Cahn Hilliard equations with memory and dynamic boundary conditions

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1 Asymptotic Analysis 71 11) DOI 1.333/ASY IOS Press Cahn Hilliard equations with memory and dynamic boundary conditions Cecilia Cavaterra a, Ciprian G. Gal b and Maurizio Grasselli c a Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, 133 Milano, Italy cecilia.cavaterra@unimi.it b Department of Mathematics, University of Missouri, Columbia, MO 6511, USA galc@missouri.edu c Dipartimento di Matematica F. Brioschi, Politecnico di Milano, 133 Milano, Italy maurizio.grasselli@polimi.it Abstract. We consider a modified Cahn Hiliard equation where the velocity of the order parameter u depends on the past history of Δμ, μ being the chemical potential with an additional viscous term αu t, α. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials e.g., glasses). In addition, the usual noflux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the socalled past history space. Existence of a variational solution is obtained. Then, in the case α>, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a smooth) global attractor as well as an exponential attractor. In the case α =, we only establish the existence of a trajectory attractor. Keywords: Cahn Hilliard equations, dynamic boundary conditions, global attractors, exponential attractors, trajectory attractors, memory relaxation 1. Introduction Let Ω R 3 be a given bounded domain with a sufficiently smooth boundary Γ. The Cahn Hilliard equation see [], cf. also [44] and references therein) with t u = Δμ, 1.1) μ = α t u Δu + fu) 1.) in Ω τ, ), τ being an arbitrary initial time, plays a basic role in Materials Science. In particular, it gives a good description of the so-called spinodal decomposition of passive) binary mixtures. This phase separation phenomenon takes place, for example, in cooling processes of metallic alloys, glasses or polymer mixtures. Here u is the relative concentration difference of the mixture components, μ is the chemical potential with an additional viscous term representing some microscopic effects see, e.g., [43], /11/$ IOS Press and the authors. All rights reserved

2 14 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions cf. also [33]). Moreover, f is the derivative of a potential function F which accounts for the presence of different phases. This potential is a logarithmic-type function defined on a bounded interval say, 1, 1)), which is often approximated by a double-well polynomial potential, namely, F y) = y 1). Such potentials, defined on the whole R, are usually known as smooth, while the logarithmic-like are named singular. Equations 1.1) and 1.) are usually subject to the so-called no-flux boundary conditions which entail the conservation of mass. In this case and, also, with other standard boundary conditions), the corresponding initial and boundary value problem has been extensively studied by many authors see [44] for some references). In particular, Cahn Hilliard equation has also been analyzed as a dissipative dynamical system see [5]) and there are satisfactory results on global and exponential attractors cf., for instance, [1]) and the convergence of solutions to steady states see, e.g., [5]). More recently, some physicists have modeled phase separation in binary systems when the dynamic interactions of the material with the walls must be taken into account cf., for instance, [11,1,37]). Mathematically speaking, this usually leads to additional terms in the free energy and then to a boundary condition characterized by the presence of t u. More precisely, such dynamic condition has the form t u Δ Γ u + n u + βu + gu) = on Γ τ, ). 1.3) Here, the constant β is positive, Δ Γ denotes the Laplace Beltrami operator on the surface Γ, while g is the derivative of a sufficiently smooth boundary potential G. In the original derivation, one has Gs) = c 1 y c,wherec 1 > accounts for a modification of the effective interaction between the components with the walls, while c orc < ) characterizes the possible preferential attraction or repulsion) of one of the components by the walls. The mathematical analysis of the Cahn Hilliard equation endowed with 1.3) and no-flux condition for μ to preserve mass conservation), that is, n μ = on Γ τ, ), 1.4) has recently been the subject of a number of papers. For instance, the case of smooth potentials has been analyzed in [5,4,47 49,54] see also [13,15,3] for nonisothermal models). The more delicate case of singular e.g., logarithmic) potentials has been considered in [5,4] cf. also [4] for nonisothermal systems). On the other hand, a modification of the Cahn Hilliard equation has proposed to account for rapid spinodal decomposition in certain materials like glasses see, for instance, [16,38]). From a general viewpoint, this modification consists in replacing Eq. 1.1) with t ut) = ks)δμt s)ds in Ω, t τ, ), 1.5) where k :, ), ) is a smooth) decreasing and summable relaxation kernel. This means that relaxation effects are described through a distributed delay. In particular, if ks) = 1 ɛ e s/ɛ,whereɛ> is a given relaxation time, Eq. 1.5) is equivalent to a purely differential equation of the form ɛ tt u + t u Δμ = in Ω τ, ). 1.6) Equation 1.6) has received lot of attention in the last years. The viscous three-dimensional case has been firstly analyzed in [1] see also [34,35]). It is worth observing that, in this case, the solutions regularize

3 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 15 in finite time as in the classical case ɛ = ). The nonviscous case is much more troublesome even in dimension two, since there is a lack of regularization effects a sort of hyperbolic behavior). Only recently, the understanding of the case α = has significantly improved see [8 3]). For the simpler one-dimensional case, the reader is referred to [1] and its references. Equation 1.5) with a more general memory kernel which does not allow to write an equivalent PDE) has been firstly studied in [] in the one dimensional nonviscous case see also [3] for no-flux boundary conditions). The corresponding dynamical system has been analyzed within the past history setting with particular regard to the stability with respect to the relaxation time. The three-dimensional nonisothermal nonviscous case has been considered in [39] and the existence of a weak) solution has been proven. In the related paper [53], well-posedness and regularity results have been obtained by taking memory kernels which are singular at. We recall that this assumption, contrary to nonsingular kernels, produces regularization effects. More recently, Eq. 1.6) with α> has been carefully analyzed in the spirit of [1]. This means that the kernel k has been rescaled with a relaxation time ɛ>and the robustness properties of the dynamical system with respect to α and ɛ have been investigated. In particular, the authors have constructed a family of exponential attractors which is continuous as α, ɛ) goes to, ), provided that ɛ is dominated by α. Of course, the latter restriction is expected due to the difficulties arising in the relaxed nonviscous case. In other words, the dynamical system in the history phase space) generated by 1.6) is close to the one corresponding to the original Cahn Hilliard equation whenever the viscosity and the relaxation time are small. A further recent result concerns 1.5) with α = in dimension two see [6]). There, using some ideas of [9], existence of a global smooth) attractor for energy solutions see [9] for this terminology) is established for small relaxation times. All the mentioned results on Eq. 1.5) or 1.6)) have been obtained for, say, standard boundary conditions no-flux, Dirichlet-type or periodic) and smooth potentials. Our goal is to study such equation subject to dynamic boundary conditions with a smooth potential. Handling singular potentials appears out of reach for the moment. More precisely, we consider 1.) 1.5) with initial condition on the whole past history of u, up to the initial time τ, ut) t τ = u t) 1.7) for some given u. We point out that, even in the case α>, the presence of a generic memory kernel implies that the dynamical system has some hyperbolic-like features related to the evolution of the past history variable see, e.g., [31]). This aspect has also been present in [8]. Nonetheless, in that paper, only Dirichlettype boundary conditions have been considered, while boundary condition 1.3) brings new additional technical difficulties even at the well-posedness level). In fact, standard techniques based on the use of fractional powers of operators cannot be exploited see, for instance, [4]). However, by employing refined energy estimates and regularity theory for suitable elliptic boundary value problems in the spirit of [4]), we can overcome such difficulties and, successfully, prove the existence of smooth) global attractors and exponential attractors for 1.) 1.5) subject to 1.7), in the viscous case. In the much harder nonviscous case we are only able to establish the existence of an energy solution which satisfies a suitable dissipative estimate. This allows us to prove the existence of a trajectory attractor according to [4] see also [3,51] for Eq. 1.6)). The construction of a family of exponential attractors which is robust with respect to α and ɛ as in [8]) seems at present rather difficult to carry out. Apparently, this is due to the lack of a certain regularity property of the solution which cannot be recovered in the case of dynamic boundary conditions see Remark 3.7).

4 16 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions The plan of this paper goes as follows. In Section we introduce the functional framework associated with our problem. Then, in Section 3, we state and prove the well-posedness result by establishing the existence of energy solutions within a suitable Galerkin approximation scheme see Theorem 3.3). In Section 4, we first demonstrate the existence of bounded absorbing sets and of a compact exponentially attracting set for energy solutions whenever α >. Hence, always in the case α >, we deduce the existence of a smooth) global attractor cf. Theorems 4.1 and 4.5). As a consequence, the global attractor contains only quasi)strong solutions. The finite dimensionality of the global attractor follows from the existence of an exponential attractor which is proven in Section 5 see Theorem 5.1). Finally, in Section 6 we show the existence of a trajectory attractor and, consequently, of a weak global attractor in the nonviscous case see Theorem 6.5). Past history formulation and functional setup In order to formulate problem 1.) 1.5), 1.7) in a proper functional setting we will follow the wellestablished past history approach see [31], cf. also [8,]). To this end, we introduce the past history variable η t s) = s Δμt y)dy.1) for any s>andt>τ. We observe that η satisfies the boundary condition: η t ) = in Ω τ, )..) If k is smooth enough and vanishing at, then an integration by parts in time, of the convolution product appearing in Eq. 1.5), leads to the following formulation of our original problem, where ν = k. Problem P. Find u, ψ, η) with ut) Γ = ψt)inτ, ) such that t u + νs)ηs)ds = inω τ, ),.3) t ψ Δ Γ ψ + n u + βψ + gψ) = on Γ τ, ),.4) μ = α t u Δu + fu) in Ω τ, ),.5) t η + s η = Δμ in Ω, ) τ, ),.6) subject to 1.4),.) and satisfying the initial conditions uτ) = u τ), ψτ) = ψ τ) in Ω, η τ = η in Ω, )..7) Note that η is defined as follows η s) = s Δμ τ y)dy in Ω, s>,

5 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 17 where μ t) = α t u t) Δu t) + fu t)), for t τ. Nevertheless, from now on η will be regarded as an independent initial datum. In other words, we will consider a more general problem with respect to the original one. However, the notion of a solution satisfying.3).7) is still too formal at this point. In order to give a rigorous definition of a solution to P cf. Definition 3.1), we need to introduce some terminology and the functional setting associated with.3).7). In the sequel, we denote by and Γ the norms on L Ω)andL Γ ), whereas the inner products in these spaces are denoted by, )and, ) Γ, respectively. Furthermore, the norms on H r Ω)andH r Γ ), r>, are indicated by H r and H r Γ ), respectively. The symbol, ) V,V stands for a duality pairing between any generic Banach spaces V and its dual V. Let us define the linear operator A N := Δ acting on DA N ) = {θ H Ω): n θ = onγ } as the realization in L Ω) of the Laplace operator endowed with Neumann boundary conditions. It is well known that A N generates a bounded analytic semigroup e A N t on L Ω) andthata N is nonnegative and self-adjoint on L Ω). Moreover, denoting by the spatial average over Ω and setting H) r Ω) = {φ Hr Ω): φ = }, H = L, we know that A 1 N : H ) Ω) H )Ω) is well defined. Henceforth, we will always refer to the following norms in H r Ω) := H r Ω)), r N +, u H r = A r/ ) N u u + u, which are equivalent to the standard ones. The Dirichlet trace map Tr D : C Ω) C Γ ), defined by Tr D u) = u Γ, extends to a linear continuous operator Tr D : H r Ω) H r 1/ Γ ), for all r>1/, which is onto for 1/ <r<3/. This map also possesses a bounded right inverse Tr 1 D : Hr 1/ Γ ) H r Ω) such that Tr D Tr 1 D ψ) = ψ, foranyψ Hr 1/ Γ ). We can thus introduce the subspaces of H r Ω) H r Γ ), V r := { u, ψ) H r Ω) H r Γ ): Tr D u) = ψ } for every r>1/, and note that we have the following dense and compact embeddings V r1 V r,for any r 1 >r > 1/. We emphasize that V r is not a product space. Let us now introduce the spaces for the memory variable η. For a nonnegative measurable function θ defined on R + and a real Hilbert space W with inner product denoted by, ) W ), let L θ R +; W )bethe Hilbert space of W -valued functions on R +, endowed with the following inner product φ 1, φ ) L θ R + ;W ) = θs) φ 1 s), φ s) ) W ds..8) Consequently, we set M σ = L ν R+ ; H σ Ω) ), M ) σ = L ν R+ ; H)Ω) σ ) for σ R. In the sequel, we will also consider Hilbert spaces of the form H k ν R + ; H σ Ω)) for k N. We conclude by observing that solutions to Problem P must also satisfy the following constraints i.e., mass conservations) ut) = u τ), η t s) =.9)

6 18 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions for t>τand s>. These can be easily derived recalling Eqs 1.4), 1.5) and using.1). For this reason, we need to introduce closed subsets of Hilbert spaces H r,σ = V r M ) σ 1 for r>1/, σ R, endowed with the scalar product induced by V r M ) σ 1, namely, H M r,σ = { u, ψ, η) V r M ) σ 1 : u M }, for some given M. Note that H M r,σ is a complete metric space with respect to the metric induced by the H r,σ -norms. We now have all the ingredients to introduce a rigorous formulation of P in the next section. 3. Variational formulation and well-posedness We begin this section by stating all the hypotheses on ν, f and g we need, even though not all of them must be satisfied at the same time. Conditions on ν: ν C 1 R + ) L 1 R + ): νs) s R +, K1) ν s) s R +, K) k = νs)ds>, ν = lim νs) <, s + K4) ν s) + λνs) for a.a. s R +, K5) for some λ>. In the rest of the paper we will take k = 1 for the sake of simplicity. Also, we recall that assumption K5) plays a role in the long term behavior only. Conditions on f and g: f, g C 1 R), N) f, g C R), N ) lim inf y) >, y + lim inf y + g y) >, N1) f y) c f 1 + y ), g y) c g 1 + y q ) y R, N) f y) c f 1 + y ), g y) c g 1 + y q 1 ) y R N ) for some positive constants c f, c g and some q [1, + ). Note that, for instance, double-well smooth potentials like F y) = y 1) are such that f = F satisfies all the above assumptions, while g can be any polynomial of odd degree with a positive leading coefficient. In order to introduce the variational formulation of P, we consider the bilinear symmetric form Υ in L Ω) L Γ )definedby Υ u, v) = Ω K3) u vdx + Γ ψ Γ ϕ ds + β ψϕ ds 3.1) Γ Γ

7 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 19 for all u = u, ψ), v = v, ϕ) V 1. Then, Υ with domain DΥ ) = V 1 V 1 ) is densely defined, closed and bounded from below in L Ω) L Γ ). Thus, by first and second representation theorems for forms see, e.g., [36], Section VI.), one concludes that there exists a unique self-adjoint operator A Γ in L Ω) L Γ ), bounded from below, associated with the form Υ, which satisfies Υ u, v) = u, A Γ v) L Ω) L Γ ) = Ω u Δv)dx + ψ Δ Γ ϕ + n v + βϕ)ds 3.) Γ for all u V 1, v DA Γ ). Moreover, DA Γ ) DA 1/ ) = DΥ ) is such that Γ DA Γ ) = { v, ϕ) V 1 : Δv L Ω), Δ Γ ϕ + n v + βϕ L Γ ) }. 3.3) Besides, from 3.1) and 3.), we deduce that A Γ is bounded from V 1 into V 1, and according to [14], Section 3, we have the following more explicit characterization of the domain for A Γ,thatis,DA Γ ) = V. Indeed, the map Ξ : u A Γ u,whenviewedasamapfromv into L Ω) L Γ ), is an isomorphism and, cf., e.g., [3], Lemma., there exists a positive constant C, independent of u, ψ), such that C 1 u, ψ) V Ξu) L Ω) L Γ ) C u, ψ) V for all u = u, ψ) V. To formulate the evolution equation for η, we need to define the linear operator T = s with domain DT) = { η M ) 1 : sη M ) 1, η) = }. 3.4) We recall that T is the infinitesimal generator of the right translation semigroup on M 1 see, e.g., [31]). We are now ready to introduce the rigorous variational) formulation of Problem P. Definition 3.1. Let T>τand set J = [τ, T ]. A triplet u, ψ, η) which fulfills u, ψ, η) L J; H 1, ), 3.5) t u L J; H 1 Ω) ), α 1/ t u L Ω J), t ψ L Γ J) 3.6) is a solution to Problem P in J with initial data u, ψ, η ) H 1, if the following identities hold almost everywhere in J: t u, v) H 1,H 1 + νs) ηs), v ) H 1,H ds = v H 1 Ω), 3.7) 1 Υ u, v) + fu), v ) + gψ), ϕ ) Γ + α tu, v) + t ψ, ϕ) Γ = μ, v) H 1,H 1 v = v, ϕ) V 1, 3.8) t η Tη, ζ) M 1 = μ, ζ) M ζ M ) As a consequence of 3.7), there holds ut) = u for a.a. t J. 3.1)

8 13 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions Remark 3.. As far as the interpretation of the initial condition is concerned, note that properties 3.5) and 3.6) imply that u = u, ψ) CJ; V 1 r )foranyr, 1/). On the other hand, it can be shown that η CJ; M 1 ) see [31]). We now state and prove the following global existence result. Note that the notion of solution we are looking for is more general than the usual notion of weak solutions. Solutions of this kind have been called energy solutions in [9]. Theorem 3.3. Let K1) K4) and N) N) hold. Then, for any given initial time τ R and u, ψ, η ) H 1,, Problem P admits at least one solution u, ψ, η) on the time interval J, according to Definition 3.1. In addition, this solution satisfies the following inequality: ut), ψt), η t) H 1, + t α t uy) + t ψy) ) Γ dy Q u, ψ, η ) H τ 1,) for t τ, for some positive monotone nondecreasing function Q independent of t and α. 3.11) Proof. The proof is divided into three steps. We make use of a Faedo Galerkin scheme similar to [13, 5]) to project Problem P on a finite dimensional space, where we can locally solve a Cauchy problem for a system of ordinary differential equations P n, having local solutions. Then, we provide some a priori estimates which guarantee that any local solution to Problem P n is actually global on J. The a priori boundedness is also used to extract a subsequence which converges weakly to a certain vector-valued function, candidate to be the solution to Problem P. Step 1: The Galerkin approximation. We begin by choosing a smooth orthonormal basis {v j }, j N +, in L Ω), which can be achieved, for instance, by taking the set {v j } of the eigenfunctions normalized in L Ω)) for the operator A N in L Ω), that is, A N v j = ϖ j v j, j N +,where{v j } DA N )and ϖ j is the eigenvalue corresponding to v j. In particular, it is well known that ϖ 1 = is simple with constant eigenfunction v 1 = Ω 1/, while all the other eigenvalues ϖ j, j, are positive and can be increasingly ordered and counted according to their multiplicities in order to form a real divergent sequence. Next, we take a smooth sequence {z j } HνR 1 + ; H) 1 Ω)) DT) forming an orthonormal basis of M ) 1.Forn N, we consider the projections O n and Q n,givenby O n : H 1 Ω) V n = span{v 1, v,..., v n }, Q n : M ) 1 Mn = span{z 1, z,..., z n } and set V = V n, M = M n. n=1 n=1 Clearly, V is a dense subspace of H 1 Ω) and thus, of L Γ ), since DA N ) is densely contained in L Γ ). Moreover, M is a dense subspace of M ) 1, respectively. Furthermore, for each v n V n CΩ), we can define elements of the form v n = v n, w n ), with w n = Tr D v n ) and note that V := span{v 1, v,..., v n } n=1

9 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 131 is dense in V 1 see, e.g., [5]). It is convenient to introduce a further projector on the trace component Õ n : H 1 Ω) Ṽ n = span{w 1, w,..., w n }. For each given n N, our approximating problem is Problem P n. Find t n τ, T ]anda k := a n k C1 [τ, t n ]; R n ), k = 1,, 3, such that u n t) = η t ns) = n a 1j t)v j, ψ n t) = j=1 n j=1 solve the system t u n, v) + a j t)z j s), μ n t) = n a 1j t)w j, 3.1) j=1 n j=1 a 3j t)v j, 3.13) νs) η n s), v ) ds =, 3.14) Υ u n, v) + fu n ), v ) + gψ n ), w ) Γ + α t u n, v ) + t ψ n, w) Γ = μ n, v), 3.15) t η n Tη n, ζ) M 1 = μ n, ζ) M 3.16) for every t τ, t n )andeveryv = v, w) X n and ζ M n, respectively, and with u n τ) = O n u, ψ n τ) = Õnu, η τ n = Q n η. 3.17) Let us write 3.14) 3.17) as a Cauchy problem for a nonlinear system of ordinary differential equations, choosing v = v i, w i ) in 3.14) and 3.15) and ζ = z i in 3.16), for i = 1,..., n. To this end, we define the n n matrices B Γ = l ij ), B = b ij ), D = d ij )andd = d ij )givenby l ij = w j, w i ) Γ, b ij = z j, v i ) M 1, d ij = Tz j, z i ) M 1, d ij = ϖ j v j, z i ) M ) Observe that B Γ is positive semidefinite by construction. After performing direct computations in 3.14) 3.16) and taking 3.18) into account, we can rewrite P n into a Cauchy problem for the following system: a 1 = Ba, 3.19) a = Da + Da 3, 3.) a 3 = Ca 1 ) + αi + B Γ )a 1, 3.1) where I is the n n identity matrix and Ca 1 ) = c i a 1 )) : R n R n is given by c i a 1 ) = n a 1j Υ v j, v i ) j=1 n ) ) n ) ) + f a 1j v j, v i + g a 1j w j, w i, i = 1,..., n. 3.) j=1 j=1 Γ

10 13 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions It is now easy to deduce from 3.19) 3.1) a system of n nonlinear ordinary differential equations in normal form for the vector-valued functions a k, k = 1,, that is, a ) 1 a ) a1 = U n, 3.3) a where U n : R n R n can be computed explicitly. On account of N), we can thus apply the Cauchy Lipschitz theorem and find a unique maximal solution u n, ψ n, η n ) C 1 [τ, t n ]; H, )top n,forsome t n τ, T ). Step : A priori estimates. We will now deduce some a priori estimates on the approximating solutions u n, ψ n, η n ). Such estimates, in particular, ensure that the solutions obtained are defined on the whole interval J = [τ, T ], for every n N. Throughout this proof, c will denote a positive generic constant, depending at most on the physical parameters of the problem, but independent of n and α. Sucha constant may vary even within the same line. Further dependencies of the constants will be pointed out if needed. We first take v = μ n t) V n and v = t u n t), t ψ n t)) X n in 3.14) and 3.15), respectively. Adding these relations together, we deduce, for all t [τ, t n ), 1 d [ u n t) + Γ ψ n t) dt Γ + β ψ n t) Γ + G ψ n t) ),1 ) Γ + F u n t) ),1 )] + α t u n t) + t ψ n t) Γ + νs) ηns), t μ n t) ) ds =. 3.4) Next, recalling that η t ns) = Tη t ns) =, take ζ = η t ns) in 3.16). This yields 1 d dt ηn t M 1 ν s) η t ns) H 1 ds = μ n, η t ns) ) M t [τ, t n ). 3.5) Set now, for all t [τ, t n ), E n t):= u n t) + Γ ψ n t) Γ + β ψ n t) Γ + ηn t M 1 + F u n t) ),1 ) + G ψ n t) ),1 ) Γ + C. 3.6) Here the constant C > is taken large enough in order to ensure that E n t) is nonnegative note that, due to assumptions N1), F and G are both bounded from below, independently of n). Adding now 3.4) and 3.5), we obtain d dt E nt) + α t u n t) + t ψ n t) Γ ν s) ηns) t H ds = 3.7) 1 for all t [τ, t n ]. Then integrating 3.7) with respect to time, on account of assumption K), we get E n t) + t [ α t u n y) + t ψ n y) Γ τ ] dy En τ) t [τ, t n ). 3.8)

11 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 133 Observe that, in light of N1) and N), there exists a positive constant c, independent of the initial data, such that c u n t), ψ n t), ηn t ) H 1, E n t) Q u n t), ψ n t), ηn) t ) H1, t [τ, t n ) 3.9) for some monotone nondecreasing function Q independent of n. In particular, the uniform bound 3.8) implies that the local solution to P n can be extended up to time T,thatis,t n = T,foreveryn. By 3.8) and 3.9), we learn that the sequence u n, ψ n, η n ) is uniformly bounded in the space H 1, by a constant depending only on the size of initial data and of the nonlinearities f, g. Integrating 3.7) over J, we get a further uniform bound for t u n and t ψ n in L Ω J) andl Γ J), respectively. Also we get a control of η n in L J; L ν R +; H 1 Ω))). Summing up, we have u n, ψ n ) L J;V 1 ) c, α t u n L Ω J) c, 3.3) η n L J;M 1 ) c, η n L J;L ν R +;H 1 Ω))) c, tψ n L Γ J) c. 3.31) Actually we can prove some more regularity properties for u n, ψ n, η n ). To this end, take v = A 1 N tu n t) and observe that t u n t) =. Exploiting the basic Hölder and Young inequalities, it is not difficult to realize, on account of K3), that t u n t) H 1 ηn t M 1 t J. 3.3) Thus, 3.31) and 3.3) yield t u n L J;H 1 Ω)) c. 3.33) Furthermore, thanks to the first bound of 3.3), we can easily control the nonlinear terms in 3.15). In particular, on account of N) and the embeddings H 1 Ω) L 6 Ω), H 1 Γ ) L p Γ ), for all p [1, ), we have fu n ) L Ω J) c, gψ n ) L q+1 Γ J) c. 3.34) Finally, by comparison in 3.15), the fact that the operator A Γ is bounded from V 1 into V 1 in particular, A Γ u n L J;V 1 ) c), combined with 3.34) also provides the following uniform bounds for μ n : μ n c, μ n L J;H 1 Ω)) c. 3.35) Step 3: Passage to the limit. On account of the above uniform inequalities, we can argue, up to subsequences, that u n, ψ n ) u, ψ) in L w J; V 1), t ψ n t ψ in L w J; L Γ ) ), 3.36) t u n t u in L w J; H 1 Ω) ), α t u n α t u in L wω J), 3.37) η n η in L ) w J; M 1, μn μ in L J; H 1 Ω) ), 3.38) η n η in L w J; L ν R+ ; H 1 Ω) )). 3.39)

12 134 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions Due to 3.36) 3.38) and classical compactness theorems, we also have u n u strongly in C J; H 1 r Ω) ), 3.4) ψ n ψ strongly in C J; H 1 r Γ ) ) 3.41) for all r, 1/) so that ut), ψt)) CJ; H 1 r Ω) H 1 r Γ )). These strong convergences entail that, up to subsequences, u n converges to u,almosteverywhereinω J,andψ n converges to ψ,almost everywhere in Γ J. Then, using the continuity of f and g, 3.4), 3.41) and 3.34), we easily deduce that, up to subsequences, fu n ) weakly converges to fu)inl Ω J), while gψ n ) weakly converges to gψ) inl q+1 Γ J), and hence in L Γ J), since q 1. Thus we can now pass to the limit in Eqs 3.14) and 3.15) and recover 3.7) and 3.8), respectively. It remains to pass to the limit in Eq. 3.16). An integration by parts and 3.38) yield, for any ζ C J; C R +; H 1 Ω))), T t ηn, y ζ ) T M dy = 1 η y n, t ζ ) T M dy 1 η y, t ζ ) M dy. 1 τ Thus, we have t η t n t η t in L w τ J; H 1 ν R+ ; H 1 Ω) )), τ and that η t CJ; Hν 1 R + ; H 1 Ω))). Moreover, on account of 3.39), we have T Tη y n, ζ ) T M dy = τ 1 ν s) ηn, y ζ ) T H dy ν s) η y, ζ ) τ 1 H dy. τ 1 Exploiting a density argument and the following distributional equality see [31]) T τ η y, t ζ ) T M dy 1 ν s) η y, ζ ) H dy = 1 τ t τ t η t Tη y, ζ ) M 1 dy, we also get 3.9) on account of 3.38). The proof is finished. In the viscous case, we can also prove the existence of smoother solutions. Theorem 3.4. Let α>. Assume K1) K4) and N ), N1), N).Ifu, ψ, η ) H 3, is such that η DT), 3.4) then, for any given initial time τ R, Problem P admits at least one solution u, ψ, η) on the time interval J, according to Definition 3.1. In addition, this solution has the following regularity properties: u, ψ, η) L J; H 3, ) W 1, J; H 1, ), 3.43) tt u L J; H 1 Ω) ), α 1/ tt u L Ω J), tt ψ L Γ J), 3.44) η L J; DT) ). 3.45)

13 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 135 Proof. We use again the approximation scheme developed in the proof of Theorem 3.3. In the sequel of this proof, c will indicate a positive constant which may depend on M, α, β, c f, c g, Ω, τ and T,butis independent of n. More precisely, referring to Problem P n, we will consider instead the approximating problem of finding u n, ψ n, μ n, η n ), of the form 3.1) and 3.13), such that tt u n, v) + νs) t η n s), v ) ds =, 3.46) Υ t u n, v) + f u n ) t u n, v ) + g ψ n ) t ψ n, w ) Γ + α tt u n, v) + tt ψ n, w) Γ = t μ n, v), 3.47) tt η n T t η n, ζ) M 1 = t μ n, ζ) M 3.48) for every t τ, t n ), v = v, w) X n and ζ M n, respectively, which satisfies initial conditions 3.17) and t u n τ) = O n ũ, t ψ n τ) = Õnũ, t η τ n = Q n η. 3.49) Here, we have set ũ := νs)η s)ds, ψ := Δ Γ ψ n u + βψ + gψ ), 3.5) η := Tη + A N μ, where μ = α νs)η s)ds Δu + fu ). 3.51) Note that, if u, ψ, η ) satisfies the assumptions of Theorem 3.4, then ũ, ψ, η ) H 1,. In fact, owing to the boundedness of the projectors O n, Õn and Q n on the corresponding subspaces, t u n τ), t ψ n τ), t ηn) τ H1, Q u, ψ, η ) H3, ), for some positive monotone nondecreasing function Q. Thus, the previous Galerkin scheme can be used. Arguing as for P n and recalling N ), for any fixed n N, we have a local unique maximal) solution u n, ψ n, η n ) C 3 [τ, t n ]; H, ). Also, if we integrate Eqs 3.46) and 3.47) with respect to time from τ to t τ, t n ), reasoning as in the proof of Theorem 3.3, we can find uniform bounds such as the ones in 3.3) and 3.31). To obtain higher-order estimates, we proceed for the sake of simplicity in a formal way by removing the index n and setting ũ = t u, ψ = t ψ, μ = t μ, η = t η,

14 136 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions so that ũ, ψ, η) solves t ũ, v) H 1,H 1 + νs) ηs), v ) H 1,H ds =, v H 1 Ω), 3.5) 1 Υ ũ, v) + f u)ũ, v ) + g ψ) ψ, w ) Γ + α tũ, v) + t ψ, w) Γ = μ, v) H 1,H 1 v V 1, 3.53) t η T η, ζ) M 1 = μ, ζ) M ζ M 1, 3.54) with initial conditions ũτ) = ũ, ψτ) = ψ, η τ = η. 3.55) We now take v = μt) in Eq. 3.5), v = t ũt) andw = t ψt) in 3.53), and ζ = η t in Eq. 3.54). Adding together the resulting identities, we obtain 1 d dtẽt) ν s) η t s) H ds + α 1 t ũt) + ψt) t Γ = f ut) ) ũt), t ũt) ) + g ψt) ) ψt), t ψt) ) Γ t J, 3.56) wherewehaveset Ẽt) = ũt) + Γ ψt) Γ + β ψt) Γ + η t M 1. By standard Hölder, Young and Sobolev inequalities, recalling N), we have f ut) ) ũt), t ũt) ) + g ψt) ) ψt), t ψt) ) Γ α t ũt) + 1 ψt) t Γ + c 1 + ut) H 1 ) ũt) H ψt) H 1 Γ )) ψt) H 1 Γ ) ) 3.57) for all t J. Then, using the known bound on u and ψ see 3.3)), on account of 3.57), integrating the inequality 3.56) with respect to time, we obtain, for all t J, Ẽt) t τ ν s) η y y) H 1 ds dy + α t τ t ũy) dy + t τ t ψy) Γ dy t Ẽτ) + c ũy) H + ψy) ) 1 H 1 Γ ) dy. 3.58) Then, the standard Gronwall lemma gives τ ũt), ψt), η t) H1, c t J. 3.59)

15 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 137 In addition, we have cf. also 3.3)) α t ũ L Ω J) + ψ t L Γ J) + η L J;L ν R +;H 1 Ω))) c. Here c also depends on u, ψ, η ) H3, and on Tη M 1. Let us now consider Eq. 3.9) or, more precisely, 3.16)), and take ζ = A N μt),wherewehaveset μt) = μt) μt). On account of 3.31) and 3.59), we obtain μt) = νs) μt), t η t s) ) ds ν s) μt), η t s) ) ds c ε t η t M 1 ν s) η t s) ) H ds + ε μt) 1 c ε 1 ν s) η t s) ) H ds 1 + ε μt) c ε + ε μt) 3.6) for all t J and every ε>, where c ε is a positive constant that depends also on ε. Then, recalling 3.35), we deduce the bound μt) H 1 Ω) c, t J. 3.61) Thus, resuming the index n, and reasoning as in the proof of Theorem 3.3, it is not difficult to realize that we can find a solution u, ψ, η) W 1, J; H 1, ) to Problem P which satisfies 3.44). Moreover, due to 3.61), we also have μ L J; H 1 Ω)). On the other hand, recalling 3.35) and 3.4), we have that ψ fulfills 3.45), thanks to the representation formula see [31]). Therefore, we are left to prove that u, ψ, η) L J; V 3 M 1 ). First of all observe that from Eq. 3.7), we deduce η t M 1 = νs) [ t ut), η t s) ) + t ut), η t s) )] ds, 3.6) that is, η L J; M 1 ). Finally, we consider the variational identity 3.8). On account of the regularity properties of t u, t ψ and μ, we can use the results in [4], Appendix A see also [3], Lemma.) to infer that u, ψ) L J; V 3 ). This concludes the proof. The uniqueness of solutions to Problem P in the viscous case follows from the following continuous dependence estimate. Lemma 3.5. Let the assumptions of Theorem 3.3 hold and assume that α>. Letu i, ψ i, η i ) be two solutions to P corresponding to the initial data u i, ψ i, η i ) H 1,, i = 1,. Then, for any t τ, the following estimate holds: u 1 u )t) H + ψ 1 1 ψ )t) H 1 Γ ) + η t 1 η t + t α t u 1 u )y) + t ψ 1 ψ )y) Γ τ M 1 Ce Ct τ) u 1 u H 1 + ψ 1 ψ H 1 Γ ) + η 1 η M 1 ), 3.63) ) dy

16 138 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions where C is a positive constant depending on the norms of the initial data in H 1,,onΩ, Γ, β, α, butis independent of time. Proof. We first prove estimate 3.63) for the smooth solutions constructed in Theorem 3.4. Thus, we assume that u i, ψ i, η i ), i = 1,, satisfy the assumptions of Theorem 3.4. Then set u = u 1 u, ψ = ψ 1 ψ, η = η 1 η, where u i, ψ i, η i ) is a solution corresponding to u i, ψ i, η i ). Then, u, ψ, η) solves t u, v) H 1,H 1 + νs) ηs), v ) H 1,H ds = v H 1 Ω), 3.64) 1 Υ u, v) + fu 1 ) fu ), v ) + gψ 1 ) gψ 1 ), w ) Γ + α tu, v) + t ψ, w) Γ = μ, v) H 1,H 1 v = v, w) V 1, 3.65) t η Tη, ζ) M 1 = μ, ζ) M ζ M 1, 3.66) with initial condition uτ) = u := u 1 u, ψτ) = ψ := ψ 1 ψ, η τ = η := η 1 η. Taking v = μt) andv = t ut), t ψt)) in 3.64) and 3.65), respectively, then adding the obtained relations together, we deduce, for all t J, 1 d ut) dt + Γ ψt) Γ + β ψt) ) Γ + α t ut) + t ψt) Γ + νs) η t s), μ ) ds = f u 1 t) ) f u t) ), t ut) ) + g ψ 1 t) ) g ψ t) ), t ψt) ) Γ. 3.67) Moreover, take ζ = η t in 3.66). This gives, for all t J, 1 d η t dt M 1 ν s) η t s) H 1 ds = Adding 3.67) to 3.68), we thus obtain the differential inequality d dt Et) + α t ut) + t ψt) Γ where Rt), Et) := ut) + Γ ψt) Γ + β ψt) Γ + η t M 1, νs) μt), η t s) ) ds. 3.68) Rt) := f u 1 t) ) f u t) ), t ut) ) + g ψ 1 t) ) g ψ t) ), t ψt) ) Γ for all t J.

17 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 139 Observe that, thanks to N), there holds fy 1 ) fy ) c f 1 + y1 + y ) y 1 y, 3.69) gy 1 ) gy ) c g 1 + y1 q + y q) y 1 y. 3.7) Thus, using standard Young, Hölder and Sobolev inequalities, we can estimate the terms on the righthand side of 3) as follows: Rt) α t ut) + t ψt) Γ + c α 1 + u 1 t) 4 L 6 + u t) 4 L 6 ) ut) H 1 + c 1 + ψ 1 t) q L 3q Γ ) + ψ t) q L 3q Γ )) ψt) H 1 Γ ) t J 3.71) for some positive constants c and c α. Thus, combining 3) with 3.71) and recalling that u i, ψ i ) satisfy 3.11) for i = 1,, we deduce, for all t J, d dt Et) + α t ut) + t ψt) Γ C ut) H + ψt) ) 1 H 1 Γ ) 3.7) for some positive constant C which depends only on the norms of the initial data in H 1,. Therefore, an application of standard Gronwall s inequality to 3.7) yields Et) Ce Ct τ) Eτ) 3.73) for all t J. Then, an integration with respect to time from τ to t of 3.7), combined with 3.73), entails the desired estimate 3.63). To conclude, it is not difficult to realize that such an estimate still holds for weak energy) solutions given by Theorem 3.3, by employing standard approximation arguments note that the constants above do not depend on the extra-assumptions of Theorem 3.4). The proof of the lemma is finished. The following is a straightforward consequence of the above results. Corollary 3.6. Let the assumptions of Lemma 3.5 hold. Then, Problem P admits a unique solution u, ψ, η) C[τ, T ]; H 1, ), for all T>τ, and the nonlinear operator St):H 1, H 1, defined by St)u, ψ, η ) = ut), ψt), η t) t τ is a strongly continuous semigroup. Remark 3.7. The constant C in 3.63) blows up as α goes to, while in [8] a similar estimate holds with a constant independent of α. This is due to the extra-regularity u L loc R; H Ω)) see [8], Lemma 4. formula 17)). It is not at all clear whether this property holds in the present case. On the other hand, the independence from α is essential if one wants to analyze the robustness properties of exponential attractors as α goes to as in [8]). Remark 3.8. We recall that uniqueness in the nonviscous case α = ) is still an open problem in dimension three see [6,8,9]).

18 14 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 4. Global attractors Our aim in this section is to prove that St) possesses the global attractor in the phase space H M 1,,for some given M. This choice is due to the constraints.9). Thus, the main result of this section is the following theorem. Theorem 4.1. Let α>. Assume K1), K3) K5) and N ), N1), N ). Then, H1, M, St)) possesses the connected global attractor A M, which is bounded in H 3,. We begin to prove the existence of a bounded absorbing set in H1, M for St). Lemma 4.. Let the assumptions of Theorem 3.3 be satisfied and, in addition, replace K) with K5). Then, for any given initial data u, ψ, η ) H1, M, there exists a solution u, ψ, η) to P which satisfies the uniform estimate ut), ψt), η t) H 1, + t+1 α t uy) + t ψy) Γ t Q u, ψ, η ) H1, ) e ρt τ) + C M, 4.1) for each t τ.here,ρ, C M are positive constants which are independent of t, α and of the initial data, while Q is a monotone nondecreasing function which is independent of t and α. Proof. We will work again within the approximation scheme used in the proof of Theorem 3.3. Consider the functional Y n t) = E n t) + ε α u n t) + ψ n t) ) Γ ε where ε, λ) is sufficiently small. Here, we have set u n t) := u n t) u n τ), ψ n t) := ψ n t) u n τ). ) dy νs) u n t), A 1 N ηt ns) ) ds, 4.) Note that u n t) = t u n t) =, for all t>τ. On account of Eqs 3.14) 3.16), we observe preliminarily that d νs) u n t), A 1 N dt ηt ns) ) ds = t u n t) H νs) u 1 n t), A 1 N tηns) t ) ds = t u n t) H ν s) u 1 n t), A 1 N ηt ns) ) ds νs) u n t), μ n t) ) ds = t u n t) H ν s) u 1 n t), A 1 N ηt ns) ) ds 1 d α un dt[ + ψ n t) ] Γ u n t) Γ ψ n t) Γ β ψ n t), ψ n t) ) Γ f u n t) ), u n t) ) g ψ n t) ), ψ n t) ) Γ. 4.3)

19 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 141 Differentiating Y n with respect to time and recalling 3.6) and 3.7), we deduce d dt Y nt) + ξy n t) ν s) ηns) t H ds = h 1 n t). 4.4) Here ξ, ε) and h n t) = α t u n t) + ε t u n t) H 1 t ψ n t) Γ ε ν s) u n t), A 1 N ηt ns) ) ds ξ [ f u n t) ) u n t) F u n t) ),1 ) + g ψ n t) ) ψ n t) G ψ n t) ),1 ) ] Γ ε ξ) [ f u n t) ), u n t) ) + g ψ n t) ), ψ n t) ) ] Γ + ξ ηn t M 1 ε ξ) u n t) + Γ ψ n t) ) Γ + ξ ψ n t) Γ εβ ψ n t), ψ n t) ) ) Γ + ξε α u n t) + ψ n t) ) Γ εξ νs) u n t), A 1 N ηt ns) ) ds + ξ C. 4.5) Let us first notice that the Hölder and Young inequalities yield, on account of the fact that u n τ) uτ) M, ξ ψ n Γ εβψ n, ψ n ) Γ εβ ξ) ψ n Γ + C M, 4.6) where C M > depends obviously on M, ξ, ε, Ω and Γ, but is independent of n, t and α. Here S denotes the N-dimensional Lebesgue measure of a set S R N. On the other hand, due to assumption N1), we have, for all y R, F y) C F,Mτ c f y M τ ) + fy)y M τ ), 4.7) 1 F y) fy)y M τ ) + C F,M τ, 4.8) Gy) C G,Mτ c g y M τ ) + gy)y M τ ), 4.9) 1 Gy) gy)y M τ ) + C G,M τ, 4.1) where all the positive constants in 4.7) 4.1) are sufficiently large and we have set M τ := uτ). Consequently, using 4.6) 4.9) and the Poincaré Wirtinger inequality i.e., u n C Ω u n ), it is possible to estimate h n as follows: h n α t u n + ε t u n H 1 t ψ n Γ ε ξ) [ F u n ),1 ) + Gψ n ),1 ) ] Γ [ εβ ξ1 + c g + ε) ] ψ n Γ [ ε ξ1 + c f C Ω + εαc Ω ) ] u n ε ξ) Γ ψ n Γ ε ν s) u n, A 1 N η ns) ) ds εξ νs) u n, A 1 N η ns) ) ds + ξ η n M 1 + C M. 4.11)

20 14 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions It remains to estimate the integral terms on the right-hand side of 4.11). Employing the Poincaré Wirtinger inequality repeatedly, in light of K4), we have ε ν s) u n, A 1 N η ns) ) ds = ε ν s) A 1/ N u n, A 1/ N η ns) ) ds ε ν un s) + C 4 ν Ων η n s) ) H ds 1 ε u n εcων 4 ν s) η n s) H ds. 4.1) 1 Moreover, on account of K3) and 3.3), it is straightforward to check that εξ νs) u n, A 1 N η ns) ) ds + ξ η n M 1 + ε t u n H 1 εξc 4 Ω u n + [ ξε + 1) + ε ] η n M ) Thus, exploiting now inequality K5), it follows from 4.4), 4.11) 4.13) that we can fix the parameters ξ, ε small enough in order to find three positive constants ξ 1, ξ, ξ 3 such that d dt Y nt) + ξ 1 Y n t) + ξ ηn t M 1 + α t u n t) + t ψ n t) Γ + ξ 3 u n t) + Γ ψ n t) Γ + ψ n t) Γ ) CM. 4.14) Note that C M depends on M, ξ, ε as well as on the other physical parameters of the problem, but is independent of time, n, α and initial data. Besides, one can easily check that C 1 un t), ψ n t), ηn t ) H 1, C Y n t) Q u n t), ψ n t), ηn) t ) H1, 4.15) for all t τ and for some positive constants C 1, C and a positive monotone nondecreasing function Q independent of n and t. Applying Gronwall s inequality see, e.g., [7], Lemma.5) to 4.14), then integrating 4.14) and using 4.15), we infer the uniform estimate u n t), ψ n t), η t n) H 1, + t+1 α t u n y) + t ψ n y) Γ t Q u, ψ, η ) H1, ) e ρt τ) + C M t τ. 4.16) ) dy Thus, by the lower semicontinuity of the norms, we easily obtain estimate 4.1). Consequently, we have the following proposition. Proposition 4.3. Let the assumptions of Lemma 3.5 hold. Then there exists R > independent of time, α and initial data) such that St) possesses an absorbing ball B M 1, R ) H M 1,, which is bounded in HM 1, independently of α. More precisely, for any bounded set B H M 1,, there exists a time t = t B) >

21 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 143 such that St)B B M 1,R ), for all t t. Moreover, for every R>, there exists C = C R), independent of α, such that, for any Θ = u, ψ, η ) B M 1,R), sup St)Θ H1, + t τ α t uy) + t ψy) Γ τ where B1, M R) denotes the ball in HM 1, of radius R, centered at. ) dy C, 4.17) In order to prove Theorem 4.1, we also need to introduce the Banach space cf. 3.4)) H := { u, ψ, η) H 3, : η DT) }, endowed with the norm u, ψ, η) H = u, ψ, η) H 3, + Tη M 1 + sup x νs) ηs) H ds. 4.18) x 1 x, ) 1 We recall that, while the embedding H 3, H 1, is only continuous, the space H is compactly embedded into H 1, see [46], Lemma 5.5). Besides, closed balls of H are compact in H 1, see [7], Proposition 8.1). Theorem 4.1 follows from the existence of a bounded exponentially attracting set in H. Theorem 4.4. Let the assumptions of Theorem 4.1 be satisfied. There exists R 1 > and a closed ball B M H R 1) H H M 1, which attracts B M 1,R ) see Proposition 4.3) exponentially fast, that is, dist H1, St)B M 1, R ), B M H R 1 ) ) Ce ρt τ) t τ 4.19) for some positive constants C and ρ. Here dist H1, denotes the nonsymmetric Hausdorff distance in H 1, and B H M 1, stands for the size of B in H M 1,. Proof. This proof is based on an argument similar to [6]. On account of Corollary 3.6, let us take Θ = u, ψ, η ) B M 1, R ) and consider the corresponding trajectory ut), ψt), η t ) = St)u, ψ, η ). From now on c will denote a generic positive constant which is independent of time and which may vary even within the same line. Further dependencies of c on other physical parameters will be pointed out as needed. In view of N1), we can find positive constants l 1, l such that f y) l 1, g y) l y R. 4.) Thus, in light of 4.) and N ), it is not difficult to realize that we can also choose sufficiently large constants γ 1 >l 1 and γ >β+ l with the following properties: 1 ζ + γ 1 l 1 ) ζ f y)ζ, ζ ), 1 Γ ζ Γ + γ l ) ζ Γ g y)ζ, ζ ) 4.1) Γ

22 144 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions for all ζ V 1 and every y R. Consequently, by setting fy) = fy) + γ 1 y, gy) = gy) + γ y,for all y R, it is immediate that both f and g are monotone nondecreasing and that they satisfy the same assumptions as the functions f and g. The following arguments are formal for the sake of simplicity, but they can be rigorously justified within the approximation scheme used in the proof of Theorem 3.3. We split a given trajectory ut), ψt), η t ) as follows see [45], cf. also [6]): ut), ψt), η t ) = u d t), ψ d t), ηd t ) + uc t), ψ c t), ηc t ), 4.) where t u d + νs)η d s)ds =, 4.3) Υ u d, v) + fu) fu c ), v ) + gψ) gψ c ), ϕ ) Γ + α t u d, v) + t ψ d, ϕ) Γ = μ d, v) H 1,H 1 v = v, ϕ) V 1, 4.4) t η d Tη d = A N μ d, 4.5) u d τ) = u u, ψ d τ) = ψ, η τ d = η 4.6) and t u c + νs)η c s)ds =, 4.7) Υ u c, v) + fu c ), v ) + gψ c ), ϕ ) Γ + α tu c, v) + t ψ c, ϕ) Γ γ 1 u, v) γ ψ, ϕ) Γ = μ c, v) H 1,H 1 v = v, ϕ) V 1, 4.8) t η c Tη c = A N μ c, 4.9) u c τ) = u, ψ c τ) =, ηc τ =. 4.3) Letusfirstshowthatu d t), ψ d t), η t d ) decays exponentially to zero with respect to the norm of H 1,. We begin by noting that, since ut), ψt)) L τ, ); V 1 )and f and g are nondecreasing, then we can easily adapt the proof of Lemma 4., to find u c t), ψ c t), η t c) H 1, + α t u c y) + t ψ c y) Γ τ which implies, on account of 4.1) and 4.), that u d t), ψ d t), η t d) H 1, + where c is obviously independent of α. α t u d y) + t ψ d y) Γ τ ) dy c t τ, 4.31) ) dy c t τ, 4.3)

23 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 145 We now take the scalar product of Eq. 4.3) with μ d t)inl Ω), and the scalar product of Eq. 4.5) with A 1 N ηt d in M. Also, we take v = t u d t), t ψ d t)) in 4.4). Adding together the resulting relationships, we find 1 d ud dt[ + Γ ψ d Γ + β ψ d Γ + η d ] M 1 + d [ fu) dt fu ) ) ] c ), u d + gψ) gψc ), ψ d Γ + α t u d + t ψ d Γ ν s) η d s) H ds τ 1 = f u) t u f ) u c ) t u c, u d + g ψ) t ψ g ψ c ) t ψ c, ψ d )Γ. 4.33) Let us consider the functional E d t) = u d t) + Γ ψ d t) Γ + β ψ d t) Γ + η t dt) M 1 + f ut) ) f u c t) ), u d t) ) f ut) ) u d t), u d t) ) + g ψt) ) g ψ c t) ), ψ d t) ) Γ g ψt) ) ψ d t), ψ d t) ) Γ 4.34) and note that, on account of 4.) and 4.1), we can find a positive constant C 1, independent of t and α, such that, for all t τ, C1 1 u d t), ψ d t), ηd) t H 1, E d t) C 1 ud t), ψ d t), ηd t Indeed, 4.1) yields ) H 1,. 4.35) fu) fu ) c ), u d f ) u)u d, u d γ1 l 1 ) u d f ) 1 u)u d, u d u d, and a similar inequality holds for the last two terms on the right-hand side of 4.34). Then, recalling K5) and performing simple computations, we can rewrite 4.33) as follows: where 1 d dt E dt) + λ ηd t M 1 1 ν s) η t s) H 1 ds Λ d t) t τ, 4.36) Λ d = f u) f u c ) ) t u c, u d ) + g ψ) g ψ c ) ) t ψ c, ψ d )Γ f u) t u,u d ) ) g ψ) t ψ,ψ d ) ) Γ. Following the proof of Lemma 4., we set ψ d = ψ d u note that u d t) = ). Then we introduce the auxiliary functional I d t) = νs) u d t), A 1 N ηt ds) ) ds, 4.37)

24 146 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions and we observe that, using Eqs 4.3) and 4.4), we have see also 4.3)) d dt I dt) = t u d t) H 1 = t u d t) H 1 ν s) u d t), A 1 N ηt ds) ) ds u d t), μ d t) ) ν s) u d t), A 1 N ηt ds) ) ds u d t) Γ ψ d t) Γ β ψ d t), ψ d t) ) Γ f ut) ) f u c t) ), u d t) ) g ψt) ) g ψ c t) ), ψ d t) ) Γ 1 d α u d t) dt[ + ψ d t) ] Γ. 4.38) Therefore, collecting 4.36) and 4.38), we find 1 d t) + εi d t) ] + λ η dt[ẽd t d M 1 1 ν s) η t s) H ds 1 ε t u d t) H + ε ν s) u 1 d t), A 1 N ηt ds) ) ds + ε u d t) + Γ ψ d t) Γ + β ψ d t) ) Γ + ε f ut) ) f u c t) ), u d t) ) + ε g ψt) ) g ψ c t) ), ψ d t) ) Γ Λ d t) + ε u g ψt) ) g ψ c t) ),1 ) Γ wherewehaveset + εβ u 1, ψ d t) ) Γ, 4.39) Ẽ d t) = E d t) + ε α u d t) + ψ d t) Γ ). 4.4) Here ε, λ) is a parameter which will be properly chosen. On account of 4.37), from 4.35), for ε> small enough, we can easily find a positive constant C = C ε, M) such that, for all t τ, C 1 u d t), ψ d t), ηd) t H 1, Ẽdt) + εi d t) C ud t), ψ d t), ηd t ) H 1,. 4.41) Let us now estimate the right-hand side of 4.39). On account of N ), 4.) and estimates 4.31) and 4.3), 4.1), from Hölder and Young inequalities, it follows that Λ d t) c t ut) u d t) H + c 1 t ψt) Γ ψ d t) H 1 Γ ) c t ut) + t ψt) ) Γ u d t) + Γ ψ d t) Γ + β ψ d t) ) Γ c t ut) + t ψt) Γ )Ẽd t) t τ. 4.4) Moreover, we have that ε u f ut) ) f u c t) ),1 ) + εβ u 1, ψ d t) ) Γ + ε u g ψt) ) g ψ c t) ),1 ) Γ c u d t) + ψ d t) Γ ) cẽdt) t τ. 4.43)

25 C. Cavaterra et al. / Cahn Hilliard equations with memory and dynamic boundary conditions 147 On the other hand, arguing as in 4.1), we find ε t u d t) H ε η t 1 d M 1, 4.44) ε ν s) u d t), A 1 N ηt ds) ) ds ε u d t) εc 4 Ων ν s) ηds) t H ds. 4.45) 1 Therefore, using the above inequalities 4.4) 4.45), from 4.39) we infer 1 d [Ẽd t) + εi d t) ] ) λ + dt ε ηd t M 1 ) 1 εc4 Ων ν s) ηds) t H ds ε u d t) + Γ ψ d t) Γ + β ψ d t) ) Γ + ε f ut) ) f u c t) ), u d t) ) + ε g ψt) ) g ψ c t) ), ψ d t) ) Γ c 1 + t ut) + t ψt) Γ )Ẽd t) 4.46) for all t τ. Thus, setting Êdt) = Ẽdt) + εi d t), choosing ε> sufficiently small and a positive constant C ε, ε/), from 4.46) we deduce that d dtêdt) + C εêdt) c 1 + t ut) + t ψt) Γ )Êd t). 4.47) It is left to notice that, owing to 4.17), we get t τ c 1 + t uy) + t ψy) Γ ) dy Cε t τ) + c α t τ, 4.48) where c is independent of α. The bound 4.48) allows us to apply a suitable Gronwall s inequality to 4.47) see, e.g., [8], Lemma 3.5), which yields Ê d t) ce Cεt τ) t τ. 4.49) Thus, recalling 4.41), 4.49) gives the desired exponential decay of u d, ψ d, η d ) H1,. Let us now obtain a bound for u c, ψ c, η c ) H. To this end, we argue as in the proof of Theorem 3.4, namely, we set ũ c = t u c, ψ c = t ψ c, η c = t η c and observe that ũ c, ψ c, η c ) solves cf. 4.7) 4.3)) t ũ c + νs) η c s)ds =, 4.5)