Non-isothermal viscous Cahn--Hilliard equation with inertial term and dynamic boundary conditions

We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality.

Here we want to extend the results proven in [27], namely, well-posedness, existence of the global attractor and convergence to a single equilibrium. More precisely, we first establish the existence and the uniqueness of global (bounded) energy and weak solutions. We recall that bounded energy solutions are more general than weak solutions (cf. Definition 2.1 below). In addition, in the present case a regularizing effect for χ is missing due to the presence of the dynamic boundary condition (1.6). This entails that the equation (1.3) must be understood in a more generalized way with respect to [27] (see Remark 2.4 below). In this case the application of the Lojasiewicz-Simon technique is also more complicated than in [27] and it seems necessary to work with weak solutions (cf. (5.28) and (5.29) below).
The plan of the paper goes as follows. In the next section the main assumptions as well as the notions of energy and weak solutions are introduced. In Section 3 some a priori energy and higher-order uniform estimates are obtained. Then, existence and uniqueness of energy and weak solutions are proven. Section 4 is devoted to establish the existence of the global attractor for the semigroup acting on the energy phase space. Finally, in Section 5 the convergence of a weak solution to a single equilibrium is analyzed. Among the open issues it is worth mentioning the existence of a family of exponential attractors and its robustness with respect to σ, ε and α (see [22] for the isothermal case).
Besides, we set and we introduce the Hilbert space L 2 div (Ω) and its inner product It is well known that if q ∈ L 2 div (Ω) then q · ν ∈ H − 1 2 (Γ) (cf. [38]). Hence we introduce the following closed subspace of L 2 div (Ω) We have W 0 ֒→ L 2 (Ω) ֒→ (W 0 ) * with dense and continuous embeddings. The Laplace operator with Neumann boundary condition and its domain are denoted by and we indicate with A 0 its restriction to H 0 . Note that A 0 is a positive linear operator. Hence, for any r ∈ R, we can define its powers A r 0 and their domains D(A r 2 0 ), setting Taking any u ∈ V * with u = 0, then v = A −1 0 u is a solution to the generalized Neumann problem for A with source u and the restriction v = 0. Hence, for any u, w ∈ V * with u = w = 0, we have We endow V * with the equivalent norm v 2 Next, we introduce the product spaces and the subspaces of H r (Ω) × H r (Γ) with the induced graph norm. We note that h = (u, v) ∈ H will be thought as a pair of functions belonging, respectively, to H and to H Γ . If we do not have additional regularity, the second component of h (i.e., v) is not necessary to be the trace of the first one (i.e. u). The elements of H r will be considered as pairs of functions (χ, χ| Γ ) such that H r is identified with a (closed) subspace of the product space H r (Ω) × H r (Γ). For r 1 > r 2 > 1 2 , the dense and compact embeddings H r 1 ֒→ H r 2 hold. Finally, we introduce the closed subspaces of H and H r as follows According to the structure of system (2.1)-(2.7), we define the product spaces endowed with the following norms It is easy to see that the continuous embedding Y ֒→ X holds.
Assumptions on the nonlinearities. Let us now list our assumptions on f and g.
Remark 2.1. Consider the potential functions F (y) = y 0 f (s)ds and G(y) = y 0 g(s)ds, y ∈ R. It is easy to check that assumptions (H2)-(H3) yield the following properties (cf. e.g., [15]): for any M 0 ∈ R, there exist c 2 , c 3 > 0 and a sufficiently large c 4 > 0 such that Similar results hold also for the potential G(y).
Remark 2.2. One can verify, for instance, that the classical double well potential F (y) = 1 4 (y 2 − 1) 2 and the corresponding function f (y) = y 3 − y satisfy (H1)-(H3) while g can be any polynomial of odd degree with a positive leading coefficient.
We are ready to introduce the variational formulation of problem (2.1)-(2.7).
Remark 2.3. We note that, due to the regularities (2.9)-(2.12), an energy solution belongs to the class C w ([0, T ]; X), where the space C w ([0, T ]; X) (X being a real Banach space) is defined as Therefore, any energy solution can be evaluated point-wisely in time and initial conditions have a well-defined meaning. The same property holds for weak solutions.
Remark 2.4. Note that in case of homogeneous Neumann boundary conditions we can recover the additional regularity χ ∈ L 2 (0, T ; H 2 (Ω)) for an energy solution (see [27, (2.18)]). Thus equation (2.15) can be written in the standard weak form (see [27, (2.6)]). However, in the present case, it seems that this regularity does not hold. On the other hand, such a property is crucial to prove that solutions to the isothermal MVCH regularize in finite time (see [2], cf. also [27] for the non-isothermal case with Fourier heat conduction). We also point out that the present notion of weak solution is a quasi-strong solution in the terminology introduced in [ The second relation (3.2) is an ODE for χ(t) , then we have It is easy to see that if χ 1 = 0, then the so-called mass conservation relation holds Based on the above observations, in order to obtain dissipative estimates of the solutions to problem (2.1)-(2.7), it is convenient to introduce the new variables which imply thatχ with the function Q 1 given by Then, system (2.1)-(2.7) can be rewritten as

9)
q · ν = ∂ νμ = 0, on Γ × (0, ∞), (3.10) Dissipative estimates. In what follows, we will derive some uniform estimates on the solutions of problem (2.1)-(2.7) which are necessary for studying the well-posedness and long-time behavior of the system. The following calculations have a formal character but they can be justified by working within a proper Faedo-Galerkin approximation scheme (see [23] and [24]).
Proof. Multiplying (3.5) and (3.6) byθ and q, respectively, and integrating over Ω, we obtain Multiplying (3.7) by A −1 0χ t and integrating over Ω, we have d dt In a similar manner, multiplying (3.7) by A −1 0χ and integrating over Ω, we get d dt Besides, using the equation (3.5) and (3.6), we deduce the identity Multiplying (3.17) and (3.18) by some small constants κ 1 , κ 2 > 0 (to be chosen later), respectively, and adding the resulting equations with (3.14)-(3.16), one deduces that d dt where K 4 depends on K 1 , K ′ 1 , κ 1 . By the Poincaré inequality, there exists C P > 0 depending on Ω such that where K 5 depends on χ 1 , |Γ| and κ 1 . Next, there exists some C Ω > 0 depending on Ω such that We now choose κ 1 , κ 2 > 0 sufficiently small so that From the above estimates we deduce the following inequality and By comparison, it is easy to verify that As a result, Since the quantities θ , χ and χ t are uniformly bounded in time, we can deduce that where the constants K 9 , K 10 > 0 may depend on Ω, θ 0 , χ 0 and χ 1 . Then, from (3.22) and (3.23) one infers estimate (3.13). The proof is complete.
Higher-order estimates. In what follows, we derive the uniform-in-time estimate in the higher-order space Y. For the sake of simplicity, from now on we shall indicate by C or C i , i ∈ N, a positive constant that may vary from line to line and also in the same line.
Proof. We (formally) differentiate (3.5)-(3.10) with respect to time and we get It is easy to verify that the initial datum can be controlled as follows where C is a constant depending on Ω and Γ. Multiplying (3.25) and (3.26) byθ t and q t , respectively, and then integrating over Ω, we obtain Multiplying (3.27) by A −1 0χ tt and integrating over Ω, we have where we have used the identities Here L ≥ c 0 + 1 is a positive constant such that f ′ (y) + L ≥ 1 and g ′ (y) + L ≥ 1 (cf. Remark 2.1). On the other hand, multiplying (3.27) by A −1 0χ t and integrating over Ω, we get Finally, using the equations (3.5) and (3.6) we obtain Multiplying (3.38), (3.39) by some small constants κ 3 , κ 4 > 0 (to be chosen later), respectively, and adding the resultants with (3.35)-(3.37), one deduces that and Using the Hölder inequality and choosing a suitable L, for κ 3 , κ 4 sufficiently small, we find where the constant C 1 , C 2 may depend on Ω, Γ, α, κ 3 , κ 4 and L.
Next, we estimate the reminder term R 1 . From Hölder inequality, Young's inequality, the Sobolev embedding theorem and the growth assumptions (H3) on f and g, it follows where Q 1 , Q 2 are certain monotone increasing functions. Taking ǫ sufficiently small, from the above estimates (3.40)-(3.46) we infer where C 3 is a small constant that may depend on Ω, Γ, α, κ 3 , κ 4 and L, but not on the solution. Besides, from the integrability of estimate (5.18) (see Section 5) we easily see that Using the dissipative estimate (3.13) (so that Q 1 ( (χ, ξ) H 1 ), Q 2 ( (χ, ξ) H 1 ) are uniformly bounded for all time) and the Gronwall-type lemma (see e.g., [25, Lemma 2.2]), then from (3.47) we infer which also easily yields that Using the estimate (3.49), from the equations (3.5)-(3.6) we deduce Applying the curl operator to (3.6), we have so that (∇ × q)(t) ≤ ∇ × q 0 , for all t ≥ 0. Combining the above estimates and (3.13), we get It remains to prove the estimate of (χ, ξ) in H 3 . To this purpose, we rewrite (2.4) and (2.6) as follows where β > 0 is a positive constant. Since µ satisfies (2.3) and (2.5), so that then from estimate (3.50) we infer As a result, we have Then, from the above estimate and Sobolev embedding theorem we infer An application of the higher-order regularity theorem [36, Corollary A.1] to the elliptic problem (4.25)-(4.26) yields Collecting all the above estimates, we have shown that (θ(t), q(t), χ(t), ξ(t), χ t ) is uniformly bounded in Y and the proof is complete.

Existence and uniqueness
Based on the uniform estimates obtained in the previous section, we are able to prove the existence and uniqueness of suitable solutions to problem (2.1)-(2.7).
Proof. (i) Based on the uniform dissipative estimate (3.24), it is standard to prove the existence of global weak solutions to problem (2.1)-(2.7) by using a Faedo-Galerkin scheme as in [23,24] for the Cahn-Hilliard equation subject to dynamic boundary conditions. The details are omitted here.
Concerning the uniqueness, it suffices to show the continuous dependence estimate for two solutions (θ (i) , q (i) , χ (i) , ξ (i) , χ (i) t ) corresponding to the two sets of data (θ 2). For this purpose, we write down the system for 57) 1 )e −t .
The regularity of weak solutions allows us to multiply (3.53) byθ, (3.54) byq and (3.55) by A −1 0χ t , respectively, and then integrate over Ω. Adding the resulting equations together, we have d dt Using the uniform estimates (3.13) for the two solutions, the growth assumption (H3) and Sobolev embedding theorems, we infer Similarly, we have Moreover, by Hölder inequality and Young inequality, we obtain Therefore, we find d dt Then, by the Gronwall lemma and the conservation properties (3.1)-(3.2), we can deduce for any t ∈ [0, T ], where C 1 , C 2 only depend on the X-norms of the initial data, α, |Ω| and |Γ|. This completes the proof for uniqueness.
(ii) We note that in the continuous dependence estimate for weak solutions (3.60), the constant C 1 , C 2 only depend on the X-norms of the initial data. This fact enables us to prove the existence and uniqueness of energy solutions to problem (2.1)-(2.7) by using the standard density argument. The details are left to the interested reader.
Remark 3.1. The estimate (3.60) provides a continuous dependence result in the (lower) Xnorm. As a consequence, S 1 (t) turns out to be a closed semigroup in the sense of [42].

Global attractor for energy solutions
In this section, we study the associated infinite-dimensional dynamical system defined by the semigroup S 2 (t) on X. More precisely, we will prove that S 2 (t) possesses the global attractor in the phase space endowed with the metric induced by the norm on X. Here M, M ′ ≥ 0 are arbitrary constants. We note that the choice of the phase space is due to the constraints (3.1), (3.2) and the decay property (3.3).
We now state the main result of this section.
Proof. For every bounded set B ⊂ X M,M ′ , consider an initial datum (θ 0 , q 0 , χ 0 , ξ 0 , where R > 0 is a constant depending on B. Besides, we observe that Thus, from the definition of Y (cf.
where R 0 may depend on M and M ′ but is independent of R and t. The proof is complete.
Next, we study the precompactness of trajectories in X.
Proof. Similar to [27], from the assumption (H2), Remark 2.1(1) and the Sobolev embedding theorem, it follows that there exists a sufficient large constant γ 0 > c 0 such that 1 2 Similarly, we can find a sufficient large constant γ 1 > c 1 such that We introducef (y) = f (y) + γ 0 y,ĝ(y) = g(y) + γ 1 y, y ∈ R. It is clear thatf ,ĝ are monotone nondecreasing functions in R. Then we split the solution to problem (2.1)-(2.7) as follows: and  In (4.6), we consider (χ, ξ) ∈ L ∞ (0, ∞; H 1 ) as given. Then, in analogy to the proof of Lemma 3.1, we can prove that problem (4.6) admits a unique global solution such that Due to (3.13) and the decomposition (4.4), we obtain similar uniform estimates for the decay part (θ d , q d , χ d , ξ d , χ d t )(t). Next, we show that (θ d , q d , χ d , ξ d , χ d t )(t) X indeed decays to zero exponentially fast as time tends to infinity. Due to the choice of initial data, it is easy to verify that As in the proof of Lemma 3.1, in (4.5) we multiply the first equation by θ d , the second equation by q d , the third equation by A −1 0 (χ d t + κ 1 χ d ) and we integrate over Ω. Then, summing up all the resulting equations and adding the functional κ 2 (q d , ∇A −1 0 θ d ) (κ 1 , κ 2 are positive constants to be determined later), we have We note thatf andĝ are monotone nondecreasing functions. Moreover, if γ 0 , γ 1 are sufficiently large, we have 11) which implies that We now estimate R d (t). First, we observe that the last three terms can be evaluated exactly as in (3.19). Using the uniform estimate (3.13), (4.7), (4.9), the Sobolev embedding inequality and the Poincaré inequality, we deduce that, for the case d = 3, there holds (the case d = 2 is similar) Similarly, we have Due to (4.1) and (4.2), we get Then, taking κ 1 and κ 2 small enough, we can find η > 1 such that Thus, from the above estimate and (4.10) we infer that there exist two positive constants K 1 , K 2 such that the following estimate holds In On account of the fact χ c t (t) = χ t (t) = χ 1 e −t , for all t ≥ 0, and the uniform estimates (3.13), (4.8), then, ∀ ǫ > 0, the right-hand side (4.13) can be evaluated as follows As a result, for any ǫ > 0, we deduce from (4.13), (4.14) and (3.48) that from which, combining with estimate (3.48), we infer, for any ǫ > 0, Then, an application of the Gronwall-type lemma (see e.g., [25,Lemma 2.2]) allows to conclude that Y d (t) ≤ CY(0)e − K 1 2 t , ∀ t ≥ 0, (4.16) which, together with (4.12), yields the exponential decay of (θ d , q d , χ d , ξ d , χ d t )(t) in X. Finally, we prove that (θ c , q c , χ c , ξ c , χ c t )(t) is bounded in a space that can be compactly embedded into X. To this aim, we (formally) differentiate (4.6) with respect to time to get We recall that Multiplying the third equation of (4.6) by A −1 0χ c tt an integrating over Ω, we obtain d dt we have (∇ × q c )(t) = 0 for all t ≥ 0. Thus, q c H 1 (Ω) ≤ C. Next, we rewrite (4.6)(4) and (4.6)(6) as follows where β > 0 is a positive constant. Since µ c satisfies we see that Using estimate (4.7) and the same argument as in Lemma 3.2, we get Collecting the estimates above, we see that (θ c , q c , χ c , ξ c , χ c t )(t) is uniformly bounded in Y, which is compactly embedded into X.
In summary, we have proved that any trajectory starting from X can be decomposed into two parts: one part decays exponentially fast to zero in X and the other part is uniformly bounded in Y. Thus, the trajectory is precompact in X. The proof is complete.
Proof of Theorem 4.1. Proposition 4.1 implies that the semigroup S 2 (t) has a bounded absorbing set in X M,M ′ . On the other hand, Proposition 4.2 yields the precompactness of the trajectory and, in particular, the existence of a compact (exponentially) attracting set (cf. Then, thanks to decomposition (4.4), one can construct a positively invariant exponential attracting set B which is bounded in Y M,M ′ . Using the same decomposition and taking the initial data in B, it is possible to prove the asymptotic compactness of the semigroup in Y. The global attractor coincides with the previous one, that is, we have a smoothness result for A. The details are left to the interested reader.

Convergence to equilibrium
In this section, we proceed to investigate the long-time behavior of single weak solution for any given initial datum (θ 0 , q 0 , χ 0 , ξ 0 , χ 1 ) ∈ Y.

Stationary problem and Lojasiewicz-Simon inequality
First, we look at the corresponding stationary problem. The steady states (θ ∞ , χ ∞ , ξ ∞ ) of problem (2.1)-(2.7) satisfy the following elliptic boundary value problem with constraints dictated by the initial data on account of the boundary conditions It is easy to see that the above system can be reduced to the following form: where µ ∞ is a constant uniquely determined by We introduce the functional for any (u, v) ∈ H 1 0 (see Section 2), where For any (u, v), (w, w Γ ) ∈ H 1 0 , we define the operator If we restrict the operator M on H 2 0 , i.e., for (u, v) ∈ H 2 0 , after integration by parts, from (5.5) we infer Here, we denote by P 0 the projection operator P 0 : H → H 0 such that P 0 u = u − u for any u ∈ H. The operator A is given by From the identities (5.5)-(5.7) it easily follows Furthermore, applying the method of minimizing sequence similar to the one used in [51], we easily prove the following Proposition 5.2. Under assumptions (H1)-(H3), the stationary problem (5.1) admits at least one solution (χ ∞ , ξ ∞ ) ∈ H 1 and θ ∞ is given by  N), provided that f, g are smooth enough.
Next, we introduce a Lojasiewicz-Simon type inequality which will be used to prove longtime behavior of global solutions to problem (2.1)-(2.7).
Lemma 5.1. Assume that f, g are real analytic and (H2), (H3) are satisfied. Let (u * , v * ) ∈ H 2 0 be a critical point of the functional Υ. Then there exist two constants ρ ∈ (0, 1 2 ) and β > 0, depending on (u * , v * ), such that, for any Proof. The proof follows from an argument similar to the one used in [45]. Here, we just point out some differences. By the assumptions, Υ is twice Fréchet differentiable with respect to the topology of H 2 . Moreover, by the Sobolev embedding H 2 (Ω) ֒→ L ∞ (Ω) (n ≤ 3), Υ is real analytic. As in [43] and using the Poincaré inequality, we can easily show that A is a strictly positive self-adjoint unbounded operator from D(A) = {(u, v) ∈ H 1 0 : A(u, v) ∈ H 0 } into H 0 . Standard spectral theory allows us to define the power A s (s ∈ R), and we infer that there exists a complete orthonormal family {(φ j , ψ j )} ∈ D(A), (j ∈ N, s ∈ R), as well as a sequence of eigenvalues 0 < λ 1 ≤ λ 2 ≤ ..., λ j → ∞ as j tends to infinity, such that (5.9) In particular, D(A 1 2 ) = H 1 0 , D(A) = H 2 0 . By a bootstrap argument, we get (φ j , ψ j ) ∈ C ∞ (Ω), for all j ∈ N. For any (u, v) ∈ H 1 0 , we have Following the idea used in [31], we now introduce the orthogonal projector P m in V 0 onto K m := span{(φ 1 , ψ 1 )..., (φ m , ψ m )} ⊂ C ∞ (Ω). As in [45], we have, for any (u, v) ∈ H 1 0 , Next, we consider the following linearized operator on In analogy to [51,Lemma 2.3], we can easily show that L(u, v) is self-adjoint on H 0 . We associate with the operator L(u * , v * ) the following bilinear form b((w 1 , w 1Γ ), (w 2 , w 2Γ )) on H 1 0 , for any ( Since (u * , v * ) ∈ V 2 , then, by the Sobolev embedding theorems, we infer that L(u * , v * ) + λ m P m is coercive in H 1 0 , provided that λ m is sufficiently large, e.g., After establishing the above framework, the proof of the extended Lojasiewicz-Simon inequality (5.8) can be reproduced taking advantage of the arguments used in [31] (see also [45,Theorem 3.1]) with minor modifications. The details are omitted here.
Remark 5.2. Recalling Remark 4.1, it can be shown that the solution converges in Y−norm to the single equilibrium.
The proof of Theorem 5.1 consists of several steps.
Step 2. Convergence to equilibrium. In the spirit of [27,31], we now consider the functional which, by the decay property (5.19) and the uniform estimate (3.13), satisfies lim t→+∞ G(t) = 0.
In summary, we have proved the conclusion of Theorem 5.1. Using the energy differential inequality (5.26), the argument developed in [32] (cf. also [49,50]) and the energy method, one can proceed to show the estimate of decay rate. The details are left to the interested readers. More precisely, the following result can be proven.
Corollary 5.1. Let the assumption of Theorem 5.1 be satisfied. Then we have for all t ≥ 0, where C is a constant depending on the X-norm of the initial datum and on the coefficients of the system, while ρ ∈ (0, 1 2 ) may depend on (χ ∞ , ξ ∞ ).