The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

Abstract

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

Keywords

Cahn–Hilliard/Allen–Cahn equations dynamic boundary conditions well-posedness global attractors fractal dimension

1. Introduction

In this article we consider the following Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions: $\begin{array}{l} (1.1) & u_{t} = Δ μ, in Ω, \\ (1.2) & μ = - Δ u + f (u + v) + f (u - v), in Ω, \\ (1.3) & v_{t} = Δ v - f (u + v) + f (u - v) - α v, in Ω, \\ (1.4) & u_{t} = δ Δ_{Γ} μ - \partial_{n} μ, on Γ, \\ (1.5) & μ = - σ Δ_{Γ} u + g (u + v) + g (u - v) + \partial_{n} u, on Γ, \\ (1.6) & v_{t} + \partial_{n} v - κ Δ_{Γ} v + g (u + v) - g (u - v) = 0, on Γ . \end{array}$

Here, $Ω \subset R^{n}$ , $n = 1, 2 or 3$ , is a smooth and bounded domain with boundary Γ corresponding to the binary material under consideration, u denotes the concentration of one of the components and is a conserved quantity, while v is an order parameter and μ represents the chemical potential. Furthermore, the parameter α reflects the location of the system within the phase diagram and may be either positive or negative. In what follows we consider, without any restriction of generality, α positive (the case α negative can be treated similarly, adapting certain a priori estimates). Moreover, δ, σ are nonnegative parameters related to the boundary diffusion and $κ > 0$ is a physical coefficient. Also, $Δ_{Γ}$ is the Laplace–Beltrami operator on Γ, $\partial_{n}$ is the outward normal derivative and the nonlinear function f is the derivative of a “bulk” double-well potential F. Finally, g is the derivative of a surface potential G and taken as an affine function. The evolution boundary value problem (1.1)–(1.6) is completed by the initial conditions $u (0) = u_{0}$ and $v (0) = v_{0}$ .

The Cahn–Hilliard/Allen–Cahn system (1.1)–(1.3) endowed with Neumann boundary conditions was introduced in [3,5] to model simultaneous order-disorder and phase separation in binary alloys on a BCC lattice in the neighborhood of the triple point. For further references on the physical pertinence of the model we refer the interested reader to [9]. The authors of [5] explored two phenomenological approaches leading to systems of coupled Allen–Cahn/Cahn–Hilliard (AC/CH) equations. Another important application of the coupled (AC/CH) equations is that under appropriate compositional conditions, ordering can be induced in a previously homogeneous material. If the composition differs slightly from these conditions, the excess composition can emerge as droplets along the boundaries between the ordered regions. This phenomena can be modeled by a coupled (AC/CH) system with degenerate mobilities. In similar applications, surface diffusion coupled with motion by mean curvature appears quite naturally. There are additional effects which are often neglected and which arguably should be included. However, the coupled motion, by itself, is not fully understood and it was thus reasonable to isolate it and study it, even given its limitations (see [8]).

In order to derive system (1.1)–(1.6), we consider the following Ginzburg–Landau free energy: $\begin{matrix} (1.7) & \begin{matrix} J (u, v) = & \int_{Ω} {F (u + v) + F (u - v) + \frac{α}{2} | v |^{2} + \frac{1}{2} (| \nabla u |^{2} + | \nabla v |^{2})} d x \\ + \int_{Γ} {G (u + v) + G (u - v) + \frac{σ}{2} | \nabla_{Γ} u |^{2} + \frac{κ}{2} | \nabla_{Γ} v |^{2}} d Σ, \end{matrix} \end{matrix}$ where $\nabla_{Γ}$ is the surface gradient. Here, we will take the double-well potential F as the regular one defined by $\begin{matrix} (1.8) & F (s) = \frac{1}{4} {(s^{2} - 1)}^{2}, \end{matrix}$ i.e., $\begin{matrix} (1.9) & f (s) = F^{'} (s) = s^{3} - s . \end{matrix}$ We note that, in the Cahn–Hilliard theory, a thermodynamically relevant potential F is the following logarithmic function which follows from a mean-field model (see [23,24,27,30]): $\begin{array}{l} F (s) = \frac{θ_{c}}{2} (1 - s^{2}) + \frac{θ}{2} [(1 - s) ln (\frac{1 - s}{2}) + (1 + s) ln (\frac{1 + s}{2})], \\ (1.10) & s \in (- 1, 1), 0 < θ < θ_{c} . \end{array}$ Furthermore, $\begin{matrix} (1.11) & f (s) (= F^{'} (s)) = - θ_{c} s + \frac{θ}{2} ln (\frac{1 + s}{1 - s}) . \end{matrix}$ This logarithmic potential F is usually approximated by the regular one defined in (1.8). It should be noted that this approximation is acceptable when the absolute temperature θ is close to the critical one $θ_{c}$ .

Furthermore, the function G is a surface potential, defined by $\begin{matrix} (1.12) & G (s) = \frac{1}{2} a s^{2} + b s, \end{matrix}$ i.e., $\begin{matrix} (1.13) & g (s) = a s + b, \end{matrix}$ where $a > 0$ accounts for a modification of the effective interaction between the components at the walls and b characterizes the possible preferential attraction of one of the components by the walls; when b vanishes, there is no preferential attraction (see [19]).

The first line in (1.7) is the Ginzburg–Landau (bulk) free energy, while the second one corresponds to the surface energy. Moreover, integrating (1.1) over Ω, we find, in view of (1.4), $\begin{matrix} \frac{d}{d t} \int_{Ω} u d x = \int_{Γ} \partial_{n} μ d Σ = - \int_{Γ} \frac{\partial u}{\partial t} d Σ, \end{matrix}$ so that we have the total (in the bulk and on the boundary) conservation of mass $\begin{matrix} (1.14) & \frac{d}{d t} (\int_{Ω} u d x + \int_{Γ} u d Σ) = 0, t ⩾ 0 . \end{matrix}$

Dynamic boundary conditions have been proposed by physicists, in the context of the Cahn–Hilliard equation, in order to account for the interactions with the walls in confined systems (cf. [10–12,19] and the references therein; see also [13] and [18] for permeable or non-permeable walls). In particular, such boundary conditions have mainly been studied for polymer mixtures (although this should also be important in other systems, such as binary metallic alloys): from a technological point of view, binary polymer mixtures are particularly interesting, since the structures occurring during the phase separation process may be frozen by a rapid quench into the glassy state; micro-structures at surfaces on very small length scales can be produced in this way.

The original Cahn–Hilliard system, endowed with dynamic boundary conditions, has been studied in [6,13–15,18,28,33,34], and [39] for regular potentials f and g. In particular, one now has satisfactory results on the existence, uniqueness and regularity of solutions and on the asymptotic behavior of the solutions. The case of logarithmic (and also more general singular) nonlinear terms has been treated in [16–18], and [30]. In that case, one feature of the problem is that we can have the nonexistence of classical (i.e., distributional) solutions.

In [3], the authors prove the well-posedness and the existence of maximal attractors and inertial sets (i.e., exponential attractors) of the coupled (AC/CH) system for the usual cubic nonlinear term $f (s) = s^{3} - β s$ in three space dimensions when Neumann boundary conditions are considered. We also refer the interested reader to [25] and [26], where the authors studied the asymptotic behaviour of the system, both for the regular and singular potential cases. The numerical study by a finite element approximation was treated in [2] for the case of a degenerate Allen–Cahn/Cahn–Hilliard system under Neumann boundary conditions.

A similar system, with a non-constant mobility, was treated in [7] where the existence of weak solutions for the Neumann problem for a degenerate parabolic system consisting of a fourth-order and a second-order equation with singular lower-order terms in one space dimension was proved. In addition, asymptotics for a similar system with a non-constant mobility, proposed as a diffuse interface model for simultaneous order-disorder and phase separation, was studied in [31]. There, A. Novick–Cohen focused on the motion in the plane. This framework yields both sharp and diffuse interface models of sintering of small grains and thermal grains boundaries grooving in polycrystalline films. This work was extended in [32], where the authors studied the partial wetting case, and their analysis accounts for motion in three space dimensions.

We also mention that several numerical methods were used in order to solve coupled (AC/CH) systems; see, e.g., [5,22,35,37,38,40]. Furthermore, a Newton–Krylov–Schwarz (NKS) method for the implicit solution of a coupled (AC/CH) system was studied in [41] and very recently, the coupled Cahn–Hilliard/Allen–Cahn system endowed with dynamic boundary conditions (1.1)–(1.6) has been studied numerically in [21], in which the authors obtained error estimates between the exact and the approximated solution and prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization.

This article is organized as follows. In Section 2, we introduce our main assumptions and notations, together with a number of tools which are needed in order to reformulate system (1.1)–(1.6). Then, in Section 3, we give the functional setting of our system. In Section 4, we derive a priori estimates, which allow us to prove in Section 5 the existence of solutions; the uniqueness is obtained by classical arguments, following [4]. In Section 6, we obtain-higher order estimates to get further regularity on the solution. Finally, adapting the proof introduced by S. Zelik in [42], in Section 7 we prove the existence of the global attractor which has finite fractal dimension.

The main technical difficulty of this work is due to the presence of dynamic boundary conditions which requires the adaptation of classical techniques in order to study the existence and uniqueness of the solutions in certain function spaces, as well as the asymptotic behavior of the model. The proofs of existence and uniqueness, as well as higher regularity of solutions, are based on a classical Galerkin approximation and a priori estimates that allow the passage to limit; we refer the interested reader to [20] and [36] for further reading. The proofs in Section 7 are based on analytical tools introduced by [1,36] and further developed by [29,42] in order to prove the existence of a global attractor of finite fractal dimension in the context of partial differential equations.

2. Notations and assumptions

First, we set $H_{Ω} = L^{2} (Ω)$ and denote by ${((\cdot, \cdot))}_{Ω}$ the scalar product on $H_{Ω}$ and by $‖ \cdot ‖_{Ω}$ the related norm. Moreover, we set $V_{Ω} = H^{1} (Ω)$ and denote by $V_{Ω}^{'}$ the (topological) dual of $V_{Ω}$ . The duality between $V_{Ω}^{'}$ and $V_{Ω}$ is indicated by ${⟨ \cdot, \cdot ⟩}_{Ω}$ .

We also set $H_{Γ} = L^{2} (Γ)$ and $V_{Γ} = H^{1} (Γ)$ and denote by ${((\cdot, \cdot))}_{Γ}$ the standard scalar product in $H_{Γ}$ , by $‖ \cdot ‖_{Γ}$ the corresponding norm, and by ${⟨ \cdot, \cdot ⟩}_{Γ}$ the duality between $V_{Γ}^{'}$ and $V_{Γ}$ . In general, $‖ \cdot ‖_{X}$ indicates the norm in the generic (real) Banach space X and ${⟨ \cdot, \cdot ⟩}_{X}$ stands for the duality between $X^{'}$ and X.

We can then define the Hilbert spaces $\begin{matrix} H = H_{Ω} \times H_{Γ} and V = \{(\begin{matrix} φ \\ ψ \end{matrix}) \in V_{Ω} \times V_{Γ}; φ |_{Γ} = ψ\}, \end{matrix}$ endowed with the natural scalar products and norms. Unless otherwise specified, in what follows, we will impose the following convention: when we write $h \in H$ , h will be interpreted as a pair of functions belonging, respectively, to $H_{Ω}$ and to $H_{Γ}$ . Analogously, when we consider $ρ \in V$ (or even $ρ \in V_{Ω}$ ), the symbol ρ will be intended, depending on the context, either as a function defined in Ω, or as a pair formed by a function in Ω and its trace on Γ.

We also define, for $ϕ = (\begin{matrix} φ \\ ψ \end{matrix}) \in H_{Ω} \times H_{Γ}$ , $\begin{matrix} ⟨ ϕ ⟩ = \frac{1}{| Ω | + | Γ |} (\int_{Ω} φ d x + \int_{Γ} ψ d Σ) . \end{matrix}$ We then set $\begin{matrix} \dot{H} = \{ϕ = (\begin{matrix} φ \\ ψ \end{matrix}) \in H; ⟨ ϕ ⟩ = 0\} and \dot{V} = V \cap \dot{H} . \end{matrix}$ In particular, the norm $‖ \cdot ‖_{\dot{V}}^{2} = ‖ \nabla \cdot ‖_{Ω}^{2} + ‖ \nabla_{Γ} \cdot ‖_{Γ}^{2}$ on $\dot{V}$ is equivalent to the usual $H^{1} (Ω) \times H^{1} (Γ)$ -norm.

We need some further discussion on the functional spaces.

We first notice that we have the chain of continuous embeddings $\begin{matrix} \dot{V} \subset V_{Ω} \subset \dot{H} \subset V_{Ω}^{'} \subset {\dot{V}}^{'} . \end{matrix}$ Indeed, it is not difficult to prove that the space $V$ is dense in $H$ .

Next, we introduce the elliptic operators $\begin{matrix} A_{η} : V \to V^{'}, η = δ, σ or κ, \end{matrix}$ defined as $\begin{matrix} {⟨ A_{η} Φ_{1}, Φ_{2} ⟩}_{V^{'}, V} = {((\nabla ϕ_{1}, \nabla ϕ_{2}))}_{Ω} + η {((\nabla_{Γ} ϕ_{1}, \nabla_{Γ} ϕ_{2}))}_{Γ}, \end{matrix}$ where $Φ_{i} = (\begin{matrix} ϕ_{i} \\ ϕ_{i} \end{matrix})$ , for $i = 1, 2$ . The operators $A_{η}$ are positive, self-adjoint and unbounded linear operators. Furthermore, we can define the domain of $A_{η}$ in $H$ , $\begin{matrix} D (A_{η}) = {Φ \in V, \exists Ξ \in H; {⟨ A_{η} Φ, Ψ ⟩}_{V^{'}, V} = {((Ξ, Ψ))}_{H} \forall Ψ \in V} . \end{matrix}$ Finally, noting that $A_{η}^{- 1}$ can be considered as a self-adjoint and compact operator on $\dot{H}$ , we can define the powers $A_{η}^{s}$ , together with their domains $D (A_{η}^{s}), s \in R$ (keeping in mind that $D (A_{η}^{0}) = H$ ), so that $A_{η}^{s}, s > 0$ , can be extended as an isomorphism from $H$ onto $D (A_{η}^{- s})$ and, more generally from $D (A_{η}^{s_{1}})$ onto $D (A_{η}^{s_{1} - s}), s_{1} \in R$ . Following exactly the same arguments as in [24] (see Proposition 5.3), one can show that $D (A_{η}) = V \cap (H^{2} (Ω) \times H^{2} (Γ))$ and the norm $‖ A_{η} \cdot ‖_{H}$ is equivalent to the usual $H^{2} (Ω) \times H^{2} (Γ)$ -norm on $D (A_{η})$ .

Furthermore, for $k \in N$ , the embedding $D (A_{η}^{k}) \subset H^{2 k} (Ω) \times H^{2 k} (Γ)$ is continuous. Also, the norm $‖ A_{η}^{k} \cdot ‖_{H}$ is equivalent to the usual $H^{2 k} (Ω) \times H^{2 k} (Γ)$ -norm on $D (A_{η}^{k})$ .

Noting that, by definition $D (A_{η}^{\frac{1}{2}}) = V$ , we see that the embedding $D (A_{η}^{k + \frac{1}{2}}) \subset H^{2 k + 1} (Ω) \times H^{2 k + 1} (Γ)$ is continuous and the norm $‖ A_{η}^{k + \frac{1}{2}} \cdot ‖_{H}$ is equivalent to the usual $H^{2 k + 1} (Ω) \times H^{2 k + 1} (Γ)$ -norm on $D (A_{η}^{k + \frac{1}{2}})$ .

Finally, we note that $D (A_{η}^{- \frac{1}{2}}) = {\dot{V}}^{'}$ . Furthermore, since the norm $‖ A_{η}^{\frac{1}{2}} \cdot ‖_{H}$ is equivalent to the $\dot{V}$ -norm, it follows that the norm $‖ \cdot ‖_{- 1} = ‖ A_{η}^{- \frac{1}{2}} \cdot ‖_{H}$ is equivalent to the usual $V^{'}$ -norm on $V^{'}$ .

We refer the interested reader to [24] for more details and proofs.

As far as the nonlinear terms f and g defined in (1.9) and (1.13) are concerned, we note that they satisfy the following properties: $\begin{array}{l} (2.1) & f^{'} ⩾ - 1, g^{'} = a > 0, \\ (2.2) & f (s) (s - m) ⩾ c_{1} F (s) - c_{2} (m) ⩾ - c_{3} (m), c_{1} > 0, c_{2}, c_{3} ⩾ 0, \\ (2.3) & g (s) (s - m) ⩾ c_{4} G (s) - c_{5} (m) ⩾ - c_{6} (m), c_{4} > 0, c_{5}, c_{6} ⩾ 0, \end{array}$ where $F (s) = \int_{0}^{s} f (ξ) d ξ; G (s) = \int_{0}^{s} g (ξ) d ξ$ and the constants $c_{i}$ , $i = 2, 3, 5, 6$ depend continuously on m.

Throughout this paper, the same letter c (and, sometimes, $c^{'}$ ) denotes constants which may change from line to line, or even in a same line.

3. Functional setting of the problem

Using the function spaces and the operators introduced in the previous section, we can rewrite (1.1)–(1.6) in the following functional form: $\begin{array}{l} (3.1) & \frac{d U}{d t} = - A_{δ} W, in V^{'}, \\ (3.2) & W = A_{σ} U + F (U + V) + F (U - V), in V^{'}, \\ (3.3) & \frac{d V}{d t} = - A_{κ} V - F (U + V) + F (U - V) - α V, in V^{'}, \end{array}$ where, with an abuse of notation, $\begin{matrix} U = (\begin{matrix} u \\ u \end{matrix}), V = (\begin{matrix} v \\ v \end{matrix}), W = (\begin{matrix} μ \\ μ \end{matrix}) \end{matrix}$ and we defined $\begin{matrix} (3.4) & F (U) = \{\begin{matrix} f (u), & in Ω, \\ g (u), & on Γ, \end{matrix} \end{matrix}$ with $U \in H$ .

Setting, whenever it makes sense, $\begin{matrix} \bar{ϕ} = ϕ - (\begin{matrix} ⟨ ϕ ⟩ \\ ⟨ ϕ ⟩ \end{matrix}), \end{matrix}$ so that $⟨ \bar{ϕ} ⟩ = 0$ , we can rewrite (3.1) in the equivalent form $\begin{matrix} A_{δ}^{- 1} \frac{d U}{d t} = - \overline{W}, in D^{'} (0, T; {\dot{V}}^{'}), \end{matrix}$ noting that $⟨ \frac{\partial U}{\partial t} ⟩ = 0$ .

Therefore, we can write the initial value problem in the following functional form: $\begin{array}{l} (3.5) & A_{δ}^{- 1} \frac{d U}{d t} = - \overline{W}, in D^{'} (0, T; {\dot{V}}^{'}), \\ (3.6) & W = A_{σ} U + F (U + V) + F (U - V), in D^{'} (0, T; V^{'}), \\ (3.7) & \frac{d V}{d t} = - A_{κ} V - F (U + V) + F (U - V) - α V, in D^{'} (0, T; V^{'}), \\ (3.8) & U |_{t = 0} = U_{0}, V |_{t = 0} = V_{0}, in V^{'}, \end{array}$ where $\begin{matrix} U_{0} = (\begin{matrix} u_{0} \\ u_{0} \end{matrix}), V_{0} = (\begin{matrix} v_{0} \\ v_{0} \end{matrix}) . \end{matrix}$ In what follows we will actually consider initial data in $V$ (see Theorem 5.1), so that the trace indeed makes sense.

4. A priori estimates

In this section we derive a priori estimates, based on which we can prove the existence and uniqueness of solutions.

First, we take the scalar product in $H$ of (3.5) by $\frac{d U}{d t}$ . We get, noting that $⟨ \frac{\partial U}{\partial t} ⟩ = 0$ , $\begin{matrix} (4.1) & {‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} = - {((W, \frac{d U}{\partial t}))}_{H} . \end{matrix}$ We also take the scalar product in $H$ of (3.6) by $\frac{d U}{d t}$ to obtain $\begin{matrix} (4.2) & \begin{matrix} {((W, \frac{d U}{d t}))}_{H} = & \frac{1}{2} \frac{d}{d t} {‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2}} + {((f (u + v) + f (u - v), \frac{\partial u}{\partial t}))}_{Ω} \\ + {((g (u + v) + g (u - v), \frac{\partial u}{\partial t}))}_{Γ} . \end{matrix} \end{matrix}$ Combining (4.1) and (4.2), we get $\begin{matrix} (4.3) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2}} + {‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} \\ + {((f (u + v) + f (u - v), \frac{\partial u}{\partial t}))}_{Ω} + {((g (u + v) + g (u - v), \frac{\partial u}{\partial t}))}_{Γ} = 0 . \end{matrix} \end{matrix}$ We take the scalar product in $H$ of (3.7) by $\frac{d V}{d t}$ to have $\begin{matrix} (4.4) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2}} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} \\ + {((f (u + v) - f (u - v), \frac{\partial v}{\partial t}))}_{Ω} + {((g (u + v) - g (u - v), \frac{\partial v}{\partial t}))}_{Γ} = 0 . \end{matrix} \end{matrix}$ Summing (4.3) and (4.4), we obtain $\begin{matrix} (4.5) & \begin{matrix} \frac{d}{d t} {‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + 2 \int_{Ω} {F (u + v) + F (u - v)} d x + 2 \int_{Γ} {G (u + v) + G (u - v)} d Σ} \\ + 2 ({‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2}) = 0 . \end{matrix} \end{matrix}$

Next, we take the scalar product in $H$ of (3.5) by $\overline{U}$ and find $\begin{matrix} (4.6) & \frac{1}{2} \frac{d}{d t} ‖ \overline{U} ‖_{- 1}^{2} = - {((W, U))}_{H} + (| Ω | + | Γ |) ⟨ W ⟩ ⟨ U_{0} ⟩ . \end{matrix}$ Taking the scalar product in $H$ of (3.6) by U, we get $\begin{matrix} (4.7) & \begin{matrix} {((W, U))}_{H} = & ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + {((f (u + v) + f (u - v), u))}_{Ω} \\ + {((g (u + v) + g (u - v), u))}_{Γ} . \end{matrix} \end{matrix}$ Thus, we have $\begin{matrix} (4.8) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ \overline{U} ‖_{- 1}^{2} + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + {((f (u + v) + f (u - v), u))}_{Ω} \\ + {((g (u + v) + g (u - v), u))}_{Γ} = (| Ω | + | Γ |) ⟨ W ⟩ ⟨ U_{0} ⟩ . \end{matrix} \end{matrix}$ We then take the scalar product in $H$ of (3.7) by V and obtain $\begin{matrix} (4.9) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ V ‖_{H}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + {((f (u + v) - f (u - v), v))}_{Ω} + {((g (u + v) - g (u - v), v))}_{Γ} = 0 . \end{matrix} \end{matrix}$ Summing (4.8) and (4.9), we find, owing to (2.2)–(2.3) $\begin{matrix} (4.10) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ \overline{U} ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}} + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + c_{1} \int_{Ω} {F (u + v) + F (u - v)} d x + c_{4} \int_{Γ} {G (u + v) + G (u - v)} d Σ \\ ⩽ (| Ω | + | Γ |) ⟨ W ⟩ ⟨ U_{0} ⟩ . \end{matrix} \end{matrix}$ We now assume that $\begin{matrix} (4.11) & | ⟨ U_{0} ⟩ | ⩽ M with M ⩾ 0 . \end{matrix}$ Thanks to the conservation of mass (1.14), assumption (4.11) implies that $| ⟨ U (t) ⟩ | ⩽ M$ , $t ⩾ 0$ . The purpose of this assumption will become clearer in Section 7 and more exactly in Theorem 7.1, where one can see that the existence of a global attractor holds on the subspace of $V^{'} \times H$ of functions of total mass equal to M.

Therefore, $\begin{matrix} (4.12) & \begin{matrix} | (| Ω | + | Γ |) ⟨ W ⟩ ⟨ U_{0} ⟩ | & ⩽ c_{M} (| ⟨ F (u + v) ⟩ | + | ⟨ F (u - v) ⟩ |) \\ ⩽ \frac{c_{1}}{2} \int_{Ω} {F (u + v) + F (u - v)} d x \\ + \frac{c_{4}}{2} \int_{Γ} {G (u + v) + G (u - v)} d Σ + c_{M}^{'} . \end{matrix} \end{matrix}$ We thus deduce from (4.10) and (4.12) that $\begin{matrix} (4.13) & \begin{matrix} \frac{d}{d t} {‖ \overline{U} ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}} + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + c_{1} \int_{Ω} {F (u + v) + F (u - v)} d x + c_{4} \int_{Γ} {G (u + v) + G (u - v)} d Σ ⩽ c_{M}^{'}, \end{matrix} \end{matrix}$ where by $c_{M}^{'}$ we denote a constant depending on M that may change at different places in the text.

Combining now (4.5) and (4.13), we have a differential inequality of the form $\begin{matrix} (4.14) & \frac{d E}{d t} + c (E + {‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2}) ⩽ c_{M}^{'}, c > 0, \end{matrix}$ where $\begin{matrix} (4.15) & \begin{matrix} E = & ‖ \overline{U} ‖_{- 1}^{2} + {⟨ U ⟩}^{2} + ‖ V ‖_{H}^{2} + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + 2 \int_{Ω} {F (u + v) + F (u - v)} d x + 2 \int_{Γ} {G (u + v) + G (u - v)} d Σ \end{matrix} \end{matrix}$ satisfies $\begin{matrix} (4.16) & E ⩾ c (‖ U ‖_{V}^{2} + ‖ V ‖_{V}^{2}) - c_{M}^{'}, c > 0 . \end{matrix}$

Next, we take the scalar product in $H$ of (3.5) by $A_{δ} \overline{U}$ and, noting that $A_{δ} \overline{U} = A_{δ} U$ , we find $\begin{matrix} (4.17) & \frac{1}{2} \frac{d}{d t} ‖ \overline{U} ‖_{H}^{2} + {‖ A_{δ}^{\frac{1}{2}} A_{σ}^{\frac{1}{2}} U ‖}_{H}^{2} + {((F (U + V), A_{δ} U))}_{H} + {((F (U - V), A_{δ} U))}_{H} = 0 . \end{matrix}$ We then take the scalar product in $H$ of (3.7) by $A_{δ} V$ to obtain $\begin{matrix} (4.18) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2} + {‖ A_{κ}^{\frac{1}{2}} A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2} + α {‖ A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2} \\ + {((F (U + V), A_{δ} V))}_{H} + {((F (U - V), A_{δ} V))}_{H} = 0 . \end{matrix} \end{matrix}$ Summing (4.17) and (4.18), we find $\begin{matrix} (4.19) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ \overline{U} ‖_{H}^{2} + {‖ A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2}} + {‖ A_{δ}^{\frac{1}{2}} A_{σ}^{\frac{1}{2}} U ‖}_{H}^{2} + {‖ A_{κ}^{\frac{1}{2}} A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2} + α {‖ A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2} \\ + {((F (U + V), A_{δ} (U + V)))}_{H} + {((F (U - V), A_{δ} (U - V)))}_{H} = 0 . \end{matrix} \end{matrix}$ Furthermore, recalling that $f (s) = s^{3} - s$ and g is affine, we have, using (2.1) $\begin{matrix} {((F (U + V), A_{δ} (U + V)))}_{H} \\ = {((\nabla f (u + v), \nabla (u + v)))}_{Ω} + δ {((\nabla_{Γ} g (u + v), \nabla_{Γ} (u + v)))}_{Γ} \\ = {((f^{'} (u + v) \nabla (u + v), \nabla (u + v)))}_{Ω} + δ {((g^{'} (u + v) \nabla_{Γ} (u + v), \nabla_{Γ} (u + v)))}_{Γ} \\ = \int_{Ω} f^{'} (u + v) {‖ \nabla (u + v) ‖}_{Ω}^{2} d x + a δ {‖ \nabla_{Γ} (u + v) ‖}_{Γ}^{2} \\ ⩾ a δ {‖ \nabla_{Γ} (u + v) ‖}_{Γ}^{2} - {‖ \nabla (u + v) ‖}_{Ω}^{2} . \end{matrix}$ Similarly, we have $\begin{matrix} {((F (U - V), A_{δ} (U - V)))}_{H} ⩾ a δ {‖ \nabla_{Γ} (u - v) ‖}_{Γ}^{2} - {‖ \nabla (u - v) ‖}_{Ω}^{2} . \end{matrix}$ We finally obtain $\begin{matrix} (4.20) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ \overline{U} ‖_{H}^{2} + {‖ A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2}} + {‖ A_{δ}^{\frac{1}{2}} A_{σ}^{\frac{1}{2}} U ‖}_{H}^{2} + {‖ A_{κ}^{\frac{1}{2}} A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2} + α {‖ A_{δ}^{\frac{1}{2}} V ‖}_{H}^{2} \\ + 2 a δ (‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla_{Γ} v ‖_{Γ}^{2}) \\ ⩽ 2 (‖ \nabla u ‖_{Ω}^{2} + ‖ \nabla v ‖_{Ω}^{2}) ⩽ 2 (‖ U ‖_{\dot{V}}^{2} + ‖ V ‖_{\dot{V}}^{2}) . \end{matrix} \end{matrix}$

Rewriting (3.5) in the equivalent form $\begin{matrix} (4.21) & W = ⟨ W ⟩ - A_{δ}^{- 1} \frac{d U}{d t}, \end{matrix}$ we have the following estimate: $\begin{matrix} (4.22) & ‖ \overline{W} ‖_{\dot{V}} ⩽ c {‖ \frac{\partial U}{\partial t} ‖}_{- 1} . \end{matrix}$ Noting that, proceeding as in (4.12), $\begin{matrix} (4.23) & {| ⟨ W ⟩ |}^{2} ⩽ c (‖ A_{σ} U ‖_{H}^{2} + {‖ f (u + v) ‖}_{Ω}^{2} + {‖ f (u - v) ‖}_{Ω}^{2} + {‖ g (u + v) ‖}_{Γ}^{2} + {‖ g (u - v) ‖}_{Γ}^{2} + 1), \end{matrix}$ we finally find $\begin{matrix} (4.24) & \begin{matrix} ‖ W ‖_{V}^{2} ⩽ & c ({‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + ‖ A_{σ} U ‖_{H}^{2} + {‖ f (u + v) ‖}_{Ω}^{2} + {‖ f (u - v) ‖}_{Ω}^{2} \\ + {‖ g (u + v) ‖}_{Γ}^{2} + {‖ g (u - v) ‖}_{Γ}^{2} + 1), \end{matrix} \end{matrix}$ which implies that $W \in L^{2} (0, T, V)$ .

5. Existence of solutions

In what follows we prove the following existence and uniqueness result:

Theorem 5.1.
We assume that $U_{0}, V_{0} \in V$ . Then, ( 3.5 )–( 3.8 ) possesses a unique weak solution $(U, V)$ such that $\begin{array}{l} U \in L^{\infty} (R^{+}, V) \cap C ([0, T]; H) \cap L^{2} (0, T; D (A_{δ}^{\frac{1}{2}} A_{σ}^{\frac{1}{2}})), \\ V \in L^{\infty} (R^{+}, V) \cap C ([0, T]; H) \cap L^{2} (0, T; D (A_{δ}^{\frac{1}{2}} A_{σ}^{\frac{1}{2}}) \cap D (A_{δ}^{\frac{1}{2}})) \end{array}$ and $\begin{matrix} (\frac{\partial U}{\partial t}, \frac{\partial V}{\partial t}) \in L^{2} (R^{+}; V^{'} \times H), \forall T > 0 . \end{matrix}$
Proof.
Existence: The proof of existence of a solution is based on the a priori estimates derived in the previous section and e.g., a standard Galerkin scheme. In particular, it follows from (4.14), (4.16) and Gronwall’s lemma, that we can construct a sequence $(U_{m}, V_{m})$ to a proper approximated problem such that $\begin{array}{l} (5.1) & (U_{m}, V_{m}) \to (U, V) weak star in L^{\infty} (0, T; V \times V), \\ (5.2) & (\frac{\partial U_{m}}{\partial t}, \frac{\partial V_{m}}{\partial t}) \to (\frac{\partial U}{\partial t}, \frac{\partial V}{\partial t}) weakly in L^{2} (0, T; V^{'} \times H) . \end{array}$ It follows from (5.1), (5.2) and the Aubin–Lions compactness theorem that $\begin{matrix} (5.3) & (U_{m}, V_{m}) \to (U, V) strongly in C ([0, T]; H \times H) . \end{matrix}$ The strong convergence given by (5.3) also implies that $\begin{matrix} (5.4) & (U_{m}, V_{m}) \to (U, V) a.e., \end{matrix}$ which allows us, using classical arguments, to pass to the limit in the nonlinear terms.

Uniqueness: Let $(U_{1}, W_{1}, V_{1})$ and $(U_{2}, W_{2}, V_{2})$ be two solutions of (3.5)–(3.8) with initial data $(U_{1, 0}, W_{1, 0}, V_{1, 0})$ and $(U_{2, 0}, W_{2, 0}, V_{2, 0})$ , respectively, such that $⟨ U_{1, 0} ⟩ = ⟨ U_{2, 0} ⟩$ . Set $\begin{matrix} (U, W, V) = (U_{1}, W_{1}, V_{1}) - (U_{2}, W_{2}, V_{2}) \end{matrix}$ and $\begin{matrix} (U_{0}, W_{0}, V_{0}) = (U_{1, 0}, W_{1, 0}, V_{1, 0}) - (U_{2, 0}, W_{2, 0}, V_{2, 0}) . \end{matrix}$ Then, $(U, W, V)$ satisfies $\begin{array}{l} (5.5) & A_{δ}^{- 1} \frac{d \overline{U}}{d t} = - \overline{W}, \\ (5.6) & W = A_{σ} U + F (U_{1} + V_{1}) - F (U_{2} + V_{2}) + F (U_{1} - V_{1}) - F (U_{2} - V_{2}), \\ (5.7) & \frac{d V}{d t} = - A_{κ} V - F (U_{1} + V_{1}) + F (U_{2} + V_{2}) - F (U_{1} - V_{1}) + F (U_{2} - V_{2}) - α V . \end{array}$ Taking the scalar product in $H$ of (5.5) by $\overline{U}$ , we get $\begin{matrix} (5.8) & \frac{1}{2} \frac{d}{d t} ‖ \overline{U} ‖_{- 1}^{2} = - {((\overline{W}, \overline{U}))}_{H} . \end{matrix}$ We note that ${((\overline{W}, \overline{U}))}_{H} = {((W, U))}_{H}$ and $\overline{U} = U$ , since $⟨ U_{0} ⟩ = 0$ .

We can now take the scalar product in $H$ of (5.6) by U and have $\begin{matrix} (5.9) & \begin{matrix} {((W, U))}_{H} = & ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), U))}_{H} \\ + {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), U))}_{H} . \end{matrix} \end{matrix}$ We thus obtain $\begin{matrix} (5.10) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ U ‖_{- 1}^{2} + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} \\ + {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), U))}_{H} + {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), U))}_{H} = 0 . \end{matrix} \end{matrix}$ Take now the scalar product in $H$ of (5.7) by V to have $\begin{matrix} (5.11) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ V ‖_{H}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), V))}_{H} + {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), V))}_{H} = 0 . \end{matrix} \end{matrix}$

Finally, summing (5.10) and (5.11), we obtain $\begin{matrix} (5.12) & \begin{matrix} \frac{1}{2} \frac{d}{d t} (‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}) + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), U + V))}_{H} + {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), U - V))}_{H} \\ = 0 . \end{matrix} \end{matrix}$

Let $p = u_{1} + v_{1}$ and $q = u_{2} + v_{2}$ . We have $\begin{matrix} (5.13) & \begin{matrix} {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), U + V))}_{H} \\ = {((f (u_{1} + v_{1}) - f (u_{2} + v_{2}), u + v))}_{Ω} + {((g (u_{1} + v_{1}) - g (u_{2} + v_{2}), u + v))}_{Γ} \\ = {(f (p) - f (q), p - q)}_{Ω} + {((g (p) - g (q), p - q))}_{Γ} \\ = {(f^{'} (ξ_{f}) (p - q), p - q)}_{Ω} + {((g^{'} (ξ_{g}) (p - q), p - q))}_{Γ} \\ ⩾ a ‖ p - q ‖_{Γ}^{2} - ‖ p - q ‖_{Ω}^{2} = a ‖ u + v ‖_{Γ}^{2} - ‖ u + v ‖_{Ω}^{2}, \end{matrix} \end{matrix}$ where $ξ_{f}$ and $ξ_{g}$ are some appropriate linear combinations of p and q.

Proceeding similarly, we obtain $\begin{matrix} (5.14) & {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), U - V))}_{H} ⩾ a ‖ u - v ‖_{Γ}^{2} - ‖ u - v ‖_{Ω}^{2} . \end{matrix}$ Therefore, using the previous inequalities, equation (5.12) yields $\begin{matrix} (5.15) & \begin{matrix} \frac{1}{2} \frac{d}{d t} (‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}) + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ + 2 a (‖ u ‖_{Γ}^{2} + ‖ v ‖_{Γ}^{2}) ⩽ 2 (‖ u ‖_{Ω}^{2} + ‖ v ‖_{Ω}^{2}) . \end{matrix} \end{matrix}$ Employing the interpolation inequality $\begin{matrix} ‖ U ‖_{H}^{2} ⩽ c ‖ U ‖_{- 1} \cdot ‖ \nabla U ‖_{H} ⩽ ε ‖ \nabla U ‖_{H}^{2} + c ‖ U ‖_{- 1}^{2}, \end{matrix}$ we deduce that $\begin{matrix} (5.16) & \begin{matrix} \frac{d}{d t} (‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}) + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ ⩽ c (‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}) . \end{matrix} \end{matrix}$ It finally follows from Gronwall’s lemma that $\begin{matrix} (5.17) & ‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2} ⩽ e^{c^{'} t} (‖ U_{0} ‖_{- 1}^{2} + ‖ V_{0} ‖_{H}^{2}), \end{matrix}$ whence the uniqueness, as well as the continuous dependence with respect to the initial data in the ${\dot{V}}^{'} \times H$ -norm. □

6. Further regularity results

The purpose of this section is to obtain higher-order a priori estimates on the solution of the Cahn–Hilliard/Allen–Cahn system.

Thanks to the regularity (4.24), we know that $W \in L^{2} (0, T; V)$ and $\overline{W} \in L^{2} (0, T; \dot{H})$ . We deduce that $A_{δ}^{- 1} \frac{d \overline{U}}{d t} \in L^{2} (0, T; \dot{H})$ , $\forall T > 0$ . Furthermore, since g is affine and $u, v \in L^{\infty} (R^{+}; H^{1} (Γ))$ , then $\begin{matrix} g (u + v), g (u - v) \in L^{\infty} (0, T; H^{1} (Γ)) . \end{matrix}$ Next, owing to the continuous embedding $H^{1} (Ω) \subset L^{6} (Ω)$ and as f is a cubic potential, we have that $\begin{matrix} f (u + v), f (u - v) \in L^{\infty} (0, T; L^{2} (Ω)) . \end{matrix}$ Noting that $\begin{array}{l} \nabla f (u + v) = f^{'} (u + v) (\nabla u + \nabla v), \\ \nabla f (u - v) = f^{'} (u - v) (\nabla u - \nabla v), \end{array}$ we thus see that $\begin{array}{l} (6.1) & {‖ \nabla f (u + v) ‖}_{Ω} ⩽ c (‖ u + v ‖_{L^{\infty} (Ω)}^{2} + 1) (‖ \nabla u ‖_{Ω} + ‖ \nabla v ‖_{Ω}), \\ (6.2) & {‖ \nabla f (u - v) ‖}_{Ω} ⩽ c (‖ u - v ‖_{L^{\infty} (Ω)}^{2} + 1) (‖ \nabla u ‖_{Ω} + ‖ \nabla v ‖_{Ω}), \end{array}$ which yields, employing Agmon’s inequality $‖ u ‖_{L^{\infty} (Ω)} ⩽ ‖ u ‖_{H^{1} (Ω)}^{\frac{1}{2}} ‖ u ‖_{H^{2} (Ω)}^{\frac{1}{2}}$ , that $\begin{matrix} (6.3) & F (U + V), F (U - V) \in L^{2} (0, T; V) . \end{matrix}$ Thanks to (4.24), we obtain $\begin{matrix} A_{σ} U = G (t) \in L^{2} (0, T; V) . \end{matrix}$ Using the elliptic regularity result in [28], we deduce that $\begin{matrix} (6.4) & U \in L^{2} (0, T; H^{3} (Ω) \times H^{3} (Γ)) . \end{matrix}$ Now, we have $\begin{matrix} A_{κ} V = - \frac{d V}{d t} - F (U + V) + F (U - V) - α V \in L^{2} (0, T; H) \end{matrix}$ and thus, by elliptic regularity, we obtain $\begin{matrix} (6.5) & V \in L^{2} (0, T; H^{2} (Ω) \times H^{2} (Γ)) . \end{matrix}$

Having these regularity estimates, we now differentiate with respect to time the following equations for $\overline{U}$ and V, respectively: $\begin{array}{l} (6.6) & A_{δ}^{- 1} \frac{d \overline{U}}{d t} + A_{σ} \overline{U} + \overline{F (U + V)} + \overline{F (U - V)} = 0, \\ (6.7) & \frac{d V}{d t} + A_{κ} V + F (U + V) - F (U - V) + α V = 0, \end{array}$ and we find $\begin{array}{l} (6.8) & A_{δ}^{- 1} \frac{d}{d t} \frac{d \overline{U}}{d t} + A_{σ} \frac{d \overline{U}}{d t} + \overline{F^{'} (U + V) (\frac{d U}{d t} + \frac{d V}{d t})} + \overline{F^{'} (U - V) (\frac{d U}{d t} - \frac{d V}{d t})} = 0, \\ (6.9) & \frac{d}{d t} \frac{d V}{d t} + A_{κ} \frac{d V}{d t} + α \frac{d V}{d t} + F^{'} (U + V) (\frac{d U}{d t} + \frac{d V}{d t}) - F^{'} (U - V) (\frac{d U}{d t} - \frac{d V}{d t}) = 0, \end{array}$ where $\begin{matrix} F^{'} (U + V) (\frac{d U}{d t} + \frac{d V}{d t}) = (\begin{matrix} f^{'} (u + v) (\frac{\partial u}{\partial t} + \frac{\partial v}{\partial t}) \\ g^{'} (u + v) (\frac{\partial u}{\partial t} + \frac{\partial v}{\partial t}) \end{matrix}) \end{matrix}$ and $\begin{matrix} F^{'} (U - V) (\frac{d U}{d t} - \frac{d V}{d t}) = (\begin{matrix} f^{'} (u - v) (\frac{\partial u}{\partial t} - \frac{\partial v}{\partial t}) \\ g^{'} (u - v) (\frac{\partial u}{\partial t} - \frac{\partial v}{\partial t}) \end{matrix}) . \end{matrix}$ Taking the scalar product in $H$ of equations (6.8) and (6.9) by $\frac{d U}{d t}$ and $\frac{d V}{d t}$ , respectively, we have $\begin{matrix} (6.10) & \begin{matrix} \frac{1}{2} \frac{d}{d t} ({‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2}) + {((F^{'} (U + V) (\frac{d U}{d t} + \frac{d V}{d t}), \frac{d U}{d t} + \frac{d V}{d t}))}_{H} \\ + {‖ \frac{\partial \overline{U}}{\partial t} ‖}_{\dot{V}}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{V}^{2} + α {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} + {((F^{'} (U - V) (\frac{d U}{d t} - \frac{d V}{d t}), \frac{d U}{d t} - \frac{d V}{d t}))}_{H} \\ ⩽ 0, \end{matrix} \end{matrix}$ which yields, owing to (2.1) and employing the interpolation inequality $\begin{matrix} {‖ \frac{\partial U}{\partial t} ‖}_{H}^{2} ⩽ {‖ \frac{\partial U}{\partial t} ‖}_{- 1} {‖ \frac{\partial U}{\partial t} ‖}_{\dot{V}}, \end{matrix}$ the differential inequality $\begin{matrix} (6.11) & \frac{d}{d t} ({‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2}) + {‖ \frac{\partial U}{\partial t} ‖}_{\dot{V}}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{V}^{2} ⩽ c ({‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2}) . \end{matrix}$ Therefore, owing to (4.14), we can apply the uniform Gronwall lemma to (6.11) and we obtain that $\begin{array}{l} (\frac{\partial U}{\partial t}, \frac{\partial V}{\partial t}) \in (L^{\infty} (r, T; V^{'}) \cap L^{2} (r, T; \dot{V})) \times (L^{\infty} (r, T; H) \cap L^{2} (r, T; V)), \\ (6.12) & \forall r > 0, \forall T > r . \end{array}$

Now, we return to (3.3), which we write in the following form: $\begin{matrix} A_{κ} V = - \frac{d V}{d t} - F (U + V) + F (U - V) - α V \in L^{2} (r, T; V) . \end{matrix}$ The elliptic regularity result in [28] yields that $\begin{matrix} (6.13) & V \in L^{2} (r, T; H^{3} (Ω) \times H^{3} (Γ)), \forall r > 0, \forall T > r . \end{matrix}$ Next, owing to Hölder’s inequality, the continuous embedding $H^{1} (Ω) \subset L^{6} (Ω)$ , and the interpolation inequality $‖ ρ ‖_{H^{1} (Ω)} ⩽ c ‖ ρ ‖_{L^{2} (Ω)}^{\frac{1}{2}} ‖ ρ ‖_{H^{2} (Ω)}^{\frac{1}{2}}, \forall ρ \in H^{2} (Ω)$ , we have $\begin{matrix} {‖ Δ f (u + v) ‖}_{Ω} \\ ⩽ c ({‖ {(u + v)}^{2} Δ (u + v) ‖}_{Ω} + ‖ Δ u ‖_{Ω} + ‖ Δ v ‖_{Ω} + {‖ (u + v) {| \nabla (u + v) |}^{2} ‖}_{Ω}) \\ ⩽ c (‖ u + v ‖_{L^{6} (Ω)}^{2} {‖ Δ (u + v) ‖}_{L^{6} (Ω)} + {‖ Δ (u + v) ‖}_{Ω} + ‖ u + v ‖_{L^{6} (Ω)} {‖ \nabla (u + v) ‖}_{L^{6} (Ω)}^{2}) \\ ⩽ c (‖ u + v ‖_{H^{1} (Ω)}^{2} ‖ u + v ‖_{H^{3} (Ω)} + ‖ u + v ‖_{H^{2} (Ω)}) \end{matrix}$ and similarly we can have $\begin{matrix} {‖ Δ f (u - v) ‖}_{Ω} ⩽ c (‖ u - v ‖_{H^{1} (Ω)}^{2} ‖ u - v ‖_{H^{3} (Ω)} + ‖ u - v ‖_{H^{2} (Ω)}) . \end{matrix}$ We note that here we have used the fact that $\begin{matrix} f^{'} (u \pm v) \sim [{(u \pm v)}^{2} + 1] and f^{″} (u \pm v) \sim [(u \pm v) + 1] . \end{matrix}$ Therefore, $\begin{matrix} (6.14) & f (u + v), f (u - v) \in L^{2} (r, T; H^{2} (Ω)) . \end{matrix}$ Now, as g is affine, we get $\begin{matrix} (6.15) & g (u + v), g (u - v) \in L^{2} (r, T; H^{2} (Γ)) . \end{matrix}$

Let us now note that (3.5) and (6.12) imply $\begin{matrix} (6.16) & \overline{W} \in L^{2} (r, T; H^{3} (Ω) \times H^{3} (Γ)), \forall r > 0 . \end{matrix}$ We have, in view of (3.6), $\begin{matrix} ⟨ W ⟩ = ⟨ F (U + V) ⟩ + ⟨ F (U - V) ⟩ . \end{matrix}$ Using the Sobolev embedding $H^{1} (Ω) \subset L^{3} (Ω)$ valid on both two and three dimensional spaces, we also know that $\begin{matrix} ⟨ F (u \pm v) ⟩ \sim \int_{Ω} [{(u \pm v)}^{3} + 1] d x \in L^{\infty} (0, T), \end{matrix}$ and thus $⟨ W ⟩ \in L^{\infty} (0, T)$ . Therefore, $\begin{matrix} (6.17) & W \in L^{2} (r, T; H^{3} (Ω) \times H^{3} (Γ)), \forall r > 0 . \end{matrix}$

Writing (3.6) as $\begin{matrix} A_{σ} U = W - F (U + V) - F (U - V) \end{matrix}$ and using $W \in L^{2} (r, T; H^{2} (Ω) \times H^{2} (Γ))$ , we see that $A_{σ} U \in L^{2} (r, T; H^{2} (Ω) \times H^{2} (Γ))$ . Taking into account the elliptic regularity, we finally obtain that $\begin{matrix} (6.18) & U \in L^{2} (r, T; H^{4} (Ω) \times H^{4} (Γ)), \forall r > 0, \forall T > 0 . \end{matrix}$ We thus deduce that $\begin{matrix} (6.19) & (U, V) \in L^{2} (r, T; H^{4} (Ω) \times H^{4} (Γ)) \times L^{2} (r, T; H^{3} (Ω) \times H^{3} (Γ)), \forall r > 0 . \end{matrix}$ Consequently, the equations $\begin{matrix} (6.20) & \{\begin{matrix} \frac{d U}{d t} + A_{δ} (A_{σ} U + F (U + V) + F (U - V)) = 0, \\ \frac{d V}{d t} + A_{κ} V + F (U + V) - F (U - V) + α V = 0, \end{matrix} \end{matrix}$ make sense in $L^{2} (r, T; H)$ .

We have thus proved the following regularity result:

Theorem 6.1.
Let the assumptions of Theorem 5.1 hold. Then we have $\begin{array}{l} (6.21) & (U, V) \in L^{2} (r, T; H^{4} (Ω) \times H^{4} (Γ)) \times L^{2} (r, T; H^{3} (Ω) \times H^{3} (Γ)), \\ (6.22) & W \in L^{2} (r, T; H^{3} (Ω) \times H^{3} (Γ)), \end{array}$ and $\begin{matrix} (6.23) & (\frac{\partial U}{\partial t}, \frac{\partial V}{\partial t}) \in (L^{\infty} (r, T; V^{'}) \cap L^{2} (r, T; \dot{V})) \times (L^{\infty} (r, T; H) \cap L^{2} (r, T; V)), \end{matrix}$ for all $r > 0$ and $T > r$ . Furthermore, the problem written under the differential operator form $\begin{matrix} (6.24) & \{\begin{matrix} \frac{d U}{d t} + A_{δ} (A_{σ} U + F (U + V) + F (U - V)) = 0, \\ \frac{d V}{d t} + A_{κ} V + F (U + V) - F (U - V) + α V = 0, \end{matrix} \end{matrix}$ makes sense in $L^{2} (r, T; H)$ .

7. Existence of global attractors

In what follows, we intend to prove the existence of global attractors for the CH/AC system. The techniques used are classical and are based on the existence of absorbing sets in suitable function spaces. For more details on the subject, we refer the interested reader to [1,29] and [36].

Using Theorem 5.1, let us first define the continuous semigroup $\begin{matrix} S (t) : Φ_{M} \to Φ_{M}, (U_{0}, V_{0}) \mapsto (U (t), V (t)), t ⩾ 0, \end{matrix}$ where $\begin{matrix} Φ_{M} : = {(U, V) \in V \times V, ⟨ U ⟩ = M} . \end{matrix}$ This family of operators forms a semigroup (i.e., $S (0) = I$ , where I denotes the identity operator, and $S (t + s) = S (t) \circ S (s), t, s ⩾ 0$ ).

Furthermore, it follows from (5.17) that we can extend, in a unique way and by continuity, this semigroup to $\begin{matrix} Φ_{M}^{'} : = {(U, V) \in V^{'} \times H, ⟨ U ⟩ = M} . \end{matrix}$

It is then standard to prove the following result:

Theorem 7.1.

The semigroup $S (t)$ possesses the global attractor $A_{M}$ on $Φ_{M}^{'}$ for the $V^{'} \times H$ -topology which is bounded in $Φ_{M}$ .

Proof.

It follows from (4.14) and Gronwall’s lemma that we have $\begin{matrix} E (t) ⩽ e^{- c t} E (0) + c^{'}, \end{matrix}$ which yields that $\begin{matrix} (7.1) & {‖ U (t) ‖}_{V^{'}}^{2} + {‖ V (t) ‖}_{H}^{2} ⩽ e^{- c t} (‖ U_{0} ‖_{V^{'}}^{2} + ‖ V_{0} ‖_{H}^{2}) + c^{'} . \end{matrix}$ Thus, $S (t)$ is dissipative in $Φ_{M}^{'}$ in the sense that it possesses a bounded absorbing set $B_{0} \subset Φ_{M}^{'}$ . Assume now that $(u_{0}, v_{0})$ belongs to a ball centered at zero and of radius R, $R > 0$ given, i.e., $‖ U_{0} ‖_{V} + ‖ V_{0} ‖_{V} ⩽ R$ . Then, since the embedding $V \times V \subset V^{'} \times H$ is continuous, $(U_{0}, V_{0})$ belongs to a bounded set in $Φ_{M}^{'}$ . Thus, there exists $t_{0} (= t_{0} (R)) ⩾ 0$ such that $t ⩾ t_{0}$ implies $S (t) (U_{0}, V_{0}) \in B_{0}$ . It then follows from (4.13) that $\begin{array}{l} (7.2) & \int_{t}^{t + 1} {‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2}} d s ⩽ c, t ⩾ t_{0}, \\ (7.3) & \int_{t}^{t + 1} \int_{Ω} {F (u + v) + F (u - v)} d x d s ⩽ c, t ⩾ t_{0}, \end{array}$ and $\begin{matrix} (7.4) & \int_{t}^{t + 1} \int_{Γ} {G (u + v) + G (u - v)} d Σ d s ⩽ c, t ⩾ t_{0}, \end{matrix}$ where the (positive) constants c depend only on $B_{0}$ and on M. We can then apply the uniform Gronwall lemma to (4.14) to deduce that $\begin{matrix} (7.5) & {‖ U (t) ‖}_{V}^{2} + {‖ V (t) ‖}_{V}^{2} ⩽ c, t ⩾ t_{0} + 1, \end{matrix}$ where the constant c depends only on $B_{0}$ and M. It follows from (7.5) that the semigroup $S (t)$ is dissipative in $Φ_{M}$ i.e., it possesses a bounded absorbing set $B_{1}$ in $Φ_{M}$ which is compact in $Φ_{M}^{'}$ .

Thus, using Theorem 2.35 in [24], we prove the existence of the global attractor $A_{M}$ in $Φ_{M}^{'}$ . □

We now give the following definition:

Definition 7.1.

Let $X \subset E$ be a relatively compact set. For $ε > 0$ , let $N_{ε} (X, E)$ be the minimal number of balls in E of radius ε which are necessary to cover X. Then, the fractal dimension of X is the quantity (which belongs to $[0, + \infty]$ ) $\begin{matrix} {dim}_{Frac} X = \underset{ε \to 0^{+}}{lim sup} \frac{{log}_{2} N_{ε} (X)}{{log}_{2} \frac{1}{ε}} (= \underset{ε \to 0^{+}}{lim sup} \frac{ln N_{ε} (X)}{ln \frac{1}{ε}}) . \end{matrix}$

Furthermore, the quantity $H_{ε} (X, E) = {log}_{2} N_{ε} (X, E)$ is called the Kolmogorov ε-entropy of X.

We then give the following general result for proving the finite fractal dimensionality of a compact set (see [42]):

Theorem 7.2.

Let X be a compact subset of E. We assume that there exists a Banach space $E_{1}$ , with norm $‖ \cdot ‖_{E_{1}}$ , such that $E_{1}$ is compactly embedded in E and a mapping $L : X \to X$ such that $L (X) = X$ and L satisfies the following smoothing property on the difference of two solutions: $\begin{matrix} (7.6) & ‖ L x_{1} - L x_{2} ‖_{E_{1}} ⩽ c ‖ x_{1} - x_{2} ‖_{E}, \forall x_{1}, x_{2} \in X, c > 0 . \end{matrix}$ Then, the fractal dimension of X is finite and satisfies $\begin{matrix} (7.7) & {dim}_{Frac} X ⩽ H_{\frac{1}{4 c}} (B_{E_{1}} (0, 1), E), \end{matrix}$ where $B_{E_{1}} (0, 1)$ is the unit ball in $E_{1}$ (note that it is relatively compact in E, so that $H_{\frac{1}{4 c}} (B_{E_{1}} (0, 1), E)$ is finite).

We now prove the finite fractal dimensionality of the global attractor $A_{M}$ using Theorem 7.2:

Theorem 7.3.

The global attractor $A_{M}$ has finite fractal dimension for the topology of $Φ_{M}^{'}$ .

Proof.

We consider two solutions $(U_{1}, V_{1})$ and $(U_{2}, V_{2})$ to the problem with initial data $(U_{1, 0}, V_{1, 0})$ and $(U_{2, 0}, V_{2, 0})$ , respectively, belonging to $A_{M}$ . We have, setting $(U, V) = (U_{1}, V_{1}) - (U_{2}, V_{2})$ and $(U_{0}, V_{0}) = (U_{1, 0}, V_{1, 0}) - (U_{2, 0}, V_{2, 0})$ (note that $⟨ U (t) ⟩ = ⟨ U_{0} ⟩, t ⩾ 0$ ), $\begin{array}{l} (7.8) & A_{δ}^{- 1} \frac{d U}{d t} + A_{σ} U + F (U_{1} + V_{1}) - F (U_{2} + V_{2}) + F (U_{1} - V_{1}) - F (U_{2} - V_{2}) = 0, \\ (7.9) & \frac{d V}{d t} + A_{κ} V + F (U_{1} + V_{1}) - F (U_{2} + V_{2}) - F (U_{1} - V_{1}) + F (U_{2} - V_{2}) + α V = 0, \\ (7.10) & U (0) = U_{0}, V (0) = V_{0} . \end{array}$

First, proceeding as in the proof of uniqueness in Theorem 5.1, we obtain $\begin{matrix} (7.11) & \begin{matrix} \frac{d}{d t} (‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}) + ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2} \\ ⩽ c (‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2}) . \end{matrix} \end{matrix}$ It follows from Gronwall’s lemma that $\begin{matrix} ‖ U ‖_{- 1}^{2} + ‖ V ‖_{H}^{2} ⩽ e^{c^{'} t} (‖ U_{0} ‖_{- 1}^{2} + ‖ V_{0} ‖_{H}^{2}), t ⩾ 0, \end{matrix}$ which yields $\begin{matrix} (7.12) & \int_{0}^{1} {‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2}} d t ⩽ c (‖ U_{0} ‖_{- 1}^{2} + ‖ V_{0} ‖_{H}^{2}) . \end{matrix}$ Next, taking the scalar product in $H$ of (7.8) by $t \frac{d U}{d t}$ , we find $\begin{matrix} (7.13) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {t (‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2})} + t {‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + t {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), \frac{d U}{d t}))}_{H} \\ + t {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), \frac{d U}{d t}))}_{H} = \frac{1}{2} (‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2}) . \end{matrix} \end{matrix}$ Taking then the scalar product in $H$ of (7.9) by $t \frac{d V}{d t}$ , we have $\begin{matrix} (7.14) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {t (‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2})} \\ + t {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} + t {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), \frac{d V}{d t}))}_{H} \\ - t {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), \frac{d V}{d t}))}_{H} \\ = \frac{1}{2} (‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2}) . \end{matrix} \end{matrix}$

Summing (7.13) and (7.14), we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {t (‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2})} + t ({‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2}) \\ + t {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), \frac{d U}{d t}))}_{H} + t {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), \frac{d U}{d t}))}_{H} \\ + t {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), \frac{d V}{d t}))}_{H} - t {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), \frac{d V}{d t}))}_{H} \\ (7.15) & = \frac{1}{2} (‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2}) . \end{array}$ Furthermore, $\begin{matrix} (7.16) & \begin{matrix} {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), \frac{d U}{d t}))}_{H} \\ ⩽ | {((A_{δ}^{\frac{1}{2}} (F (U_{1} + V_{1}) - F (U_{2} + V_{2})), A_{δ}^{- \frac{1}{2}} \frac{d U}{d t}))}_{H} | \\ ⩽ c {‖ \frac{\partial U}{\partial t} ‖}_{- 1} {{‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖}_{Ω}^{2} + δ {‖ \nabla_{Γ} (g (u_{1} + v_{1}) - g (u_{2} + v_{2})) ‖}_{Γ}^{2}}^{\frac{1}{2}} \end{matrix} \end{matrix}$ and similarly $\begin{matrix} (7.17) & \begin{matrix} {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), \frac{d U}{d t}))}_{H} \\ ⩽ c {‖ \frac{\partial U}{\partial t} ‖}_{- 1} {{‖ \nabla (f (u_{1} - v_{1}) - f (u_{2} - v_{2})) ‖}_{Ω}^{2} + δ {‖ \nabla_{Γ} (g (u_{1} - v_{1}) - g (u_{2} - v_{2})) ‖}_{Γ}^{2}}^{\frac{1}{2}} . \end{matrix} \end{matrix}$ We now have $\begin{matrix} (7.18) & \begin{matrix} {‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖}_{Ω} \\ ⩽ {‖ \nabla (f^{'} (ξ_{u_{1}, u_{2}, v_{1}, v_{2}}) (u + v)) ‖}_{Ω} \\ ⩽ {‖ f^{″} (ξ_{u_{1}, u_{2}, v_{1}, v_{2}}) \nabla ξ_{u_{1}, u_{2}, v_{1}, v_{2}} (u + v) ‖}_{Ω} + {‖ f^{'} (ξ_{u_{1}, u_{2}, v_{1}, v_{2}}) \nabla (u + v) ‖}_{Ω} \\ ⩽ c {‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} \nabla ξ_{u_{1}, u_{2}, v_{1}, v_{2}} (u + v) ‖}_{Ω} + c {‖ (ξ_{u_{1}, u_{2}, v_{1}, v_{2}}^{2} + 1) \nabla (u + v) ‖}_{Ω}, \end{matrix} \end{matrix}$ where $ξ_{u_{1}, u_{2}, v_{1}, v_{2}}$ is an appropriate linear combination of $u_{1}$ , $v_{1}$ , $u_{2}$ , $v_{2}$ .

Concerning the first term on the right-hand side, we see that $\begin{matrix} {‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} \nabla ξ_{u_{1}, u_{2}, v_{1}, v_{2}} (u + v) ‖}_{Ω} & ⩽ c ‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{L^{\infty} (Ω)} ‖ \nabla ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{Ω} ‖ u + v ‖_{L^{6} (Ω)} \\ ⩽ c ‖ ξ ‖_{H^{1} (Ω)}^{\frac{1}{2}} ‖ ξ ‖_{H^{2} (Ω)}^{\frac{1}{2}} ‖ \nabla ξ ‖_{H^{\frac{1}{2}} (Ω)} ‖ u + v ‖_{H^{1} (Ω)} \\ ⩽ c ‖ ξ ‖_{H^{1} (Ω)} ‖ ξ ‖_{H^{2} (Ω)} ‖ u + v ‖_{H^{1} (Ω)} . \end{matrix}$ Using the interpolation inequality $‖ ξ ‖_{H^{\frac{1}{2}} (Ω)} ⩽ c ‖ ξ ‖_{Ω}^{\frac{1}{2}} ‖ ξ ‖_{H^{1} (Ω)}^{\frac{1}{2}}$ , we see that $\begin{matrix} ξ_{u_{1}, u_{2}, v_{1}, v_{2}} \in L^{\infty} (R^{+}, V) \end{matrix}$ and thus $\begin{matrix} {‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} \nabla ξ_{u_{1}, u_{2}, v_{1}, v_{2}} (u + v) ‖}_{Ω} ⩽ c ‖ ξ_{+} ‖_{H^{2} (Ω)} ‖ u + v ‖_{H^{1} (Ω)} . \end{matrix}$ Concerning the second term on the right-hand side of (7.18), we proceed similarly, $\begin{matrix} {‖ (ξ_{u_{1}, u_{2}, v_{1}, v_{2}}^{2} + 1) \nabla (u + v) ‖}_{Ω} & ⩽ c (‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{L^{\infty} (Ω)}^{2} + 1) ‖ u + v ‖_{H^{1} (Ω)} \\ ⩽ c (‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{H^{1} (Ω)} ‖ ξ_{+} ‖_{H^{2} (Ω)} + 1) ‖ u + v ‖_{H^{1} (Ω)} \\ ⩽ c (‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{H^{2} (Ω)} + 1) ‖ u + v ‖_{H^{1} (Ω)} . \end{matrix}$ We thus obtain $\begin{matrix} (7.19) & {‖ \nabla (f (u_{1} + v_{1}) - f (u_{2} + v_{2})) ‖}_{Ω} ⩽ c (‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{H^{2} (Ω)} + 1) ‖ u + v ‖_{H^{1} (Ω)} \end{matrix}$ and, similarly, $\begin{matrix} (7.20) & {‖ \nabla (f (u_{1} - v_{1}) - f (u_{2} - v_{2})) ‖}_{Ω} ⩽ c (‖ {\tilde{ξ}}_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{H^{2} (Ω)} + 1) ‖ u - v ‖_{H^{1} (Ω)}, \end{matrix}$ with ${\tilde{ξ}}_{u_{1}, u_{2}, v_{1}, v_{2}} \in L^{\infty} (R^{+}, V)$ . Since g is affine, we get $\begin{matrix} (7.21) & {‖ \nabla_{Γ} (g (u_{1} + v_{1}) - g (u_{2} + v_{2})) ‖}_{Γ}^{2} ⩽ a {‖ \nabla_{Γ} (u + v) ‖}_{Γ}^{2} \end{matrix}$ and $\begin{matrix} (7.22) & {‖ \nabla_{Γ} (g (u_{1} - v_{1}) - g (u_{2} - v_{2})) ‖}_{Γ}^{2} ⩽ a {‖ \nabla_{Γ} (u - v) ‖}_{Γ}^{2} . \end{matrix}$ Furthermore, $\begin{matrix} (7.23) & \begin{matrix} | t {((F (U_{1} + V_{1}) - F (U_{2} + V_{2}), \frac{d V}{d t}))}_{H} | \\ ⩽ t {‖ F (U_{1} + V_{1}) - F (U_{2} + V_{2}) ‖}_{H} {‖ \frac{\partial V}{\partial t} ‖}_{H} \\ ⩽ \frac{t}{4} {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} + t {‖ F (U_{1} + V_{1}) - F (U_{2} + V_{2}) ‖}_{H}^{2} \\ ⩽ \frac{t}{4} {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} + t {‖ f (u_{1} + v_{1}) - f (u_{2} + v_{2}) ‖}_{Ω}^{2} + t {‖ g (u_{1} + v_{1}) - g (u_{2} + v_{2}) ‖}_{Γ}^{2} \\ ⩽ \frac{t}{4} {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} + t {‖ f (u_{1} + v_{1}) - f (u_{2} + v_{2}) ‖}_{Ω}^{2} + t a ‖ u + v ‖_{Γ}^{2} \end{matrix} \end{matrix}$ and, similarly, $\begin{matrix} (7.24) & \begin{matrix} | t {((F (U_{1} - V_{1}) - F (U_{2} - V_{2}), \frac{d V}{d t}))}_{H} | \\ ⩽ \frac{t}{4} {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} + t {‖ F (U_{1} - V_{1}) - F (U_{2} - V_{2}) ‖}_{H}^{2} \\ ⩽ \frac{t}{4} {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2} + t {‖ f (u_{1} - v_{1}) - f (u_{2} - v_{2}) ‖}_{Ω}^{2} + t a ‖ u - v ‖_{Γ}^{2} . \end{matrix} \end{matrix}$ We have $\begin{matrix} (7.25) & \begin{matrix} {‖ f (u_{1} + v_{1}) - f (u_{2} + v_{2}) ‖}_{Ω}^{2} & = \int_{Ω} {| f^{'} (ξ_{u_{1}, u_{2}, v_{1}, v_{2}}) (u + v) |}^{2} d x \\ ⩽ c \int_{Ω} {(ξ_{u_{1}, u_{2}, v_{1}, v_{2}}^{2} + 1)}^{2} {(u + v)}^{2} d x \\ ⩽ c {[\int_{Ω} {(ξ_{u_{1}, u_{2}, v_{1}, v_{2}}^{2} + 1)}^{3} d x]}^{\frac{2}{3}} ‖ u + v ‖_{L^{6} (Ω)}^{2} . \end{matrix} \end{matrix}$ Noting that $u_{1}, u_{2}, v_{1}, v_{2} \in L^{\infty} (R^{+}, V)$ , we see, owing to the triangular inequality in $L^{3} (Ω)$ and the continuous embedding $L^{6} (Ω) \subset H^{1} (Ω)$ , that $\begin{matrix} {[\int_{Ω} {(u + v)}^{3} d x]}^{\frac{2}{3}} ‖ u + v ‖_{L^{6} (Ω)}^{2} & = {‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}}^{2} + 1 ‖}_{L^{3} (Ω)}^{2} ‖ u + v ‖_{L^{6} (Ω)}^{2} \\ ⩽ {[‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{L^{6} (Ω)} + 1]}^{2} ‖ u + v ‖_{H^{1} (Ω)}^{2} \\ ⩽ c ‖ u + v ‖_{H^{1} (Ω)}^{2} . \end{matrix}$ Thus, $\begin{matrix} (7.26) & {‖ f (u_{1} + v_{1}) - f (u_{2} + v_{2}) ‖}_{Ω}^{2} ⩽ c ‖ u + v ‖_{H^{1} (Ω)}^{2} \end{matrix}$ and, similarly, $\begin{matrix} (7.27) & {‖ f (u_{1} - v_{1}) - f (u_{2} - v_{2}) ‖}_{Ω}^{2} ⩽ c ‖ u - v ‖_{H^{1} (Ω)}^{2} . \end{matrix}$

Therefore, gathering all these estimates, we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {t (‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2})} + c t ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial v}{\partial t} ‖}_{H}^{2}) \\ ⩽ \frac{1}{2} (‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2}) \\ (7.28) & + c t (‖ u ‖_{H^{1} (Ω)}^{2} + ‖ v ‖_{H^{1} (Ω)}^{2} + ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla_{Γ} v ‖_{Γ}^{2} + h (t) ‖ u ‖_{H^{1} (Ω)}^{2} + h (t) ‖ v ‖_{H^{1} (Ω)}^{2}), \end{array}$ where $\begin{matrix} h (t) = ‖ ξ_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{H^{2} (Ω)}^{2} + ‖ {\tilde{ξ}}_{u_{1}, u_{2}, v_{1}, v_{2}} ‖_{H^{2} (Ω)}^{2}, \end{matrix}$ with $\begin{matrix} ξ_{u_{1}, u_{2}, v_{1}, v_{2}}, {\tilde{ξ}}_{u_{1}, u_{2}, v_{1}, v_{2}} \in L^{2} (0, T; H^{2} (Ω)), \end{matrix}$ since $u_{1}, u_{2}, v_{1}, v_{2} \in L^{2} (0, T; H^{2} (Ω))$ thanks to Theorem 5.1.

Thus, denoting $\begin{matrix} (7.29) & E_{1} (t) = ‖ \nabla u ‖_{Ω}^{2} + σ ‖ \nabla_{Γ} u ‖_{Γ}^{2} + ‖ \nabla v ‖_{Ω}^{2} + κ ‖ \nabla_{Γ} v ‖_{Γ}^{2} + α ‖ V ‖_{H}^{2}, \end{matrix}$ we obtain the following differential inequality: $\begin{matrix} (7.30) & \frac{d}{d t} (t E_{1} (t)) + t ({‖ \frac{\partial U}{\partial t} ‖}_{- 1}^{2} + {‖ \frac{\partial V}{\partial t} ‖}_{H}^{2}) ⩽ c (h (t) t + 1) E_{1} (t), \end{matrix}$ where $h \in L^{1} (0, T)$ . In particular, $\begin{matrix} (7.31) & \frac{d}{d t} (t E_{1} (t)) ⩽ c h (t) (t E_{1} (t)) + c E_{1} (t) . \end{matrix}$ Using Gronwall’s lemma, it follows that $\begin{matrix} (7.32) & t E_{1} (t) ⩽ c e^{\int_{0}^{t} h (s) d s} \int_{0}^{t} e^{- \int_{0}^{τ} h (τ) d τ} E_{1} (s) d s ⩽ c^{'} \int_{0}^{t} E_{1} (s) d s \end{matrix}$ and, taking $t = 1$ , we have $\begin{matrix} (7.33) & E_{1} (1) ⩽ c^{'} \int_{0}^{1} E_{1} (s) d s . \end{matrix}$ From (7.12), we also obtain $\begin{matrix} (7.34) & \int_{0}^{1} E_{1} (s) d s ⩽ c (‖ U_{0} ‖_{- 1}^{2} + ‖ V_{0} ‖_{H}^{2}) . \end{matrix}$ Consequently, $\begin{matrix} (7.35) & E_{1} (1) ⩽ c (‖ U_{0} ‖_{- 1}^{2} + ‖ V_{0} ‖_{H}^{2}) \end{matrix}$ and we conclude the proof, thanks to Theorem 7.2 with $L = S (1)$ . □

Footnotes

Acknowledgement

The authors wish to thank an anonymous referee for her/his careful reading of the manuscript and many useful comments which helped improve it.

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