

([0,+∞) ; X2

a

).

Moreover, there holds for all t ≥ 0(i) 〈φ (t)〉Ω = 〈φ0〉Ω and, in addition, 〈θ (t)〉Ω = 〈θ0〉Ω , if K = N and(ii) 〈φ (t)〉Ω = 〈φ0〉Ω and, in addition, 〈〈θ (t)〉〉 = 〈〈θ0〉〉 , if K = W0.

The following continuous-dependence result follows:

Global well-posedness 1267

Theorem 10. Let (ψi, θi) be the solution to (4.1)–(4.7) corresponding to theinitial data (ψ0i, θ0i) , i = 1, 2. Then, for any t ≥ 0, the following estimateholds:

‖ψ1 (t)− ψ2 (t)‖2V1+ ‖θ1 (t)− θ2 (t)‖2X2

a

+∫ t

0

[‖∇u (s)‖22 + ‖ψt (s)‖22,Γ + ‖∇θ (s)‖22 +

c

b

∥∥θ|Γ (s)∥∥2

2,Γ

]ds

≤ CeLt(‖ψ01 − ψ02‖2V1

+ ‖θ01 − θ02‖2X2a

),

where C and L are positive constants depending on the norms of the initialdata in V3 × ZK , on Ω and on the parameters of the problem, but are bothindependent of t and ε .

Proof. Let φ1 ∈ V1 be such that u0 ∈ H1 (Ω) as in Section 3 (see (3.7) and(3.8)) such that

µ1|Ω =∆φ0 + u0 − F

′1 (φ0) + λ0θ0

ε= φ1|Ω,

φ1|Γ = d(α∆Γφ0 − ∂nφ0 − βφ0 − F

′2 (φ0)

). (4.8)

We begin by recalling that problem(PKε)

is given by εut −∆u = −φt in Ω,∂nu = 0 on Γ,u (0, ·) = u0,

(4.9)

εφt −∆φ = u− F ′1 (φ)− λ0θ in Ω,

α∆Γφ− ∂nφ− βφ− φtd = F

′2 (φ) on Γ,

φ (0, ·) = φ0

(4.10)

and θt −∆θ = −λ0φt, in Ω,aθt + b∂nθ + cθ = 0, on Γ,

θ (0, ·) = θ0

(4.11)

andµ0|Ω = −∆φ0 + F

′(φ0)− λ0θ0, u0 = µ0 + εµ1. (4.12)

It easy to see from (4.8) , (4.12) that the conditions of Theorem 6 are satisfied;thus, problem (4.9)–(4.11) has a unique global solution (φε, uε, θε) . Also,recall that by (3.58), we have, for every 0 < ε ≤ 1,

‖φ (t)‖2V2≤ C,

1268 Ciprian G. Gal

where C > 0 is independent of t and ε. Moreover, by the a priori estimatesin the previous section (3.42)–(3.69), for any T > 0, we have

φε uniformly bounded in C ([0, T ] ; V2) ∩ L2 ([0, T ] ; V3) , (4.13)

φεt uniformly bounded in L2 ([0, T ] ; V1) , (4.14)√εφεt uniformly bounded in C ([0, T ] ; V1) , (4.15)

uε uniformly bounded in C([0, T ] ;H1 (Ω)

), (4.16)

√εuεt uniformly bounded in L2

([0, T ] ;L2 (Ω)

), (4.17)

∆uε uniformly bounded in L2([0, T ] ;L2 (Ω)

), (4.18)

AKθε, θεt uniformly bounded in L2

([0, T ] ; X2

a

), (4.19)

θε uniformly bounded in C ([0, T ] ;ZK) . (4.20)Hence, we have a subsequence ε and φε, uε, and θε which we still denote byε, φε, uε, θε and ψ, µ, and θ such that, as ε→ 0+, we have

φε → ψ weak∗ L∞ ([0, T ] ; V2) , (4.21)

φε → ψ weakly in L2 ([0, T ] ; V3) , (4.22)

φεt → ψt weakly in L2 ([0, T ] ; V1) , (4.23)

uε → µ weak∗ in L∞([0, T ] ;H1 (Ω)

), (4.24)

εuεt → 0 strongly in L2([0, T ] ;L2 (Ω)

), (4.25)

εφεt → 0 strongly in C ([0, T ] ; V1) , (4.26)

∆uε → ∆µ weakly in L2([0, T ] ;L2 (Ω)

), (4.27)

AKθε → AKθ, θ

εt → θt weakly in L2

([0, T ] ; X2

a

), (4.28)

θε → θ weak∗ in L∞ ([0, T ] ;ZK) . (4.29)It follows from (4.21) , (4.23) and Lemma 8, (ii) that

φε → ψ strongly in C ([0, T ] ; V2−ω) (4.30)

and ψ ∈ C ([0, T ] ; V2−ω) , ψ (0, ·) = ψ0, with ω > 0 sufficiently small. More-over, due to the embedding H2−$ ⊂ C, for $ ∈ (0, 1/2) , since n ≤ 3, theconvergence (4.30) yields

φε → ψ strongly in C([0, T ] ;C

(Ω)).

By the well-known Aubin compactness theorem (Lemma 8, (i)), we deducefrom (4.22) , (4.23) that

φε → ψ strongly in L2 ([0, T ] ; V3−ω) . (4.31)

Global well-posedness 1269

Since Fi ∈ C2 (R,R) , there is a positive constant Ci > 0 (independent ofs1, s2) such that∣∣∣F ′i (s1)− F ′i (s2)

∣∣∣ ≤ Ci |Li (s1, s2)| |s1 − s2| ,

with

Li (s1, s2) =∫ 1

0F′′i (τs1 + (1− τ) s2) dτ.

Then, Holder and proper interpolation inequalities (for instance, H2−$ ⊂ C,for sufficiently small $ ∈ (0, 1/2)) yield∥∥∥F ′1 (φε)− F ′1 (ψ)

∥∥∥2

2≤ C ‖L (φε, ψ)‖2∞ ‖φ

ε − ψ‖22

≤ CQ (‖φε‖H2−$ + ‖ψ‖H2−$) ‖φε − ψ‖22 . (4.32)Consequently, it follows from (4.30) and (4.32) that

F′1 (φε)→ F

′1 (ψ) strongly in C

([0, T ] ;L2 (Ω)

). (4.33)

An analogous estimate holds for F′2; that is, we have

F′2 (φε)→ F

′2 (ψ) strongly in C

([0, T ] ;L2 (Γ)

).

Finally, (4.28) and (4.29) yield θε → θ strongly in C([0, T ]; (ZK ,X2a)ξ), where

(, )ξ denotes the interpolation space. Taking the weak limit in both (4.9)and (4.10) yields that (4.1) holds in L2([0, T ];L2(Ω)) and (4.3) holds inL2([0, T ];ZK). Moreover, the equation (4.2) holds in L2([0, T ];H1(Ω)). Theboundary conditions (4.4) and (4.5) respectively, hold in L2([0, T ];H−1/2(Γ))and respectively, in L2

([0, T ] ;H1 (Γ)

). Finally, the boundary condition (4.6)

holds in L2([0, T ] ;H−1/2 (Γ)

).

Since ψt ∈ L2 ([0, T ] ; V1), by regularity of elliptic equations, we deducethat µ ∈ L2

([0, T ] ;H3 (Ω)

). Therefore, the existence result follows. Since

(φε, uε, θε) is a solution of(PKε)

that also satisfies the conservation laws (i),(ii) of Theorem 6, we may pass to limit as ε → 0+ in these equations. Itfollows that the solution (ψ, θ) of (4.1)–(4.7) satisfies (i) and (ii) of Theo-rem 9.

It remains to prove the uniqueness. For this purpose, let ψ1, θ1 and ψ2, θ2

be two solutions and let ψ := ψ1 − ψ2 and θ := θ1 − θ2. Then the functionsψ and θ and the corresponding µ satisfy

ψt = ∆µ , in Ω× (0,∞) , (4.34)

µ = −∆ψ + F′1 (ψ1)− F ′1 (ψ2)− λ0θ, in Ω× (0,∞) , (4.35)

1270 Ciprian G. Gal

θt −∆θ = −λ0ψt, in Ω× (0,∞) , (4.36)with the boundary conditions

∂nµ = 0, on Γ× (0,∞) ,1dψt = α∆Γψ − ∂nψ − βψ −

(F′2 (ψ1)− F ′2 (ψ2)

), on Γ× (0,∞) ,

andaθt + b∂nθ + cθ = 0, on Γ× (0,∞) ,

ψ|t=0 = ψ0, θ|t=0 = θ0.Multiplying equation (4.34) by µ, equation (4.35) by ψt and (4.36) by θrespectively, integrating over Ω and using the boundary conditions, we obtain

12d

dt

[‖ψ (t)‖2V1

+ ‖θ (t)‖2X2a

]+ ‖∇µ (t)‖22 + ‖∇θ (t)‖22 +

c

b

∥∥θ|Γ (t)∥∥2

2,Γ

+1d‖ψt (t)‖22,Γ + +

⟨F′1 (ψ1 (t))− F ′1 (ψ2 (t)) , ψt (t)

⟩2

+⟨F′2 (ψ1 (t))− F ′2 (ψ2 (t)) , ψt (t)

⟩2,Γ

= 0. (4.37)

Let us first define the following Neumann boundary-value problem asso-ciated with the Laplace equation. Let f ∈ L2 (Ω) be such that∫

Ωf dx = 0. (4.38)

The unique solution of −∆w = f in Ω∂nw = 0 on Γ, (4.39)

will be denoted by w = (−∆N )−1 f . Note that (4.38) is a compatibility con-dition which is necessary for the problem (4.39) to be well defined. Considernow the space H−1 endowed with the norm

‖v‖2H−1 =⟨

(−∆N )−1 (v − 〈v〉Ω) , (v − 〈v〉Ω)⟩

2+ 〈v〉2Ω .

Notice that when f := ψt (t) , it can be easily verified that the constraint(4.38) remains true for such choice of the function f since 〈ψt (t)〉Ω = 0.Consequently, we have

‖ψt (t)‖2H−1 = − (µ,∆µ)2 = ‖∇µ‖22 . (4.40)

We estimate the last two terms in relation (4.37) , using the assumptions onFi as follows:⟨

F′1 (ψ1)− F ′1 (ψ2) , ψt

⟩2

+⟨F′2 (ψ1)− F ′2 (ψ2) , ψt

⟩2,Γ

Global well-posedness 1271

≤ Q(‖ψ1‖V2−$

+ ‖ψ2‖V2−$

) [|〈ψ,ψt〉2|+

∣∣∣〈ψ,ψt〉2,Γ∣∣∣]≤ ρ

(‖ψt‖2H−1 + ‖ψt‖22,Γ

)+ Cρ

(‖ψ‖2H1 + ‖ψ‖22,Γ

), (4.41)

for a sufficiently small ρ > 0 and positive constants C and Cρ independentof t and ε. Here Q is a monotonic function that depends at most on the sizeof F1 and F2, and is independent of t and ε. Finally, applying a suitableversion of Gronwall’s inequality to (4.37) and using (4.40) and (4.41) yieldsthe continuous dependence result of Theorem 10. Consequently, it followsthat ψ (t) = 0 and θ (t) = 0 on [0, T ] . We have completed the proof ofTheorem 9.

5. The one-dimensional case

In the one-dimensional case, the surface diffusion term ∆Γ does notappear in the boundary conditions (2.5) of the approximate problem (Pε).Therefore, some changes have to be made in the previous sections. Forinstance, in Section 2, we should replace the space Vs by Hs (Ω) (for ev-ery s ≥ 0) and its corresponding inner product. Moreover, since the Hs-regularity result (i.e., Lemma 2) does not hold for the auxiliary linear prob-lem (2.6)–(2.8), instead we should use the one-dimensional elliptic theory forRobin boundary conditions (compare to (2.5)). Nevertheless, estimates ofTheorem 3 hold in this case as well.

Consequently, with these remarks in mind, we have the correspondingexistence result for the system of equations (4.1)–(4.7) for one-dimensionalbounded domains.

Theorem 11. Suppose ψ0 ∈ H3 (Ω) , θ ∈ ZK , K ∈ D,N,R,W0,W1 .Then the initial-value problem (4.1)–(4.7) admits a unique global solution(ψ, θ) such that for any ω > 0, we have

ψ ∈ C([0,+∞) ;H2−ω (Ω)

),

ψt ∈ L2loc

([0,+∞) ;H1 (Ω)

), ψ ∈ L2

loc

([0,+∞) ;H3−ω (Ω)

),

µ ∈ L2loc

([0,+∞) ;H3 (Ω)

)∩ L∞loc

([0,+∞) ;H1 (Ω)

),

θ ∈ C([0,+∞) ;H1−ω (Ω)

)∩ L2

loc ([0, T ] ;D (AK)) ,

θt ∈ L2loc ([0,+∞) ; Xa

2) .Moreover, the solution (ψ, θ) satisfies the same conservation laws of Theo-rem 9 and the continuous-dependence result of Theorem 10.

1272 Ciprian G. Gal

References

[1] S. Aizicovici and H. Petzeltova, Asymptotic behaviour of solutions of a conservedphase-field system with memory, Journal of Integral Eqn. and Appl., 15 (2003), 217–240.

[2] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Diff.Eqns., 72 (1988), 201–269.

[3] H. Amann and J. Escher, Strongly continuous dual semigroups, Ann. Mat. Pura Appl.(IV) CLXXI (1996), 41–62.

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

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

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

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

[8] J.W. Cahn and E. Hilliard, Free energy of a nonuniform system, J. Chem. Phys., 28(1958), 258–367.

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

[10] The dynamics of a conserved phase-field system: Stefan-like, Hele–Shaw, and Cahn–Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77–94.

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

[12] L. Cherfils and A. Miranville, Some remarks on the asymptotic behavior of the Cagi-nalp system with singular potentials, Adv. Math. Sci. Appl., to appear.

[13] P. Colli, G. Gilardi, M. Grasselli, and G. Schimperna, Global existence for the con-served phase field model with memory and quadratic nonlinearity, Portugaliae Math-ematica, 58 (2001).

[14] P. Colli and P. Laurencot, Uniqueness of weak solutions to the phase-field model withmemory, J. Math. Sci. Univ. Tokyo, 5 (1998), 459–476.

[15] A. Damlamian, N. Kenmochi, and N. Sato, Subdifferential operator approach to a classof nonlinear systems for Stefan problems with phase relaxation, Nonlinear Anal., 23(1994), 115–142.

[16] M. Efendiev, A. Miranville, and S. Zelik, Exponential attractors for a singularly per-turbed Cahn–Hilliard system, Math. Nachr., 272 (2004), 11–31.

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

[18] C.M. Elliott and S. Zheng, On the Cahn–Hilliard equation, Arch. Rational Mech.Anal., 96 (1986), 339–357.

[19] H.P. Fischer, Ph. Maass, and W. Dieterich, Novel surface modes of spinodal decom-position, Phys. Rev. Letters, 79 (1997), 893–896.

Global well-posedness 1273

[20] H.P. Fischer, Ph. Maass, and W. Dieterich, Diverging time and length scales of spin-odal decomposition modes in thin flows, Europhys. Letters, 62 (1998), 49–54.

[21] A. Favini, G.R. Goldstein, J.A. Goldstein, and S. Romanelli, The heat equation withgeneral Wentzell boundary conditions, J. Evol. Eq., 2 (2002), 1–19.

[22] A. Favini, G. Ruiz Goldstein, J.A. Goldstein, and S. Romanelli, The heat equationwith nonlinear general Wentzell boundary condition, Adv. Differential Equations, 11(2006), 481–510.

[23] C.G. Gal, Exponential attractors for a non-isothermal viscous Cahn–Hilliard modelequation dynamic boundary conditions, submitted.

[24] C.G. Gal, A Cahn–Hilliard model in bounded domains with permeable walls, Math.Meth. Appl. Sci., 29 (2006), 2009–2036.

[25] C.G. Gal, Exponential attractors for a Cahn–Hilliard model in bounded domains withpermeable walls, Electronic J. of Diff. Equations, (2006).

[26] G. Ruiz Goldstein, Derivation of dynamical boundary conditions, Adv. DifferentialEquations, 11 (2006), 457–480.

[27] M. Grasselli, H. Petzeltova, and G. Schimperna, Asymptotic behaviour of a non-isothermal viscous Cahn–Hilliard equation with inertial term, submitted.

[28] L. Hormander, Linear partial differential operators, Grundlehren Math. Wiss. 116,Springer-Verlag, Berlin (1976).

[29] R. Kenzler, F. Eurich, Ph. Maass, B. Rinn, J. Schropp, E. Bohl, and W. Dieterich,Phase separation in confined geometries: Solving the Cahn–Hilliard equation withgeneric boundary conditions, Computer Phys. Comm., 133 (2001), 139–157.

[30] N. Kenmochi, M. Niezgodka, and I. Pawlow, Subdifferential operator approach to theCahn–Hilliard equations with constraint, J. Diff. Equations, 117 (1995), 320–356.

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

[32] A. Miranville and S. Zelik, Exponential attractors for the Cahn–Hilliard equation withdynamical boundary conditions, Math. Models Appl. Sci., 28 (2005), 709–735.

[33] A. Novick–Cohen, The Cahn–Hilliard equation: Mathematical and modeling perspec-tives, Adv. Math. Sci. Appl., 8 (1998), 965–985.

[34] A. Novick–Cohen, On the viscous Cahn–Hilliard equation, in “Material instabilities incontinuum mechanics (Edinburgh, 1985–1986),” Oxford Sci. Publ., 329–342, OxfordUniv. Press, New York, 1988.

[35] J. Pruss, Maximal regularity for abstract parabolic problems with inhomogeous bound-ary data in Lp spaces, Proceedings “Equadiff, 10, Prague, August 27–31” (2001).

[36] J. Peetre, Another approach to elliptic boundary value problems, Comm. Pure Appl.Math., 14 (1961), 711–731.

[37] J. Pruss, R. Racke, and S. Zheng, Maximal regularity and asymptotic behavior ofsolutions for the Cahn–Hilliard equation with dynamic boundary conditions, Ann. Mat.Pura Appl., 185 (2006), 627–648.

[38] J. Pruss and M. Wilke, Maximal Lp regularity for the Cahn–Hilliard equation withnon-constant temperature and dynamic boundary conditions, in “Partial Differen-tial Equations and Functional Analysis,” Oper. Theory Adv. Appl. 168, 209–236,Birkhauser, Basel, 2006.

[39] R. Racke and S. Zheng, The Cahn–Hilliard equation with dynamical boundary condi-tions, Adv. Differential Equations, 8 (2003), 83–110.

1274 Ciprian G. Gal

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

[41] G. Schimperna, Abstract approach to evolution equations of phase field type and ap-plications, J. Differential Equations, 164 (2000), 395–430.

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

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

[44] M.I. Visik, On general boundary problems for elliptic differential equations (in Rus-sian), Trudy Moskow. Math. Obsc., 1 (1952), 187–246.

[45] S. Zheng, Asymptotic behavior of solutions to the Cahn–Hilliard equations, Appl.Anal., 3 (1986), 165–184.

[46] S. Zheng, Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, Pit-man Monographs Surv. Pure Appl. Math. 76, Longman; John Wiley & Sons, NewYork (1995).