Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn

Abstract

We consider a class of Cahn–Hilliard equation that models phase separation process of binary mixtures involving nontrivial boundary interactions in a bounded domain with non-permeable wall. The system is characterized by certain dynamic type boundary conditions and the total mass, in the bulk and on the boundary, is conserved for all time. For the case with physically relevant singular (e.g., logarithmic) potential, global regularity of weak solutions is established. In particular, when the spatial dimension is two, we show the instantaneous strict separation property such that for arbitrary positive time any weak solution stays away from the pure phases $\pm 1$ , while in the three dimensional case, an eventual separation property for large time is obtained. As a consequence, we prove that every global weak solution converges to a single equilibrium as $t \to \infty$ , by the usage of an extended Łojasiewicz–Simon inequality.

Keywords

Cahn–Hilliard equation dynamic boundary condition singular potential separation from pure states convergence to equilibrium

1. Introduction

In this paper, we consider the Cahn–Hilliard equation: $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t} u - Δ μ = 0, \\ μ = - Δ u + F^{'} (u), \end{matrix} in (0, \infty) \times Ω, \end{matrix}$ subject to the following dynamic boundary conditions: $\begin{matrix} (1.2) & \{\begin{matrix} \partial_{t} u + \partial_{ν} μ - σ Δ_{Γ} μ + κ μ = 0, \\ μ = \partial_{ν} u - χ Δ_{Γ} u + F_{Γ}^{'} (u), \end{matrix} on (0, \infty) \times Γ, \end{matrix}$ and the initial condition $\begin{matrix} (1.3) & u (0) = u_{0} in Ω . \end{matrix}$ Here, $Ω \subset R^{n}$ ( $n = 2, 3$ ) is a bounded domain with smooth boundary $Γ : = \partial Ω$ , $ν = ν (x)$ is the unit outer normal vector on Γ and $\partial_{ν}$ denotes the outward normal derivative on the boundary. The symbol Δ denotes the usual Laplace operator in Ω and $Δ_{Γ}$ stands for the Laplace–Beltrami operator on Γ. F and $F_{Γ}$ denote the bulk and boundary potentials, respectively. The constant $κ ⩾ 0$ is related to possible mass exchange to the environment and σ, χ are some given nonnegative constants that account for possible boundary diffusion. When $σ, χ > 0$ , system (1.1)–(1.3) can be regarded as equation and dynamic boundary condition of Cahn–Hilliard type.

The Cahn–Hilliard equation (1.1) is a fundamental diffuse interface model for multi-phase systems. It was first proposed in materials science to describe the pattern formation evolution of micro-structures during the phase separation process in binary alloys [4,51]. In recent years, it has been extended to many areas of scientific research, for instance, diblock copolymer, image inpainting, and multiphase fluid flows. When the evolution is confined in a bounded domain Ω, suitable boundary conditions should be taken into account for equation (1.1). Classical choices are the homogeneous Neumann boundary conditions: $\begin{matrix} \partial_{ν} μ = \partial_{ν} u = 0, on (0, \infty) \times Γ . \end{matrix}$ The corresponding initial boundary value problem has been well-understood and fairly complete results on its mathematical analysis (e.g., well-posedness, regularity of solutions and long-time behavior) have been obtained in the literature. We refer to, for instance, [1,17,28,34,39,54] and the references therein, for further details, see the recent review paper [44].

In recent studies, the so-called dynamic boundary conditions have been proposed in order to describe certain effective short-range interactions between the mixture and the solid wall (i.e., the boundary) [18,35]. In this case, the evolution of binary mixtures is characterized by the total free energy of the following typical form: $\begin{matrix} (1.4) & E (u) : = \int_{Ω}^{} (\frac{1}{2} | \nabla u |^{2} + F (u)) d x + \int_{Γ}^{} (\frac{χ}{2} | \nabla_{Γ} u |^{2} + F_{Γ} (u)) d Γ, \end{matrix}$ that is, the sum of a Ginzburg–Landau (bulk) free energy and a surface free energy. The potential function F usually has a double-well structure and a thermodynamically relevant case is given by the so-called logarithmic potential: $\begin{matrix} (1.5) & F (r) = (1 + r) ln (1 + r) + (1 - r) ln (1 - r) - c r^{2}, for r \in (- 1, 1), \end{matrix}$ where the constant $c > 0$ is large enough such that F is nonconvex and has local minima at $r = \pm r_{*}$ , where $- 1 < - r_{*} < 0 < r_{*} < 1$ . This potential function is viewed as a singular one since its derivative $F^{'} : = β + π$ with $\begin{matrix} (1.6) & β (r) : = ln (\frac{1 + r}{1 - r}), for all r \in (- 1, 1), π (r) = - 2 c r, for all r \in [- 1, 1], \end{matrix}$ satisfies ${lim}_{r \to \pm 1} β (r) sgn r = \infty$ . In applications, it is often approximated by certain regular potentials with the prototype given by $F (r) = (1 / 4) {(r^{2} - 1)}^{2}$ on the extended domain $R$ . Based on the energy functional (1.4), different types of dynamic boundary conditions for the Cahn–Hilliard equation have been derived and analyzed in the literature, see for instance, [7,10,12,15,20,21,25–27,29,37,38,42,43,45,47,52,53,60]. In particular, concerning the dynamic boundary condition (1.2) that we are going to investigate in this paper, it was first introduced in [21] (with $σ = 0$ , $χ > 0$ , $κ > 0$ , referred to as the Wentzell boundary condition) and then derived in a slightly different form by [29] (with $σ ⩾ 0$ , $χ ⩾ 0$ , $κ = 0$ ). This type of boundary condition describes the bulk-surface phase separation process in a binary mixture confined to a bounded region with porous walls such that possible mass fluxes between the bulk and the boundary are allowed. The parameter κ distinguishes the cases of permeable wall ( $κ > 0$ ) and non-permeable wall ( $κ = 0$ ), which is related to the property on conservation of total (i.e., bulk plus boundary) mass such that $\begin{matrix} (1.7) & \frac{d}{d t} (\int_{Ω} u d x + \int_{Γ} u d Γ) = - κ \int_{Γ} μ d Γ, in (0, \infty) . \end{matrix}$ On the other hand, under condition (1.2), the system preserves the dissipation of total free energy $E (u)$ provided that $σ, κ ⩾ 0$ : $\begin{matrix} (1.8) & \frac{d}{d t} E (u) + \int_{Ω} | \nabla μ |^{2} d x + \int_{Γ} (σ | \nabla_{Γ} μ |^{2} + κ | μ |^{2}) d Γ = 0, in (0, \infty) . \end{matrix}$

The initial boundary value problem (1.1)–(1.3) with regular potentials F and $F_{Γ}$ has been studied extensively in the literature. When $σ = 0$ , $χ > 0$ , existence, uniqueness and regularity of solutions were proved in [21,22,33] ( $κ > 0$ ) and [23,33] ( $κ = 0$ ) by different approaches; long-time behavior of global solutions were investigated in [22,58] ( $κ > 0$ ) and [23,24] ( $κ = 0$ ), proving the existence of global and exponential attractors as well as convergence of global solutions to single steady states as $t \to \infty$ . Concerning the problem with general (singular) potentials and a non-permeable wall ( $κ = 0$ ), existence and uniqueness of global weak solutions and their long-time behavior were studied in [8] ( $σ ⩾ 0$ , $χ ⩾ 0$ ) and [29] ( $σ = χ = 1$ ), see also [11] in which the double obstacle potential was handled and recent works [14,15,27] for the system with additional convection and viscous terms. Last but not least, we refer to [9,20] for numerical studies, to [19] for the associated optimal boundary control problem, and to [49] for the existence of time periodic solutions.

In this paper, we consider the initial boundary value problem (1.1)–(1.3) with $\begin{matrix} κ = 0, χ > 0 and σ ⩾ 0, \end{matrix}$ namely, imposing the evolution problem in a bounded domain with non-permeable wall and keeping the contributation of boundary diffusion in the free energy (below we just take $χ = 1$ without loss of generality). In particular, we are interested in the regularity of global weak solutions and their long-time behavior when the potential F is allowed to be singular (e.g., (1.5)). As it has been pointed out in [8,48], the Cahn–Hilliard equation with dynamic boundary condition and singular potential is mathematically difficult, since the interplay between them may allow the solution to reach the pure states $\pm 1$ in regions with nonzero measure. To handle this, several attempts have been made in the literature. In [29], the authors obtained the regularity and long-time behavior of solutions under certain growth restrictions on F, which unfortunately exclude the thermodynamically relevant logarithmic function. Later in [8], the authors introduced a variational inequality that enables them to prove the existence of finite-dimensional attractors for variational solutions, including the case of logarithmic nonlinearities (cf. [48] for the case with a different type of dynamic boundary condition). We note that in those works, the boundary potential $F_{Γ}$ was assume to be a $C^{2}$ function with at most quadratic growth. On the other hand, under a different assumption that the boundary potential $F_{Γ}$ somehow dominates the bulk potential F, the authors of [11] could prove the existence of global weak as well as strong solutions for a general class of nonlinearities.

Below we choose to work with singular potentials in a setting similar to [11]. In this case, the bulk and boundary potentials in (1.4) are decomposed as $\begin{matrix} F (r) = \hat{β} (r) + \hat{π} (r), F_{Γ} (r) = {\hat{β}}_{Γ} (r) + {\hat{π}}_{Γ} (r), \end{matrix}$ where $\hat{β}, {\hat{β}}_{Γ} : R \to [0, \infty]$ are some convex, proper and l.s.c. functions and $\hat{π}, {\hat{π}}_{Γ} : R \to R$ are of class $C^{2}$ with Lipschitz continuous first derivatives. The associated subdifferentials are denoted by $β = \partial \hat{β}$ , $β_{Γ} = \partial {\hat{β}}_{Γ}$ , respectively, which are maximal monotone operators with domains $D (β)$ , $D (β_{Γ})$ . Under suitable assumptions on these nonlinearities (see (A1)–(A3) in Section 2 for details) that in particular are fulfilled by the physically relevant logarithmic potential (1.5), the following results can be established for problem (1.1)–(1.3).

(I) Regularity of global weak solutions. More precisely, we show the so-called strict separation property provided that the initial datum is not a pure state $\pm 1$ (see Theorem 2.1): in both two and three dimensions, the global weak solution will be regular and stay uniformly away from $\pm 1$ after a sufficient large time; while in dimension two, the strict separation indeed happens instantaneously, with a uniform distance (with respect to the initial energy and total mass) for all $t ⩾ η$ ( $η > 0$ is an arbitrary but fixed constant). Our result gives a first example on the instantaneous separation property of weak solutions to the Cahn–Hilliard equation subject to dynamic boundary conditions in two dimension. It also extends the existing literature, for instance, [28,46] for Cahn–Hilliard type equations with logarithmic potential as well as classical Neumann boundary conditions, and [29] for the case with dynamic boundary condition in which the eventually separation property was obtained under certain stronger assumptions on the bulk potential that excludes the logarithmic potential (1.5).

(II) Long-time behavior. Once the separation property is proven, the Cahn–Hilliard equation with singular potentials can be regarded as an equation with globally Lipschitz nonlinearities from a certain time on. Thus, we are able to study the long-time behavior of global solutions just like the case with regular potentials [24,58]. More precisely, assuming in addition that the potentials F, $F_{Γ}$ are real analytic, we prove the convergence of any global weak solution to a single equilibrium as $t \to \infty$ (see Theorem 2.2). The same subject was treated in [29, Theorem 3.22] by applying an extended Łojasiewicz–Simon inequality. However, the result therein was obtained only on a restricted situation for F, excluding the logarithmic potential (1.5) (see [29, Remark 3.8]). The proof of convergence to equilibrium relies on the celebrated Łojasiewicz–Simon approach, see e.g., [30,32] for a simplified illustration. It has been successfully applied to the study of Cahn–Hilliard type equations, for instance, we can refer to [3,10,24,42,52,54,58,60] for the case of regular potentials and to [1,28] for the case of the logarithmic potential (1.5). See also [40,56,59] for related results on the second-order Allen–Cahn type equations under dynamic boundary conditions.

The remaining part of this paper is organized as follows. In Section 2, we introduce the function spaces and necessary assumptions, and state the main results. In Section 3, we derive some uniform estimates and give a preliminary result on the regularity of global weak solutions. In Section 4, we prove our main result Theorem 2.1 on the separation property. Section 5 is devoted to the proof of Theorem 2.2 on the convergence to equilibrium. In the Appendix, we report some technical lemmas that have been used in this paper.

2. Preliminaries and main results

In this section, we set up our target problem and state the main results.

2.1. Notation

If X is a (real) Banach space and $X^{*}$ is its topological dual, then $‖ \cdot ‖_{X}$ indicates the norm of X and ${⟨ \cdot, \cdot ⟩}_{X^{*}, X}$ denotes the corresponding duality product. We assume that $Ω \subset R^{n}$ ( $n = 2, 3$ ) is a bounded domain with smooth boundary $Γ : = \partial Ω$ . Then we denote by $L^{p} (Ω)$ and $L^{p} (Γ)$ $(p ⩾ 1)$ the standard Lebesgue spaces. When $p = 2$ , the inner products in the Hilbert spaces $L^{2} (Ω)$ and $L^{2} (Γ)$ will be denoted by ${(\cdot, \cdot)}_{L^{2} (Ω)}$ and ${(\cdot, \cdot)}_{L^{2} (Γ)}$ , respectively. For $s \in R$ , $p ⩾ 1$ , $W^{s, p} (Ω)$ and $W^{s, p} (Γ)$ stand for the Sobolev spaces. If $p = 2$ , we denote $W^{s, p} (Ω) = H^{s} (Ω)$ and $W^{s, p} (Γ) = H^{s} (Γ)$ . For simplicity, we denote $\begin{array}{l} H : = L^{2} (Ω), V : = H^{1} (Ω), W : = H^{2} (Ω), \\ H_{Γ} : = L^{2} (Γ), V_{Γ} : = H^{1} (Γ), W_{Γ} : = H^{2} (Γ), \end{array}$ with standard norms and inner products indicated above. Next, we define the Hilbert spaces $\begin{array}{l} H : = H \times H_{Γ}, \\ V : = {z \in V \times V_{Γ} : z = (z, z_{Γ}) and z_{Γ} = z_{|_{Γ}} a.e. on Γ}, \\ W : = (W \times W_{Γ}) \cap V, \end{array}$ endowed with natural inner products and related norms. Here, $z_{|_{Γ}}$ stands for the trace of a function z defined in Ω. Hereafter, we use a bold letter like $z$ to denote the corresponding pair $(z, z_{Γ})$ . Let us restate that if $z : = (z, z_{Γ}) \in V$ then $z_{Γ}$ means exactly the trace of z on Γ, while if $z : = (z, z_{Γ}) \in H$ , then $z \in H$ and $z_{Γ} \in H_{Γ}$ are actually independent. From the definition, we easy see that $V$ is dense in $H$ and the chain of continuous embeddings holds $V \subset V \subset H \subset V^{*} \subset V^{*}$ . Indeed, for each $z \in V$ we see that $(z, z_{|_{Γ}}) \in H$ , and for each $z^{*} \in V^{*}$ we can define $z^{*} \in V^{*}$ by ${⟨ z^{*}, z ⟩}_{V^{*}, V} : = {⟨ z^{*}, z ⟩}_{V^{*}, V}$ for all $z = (z, z_{Γ}) \in V$ . In what follows, we set for $σ ⩾ 0$ $\begin{array}{l} V_{σ} : = V, W_{σ} : = W, if σ > 0, \\ V_{σ} : = V, W_{σ} : = W, if σ = 0 . \end{array}$ For any $z^{*} \in V_{σ}^{*}$ , we define the generalized mean value by setting $\begin{matrix} m (z^{*}) : = \frac{1}{| Ω | + | Γ |} {⟨ z^{*}, 1 ⟩}_{V_{σ}^{*}, V_{σ}}, \end{matrix}$ where $| Ω | : = \int_{Ω}^{} 1 d x$ and $| Γ | : = \int_{Γ}^{} 1 d Γ$ . It leads to the usual mean value function when applied to elements of $H$ , i.e., $m : H \to R$ such that $\begin{matrix} m (z) : = \frac{1}{| Ω | + | Γ |} (\int_{Ω}^{} z d x + \int_{Γ}^{} z_{Γ} d Γ), for all z \in H . \end{matrix}$ Next, we introduce the subspace $H_{0}$ of $H$ by $\begin{matrix} H_{0} : = {z \in H : m (z) = 0}, \end{matrix}$ and define $V_{0} : = V \cap H_{0}$ , $W_{0} : = W \cap H_{0}$ , respectively. Then the dense and compact embedding $V_{0} ↪ ↪ H_{0}$ holds (see for instance, [11, Lemma B]). Moreover, for $σ ⩾ 0$ , we also set $\begin{matrix} V_{σ, 0} : = V_{σ} \cap H_{0} . \end{matrix}$ The equivalent norms in spaces $H_{0}$ , $V_{σ, 0}$ are given by $‖ z ‖_{H_{0}} : = ‖ z ‖_{H}$ for all $z \in H_{0}$ and $\begin{matrix} ‖ z ‖_{V_{σ, 0}} : = {(\int_{Ω}^{} | \nabla z |^{2} d x + σ \int_{Γ}^{} | \nabla_{Γ} z_{Γ} |^{2} d Γ)}^{1 / 2}, for all z \in V_{σ, 0}, \end{matrix}$ thanks to the generalized Poincaré inequality (A.1). For $σ ⩾ 0$ , we define the following bilinear form: $\begin{matrix} a_{σ} (z, \tilde{z}) : = \int_{Ω}^{} \nabla z \cdot \nabla \tilde{z} d x + σ \int_{Γ}^{} \nabla_{Γ} z_{Γ} \cdot \nabla_{Γ} {\tilde{z}}_{Γ} d Γ, for all z, \tilde{z} \in V_{σ}, \end{matrix}$ and the duality mapping $A_{σ} : V_{σ, 0} \to V_{σ, 0}^{*}$ given by $\begin{matrix} {⟨ A_{σ} z, \tilde{z} ⟩}_{V_{σ, 0}^{*}, V_{σ, 0}} = a_{σ} (z, \tilde{z}), for all z, \tilde{z} \in V_{σ, 0} . \end{matrix}$ Then from [23] (for $σ = 0$ ) and [11] (for $σ > 0$ ), we infer that the operator $A_{σ}$ is a linear isomorphism and its inverse operator $A_{σ}^{- 1} : V_{σ, 0}^{*} \to V_{σ, 0}$ is compact on $H_{0}$ . Besides, we can define the inner product in $V_{σ, 0}^{*}$ by $\begin{matrix} (2.1) & {(z^{*}, {\tilde{z}}^{*})}_{V_{σ, 0}^{*}} : = {⟨ z^{*}, A_{σ}^{- 1} {\tilde{z}}^{*} ⟩}_{V_{σ, 0}^{*}, V_{σ, 0}}, for all z^{*}, {\tilde{z}}^{*} \in V_{σ, 0}^{*} . \end{matrix}$ Finally, we denote by $P : H \to H_{0}$ the projection $\begin{matrix} P z : = z - m (z) 1 = (z - m (z), z_{Γ} - m (z)), for all z \in H . \end{matrix}$ Then for any $z^{*} \in V_{σ, 0}^{*}$ , one can define ${⟨ z^{*}, z ⟩}_{V_{σ}^{*}, V_{σ}} : = {⟨ z^{*}, P z ⟩}_{V_{σ, 0}^{*}, V_{σ, 0}}$ for all $z \in V_{σ}$ . Furthermore, one can identify the dual space $V_{σ, 0}^{*}$ by ${z^{*} \in V_{σ}^{*} : {⟨ z^{*}, 1 ⟩}_{V_{σ}^{*}, V_{σ}} = 0}$ .

2.2. The initial boundary value problem

Hereafter, for each $T \in (0, \infty)$ we denote $\begin{matrix} Q_{T} : = (0, T) \times Ω, Σ_{T} : = (0, T) \times Γ, Q : = (0, \infty) \times Ω, Σ : = (0, \infty) \times Γ . \end{matrix}$ The original problem (1.1)–(1.3) can be viewed as a sort of transmission problem that consists of a Cahn–Hilliard equation in the bulk and another evolution equation on the boundary as a dynamic boundary condition (cf. [47]). To this end, introducing two new variables on Γ: $\begin{matrix} u_{Γ} = u_{|_{Γ}}, μ_{Γ} = μ_{|_{Γ}}, \end{matrix}$ we can reformulate the target problem (1.1)–(1.3) as follows (recalling that we take $κ = 0$ , $χ = 1$ ): find $u, μ : Q \to R$ and $u_{Γ}, μ_{Γ} : Σ \to R$ satisfying $\begin{matrix} (2.2) & \{\begin{matrix} \partial_{t} u - Δ μ = 0, & a.e. in Q, \\ μ = - Δ u + β (u) + π (u), & a.e. in Q, \\ u_{|_{Γ}} = u_{Γ}, μ_{|_{Γ}} = μ_{Γ}, & a.e. on Σ, \\ \partial_{t} u_{Γ} + \partial_{ν} μ - σ Δ_{Γ} μ_{Γ} = 0, & a.e. on Σ, \\ μ_{Γ} = \partial_{ν} u - Δ_{Γ} u_{Γ} + β_{Γ} (u_{Γ}) + π_{Γ} (u_{Γ}), & a.e. on Σ, \\ u (0) = u_{0}, & a.e. in Ω, \\ u_{Γ} (0) = u_{0 Γ}, & a.e. on Γ . \end{matrix} \end{matrix}$ In this manner, the original Cahn–Hilliard equation (1.1) subject to those nontrivial boundary conditions (1.2) can be viewed as a bulk-surface coupled system such that the bulk unknown variables $(u, μ)$ now satisfy (standard) nonhomogeneous Dirichlet boundary conditions that are determined through a surface evolution system for the boundary variables $(u_{Γ}, μ_{Γ})$ .

Next, we present our basic hypotheses on the nonlinear terms and initial data.

$β, β_{Γ} \in C^{1} (- 1, 1)$ are monotone increasing functions with $\begin{array}{l} lim_{r ↘ - 1} β (r) = - \infty, lim_{r ↘ - 1} β_{Γ} (r) = - \infty, \\ lim_{r ↗ 1} β (r) = \infty, lim_{r ↗ 1} β_{Γ} (r) = \infty . \end{array}$ Their primitive denoted by $\hat{β}$ , ${\hat{β}}_{Γ}$ , respectively, satisfy $\hat{β}$ , ${\hat{β}}_{Γ} \in C ([- 1, 1]) \cap C^{2} (- 1, 1)$ . The derivatives $β^{'}$ , $β_{Γ}^{'}$ are convex and $\begin{matrix} β^{'} (r) ⩾ γ, β_{Γ}^{'} (r) ⩾ γ, for all r \in (- 1, 1) \end{matrix}$ for some positive constant $γ > 0$ . Without loss of generality, we set $\hat{β} (0) = {\hat{β}}_{Γ} (0) = β (0) = β_{Γ} (0) = 0$ and make the extension $\hat{β} (r) = \infty$ , ${\hat{β}}_{Γ} (r) = \infty$ for $| r | > 1$ .

There exist positive constants $c_{1}$ , $c_{2}$ such that $\begin{matrix} | β (r) | ⩽ c_{1} | β_{Γ} (r) | + c_{2}, for all r \in (- 1, 1) . \end{matrix}$

$π, π_{Γ} \in W^{1, \infty} (R)$ such that $\begin{matrix} | π^{'} (r) | ⩽ L, | π_{Γ}^{'} (r) | ⩽ L, for all r \in R, \end{matrix}$ with $L > 0$ being a certain given constant.

$u_{0} : = (u_{0}, u_{0 Γ}) \in V$ , $m (u_{0}) = m_{0}$ for some constant $m_{0} \in (- 1, 1)$ and the compatibility conditions $\hat{β} (u_{0}) \in L^{1} (Ω)$ , ${\hat{β}}_{Γ} (u_{0 Γ}) \in L^{1} (Γ)$ hold.

Remark 2.1.

The physically relevant logarithmic potential with Lipschitz perturbations (1.5), serves as a typical example that satisfies assumptions $(A 1)$ – $(A 3)$ . The assumption $m_{0} \in (- 1, 1)$ in (A4) indicates that the initial datum is not allowed to be a pure state (i.e., $\pm 1$ ). On the other hand, if the initial datum is a pure state then no separation process will take place.

Assumption (A2) can be viewed as a compatibility condition for the bulk/boundary potentials, which implies that the boundary potential plays a dominant role. This choice was first proposed in [5] for the Allen–Cahn equation with dynamic boundary condition and then used in [12,13] for the Cahn–Hilliard equation. We remark that an assumption on the opposite direction (i.e., a dominant bulk potential) is also possible (see for instance, the one postulated in [26] for the Cahn–Hilliard equation with a dynamic boundary condition of Allen–Cahn type).

As a preliminary result, we have the following conclusion on existence and uniqueness of global weak solutions to problem (2.2).
Proposition 2.1 (Global weak solutions).

Suppose that $Ω \subset R^{n}$ $(n = 2, 3)$ is a bounded domain with smooth boundary Γ and $σ ⩾ 0$ . For arbitrary $T \in (0, \infty)$ , under the assumptions (A1)–(A4), problem ( 2.2 ) admits a global weak solution $(u, μ) = (u, u_{Γ}, μ, μ_{Γ})$ in the following sense: $\begin{array}{l} (2.3) & v \in H^{1} (0, T; V_{σ, 0}^{*}) \cap L^{\infty} (0, T; V_{0}) \cap L^{2} (0, T; W_{0}), \\ (2.4) & μ \in L^{2} (0, T; V_{σ}), \\ (2.5) & (β (u), β_{Γ} (u_{Γ})) \in L^{2} (0, T; H), \end{array}$ with $v = u - m_{0} 1$ such that $\begin{matrix} (2.6) & {⟨ u^{'} (t), z ⟩}_{V_{σ}^{*}, V_{σ}} + a_{σ} (μ (t), z) = 0, for all z \in V_{σ}, a.a. t \in (0, T) \end{matrix}$ and $\begin{array}{l} (2.7) & μ = - Δ u + β (u) + π (u), a.e. in Q_{T}, \\ (2.8) & μ_{Γ} = \partial_{ν} u - Δ_{Γ} u_{Γ} + β_{Γ} (u_{Γ}) + π_{Γ} (u_{Γ}), a.e. on Σ_{T}, \\ (2.9) & u (0) = u_{0}, a.e. in Ω, u_{Γ} (0) = u_{0 Γ}, a.e. on Γ . \end{array}$ Moreover, the function $u$ is unique and we have $\begin{matrix} (2.10) & {‖ u^{(1)} (t) - u^{(2)} (t) ‖}_{V_{σ, 0}^{*}} ⩽ C_{T} {‖ u_{0}^{(1)} - u_{0}^{(2)} ‖}_{V_{σ, 0}^{*}}, for all t \in [0, T], \end{matrix}$ where for $i = 1, 2$ , $u^{(i)}$ is the weak solution corresponding to the initial datum $u_{0}^{(i)}$ , $C_{T}$ is a positive constant only depending on L and T.

The proof of Proposition 2.1 with $σ > 0$ follows the same arguments as [11, Theorems 2.1, 2.2, and Remarks 1, 2], while the case $σ = 0$ can be treated in a similar way with minor modifications on function spaces and energy estimates. Here, we only sketch the strategy of its proof. For each $ε > 0$ , we consider the following viscous Cahn–Hilliard system: $\begin{array}{l} (2.11) & v_{ε}^{'} (t) + A_{σ} P μ_{ε} (t) = 0 in V_{σ, 0}^{*}, for a.a. t \in (0, T), \\ (2.12) & μ_{ε} (t) = ε v_{ε}^{'} (t) + \partial φ (v_{ε} (t)) + β_{ε} (u_{ε} (t)) + π (u_{ε} (t)) in H, for a.a. t \in (0, T), \\ (2.13) & v_{ε} (0) = v_{0} in H_{0}, \end{array}$ where $v_{ε} = u_{ε} - m_{0} 1$ , $π (z) : = (π (z), π_{Γ} (z_{Γ}))$ and $β_{ε} (z) : = (β_{ε} (z), β_{Γ, c_{1} ε} (z_{Γ}))$ such that $β_{ε}$ , $β_{Γ, c_{1} ε}$ are the standard Yosida approximation of β, $β_{Γ}$ , respectively (see e.g., [6] and [11, Section 4]). In equation (2.12), $φ : H_{0} \to [0, \infty]$ is a proper, lower semi-continuous and convex functional defined by $\begin{matrix} φ (z) : = \{\begin{matrix} \frac{1}{2} \int_{Ω}^{} | \nabla z |^{2} d x + \frac{1}{2} \int_{Γ}^{} | \nabla_{Γ} z_{Γ} |^{2} d Γ, & if z \in V_{0}, \\ + \infty, & otherwise . \end{matrix} \end{matrix}$ The subdifferential $\partial φ$ of φ is given by $\partial φ (z) = (- Δ z, \partial_{ν} z - Δ_{Γ} z_{Γ})$ with $D (\partial φ) = V_{0} \cap W$ (see, e.g., [11, Lemma C]). Then, by the abstract theory of doubly nonlinear evolution inclusions [16], we can prove the existence and uniqueness of an approximate solution $v_{ε} \in H^{1} (0, T; H_{0}) \cap C ([0, T]; V_{0}) \cap L^{2} (0, T; W_{0})$ with $μ_{ε} \in L^{2} (0, T; V_{σ})$ to problem (2.11)–(2.13). From the definition of $v_{ε}$ , we also know that $u_{ε} \in H^{1} (0, T; H) \cap C ([0, T]; V) \cap L^{2} (0, T; W)$ . Then one can show that the family of approximating solutions $(u_{ε}, μ_{ε})$ satisfy sufficient a priori estimates that are uniform with respect to the approximation parameter ε. Hence, by taking the limit as $ε \to 0$ (up to a subsequence), the limit function $(u, μ)$ is indeed our target solution to problem (2.2) satisfying properties (2.3)–(2.9). Uniqueness of the solution can be easily obtained by using the energy method. For further details, we refer to [11, Sections 3,4].

2.3. Main results

We are now in a position to state the main results of this paper. The first theorem is related to the separation property.

Theorem 2.1 (Separation from pure states).

Suppose that the domain $Ω \subset R^{n}$ $(n = 2, 3)$ is bounded with smooth boundary Γ and $σ ⩾ 0$ , besides, assumptions (A1)–(A4) are satisfied. Let $(u, μ) = (u, u_{Γ}, μ, μ_{Γ})$ be the global weak solution to problem ( 2.2 ) obtained in Proposition 2.1 .

There exist a constant $δ_{1} \in (0, 1)$ and a large time $T_{1} > 0$ such that $\begin{matrix} (2.14) & {‖ u (t) ‖}_{L^{\infty} (Ω)} ⩽ 1 - δ_{1}, {‖ u_{Γ} (t) ‖}_{L^{\infty} (Γ)} ⩽ 1 - δ_{1}, for all t ⩾ T_{1}, \end{matrix}$ where the constant $δ_{1}$ may depend on $m_{0}$ but is independent of $u_{0}$ .

If $n = 2$ , $σ > 0$ and in addition, there is a positive constant $c_{0}$ such that $\begin{matrix} (2.15) & | β^{'} (r) | ⩽ e^{c_{0} | β (r) | + c_{0}}, for all r \in (- 1, 1), \end{matrix}$ then for any given $η > 0$ , there exists $δ_{2} \in (0, 1)$ depending on η, $m_{0}$ and $E (u_{0})$ such that $\begin{matrix} (2.16) & {‖ u (t) ‖}_{L^{\infty} (Ω)} ⩽ 1 - δ_{2}, {‖ u_{Γ} (t) ‖}_{L^{\infty} (Γ)} ⩽ 1 - δ_{2}, for all t ⩾ η . \end{matrix}$

Remark 2.2.

The estimate (2.14) implies that the value of $u$ will be strictly separated from the pure states $\pm 1$ at least after a certain large positive time by a uniform distance. As a consequence, the singular potentials β, $β_{Γ}$ and their derivatives will no longer blow up along the evolution and they turn out to be Lipschitz continuous and bounded functions. This fact leads to further higher-order regularity of global weak solutions and will be helpful for the study of long-time behavior of problem (2.2).

It is straightforward to verify that the additional assumption (2.15) is satisfied in the case of logarithmic potential (1.5).

We note that when $σ > 0$ , the term $σ Δ_{Γ} μ_{Γ}$ (accounting for possible boundary diffusion) yields a regularizing effect on the boundary. It remains an open question whether the conclusion (2.16) still holds when $σ = 0$ .

Our second result concerns the long-time behavior of problem (2.2), more precisely, we prove the uniqueness of asymptotic limit of any global weak solution as $t \to \infty$ . Theorem 2.2 (Convergence to equilibrium).

Suppose that the domain $Ω \subset R^{n}$ $(n = 2, 3)$ is bounded with smooth boundary Γ and $σ ⩾ 0$ , besides, assumptions (A1)–(A4) are satisfied. In addition, we assume that β, $β_{Γ}$ are real analytic on $(- 1, 1)$ and π, $π_{Γ}$ are real analytic on $R$ . Let $u$ be the global weak solution to problem ( 2.2 ) obtained in Proposition 2.1 , we have $\begin{matrix} lim_{t \to \infty} {‖ u (t) - u_{\infty} ‖}_{W} = 0, \end{matrix}$ where $u_{\infty} : = (u_{\infty}, u_{Γ, \infty})$ is a steady state to problem ( 2.2 ) that satisfies the nonlocal elliptic problem $\begin{matrix} \{\begin{matrix} - Δ u_{\infty} + β (u_{\infty}) + π (u_{\infty}) = μ_{\infty}, & a.e. in Ω, \\ \partial_{ν} u_{\infty} - Δ_{Γ} u_{Γ, \infty} + β_{Γ} (u_{Γ, \infty}) + π_{Γ} (u_{Γ, \infty}) = μ_{\infty}, & a.e. on Γ, \\ m (u_{\infty}) = m_{0}, \end{matrix} \end{matrix}$ with a constant $μ_{\infty}$ given by $\begin{matrix} μ_{\infty} = \frac{1}{| Ω | + | Γ |} [\int_{Ω} (β (u_{\infty}) + π (u_{\infty})) d x + \int_{Γ} (β_{Γ} (u_{Γ, \infty}) + π_{Γ} (u_{Γ, \infty})) d Γ] . \end{matrix}$ Moreover, $\begin{matrix} {‖ u (t) - u_{\infty} ‖}_{W} ⩽ C_{η} {(1 + t)}^{- \frac{θ^{*}}{1 - 2 θ^{*}}}, for all t ⩾ η > 0, \end{matrix}$ where $θ^{*} \in (0, 1 / 2)$ is a constant depending on $u_{\infty}$ , the positive constant $C_{η}$ may depend on $E (u_{0})$ , $u_{\infty}$ , $m_{0}$ , Ω, Γ, and η.

Remark 2.3.
The results of Theorem 2.1 and Theorem 2.2 can be extended to the case with permeable walls (i.e., $κ > 0$ ) with minor changes in function spaces and estimates, keeping in mind that the mass conservation property no longer holds (see (1.7)) and on the other hand, there exists an extra boundary dissipation term in the energy equality (see (1.8)). This will compensate the generalized Poincaré inequality (see Lemma A.3) to recover the $H^{1}$ -norm of μ (cf. [24,58]).

3. Regularity of global weak solutions

In this section, we prove some basic properties and preliminary regularity results for the global weak solution $(u, μ)$ to problem (2.2).

3.1. Mass conservation and energy equality

Hereafter, let $(u, μ)$ be the unique global weak solution obtained in Proposition 2.1. Taking $z = 1$ as the test function in the weak form (2.6), we easily deduce the mass conservation property for problem (2.2) (see also [11, Remark 2]):

Lemma 3.1.
For all $t ⩾ 0$ , it holds $\begin{matrix} (3.1) & m (u (t)) = m_{0} . \end{matrix}$

Next, we define the free energy of the system (recall (1.4)): $\begin{array}{l} E (z) : = & \frac{1}{2} \int_{Ω}^{} | \nabla z |^{2} d x + \frac{1}{2} \int_{Γ}^{} | \nabla_{Γ} z_{Γ} |^{2} d Γ + \int_{Ω}^{} (\hat{β} (z) + \hat{π} (z)) d x \\ (3.2) & + \int_{Γ}^{} ({\hat{β}}_{Γ} (z_{Γ}) + {\hat{π}}_{Γ} (z_{Γ})) d Γ, \end{array}$ for all $z : = (z, z_{Γ}) \in V$ with $\hat{β} (z) \in L^{1} (Ω)$ , $\hat{β} (z_{Γ}) \in L^{1} (Γ)$ . Here, the primitives $\hat{π}$ , ${\hat{π}}_{Γ}$ are given by $\begin{matrix} \hat{π} (r) = \int_{0}^{r} π (s) d s, {\hat{π}}_{Γ} (r) = \int_{0}^{r} π_{Γ} (s) d s, for r \in [- 1, 1] . \end{matrix}$ Then we can derive a basic energy inequality for problem (2.2) that yields uniform in time estimate for global weak solutions:
Lemma 3.2.
For almost all $t ⩾ 0$ , it holds $\begin{matrix} (3.3) & E (u (t)) + \int_{0}^{t} {‖ u^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s ⩽ E (u_{0}) . \end{matrix}$ Then there exists a positive constant* $M_{1}$ such that $\begin{array}{l} (3.4) & ‖ u ‖_{L^{\infty} (0, \infty; V)} + \int_{0}^{\infty} {‖ u^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s ⩽ M_{1}, \\ (3.5) & \int_{0}^{\infty} {‖ P μ (s) ‖}_{V_{σ, 0}}^{2} d s ⩽ M_{1} . \end{array}$
Proof.
The conclusion can be draw by working with the approximate solutions of (2.11)–(2.13) and then passing to the limit. Using the fact $v_{ε}^{'} \in L^{2} (0, T; H_{0})$ , the chain rule of the subdifferential (see, e.g., [55, Lemma 4.3, Section IV]) and (2.1), we see that $φ (v_{ε} (\cdot))$ is absolutely continuous on $[0, T]$ and $\begin{array}{l} \int_{0}^{t} {‖ u_{ε}^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s \\ = \int_{0}^{t} {⟨ v_{ε}^{'} (s), A_{σ}^{- 1} (- A_{σ} P μ_{ε} (s)) ⟩}_{V_{σ, 0}^{}, V_{σ, 0}} d s \\ = - \int_{0}^{t} {(v_{ε}^{'} (s), μ_{ε} (s))}_{H} d s \\ = - \int_{0}^{t} {(v_{ε}^{'} (s), ε v_{ε}^{'} (s) + \partial φ (v_{ε} (s)))}_{H_{0}} d s - \int_{0}^{t} {(u_{ε}^{'} (s), β_{ε} (u_{ε} (s)) + π (u_{ε} (s)))}_{H} d s \\ = - ε \int_{0}^{t} {‖ v_{ε}^{'} (s) ‖}_{H_{0}}^{2} d s - \int_{0}^{t} \frac{d}{d s} E_{ε} (u_{ε} (s)) d s \\ (3.6) & = - ε \int_{0}^{t} {‖ v_{ε}^{'} (s) ‖}_{H_{0}}^{2} d s - E_{ε} (u_{ε} (t)) + E_{ε} (u_{0}), \end{array}$ for all $t \in [0, T]$ , where $\begin{array}{l} E_{ε} (z) : = & \frac{1}{2} \int_{Ω}^{} | \nabla z |^{2} d x + \frac{1}{2} \int_{Γ}^{} | \nabla_{Γ} z_{Γ} |^{2} d Γ + \int_{Ω}^{} ({\hat{β}}_{ε} (z) + \hat{π} (z)) d x \\ (3.7) & + \int_{Γ}^{} ({\hat{β}}_{Γ, ε} (z_{Γ}) + {\hat{π}}_{Γ} (z_{Γ})) d Γ, for all z \in V, \end{array}$ with ${\hat{β}}_{ε} (r) = \int_{0}^{r} β_{ε} (s) d s$ , ${\hat{β}}_{Γ, ε} (r) = \int_{0}^{r} β_{Γ, ε} (s) d s$ .

Using the fact of weak and strong convergences (see, [11, Section 4.3] for $σ > 0$ and the case $σ = 0$ can be treated similarly by changing the corresponding function spaces) $\begin{array}{l} v_{ε} \to v weakly in H^{1} (0, T; V_{σ, 0}^{}), with v = u - m_{0} 1, \\ ε v_{ε} \to 0 strongly in H^{1} (0, T; H_{0}), \\ u_{ε} \to u strongly in C ([0, T]; H) \cap L^{\infty} (0, T; V), \end{array}$ the lower semicontinuity of norms and the maximal monotonicity of β, $β_{Γ}$ , we obtain $\begin{array}{l} \underset{ε \to 0}{lim inf} \int_{0}^{t} {‖ u_{ε}^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s ⩾ \int_{0}^{t} {‖ u^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s, \\ \underset{ε \to 0}{lim inf} E_{ε} (u_{ε} (t)) = lim_{ε \to 0} E_{ε} (u_{ε} (t)) = E (u (t)), for a.a. t \in (0, T) . \end{array}$ Therefore, taking lim inf as $ε \to 0$ in (3.6) (noting that $T > 0$ is arbitrary), we conclude the energy inequality (3.3). We recall that from assumptions (A1)–(A2), there exist a nonnegative constant $c_{3}$ such that the primitives $\hat{β}$ and ${\hat{β}}_{Γ}$ satisfy $\begin{matrix} (3.8) & \hat{β} (r) + \hat{π} (r) ⩾ - c_{3}, {\hat{β}}_{Γ} (r) + {\hat{π}}_{Γ} (r) ⩾ - c_{3}, for all r \in [- 1, 1] . \end{matrix}$ Hence, we have $\begin{matrix} E (u (t)) ⩾ \frac{1}{2} \int_{Ω}^{} {| \nabla u (t) |}^{2} d x + \frac{1}{2} \int_{Γ}^{} {| \nabla_{Γ} u_{Γ} (t) |}^{2} d Γ - c_{3} (| Ω | + | Γ |), \end{matrix}$ for a.a. $t ⩾ 0$ . Therefore, recalling (3.1) and the generalized Poincaré inequality (A.1), we obtain $\begin{matrix} \int_{0}^{t} {‖ u^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s ⩽ E (u_{0}) + c_{3} (| Ω | + | Γ |) \end{matrix}$ and $\begin{array}{l} {‖ u (t) ‖}_{V}^{2} & = {‖ u (t) ‖}_{H}^{2} + \int_{Ω}^{} {| \nabla u (t) |}^{2} d x + \int_{Γ}^{} {| \nabla_{Γ} u_{Γ} (t) |}^{2} d Γ \\ = {‖ u (t) - m (u (t)) ‖}_{H}^{2} + (| Ω | + | Γ |) {| m (u (t)) |}^{2} + {‖ u (t) - m (u (t)) ‖}_{V_{0}}^{2} \\ ⩽ (C_{P} + 1) {‖ u (t) - m_{0} 1 ‖}_{V_{0}}^{2} + (| Ω | + | Γ |) | m_{0} |^{2} \\ ⩽ 2 (C_{P} + 1) E (u_{0}) + 2 (C_{P} + 1) c_{3} (| Ω | + | Γ |) + (| Ω | + | Γ |) | m_{0} |^{2}, \end{array}$ for a.a. $t ⩾ 0$ . Since the right hand side is independent of t, then using the Lebesugue monotone convergence theory, we obtain the estimate (3.4). On the other hand, by the comparison of the equation (2.11), we also get (3.5).

The proof is complete. □

Next, we show that weak solutions to problem (2.2) satisfy an energy equality, which is a standard structure of the Cahn–Hilliard system. Lemma 3.3.
For any* $η > 0$ , the mapping $t \mapsto E (u (t))$ is absolutely continuous for all $t ⩾ η$ and $\begin{matrix} (3.9) & \frac{d}{d t} E (u (t)) + {‖ u^{'} (t) ‖}_{V_{σ, 0}^{}}^{2} = 0, for a.a. t ⩾ η . \end{matrix}$
Proof.
Firstly, for any given $η > 0$ , we show that there exists a positive constant $M_{2}$ such that $\begin{matrix} (3.10) & {‖ u^{'} ‖}_{L^{\infty} (η, \infty; V_{σ, 0}^{})} + \int_{t}^{t + 1} {‖ u^{'} (s) ‖}_{V}^{2} d s ⩽ M_{2}, for all t ⩾ η . \end{matrix}$ Hereafter, for each $h > 0$ , we use the symbol of difference quotient $\partial_{t}^{h} v (t) : = (v (t + h) - v (t)) / h$ with respect to the time variable t. Taking the difference of (2.11) at $t = s + h$ and $t = s$ , we have $\begin{matrix} \partial_{t}^{h} v_{ε}^{'} (s) + A_{σ} P (\partial_{t}^{h} μ_{ε} (s)) = 0 in V_{σ, 0}^{} . \end{matrix}$ This is equivalent to $\begin{matrix} (3.11) & ε \partial_{t}^{h} v_{ε}^{'} (s) + A_{σ}^{- 1} \partial_{t}^{h} v_{ε}^{'} (s) + \partial φ (\partial_{t}^{h} v_{ε} (s)) = P (- \partial_{t}^{h} β_{ε} (u_{ε} (s)) - \partial_{t}^{h} π (u_{ε} (s))) in H_{0}, \end{matrix}$ for a.a. $s ⩾ 0$ . Multiplying (3.11) by $\partial_{t}^{h} v_{ε} (s) \in H_{0}$ , using (2.1), and applying the Ehrling lemma given by Lemma A.1, we obtain that $\begin{array}{l} \frac{ε}{2} \frac{d}{d s} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{H_{0}}^{2} + \frac{1}{2} \frac{d}{d s} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{σ, 0}^{}}^{2} + {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{0}}^{2} \\ ⩽ \frac{1}{h^{2}} {(π (u_{ε} (s + h)) - π (u_{ε} (s)), u_{ε} (s + h) - u_{ε} (s))}_{H} \\ + \frac{1}{h^{2}} {(π_{Γ} (u_{Γ, ε} (s + h)) - π_{Γ} (u_{Γ, ε} (s)), u_{Γ, ε} (s + h) - u_{Γ, ε} (s))}_{H_{Γ}} \\ ⩽ L {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{H_{0}}^{2} \\ ⩽ \frac{1}{2} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{0}}^{2} + M_{2}^{'} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{0}^{}}^{2} \\ (3.12) & ⩽ \frac{1}{2} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{0}}^{2} + M_{2}^{″} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{σ, 0}^{}}^{2}, for a.a. s ⩾ 0, \end{array}$ since β, $β_{Γ}$ are monotone. The constants $M_{2}^{'}$ and $M_{2}^{″}$ in (3.12) may depend on L. Then for any fixed $η > 0$ , applying the uniform Gronwall type inequality given by Lemma A.2 with $t_{0} : = 0$ and $r : = η$ , we deduce $\begin{array}{l} ε {‖ \partial_{t}^{h} v_{ε} (t + η) ‖}_{H_{0}}^{2} + {‖ \partial_{t}^{h} v_{ε} (t + η) ‖}_{V_{σ, 0}^{}}^{2} \\ (3.13) & ⩽ \frac{1}{η} (ε \int_{t}^{t + η} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{H_{0}}^{2} d s + \int_{t}^{t + η} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{σ, 0}^{}}^{2} d s) e^{2 M_{2}^{″} η}, for all t ⩾ 0 . \end{array}$ On the other hand, we infer from (3.6) that $v_{ε} \in L^{2} (0, + \infty; V_{σ, 0}^{})$ . Hence, for a.a. $s > 0$ , it holds $\begin{matrix} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{σ, 0}^{}} ⩽ \frac{1}{h} \int_{s}^{s + h} {‖ v_{ε}^{'} (τ) ‖}_{V_{σ, 0}^{}} d τ \to {‖ v_{ε}^{'} (s) ‖}_{V_{σ, 0}^{}}, as h \to 0^{+}, \end{matrix}$ and thus $\begin{matrix} \int_{t}^{t + η} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{σ, 0}^{}}^{2} d s ⩽ \int_{t}^{t + η + h} {‖ v_{ε}^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s, for all t ⩾ 0 . \end{matrix}$ In a similar manner, we get $\begin{matrix} ε \int_{t}^{t + η} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{H_{0}}^{2} d s ⩽ ε \int_{t}^{t + η + h} {‖ v_{ε}^{'} (s) ‖}_{H_{0}}^{2} d s, for all t ⩾ 0 . \end{matrix}$ As a consequence, using (3.6) and (3.8) we have in the right hand side of (3.13) $\begin{array}{l} \frac{ε}{η} \int_{t}^{t + η} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{H_{0}}^{2} d s + \frac{1}{η} \int_{t}^{t + η} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{σ, 0}^{}}^{2} d s \\ ⩽ \frac{ε}{η} \int_{t}^{t + η + h} {‖ v_{ε}^{'} (s) ‖}_{H_{0}}^{2} d s + \frac{1}{η} \int_{t}^{t + η + h} {‖ v_{ε}^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s \\ ⩽ \frac{1}{η} [E_{ε} (u_{0}) - E_{ε} (u_{ε} (t + η + h))] \\ (3.14) & ⩽ \frac{1}{η} [E_{ε} (u_{0}) + c_{3} (| Ω | + | Γ |)], \end{array}$ for all $0 < h ≪ 1$ . Recalling (3.13) and changing the variable $τ : = t + η$ , we get $\begin{array}{l} ε {‖ \partial_{t}^{h} v_{ε} (τ) ‖}_{H_{0}}^{2} + {‖ \partial_{t}^{h} v_{ε} (τ) ‖}_{V_{σ, 0}^{}}^{2} \\ ⩽ \frac{1}{η} [E_{ε} (u_{0}) + c_{3} (| Ω | + | Γ |)] e^{2 M_{2}^{″} η}, for all τ ⩾ η, 0 < h ≪ 1, \end{array}$ where the right-hand side is independent of h. Thus, letting $h \to 0$ , we see that $\begin{array}{l} \sqrt{ε} \partial_{t}^{h} v_{ε} \to \sqrt{ε} v_{ε}^{'} weakly star in L^{\infty} (η, \infty; H_{0}), \\ \partial_{t}^{h} v_{ε} \to v_{ε}^{'} weakly star in L^{\infty} (η, \infty; V_{σ, 0}^{}) \end{array}$ as $h \to 0$ with the following estimate $\begin{matrix} (3.15) & ε {‖ v_{ε}^{'} (τ) ‖}_{H_{0}}^{2} + {‖ v_{ε}^{'} (τ) ‖}_{V_{σ, 0}^{}}^{2} ⩽ \frac{1}{η} [E_{ε} (u_{0}) + c_{3} (| Ω | + | Γ |)] e^{2 M_{2}^{″} η} \end{matrix}$ for all $τ ⩾ η$ . Since ${lim}_{ε \to 0} E_{ε} (u_{0}) = E (u_{0})$ , the right hand side of (3.15) can be bounded by a constant $M_{2}^{‴}$ independent of ε. This implies that as $ε \to 0$ it holds $\begin{array}{l} ε v_{ε}^{'} \to 0 strongly in L^{\infty} (η, \infty; H_{0}), \\ v_{ε}^{'} \to v^{'} weakly star in L^{\infty} (η, \infty; V_{σ, 0}^{}) \end{array}$ with the same estimate $\begin{matrix} (3.16) & {‖ u^{'} (τ) ‖}_{V_{σ, 0}^{}}^{2} = {‖ v^{'} (τ) ‖}_{V_{σ, 0}^{}}^{2} ⩽ M_{2}^{‴}, for all τ ⩾ η . \end{matrix}$ Finally, integrating (3.12) over $[t, t + 1]$ with respect to time s, we get for each $t ⩾ η$ $\begin{array}{l} \int_{t}^{t + 1} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{0}}^{2} d s & ⩽ ε {‖ \partial_{t}^{h} v_{ε} (t) ‖}_{H_{0}}^{2} + {‖ \partial_{t}^{h} v_{ε} (t) ‖}_{V_{σ, 0}^{}}^{2} + 2 M_{2}^{″} \int_{t}^{t + 1} {‖ \partial_{t}^{h} v_{ε} (s) ‖}_{V_{σ, 0}^{}}^{2} d s \\ ⩽ (\frac{1}{η} + 2 M_{2}^{″}) [E_{ε} (u_{0}) + c_{3} (| Ω | + | Γ |)] e^{2 M_{2}^{″} η} \end{array}$ for all $0 < h ≪ 1$ . Thus, letting $h \to 0$ and $ε \to 0$ again, we see that $\begin{matrix} (3.17) & \int_{t}^{t + 1} {‖ u^{'} (s) ‖}_{V}^{2} d s ⩽ C_{P} \int_{t}^{t + 1} {‖ v^{'} (s) ‖}_{V_{0}}^{2} d s ⩽ C_{P} (\frac{1}{η} + 2 M_{2}^{″}) M_{2}^{‴}, for all t ⩾ η . \end{matrix}$ Combining (3.16) and (3.17) we arrive at the estimate (3.10).

By virtue of the Sobolev embedding theorem, we infer from (3.4) and (3.10) the additional continuity $\begin{matrix} (3.18) & u \in C ([η, \infty); V) . \end{matrix}$ Moreover, going back to the proof of Lemma 3.2, we have $\begin{array}{l} - \int_{s}^{t} {‖ u^{'} (τ) ‖}_{V_{σ, 0}^{*}}^{2} d τ \\ = \int_{s}^{t} {(v^{'} (τ), μ (τ))}_{H} d τ \\ = \int_{s}^{t} {(v^{'} (τ), \partial φ (v (τ)))}_{H_{0}} d τ + \int_{s}^{t} {(u^{'} (τ), β (u (τ)) + π (u (τ)))}_{H} d τ \\ = \int_{s}^{t} \frac{d}{d τ} E (u (τ)) d τ \\ = E (u (t)) - E (u (s)) for all s, t ⩾ η, \end{array}$ that is, the mapping $t \mapsto E (u (t))$ is absolutely continuous for all $t ⩾ η$ and the energy equality (3.9) holds for a.a. $t ⩾ η$ . □

3.2. Higher-order estimates

We proceed to derive some higher-order estimates for the global weak solutions.

Lemma 3.4.
For any $η > 0$ , there exists a positive constant $M_{3}$ such that $\begin{array}{l} (3.19) & \begin{aligned} {‖ β (u) ‖}_{L^{\infty} (η, \infty; L^{1} (Ω))} + {‖ β_{Γ} (u_{Γ}) ‖}_{L^{\infty} (η, \infty; L^{1} (Γ))} ⩽ M_{3}, \\ ‖ μ ‖_{L^{\infty} (η, \infty; V_{σ})} ⩽ M_{3} . \end{aligned} \end{array}$
Proof.
From the comparison in equation (2.11) with the estimate (3.10), we see that $\begin{matrix} (3.20) & ‖ P μ ‖_{L^{\infty} (η, \infty; V_{σ, 0})} = {‖ v^{'} ‖}_{L^{\infty} (η, \infty; V_{σ, 0}^{})} ⩽ M_{2} . \end{matrix}$ Next, we estimate the mean value $m (μ)$ for the chemical potential $μ$ . Taking $z : = A_{σ}^{- 1} v (s)$ in the weak form (2.6) at $t = s$ , using (2.1) and (2.7)–(2.8) we get $\begin{array}{l} (3.21) & {(v^{'} (s), v (s))}_{V_{σ, 0}^{}} + {‖ v (s) ‖}_{V_{0}}^{2} + {(β (u (s)), v (s))}_{H} + {(π (u (s)), v (s))}_{H} = 0, \end{array}$ for a.a. $s ⩾ 0$ . From assumption (A1) and (A4), we see that there exist positive constants $c_{4}$ and $c_{5}$ such that (cf. [46, Proposition A.1], also [26, Section 5]) $\begin{array}{l} (3.22) & β (r) (r - m_{0}) ⩾ c_{4} | β (r) | - c_{5} for all r \in (- 1, 1), \\ (3.23) & β_{Γ} (r) (r - m_{0}) ⩾ c_{4} | β_{Γ} (r) | - c_{5} for all r \in (- 1, 1) . \end{array}$ Recalling the definition $v = u - m_{0} 1$ , and using (3.22)–(3.23), we can deduce from (3.21) and (A2) that $\begin{array}{l} c_{4} \int_{Ω}^{} | β (u (s)) | d x + c_{4} \int_{Γ}^{} | β_{Γ} (u_{Γ} (s)) | d Γ \\ ⩽ c_{5} (| Ω | + | Γ |) + {‖ π (u (s)) ‖}_{H} {‖ v (s) ‖}_{H_{0}} + {‖ v^{'} (s) ‖}_{V_{σ, 0}^{}} {‖ v (s) ‖}_{V_{σ, 0}^{}} \\ ⩽ c_{5} (| Ω | + | Γ |) + (L {‖ u (s) ‖}_{H} + {‖ π (0) ‖}_{H}) [{‖ u (s) ‖}_{H} + | m_{0} | (| Ω |^{1 / 2} + | Γ |^{1 / 2})] \\ + C {‖ u^{'} (s) ‖}_{V_{σ, 0}^{}} {‖ u (s) ‖}_{H}, for a.a. s ⩾ η . \end{array}$ Therefore, using equations (2.7), (2.8) and applying estimates (3.4), (3.10), we see that there exists a positive constant $M_{3}^{'}$ , depending on $M_{1}$ , $M_{2}$ , $c_{4}$ , $c_{5}$ , Ω, Γ, L, and $m_{0}$ such that $\begin{array}{l} | m (μ (s)) | \\ ⩽ \frac{1}{| Ω | + | Γ |} [\int_{Ω}^{} | β (u (s)) | d x + \int_{Γ}^{} | β_{Γ} (u_{Γ} (s)) | d Γ + \int_{Ω}^{} | π (u (s)) | d x + \int_{Γ}^{} | π_{Γ} (u_{Γ} (s)) | d Γ] \\ (3.24) & ⩽ M_{3}^{'}, for a.a. s ⩾ η . \end{array}$ Combining this estimate with (3.20), we can apply the generalized Poincaré inequality given by Lemma A.3 to achieve the conclusion (3.19). □
Remark 3.1.
Thanks to (3.10) and (3.19), we are able to rewrite (2.6) as $\begin{matrix} {(u^{'} (t), z)}_{H} + a_{σ} (μ (t), z) = 0, for all z \in V_{σ} and a.a. t \in [η, \infty) . \end{matrix}$ Regarding this as an elliptic problem for $μ$ , then from the elliptic regularity theory (for $σ > 0$ , see Lemma A.6), (3.10) and (3.19), we see that for any $T > η > 0$ , $\begin{matrix} μ \in L^{2} (η, T; W_{σ}) . \end{matrix}$ This allows us to obtain the strong form of equation (2.6): $\begin{array}{l} (3.25) & \partial_{t} u - Δ μ = 0, a.e. in (η, \infty) \times Ω, \\ (3.26) & \partial_{t} u_{Γ} + \partial_{ν} μ - σ Δ_{Γ} μ_{Γ} = 0, a.e. on (η, \infty) \times Γ . \end{array}$ See, e.g., [11, Section 4] for related discussions when $σ = 1$ .

The following lemma is a generalization of [28, Corollary 4.3] from the case of homogeneous Neumann boundary condition to the current higher-order nonlinear boundary condition (2.8) with singular term.
Lemma 3.5.
For any* $η > 0$ , there exists a positive constant $M_{4}$ such that $\begin{array}{l} (3.27) & {‖ β (u) ‖}_{L^{\infty} (η, \infty; L^{p} (Ω))} + {‖ β (u_{Γ}) ‖}_{L^{\infty} (η, \infty; L^{p} (Γ))} ⩽ M_{4}, \\ (3.28) & {‖ β_{Γ} (u_{Γ}) ‖}_{L^{\infty} (η, \infty; H_{Γ})} ⩽ M_{4}, \\ (3.29) & ‖ u ‖_{L^{\infty} (η, \infty; W)} ⩽ M_{4} . \end{array}$ In ( 3.27 ), when $n = 3$ , $p \in [1, 6]$ if $σ > 0$ and $p \in [1, 4]$ if $σ = 0$ ; when $n = 2$ , $p \in [1, \infty)$ .
Proof.
As in [28], for each $k \in N ∖ {1}$ , we define the Lipschitz continuous function $h_{k} : R \to R$ by $\begin{matrix} (3.30) & h_{k} (r) : = \{\begin{matrix} - 1 + \frac{1}{k} & if r < - 1 + \frac{1}{k}, \\ r & if - 1 + \frac{1}{k} ⩽ r ⩽ 1 - \frac{1}{k}, \\ 1 - \frac{1}{k} & if r > 1 - \frac{1}{k} . \end{matrix} \end{matrix}$ For $s ⩾ η$ , define $\begin{array}{l} (3.31) & \begin{aligned} u_{k} (s) : = h_{k} \circ u (s), u_{Γ, k} (s) : = h_{k} \circ u_{Γ} (s) . \end{aligned} \end{array}$ Then we have $u_{k} : = (u_{k}, u_{Γ, k}) \in C ([η, \infty); V)$ for any $η > 0$ and $\begin{matrix} \nabla u_{k} = \nabla u χ_{[- 1 + \frac{1}{k}, 1 - \frac{1}{k}]} (u), \nabla_{Γ} u_{Γ, k} = \nabla_{Γ} u_{Γ} χ_{[- 1 + \frac{1}{k}, 1 - \frac{1}{k}]} (u_{Γ}), \end{matrix}$ see, e.g., [36, Corollary A.6, Chapter II]. For any $k \in N ∖ {1}$ and $p ⩾ 2$ , we see that $β_{k} : = | β (u_{k}) |^{p - 2} β (u_{k}) \in C ([η, \infty); V)$ is well-defined and $\begin{matrix} \nabla β_{k} = (p - 1) {| β (u_{k}) |}^{p - 2} β^{'} (u_{k}) \nabla u_{k} . \end{matrix}$ Besides, we note that ${(β_{k})}_{|_{Γ}} = | β (u_{Γ, k}) |^{p - 2} β (u_{Γ, k}) \in C ([η, \infty); V_{Γ})$ . Denote $\begin{matrix} \tilde{μ} : = μ - π (u), {\tilde{μ}}_{Γ} : = μ_{Γ} - π_{Γ} (u_{Γ}) . \end{matrix}$ Then one can see that $\tilde{μ} \in L^{\infty} (η, \infty; V)$ , and ${\tilde{μ}}_{Γ} \in L^{\infty} (η, \infty; V_{Γ})$ if $σ > 0$ , ${\tilde{μ}}_{Γ} \in L^{\infty} (η, \infty; H^{1 / 2} (Γ))$ if $σ = 0$ (see Lemma 3.4).

We start to estimate the singular terms $β (u)$ and $β (u_{Γ})$ . For this purpose, multiplying the equation (cf. (2.7)) $\begin{matrix} (3.32) & - Δ u + β (u) = \tilde{μ}, \end{matrix}$ by $β_{k}$ , integrating over Ω, using integration by parts and (2.8) for the term $\partial_{ν} u$ , we get $\begin{array}{l} \int_{Ω} {| β (u_{k} (s)) |}^{p - 2} β (u_{k} (s)) β (u (s)) d x + \int_{Γ} {| β (u_{Γ, k} (s)) |}^{p - 2} β (u_{Γ, k} (s)) β_{Γ} (u_{Γ} (s)) d Γ \\ = - (p - 1) \int_{Ω}^{} {| β (u_{k} (s)) |}^{p - 2} β^{'} (u_{k} (s)) \nabla u_{k} (s) \cdot \nabla u (s) d x \\ - (p - 1) \int_{Γ}^{} {| β (u_{Γ, k} (s)) |}^{p - 2} β^{'} (u_{Γ, k} (s)) \nabla_{Γ} u_{Γ, k} (s) \cdot \nabla_{Γ} u_{Γ} (s) d Γ \\ + \int_{Ω} \tilde{μ} (s) {| β (u_{k} (s)) |}^{p - 2} β (u_{k} (s)) d x + \int_{Γ} {\tilde{μ}}_{Γ} (s) {| β (u_{Γ, k} (s)) |}^{p - 2} β (u_{Γ, k} (s)) d Γ \\ (3.33) & = : I_{1} + I_{2} + I_{3} + I_{4}, \end{array}$ for a.a. $s ⩾ η$ . From assumption (A1), we infer that $\begin{matrix} I_{1} ⩽ 0, I_{2} ⩽ 0 . \end{matrix}$ Next, by the Sobolev embedding theorem, the Hölder and Young inequalities, we obtain $\begin{matrix} I_{3} ⩽ \frac{1}{2} {‖ β (u_{k}) ‖}_{L^{p} (Ω)}^{p} + C ‖ \tilde{μ} ‖_{L^{p} (Ω)}^{p} ⩽ \frac{1}{2} {‖ β (u_{k}) ‖}_{L^{p} (Ω)}^{p} + C ‖ \tilde{μ} ‖_{V}^{p}, \end{matrix}$ and $\begin{array}{l} I_{4} & ⩽ \frac{1}{2} {‖ β (u_{Γ, k}) ‖}_{L^{q} (Γ)}^{q} + C ‖ {\tilde{μ}}_{Γ} ‖_{L^{q} (Γ)}^{q} \\ ⩽ \{\begin{matrix} \frac{1}{2} ‖ β (u_{Γ, k}) ‖_{L^{q} (Γ)}^{q} + C ‖ {\tilde{μ}}_{Γ} ‖_{V_{Γ}}^{q}, & if σ > 0, \\ \frac{1}{2} ‖ β (u_{Γ, k}) ‖_{L^{q} (Γ)}^{q} + C ‖ {\tilde{μ}}_{Γ} ‖_{H^{1 / 2} (Γ)}^{q}, & if σ = 0, \end{matrix} \end{array}$ where in the above estimates, when $n = 3$ , $p \in [2, 6]$ , and $q \in [2, \infty)$ if $σ > 0$ , $q \in [2, 4]$ if $σ = 0$ ; when $n = 2$ , $p \in [2, \infty)$ and $q \in [2, \infty)$ . As a remark, throughout this proof, the reader should keep in mind that the meaning of the generic constant C may change from line to line and even within the same chain of inequalities.

Now from assumption (A1), we see that $β (r)$ , $β_{Γ} (r)$ as well as $β (h_{k} (r))$ have the same sign for all $r \in (- 1, 1)$ , this fact combined with (3.30) yields that for any $k \in N ∖ {1}$ $\begin{matrix} (3.34) & β {(u_{k})}^{2} ⩽ β (u_{k}) β (u), β {(u_{Γ, k})}^{2} ⩽ β (u_{Γ, k}) β (u_{Γ}) . \end{matrix}$ Moreover, it follows from assumption (A2) that $\begin{array}{l} (3.35) & 0 ⩽ β (u_{Γ, k}) β (u_{Γ}) ⩽ β (u_{Γ, k}) [c_{1} β_{Γ} (u_{Γ}) + c_{2}], if u_{Γ} > 0, \\ (3.36) & 0 < β (u_{Γ, k}) β (u_{Γ}) ⩽ - β (u_{Γ, k}) [- c_{1} β_{Γ} (u_{Γ}) + c_{2}], if u_{Γ} < 0 . \end{array}$ As a consequence, for any $p ⩾ 2$ , we obtain $\begin{matrix} {‖ β (u_{k}) ‖}_{L^{p} (Ω)}^{p} ⩽ \int_{Ω} {| β (u_{k}) |}^{p - 2} β (u_{k}) β (u) d x, \end{matrix}$ and $\begin{array}{l} {‖ β (u_{Γ, k}) ‖}_{L^{p} (Γ)}^{p} & ⩽ \int_{Γ} {| β (u_{Γ, k}) |}^{p - 2} β (u_{Γ, k}) β (u_{Γ}) d Γ \\ ⩽ c_{1} \int_{Γ} {| β (u_{Γ, k}) |}^{p - 2} β (u_{Γ, k}) β_{Γ} (u_{Γ}) d Γ + c_{2} \int_{Γ} {| β (u_{Γ, k}) |}^{p - 1} d Γ \\ ⩽ c_{1} \int_{Γ} {| β (u_{Γ, k}) |}^{p - 2} β (u_{Γ, k}) β_{Γ} (u_{Γ}) d Γ + \frac{1}{2} {‖ β (u_{Γ, k}) ‖}_{L^{p} (Γ)}^{p} + C, \end{array}$ a.e. on $[η, \infty)$ , where C is a positive constant that only depends on $c_{2}$ , $| Γ |$ , and p.

Combining the above estimates, we deduce from (3.33) that $\begin{array}{l} {‖ β (u_{k}) ‖}_{L^{p} (Ω)}^{p} + {‖ β (u_{Γ, k}) ‖}_{L^{p} (Γ)}^{p} ⩽ \{\begin{matrix} C ‖ \tilde{μ} ‖_{V}^{p} + C ‖ {\tilde{μ}}_{Γ} ‖_{V_{Γ}}^{p} + C, & if σ > 0, \\ C ‖ \tilde{μ} ‖_{V}^{p} + C ‖ {\tilde{μ}}_{Γ} ‖_{H^{1 / 2} (Γ)}^{p} + C, & if σ = 0, \end{matrix} \end{array}$ when $n = 3$ , $p \in [2, 6]$ if $σ > 0$ , $p \in [2, 4]$ if $σ = 0$ ; when $n = 2$ , $p \in [2, \infty)$ , the constant C may depend on $c_{1}$ , $c_{2}$ , Ω, Γ, p but is independent of k. Passing to the limit as $k \to \infty$ , owing to Fatou’s lemma, we conclude from (3.4) and (3.19) that $\begin{matrix} (3.37) & {‖ β (u (s)) ‖}_{L^{p} (Ω)} + {‖ β (u_{Γ} (s)) ‖}_{L^{p} (Γ)} ⩽ C, for a.a. s ⩾ η . \end{matrix}$ The case $p \in [1, 2)$ can be easily handled by the Hölder inequality.

Next, we estimate the boundary potential $β_{Γ} (u_{Γ})$ . Multiplying the equation (cf. (2.8)) $\begin{matrix} (3.38) & \partial_{ν} u - Δ_{Γ} u_{Γ} + β_{Γ} (u_{Γ}) = μ_{Γ} - π_{Γ} (u_{Γ}) = : {\tilde{μ}}_{Γ} a.e. on Σ, \end{matrix}$ by $β_{Γ} (u_{Γ, k}) \in C ([η, \infty); V_{Γ})$ , integrating over Γ, after integration by parts, we get $\begin{array}{l} \int_{Γ} β_{Γ} (u_{Γ, k} (s)) β_{Γ} (u_{Γ} (s)) d Γ \\ = - \int_{Γ}^{} β_{Γ}^{'} (u_{Γ, k} (s)) \nabla_{Γ} u_{Γ, k} (s) \cdot \nabla_{Γ} u_{Γ} (s) d Γ + \int_{Γ} {\tilde{μ}}_{Γ} (s) β_{Γ} (u_{Γ, k} (s)) d Γ \\ - \int_{Γ} \partial_{ν} u (s) β_{Γ} (u_{Γ, k} (s)) d Γ \\ (3.39) & = : I_{5} + I_{6} + I_{7}, \end{array}$ for a.a. $s ⩾ η$ . Similar to $I_{1}$ , we see that $I_{5} ⩽ 0$ . The other two terms $I_{6}$ and $I_{7}$ can be estimated as follows: $\begin{matrix} I_{6} ⩽ \frac{1}{4} {‖ β_{Γ} (u_{Γ, k}) ‖}_{H_{Γ}}^{2} + ‖ {\tilde{μ}}_{Γ} ‖_{H_{Γ}}^{2}, I_{7} ⩽ \frac{1}{4} {‖ β_{Γ} (u_{Γ, k}) ‖}_{H_{Γ}}^{2} + ‖ \partial_{ν} u ‖_{H_{Γ}}^{2} . \end{matrix}$ Besides, by the trace theorem (see e.g., [2]), Lemma A.1 and Young’s inequality, we see that for some $r \in (3 / 2, 2)$ , it holds $\begin{matrix} ‖ \partial_{ν} u ‖_{H_{Γ}}^{2} ⩽ C ‖ u ‖_{H^{r} (Ω)}^{2} ⩽ ζ ‖ u ‖_{H^{2} (Ω)}^{2} + C_{ζ} ‖ u ‖_{H}^{2}, \end{matrix}$ for any $ζ > 0$ . Similar to (3.34), using the fact $\begin{matrix} β_{Γ} {(u_{Γ, k})}^{2} ⩽ β_{Γ} (u_{Γ, k}) β_{Γ} (u_{Γ}), \end{matrix}$ we deduce from (3.39) that $\begin{matrix} {‖ β_{Γ} (u_{Γ, k}) ‖}_{H_{Γ}}^{2} ⩽ 2 ‖ {\tilde{μ}}_{Γ} ‖_{H_{Γ}}^{2} + 2 ζ ‖ u ‖_{H^{2} (Ω)}^{2} + 2 C_{ζ} ‖ u ‖_{H}^{2} . \end{matrix}$ Passing to the limit as $k \to \infty$ , it follows that $\begin{matrix} (3.40) & {‖ β_{Γ} (u_{Γ}) ‖}_{H_{Γ}}^{2} ⩽ 2 ‖ {\tilde{μ}}_{Γ} ‖_{H_{Γ}}^{2} + 2 ζ ‖ u ‖_{H^{2} (Ω)}^{2} + 2 C_{ζ} ‖ u ‖_{H}^{2} . \end{matrix}$ From the elliptic regularity theory Lemma A.6, we have $\begin{array}{l} ‖ u ‖_{W} ⩽ C ({‖ β (u) ‖}_{H} + {‖ π (u) ‖}_{H} + ‖ μ ‖_{H} + {‖ β_{Γ} (u_{Γ}) ‖}_{H_{Γ}} + {‖ π_{Γ} (u_{Γ}) ‖}_{H_{Γ}} + ‖ μ_{Γ} ‖_{H_{Γ}} + ‖ u ‖_{H}), \end{array}$ a.e. on $[η, \infty)$ . In view of (3.4), (3.19), (3.37) and taking the coefficient ζ sufficiently small in (3.40), we get $\begin{matrix} (3.41) & {‖ u (s) ‖}_{W} ⩽ \frac{1}{2} {‖ u (s) ‖}_{W} + C, for a.a. s ⩾ η, \end{matrix}$ which together with (3.4), (3.19) and (3.40) further implies $\begin{matrix} (3.42) & {‖ β_{Γ} (u_{Γ} (s)) ‖}_{H_{Γ}} ⩽ C, for a.a. s ⩾ η . \end{matrix}$

Collecting the above estimates (3.37), (3.41) and (3.42), after choosing the constant $M_{4}$ suitably large, we arrive at our conclusions (3.27)–(3.29). The proof is complete. □

In summary, thanks to Lemmas 3.4, 3.5 and Remark 3.1, since $η > 0$ is arbitrary, we see that every global weak solution $(u, μ)$ to problem (2.2) becomes a global strong solution instantaneously when $t > 0$ .
4. Separation from pure states

In this section, we prove Theorem 2.1 which yields the separation property of global weak solutions to problem (2.2).

4.1. Eventual separation from pure states

The eventual separation property for sufficiently large time is obtained by a dynamical approach (see e.g., [1,26]).

For any given number $a \in (- 1, 1)$ , we introduce the phase space (cf. (A4)) $\begin{matrix} Φ_{a} : = {u = (u, u_{Γ}) \in V : \hat{β} (u) \in L^{1} (Ω), {\hat{β}}_{Γ} (u_{Γ}) \in L^{1} (Γ), m (u) = a} . \end{matrix}$ The distance on $Φ_{a}$ is defined as follows $\begin{array}{l} {dist}_{Φ_{a}} (u^{(1)}, u^{(2)}) : = & {‖ (u^{(1)}, u_{Γ}^{(1)}) - (u^{(1)}, u_{Γ}^{(1)}) ‖}_{V} + {| \int_{Ω} \hat{β} (u^{(1)}) d x - \int_{Ω} \hat{β} (u^{(2)}) d x |}^{\frac{1}{2}} \\ + {| \int_{Γ} {\hat{β}}_{Γ} (u_{Γ}^{(1)}) d Γ - \int_{Γ} {\hat{β}}_{Γ} (u_{Γ}^{(2)}) d Γ |}^{\frac{1}{2}} . \end{array}$

Then we have

Proposition 4.1.
Assume that the assumptions in Proposition 2.1 are satisfied. The initial boundary value problem ( 2.2 ) defines a strongly continuous semigroup $S (t) : Φ_{m_{0}} \mapsto Φ_{m_{0}}$ such that $\begin{matrix} S (t) u_{0} = u (t), for all t ⩾ 0, \end{matrix}$ where $u$ is the unique global weak solution to problem ( 2.2 ) subject to the initial datum $u_{0} \in Φ_{m_{0}}$ .
Proof.
We infer from (2.3) that $u \in C ([0, \infty); H)$ . Thanks to (A1), $\hat{β}$ , ${\hat{β}}_{Γ}$ are proper, convex and lower semi-continuous functionals on H, $H_{Γ}$ , respectively. Hence, from this fact, (A3) and the strong convergence ${lim}_{t \to 0} u (t) = u_{0}$ in $H$ , we get $\begin{array}{l} (4.1) & \begin{aligned} lim_{t \to 0} \int_{Ω} \hat{β} (u (t)) d x = \int_{Ω} \hat{β} (u_{0}) d x, lim_{t \to 0} \int_{Γ} {\hat{β}}_{Γ} (u_{Γ} (t)) d Γ = \int_{Γ} {\hat{β}}_{Γ} (u_{0 Γ}) d Γ, \\ lim_{t \to 0} \int_{Ω} \hat{π} (u (t)) d x = \int_{Ω} \hat{π} (u_{0}) d x, lim_{t \to 0} \int_{Γ} {\hat{π}}_{Γ} (u_{Γ} (t)) d Γ = \int_{Γ} {\hat{π}}_{Γ} (u_{0 Γ}) d Γ . \end{aligned} \end{array}$ On the other hand, recall (3.9), since $η > 0$ is arbitrary, we deduce the energy equality $\begin{matrix} (4.2) & E (u (t)) + \int_{0}^{t} {‖ u^{'} (s) ‖}_{V_{σ, 0}^{}}^{2} d s = E (u (0)) = E (u_{0}), for all t > 0 . \end{matrix}$ Then it holds ${lim}_{t \to 0} ‖ u (t) ‖_{V} = ‖ u_{0} ‖_{V}$ . Since $u \in C_{w} ([0, \infty); V)$ due to (2.3), then we obtain the strong convergence ${lim}_{t \to 0} ‖ u (t) - u_{0} ‖_{V} = 0$ . This combined with (3.18) further implies that $u \in C ([0, \infty); Φ_{m_{0}})$ .

On the other hand, let $u^{(i)}$ be the weak solution corresponding to the initial datum $u_{0}^{(i)} \in Φ_{m_{0}}$ for $i = 1, 2$ . From (2.10), (3.29) we infer that for any fixed $t > 0$ , it holds $\begin{array}{l} {‖ u^{(1)} (t) - u^{(2)} (t) ‖}_{V} & ⩽ C {‖ u^{(1)} (t) - u^{(2)} (t) ‖}_{W}^{\frac{2}{3}} {‖ u^{(1)} (t) - u^{(2)} (t) ‖}_{V^{}}^{\frac{1}{3}} \\ ⩽ C (t) {‖ u^{(1)} (t) - u^{(2)} (t) ‖}_{V_{σ, 0}^{}}^{\frac{1}{3}} \\ ⩽ C (t) {‖ u_{0}^{(1)} - u_{0}^{(2)} ‖}_{V_{σ, 0}^{}}^{\frac{1}{3}} ⩽ C (t) {‖ u_{0}^{(1)} - u_{0}^{(2)} ‖}_{V}^{\frac{1}{3}} . \end{array}$ Then by an argument similar to (4.1) and noting that $S (0) = I$ , we can easily conclude that $S (t) \in C (Φ_{m_{0}}, Φ_{m_{0}})$ for all $t ⩾ 0$ . □

Next, we consider the stationary problem corresponding to (2.2), which can be (formally) obtained by neglecting those time derivatives. $\begin{matrix} (4.3) & \{\begin{matrix} Δ μ_{s} = 0, & a.e. in Ω, \\ μ_{s} = - Δ u_{s} + β (u_{s}) + π (u_{s}), & a.e. in Ω, \\ u_{s |_{Γ}} = u_{Γ s}, μ_{s |_{Γ}} = μ_{Γ s}, & a.e. on Γ, \\ \partial_{ν} μ_{s} - σ Δ_{Γ} μ_{Γ_{s}} = 0, & a.e. on Γ, \\ μ_{Γ s} = \partial_{ν} u_{s} - Δ_{Γ} u_{Γ s} + β_{Γ} (u_{Γ s}) + π_{Γ} (u_{Γ s}), & a.e. on Γ . \end{matrix} \end{matrix}$ It is straightforward to check that, if a pair $μ_{s} = (μ_{s}, μ_{Γ s})$ is a solution to problem (4.3), then $μ_{s} = μ_{Γ s}$ must be a constant. Thus, system (4.3) simply reduces to a nonlocal elliptic boundary value problem for $u_{s} = (u_{s}, u_{Γ s})$ : $\begin{matrix} (4.4) & \{\begin{matrix} μ_{s} = - Δ u_{s} + β (u_{s}) + π (u_{s}), & a.e. in Ω, \\ u_{s |_{Γ}} = u_{Γ s}, & a.e. on Γ, \\ μ_{s} = \partial_{ν} u_{s} - Δ_{Γ} u_{Γ s} + β_{Γ} (u_{Γ s}) + π_{Γ} (u_{Γ s}), & a.e. on Γ, \end{matrix} \end{matrix}$ where $\begin{matrix} (4.5) & μ_{s} = \frac{1}{| Ω | + | Γ |} [\int_{Ω} (β (u_{s}) + π (u_{s})) d x + \int_{Γ} (β_{Γ} (u_{Γ s}) + π_{Γ} (u_{Γ s})) d Γ] . \end{matrix}$ More precisely, we introduce the following notion.
Definition 4.1.
A pair $u_{s} = (u_{s}, u_{Γ s})$ is called a steady state of problem (2.2), if $u_{s} = (u_{s}, u_{Γ s}) \in Φ_{a}$ for some $a \in (- 1, 1)$ , $β (u_{s}) \in L^{2} (Ω)$ , $β_{Γ} (u_{Γ s}) \in L^{2} (Γ)$ and $\begin{array}{l} \int_{Ω} \nabla u_{s} \cdot \nabla z d x + \int_{Ω} (β (u_{s}) + π (u_{s}) - μ_{s}) z d x + \int_{Γ} \nabla_{Γ} u_{Γ s} \cdot \nabla_{Γ} z_{Γ} d Γ \\ (4.6) & + \int_{Γ} (β_{Γ} (u_{Γ s}) + π_{Γ} (u_{Γ s}) - μ_{s}) z_{Γ} d Γ = 0, for all z = (z, z_{Γ}) \in V, \end{array}$ with the constant $μ_{s}$ given by (4.5).
Remark 4.1.
The constraint $m (u_{s}) = a$ for some given $a \in (- 1, 1)$ in Definition 4.1 is not necessary for the stationary problem. It will play a role when we connect problem (4.4)–(4.5) to the corresponding evolution problem (2.2), due to the mass conservation property (3.1) such that we need to set $a = m (u_{0}) = m_{0}$ (cf. (A4)).

The following lemma provides a useful characterization on the steady states.
Lemma 4.1.
Assume that $a \in (- 1, 1)$ and assumptions (A1)–(A3) are satisfied. We denote the set of steady states by $S_{a}$ . There exist uniform constants $M_{a} > 0$ and $δ_{a} \in (0, 1)$ such that every steady state $u_{s} = (u_{s}, u_{Γ s}) \in S_{a}$ and its associated constant $μ_{s}$ satisfy $\begin{array}{l} (4.7) & - 1 + δ_{a} ⩽ u_{s} ⩽ 1 - δ_{a}, in Ω, \\ (4.8) & - 1 + δ_{a} ⩽ u_{Γ s} ⩽ 1 - δ_{a}, on Γ, \\ (4.9) & | μ_{s} | ⩽ M_{a} . \end{array}$ Moreover, the set $S_{a}$ is bounded in $H^{3} (Ω) \times H^{3} (Γ)$ .
Proof.
The proof follows from the idea in [26, Lemma 6.1]. Since $u_{s} = (u_{s}, u_{Γ s}) \in S_{a}$ , then by (A1) we have $‖ u_{s} ‖_{L^{\infty} (Ω)} ⩽ 1$ , $‖ u_{Γ s} ‖_{L^{\infty} (Γ)} ⩽ 1$ . Taking $z = u_{s} - a 1$ in (4.6), we have $\begin{array}{l} \int_{Ω} {| \nabla (u_{s} - a) |}^{2} d x + \int_{Ω} β (u_{s}) (u_{s} - a) d x + \int_{Γ} {| \nabla_{Γ} (u_{Γ s} - a) |}^{2} d Γ \\ + \int_{Γ} β_{Γ} (u_{Γ s}) (u_{Γ s} - a) d Γ \\ = - \int_{Ω} π (u_{s}) (u_{s} - a) d x - \int_{Γ} π_{Γ} (u_{Γ s}) (u_{Γ s} - a) d Γ \\ ⩽ C, \end{array}$ where C may depend on Ω, Γ, L and a. From the above estimate and (3.22), (3.23), (4.5), we can easily conclude (4.9). In particular, we note that $M_{a}$ is independent of $u_{s}$ .

Using (A1) again, there exists $δ_{a} \in (0, 1)$ such that $\begin{array}{l} β (r) + π (r) - M_{a} ⩾ 1, β_{Γ} (r) + π_{Γ} (r) - M_{a} ⩾ 1, for all r \in [1 - δ_{a}, 1], \\ β (r) + π (r) + M_{a} ⩽ - 1, β_{Γ} (r) + π_{Γ} (r) + M_{a} ⩽ - 1, for all r \in [- 1, - 1 + δ_{a}], \end{array}$ and the constant $δ_{a}$ is also independent of $u_{s}$ . Taking the test function in (4.6) as $z = {[u_{s} - (1 - δ_{a}) 1]}^{+}$ , $z = {[u_{s} + (1 - δ_{a}) 1]}^{-}$ , respectively, where ${[u]}^{+} : = ({[u]}^{+}, {[u_{Γ}]}^{+}) : = (max {0, u}, max {0, u_{Γ}})$ , ${[u]}^{-} : = ({[u]}^{-}, {[u_{Γ}]}^{-}) : = (- min {0, u}, - min {0, u_{Γ}})$ for $u : = (u, u_{Γ})$ . Then, we infer that $\begin{array}{l} \int_{Ω} {[u_{s} - (1 - δ_{a})]}^{+} d x + \int_{Γ} {[u_{Γ s} - (1 - δ_{a})]}^{+} d Γ \\ + \int_{Ω} {| \nabla {[u_{s} - (1 - δ_{a})]}^{+} |}^{2} d x + \int_{Γ} {| \nabla_{Γ} {[u_{Γ s} - (1 - δ_{a})]}^{+} |}^{2} d Γ ⩽ 0, \\ \int_{Ω} {[u_{s} + (1 - δ_{a})]}^{-} d x + \int_{Γ} {[u_{Γ s} + (1 - δ_{a})]}^{-} d Γ \\ + \int_{Ω} {| \nabla {[u_{s} + (1 - δ_{a})]}^{-} |}^{2} d x + \int_{Γ} {| \nabla_{Γ} {[u_{Γ s} + (1 - δ_{a})]}^{-} |}^{2} d Γ ⩽ 0, \end{array}$ which yield the conclusion (4.7)–(4.8).

Finally, the separation property and assumptions (A1)–(A2) enable us to apply the elliptic regularity theory (see Lemma A.6) to conclude that $u_{s} \in H^{3} (Ω) \times H^{3} (Γ)$ . The proof is complete. □

Returning to the evolution problem (2.2), for any initial datum $u_{0}$ satisfying (A4), we define the ω-limit set $ω (u_{0})$ as follows: $\begin{matrix} ω (u_{0}) : = \{u_{\infty} \in [H^{2 r} (Ω) \times H^{2 r} (Γ)] \cap Φ_{m_{0}} : \begin{matrix} there exists {t_{n}}_{n \in N} with t_{n} ↗ \infty \\ such that u (t_{n}) \to u_{\infty} \\ in H^{2 r} (Ω) \times H^{2 r} (Γ) as n \to \infty \end{matrix}\}, \end{matrix}$ for some $r \in [1 / 2, 1)$ .

Then we show the relationship between $ω (u_{0})$ and the set of steady states $S_{m_{0}}$ .
Lemma 4.2.
Under the assumptions (A1)–(A4), $ω (u_{0})$ is a non-empty, connected and compact set in $H^{2 r} (Ω) \times H^{2 r} (Γ)$ for $r \in [1 / 2, 1)$ . Moreover, $\begin{array}{l} (4.10) & ω (u_{0}) \subset S_{m_{0}} \end{array}$ such that every element $u_{\infty} : = (u_{\infty}, u_{Γ, \infty}) \in ω (u_{0})$ is a strong solution to the elliptic boundary value problem ( 4.3 ) with the associated constant $μ_{\infty}$ determined by ( 4.5 ).
Proof.
From the estimate (3.29), we see that the orbit ${u (t)}_{t ⩾ η}$ is relatively compact in $H^{2 r} (Ω) \times H^{2 r} (Γ)$ for any $r \in [1 / 2, 1)$ . On the other hand, the free energy $E (u)$ defined by (3.2) serves as a strict Lyapunov function for the semigroup $S (t)$ (see (4.2)). Therefore, the conclusion of the present lemma follows from the well-known results in the dynamical system (see e.g., [31, Theorem 4.3.3]) and Lemma 4.1. We also refer to [29, Theorem 3.15] for an alternative proof with minor modifications due to the assumptions on β, $β_{Γ}$ . □

Lemmas 4.1 and 4.2 yield the property of uniform separation from pure states $\pm 1$ for any element of the ω-limit set $ω (u_{0})$ (see (4.7), (4.8) and (4.10)). This essential fact enables us to prove the eventual separation property for global weak solutions to problem (2.2). Proof of Theorem 2.1. Part (1).
It follows from the definition of $ω (u_{0})$ that $\begin{matrix} lim_{t \to \infty} dist (S (t) u_{0}, ω (u_{0})) = 0 in H^{2 r} (Ω) \times H^{2 r} (Γ), \end{matrix}$ where the above distance is given by $dist (z, ω (u_{0})) = {inf}_{y \in ω (u_{0})} ‖ z - y ‖_{H^{2 r} (Ω) \times H^{2 r} (Γ)}$ . By the Sobolev embedding theorem, we see that $H^{2 r} (Ω) \times H^{2 r} (Γ) ↪ C (\overline{Ω}) \times C (Γ)$ when $r \in (n / 4, 1)$ ( $n = 2, 3$ ). Then thanks to Lemmas 4.1 and 4.2, we can conclude (2.14) with the choice $\begin{matrix} δ_{1} = \frac{1}{2} δ_{m_{0}}, \end{matrix}$ where the constant $δ_{m_{0}}$ is determined as in Lemma 4.1. □

4.2. Instantaneous separation from pure states in two dimensional case

The improved instantaneous separation property can be achieved by some further higher-order estimates for global weak solutions that only depend on an upper bound for the initial energy $E (u_{0})$ and on the average of the total mass $m_{0}$ . In this case, the spatial dimension ( $n = 2$ ) and the regularization effect of the surface diffusion ( $σ > 0$ ) turn out to be crucial due to the Trudinger–Moser inequality (see Lemma A.5) and the available regularity on the chemical potential $μ$ (see (3.19)).

Lemma 4.3.
Let $n = 2$ , $σ > 0$ . For any $1 ⩽ p < \infty$ and $η > 0$ , there exists a positive constant $C (p, η)$ such that $\begin{matrix} (4.11) & {‖ β^{'} (u) ‖}_{L^{p} (t, t + 1; L^{p} (Ω))} + {‖ β^{'} (u_{Γ}) ‖}_{L^{p} (t, t + 1; L^{p} (Γ))} ⩽ C (p, η), for all t ⩾ η . \end{matrix}$
Proof.
The proof relies on the idea of [46, Lemma 7.1] for the Cahn–Hilliard equation with homogeneous Neumann boundary conditions (see, also [28, Lemma 5.1]). Nevertheless, necessary modifications have to be made in order to handle the current complicated boundary conditions.

For any $k \in N ∖ {1}$ and $K > 0$ , let $(u_{k}, u_{Γ, k})$ be defined as (3.31). Because $u_{k}$ belongs to the bounded interval $[- 1 + 1 / k, 1 - 1 / k]$ , then we see from (A1) and the assumption (2.15) that $\begin{array}{l} \nabla (β (u_{k}) e^{K | β (u_{k}) |}) = β^{'} (u_{k}) (1 + K | β (u_{k}) |) e^{K | β (u_{k}) |} \nabla u_{k} \in C ([η, \infty); H), \end{array}$ and ${(β (u_{k}) e^{K | β (u_{k}) |})}_{|_{Γ}} = β (u_{Γ, k} (s)) e^{K | β (u_{Γ, k} (s)) |}$ . Therefore, testing the equation (3.32) by $β (u_{k}) e^{K | β (u_{k}) |}$ , we get $\begin{array}{l} \int_{Ω}^{} (\nabla u (s) \cdot \nabla u_{k} (s)) β^{'} (u_{k} (s)) (1 + K | β (u_{k} (s)) |) e^{K | β (u_{k} (s)) |} d x \\ - \int_{Γ}^{} \partial_{ν} u (s) β (u_{Γ, k} (s)) e^{K | β (u_{Γ, k} (s)) |} d Γ + \int_{Ω}^{} β (u (s)) β (u_{k} (s)) e^{K | β (u_{k} (s)) |} d x \\ (4.12) & = \int_{Ω}^{} \tilde{μ} (s) β (u_{k} (s)) e^{K | β (u_{k} (s)) |} d x, for a.a. s ⩾ η . \end{array}$ Next, we note that $\begin{array}{l} \nabla_{Γ} (β (u_{Γ, k}) e^{K | β (u_{Γ, k}) |}) & = β^{'} (u_{Γ, k}) (1 + K | β (u_{Γ, k}) |) e^{K | β (u_{Γ, k}) |} \nabla_{Γ} u_{Γ, k} \\ \in C ([η, \infty); H_{Γ}), \end{array}$ Then, testing (3.38) by $β (u_{Γ, k}) e^{K | β (u_{Γ, k}) |}$ , we get $\begin{array}{l} \int_{Γ}^{} \partial_{ν} u (s) β (u_{Γ, k} (s)) e^{K | β (u_{Γ, k} (s)) |} d Γ \\ + \int_{Γ}^{} (\nabla_{Γ} u_{Γ} (s) \cdot \nabla_{Γ} u_{Γ, k} (s)) β^{'} (u_{Γ, k} (s)) (1 + K | β (u_{Γ, k} (s)) |) e^{K | β (u_{Γ, k} (s)) |} d Γ \\ + \int_{Γ}^{} β_{Γ} (u_{Γ} (s)) β (u_{Γ, k} (s)) e^{K | β (u_{Γ, k} (s)) |} d Γ \\ (4.13) & = \int_{Γ}^{} {\tilde{μ}}_{Γ} (s) β (u_{Γ, k} (s)) e^{K | β (u_{Γ, k} (s)) |} d Γ, for a.a. s ⩾ η . \end{array}$ From (A1), we see that the first term on the left hand side of (4.12) and the second term on the left hand side of (4.13) are nonnegative. Then adding (4.12) and (4.13) together, we infer from (3.34), (3.35) and (3.36) that $\begin{array}{l} \int_{Ω}^{} β {(u_{k} (s))}^{2} e^{K | β (u_{k} (s)) |} d x + \frac{1}{c_{1}} \int_{Γ}^{} β {(u_{Γ, k} (s))}^{2} e^{K | β (u_{Γ, k} (s)) |} d Γ \\ ⩽ \int_{Ω}^{} | \tilde{μ} (s) | | β (u_{k} (s)) | e^{K | β (u_{k} (s)) |} d x + \int_{Γ}^{} (| {\tilde{μ}}_{Γ} (s) | + \frac{c_{2}}{c_{1}}) | β (u_{Γ, k} (s)) | e^{K | β (u_{Γ, k} (s)) |} d Γ \\ (4.14) & = : J_{1} + J_{2}, for a.a. s ⩾ η . \end{array}$ Applying the generalized Young inequality given by Lemma A.4, there exist positive constants N and $M_{5}$ that may depend on K, $c_{1}$ but are independent of k such that $\begin{matrix} | \tilde{μ} (s) | β (u_{k} (s)) e^{K | β (u_{k} (s)) |} ⩽ e^{N | \tilde{μ} (s) |} + \frac{1}{2} β {(u_{k} (s))}^{2} e^{K | β (u_{k} (s)) |} + M_{5}, \end{matrix}$ and $\begin{array}{l} (| {\tilde{μ}}_{Γ} (s) | + \frac{c_{2}}{c_{1}}) | β (u_{Γ, k} (s)) | e^{K | β (u_{Γ, k} (s)) |} \\ (4.15) & ⩽ e^{N (| {\tilde{μ}}_{Γ} (s) | + \frac{c_{2}}{c_{1}})} + \frac{1}{2 c_{1}} β {(u_{Γ, k} (s))}^{2} e^{K | β (u_{Γ, k} (s)) |} + M_{5}, \end{array}$ for a.a. $s ⩾ η$ . Then we can control $J_{1}$ as follows (see [46, Lemma 7.1]): in view of the estimate (3.19), we can employ the Trudinger–Moser type inequality given by Lemma A.5 to conclude that there exists a positive constant ${\tilde{C}}_{TM}$ depending on $C_{TM}$ and N such that $\begin{matrix} \int_{Ω}^{} e^{N | \tilde{μ} (s) |} d x ⩽ {\tilde{C}}_{TM} e^{{\tilde{C}}_{TM} ‖ \tilde{μ} (s) ‖_{V}^{2}}, for a.a. s ⩾ η . \end{matrix}$ Hence, $\begin{array}{l} (4.16) & J_{1} ⩽ {\tilde{C}}_{TM} e^{{\tilde{C}}_{TM} ‖ \tilde{μ} (s) ‖_{V}^{2}} + \frac{1}{2} \int_{Ω} β {(u_{k} (s))}^{2} e^{K | β (u_{k} (s)) |} d x + M_{5} | Ω | . \end{array}$ Next, we deduce from the embedding $V_{Γ} ↪ C (Γ)$ that $\begin{array}{l} \int_{Γ} e^{N (| {\tilde{μ}}_{Γ} (s) | + \frac{c_{2}}{c_{1}})} d Γ ⩽ | Γ | e^{N (‖ {\tilde{μ}}_{Γ} (s) ‖_{V_{Γ}} + \frac{c_{2}}{c_{1}})}, for a.a. s ⩾ η, \end{array}$ which together with (4.15) implies $\begin{array}{l} (4.17) & J_{2} & ⩽ | Γ | e^{N (‖ {\tilde{μ}}_{Γ} (s) ‖_{V_{Γ}} + \frac{c_{2}}{c_{1}})} + \frac{1}{2 c_{1}} \int_{Γ}^{} β {(u_{Γ, k} (s))}^{2} e^{K | β (u_{Γ, k} (s)) |} d Γ + M_{5} | Γ | . \end{array}$ Combining (4.14), (4.16) and (4.17), we arrive at $\begin{array}{l} \int_{Ω}^{} β {(u_{k} (s))}^{2} e^{K | β (u_{k} (s)) |} d x + \frac{1}{c_{1}} \int_{Γ}^{} β {(u_{Γ, k} (s))}^{2} e^{K | β (u_{Γ, k} (s)) |} d Γ \\ (4.18) & ⩽ 2 {\tilde{C}}_{TM} e^{{\tilde{C}}_{TM} ‖ \tilde{μ} (s) ‖_{V}^{2}} + 2 | Γ | e^{N (‖ {\tilde{μ}}_{Γ} (s) ‖_{V_{Γ}} + \frac{c_{2}}{c_{1}})} + 2 M_{5} (| Ω | + | Γ |), \end{array}$ for a.a. $s ⩾ η$ .

For any fixed $p ⩾ 1$ , we take $K : = p c_{0}$ , where the constant $c_{0}$ is given in (2.15). Then we deduce from the assumptions (2.15), $σ > 0$ and estimates (3.19), (4.18) that $\begin{array}{l} \int_{t}^{t + 1} \int_{Ω}^{} {| β^{'} (u_{k} (s)) |}^{p} d x d s + \frac{1}{c_{1}} \int_{t}^{t + 1} \int_{Γ}^{} {| β^{'} (u_{Γ, k} (s)) |}^{p} d Γ d s \\ ⩽ \int_{t}^{t + 1} \int_{Ω}^{} {(e^{c_{0} | β (u_{k} (s)) | + c_{0}})}^{p} d x d s + \frac{1}{c_{1}} \int_{t}^{t + 1} \int_{Γ}^{} {(e^{c_{0} | β (u_{Γ, k} (s)) | + c_{0}})}^{p} d Γ d s \\ = C \int_{t}^{t + 1} (\int_{Ω}^{} e^{K | β (u_{k} (s)) |} d x + \frac{1}{c_{1}} \int_{Γ}^{} e^{K | β (u_{Γ, k} (s)) |} d Γ) d s \\ ⩽ C \int_{t}^{t + 1} (e^{K} | Ω | + \int_{Ω}^{} β {(u_{k} (s))}^{2} e^{K | β (u_{k} (s)) |} d x + \frac{e^{K}}{c_{1}} | Γ | \\ + \frac{1}{c_{1}} \int_{Γ}^{} β {(u_{Γ, k} (s))}^{2} e^{K | β (u_{Γ, k} (s)) |} d Γ) d s \\ ⩽ 2 C {\tilde{C}}_{TM} e^{{\tilde{C}}_{TM} M_{3}^{2}} + 2 C | Γ | e^{N (M_{3} + \frac{c_{2}}{c_{1}})} + 2 C M_{5} (| Ω | + | Γ |) \\ (4.19) & + C e^{K} (| Ω | + c_{1}^{- 1} | Γ |), for all t ⩾ η, \end{array}$ where we have used the fact that, if $r < 1$ then $e^{K r} ⩽ e^{K}$ , and if $r ⩾ 1$ then $e^{K r} ⩽ r^{2} e^{K r}$ , and the positive constant C is independent of k.

Since we already know that $| u | < 1$ a.e. in Q and $| u_{Γ} | < 1$ a.e. on Σ, then $u_{k} \to u$ a.e. in Q and $u_{Γ, k} \to u_{Γ}$ a.e. on Σ as $k \to \infty$ , which imply $\begin{array}{l} β^{'} (u_{k}) \to β^{'} (u) a.e. in Q, \\ β^{'} (u_{Γ, k}) \to β^{'} (u_{Γ}) a.e. on Σ . \end{array}$ On the other hand, by virtue of the Lions lemma [41, Lemme 1.3, Chapitre 1] and the uniform estimate (4.19), we see that $\begin{array}{l} β^{'} (u_{k}) \to β^{'} (u) weakly in L^{p} (t, t + 1; L^{p} (Ω)), \\ β^{'} (u_{Γ, k}) \to β^{'} (u_{Γ}) weakly in L^{p} (t, t + 1; L^{p} (Γ)), \end{array}$ as $k \to \infty$ , for all $t ⩾ η$ . Finally, taking ${lim inf}_{k \to \infty}$ in (4.19) we easily arrive at the conclusion (4.11). □

Lemma 4.3 enables us to obtain some improved estimates on the time derivative of weak solution.
Lemma 4.4.
Let $n = 2$ , $σ > 0$ . For any $η > 0$ , there exists a positive constant $M_{6}$ such that $\begin{array}{l} (4.20) & {‖ u^{'} ‖}_{L^{\infty} (2 η, \infty; H)} ⩽ M_{6}, \\ (4.21) & \int_{t}^{t + 1} {‖ u^{'} (s) ‖}_{W}^{2} d s ⩽ M_{6}, for all t ⩾ 2 η . \end{array}$
Proof.
Taking the difference of (3.25) at $t = s$ and $t = s + h$ , multiplying the resultant by $1 / h$ , we have $\begin{matrix} (4.22) & \partial_{t} (\partial_{t}^{h} u (s)) - Δ \partial_{t}^{h} μ (s) = 0 a.e. in Ω, \end{matrix}$ for a.a. $s ⩾ η$ . Analogously, we obtain from (3.26) that $\begin{matrix} (4.23) & \partial_{t} (\partial_{t}^{h} u_{Γ} (s)) + \partial_{ν} \partial_{t}^{h} μ (s) - σ Δ_{Γ} \partial_{t}^{h} μ_{Γ} (s) = 0 a.e. on Γ, \end{matrix}$ for a.a. $s ⩾ η$ . Multiplying (4.22) by $\partial_{t}^{h} u (s)$ and integrating over Ω, we get $\begin{array}{l} \frac{1}{2} \frac{d}{d s} {‖ \partial_{t}^{h} u (s) ‖}_{H}^{2} - \int_{Ω}^{} \partial_{t}^{h} μ (s) Δ \partial_{t}^{h} u (s) d x - \int_{Γ}^{} \partial_{ν} \partial_{t}^{h} μ (s) \partial_{t}^{h} u_{Γ} (s) d Γ \\ (4.24) & + \int_{Γ} \partial_{t}^{h} μ_{Γ} (s) \partial_{ν} \partial_{t}^{h} u (s) d Γ = 0 . \end{array}$ Next, multiplying (4.23) by $\partial_{t}^{h} u_{Γ} (s)$ and integrating over Γ, we get $\begin{array}{l} (4.25) & \frac{1}{2} \frac{d}{d s} {‖ \partial_{t}^{h} u_{Γ} (s) ‖}_{H_{Γ}}^{2} + \int_{Γ}^{} \partial_{ν} \partial_{t}^{h} μ (s) \partial_{t}^{h} u_{Γ} (s) d Γ + σ \int_{Γ}^{} \nabla_{Γ} \partial_{t}^{h} μ_{Γ} (s) \cdot \nabla_{Γ} \partial_{t}^{h} u_{Γ} (s) d Γ = 0 . \end{array}$ On the other hand, taking differences of equations (2.7) and (2.8) at $t = s$ and $t = s + h$ , respectively, multiplying the resultants by $1 / h$ , we have $\begin{array}{l} (4.26) & \partial_{t}^{h} μ (s) = - Δ \partial_{t}^{h} u (s) + \partial_{t}^{h} β (u (s)) + \partial_{t}^{h} π (u (s)) a.e. in Ω, \\ (4.27) & \partial_{t}^{h} μ_{Γ} (s) = \partial_{ν} \partial_{t}^{h} u (s) - Δ_{Γ} \partial_{t}^{h} u_{Γ} (s) + \partial_{t}^{h} β_{Γ} (u_{Γ} (s)) + \partial_{t}^{h} π_{Γ} (u_{Γ} (s)) a.e. on Γ, \end{array}$ for a.a. $s ⩾ η$ . Multiplying (4.27) by $σ \partial_{t}^{h} μ_{Γ} (s)$ , integrating over Γ, we get $\begin{array}{l} σ {‖ \partial_{t}^{h} μ_{Γ} (s) ‖}_{H_{Γ}}^{2} = & σ \int_{Γ}^{} \partial_{ν} \partial_{t}^{h} u (s) \partial_{t}^{h} μ_{Γ} (s) d Γ + σ \int_{Γ}^{} \nabla_{Γ} \partial_{t}^{h} u_{Γ} (s) \cdot \nabla_{Γ} \partial_{t}^{h} μ_{Γ} (s) d Γ \\ + σ \int_{Γ}^{} \partial_{t}^{h} (β_{Γ} (u_{Γ} (s)) + π_{Γ} (u_{Γ} (s))) \partial_{t}^{h} μ_{Γ} (s) d Γ, \end{array}$ for a.a. $s ⩾ η$ . Adding this with (4.24) and (4.25), using (4.26) and Young’s inequality, we obtain that $\begin{array}{l} \frac{1}{2} \frac{d}{d s} ({‖ \partial_{t}^{h} u (s) ‖}_{H}^{2} + {‖ \partial_{t}^{h} u_{Γ} (s) ‖}_{H_{Γ}}^{2}) + {‖ \partial_{t}^{h} Δ u (s) ‖}_{H}^{2} + σ {‖ \partial_{t}^{h} μ_{Γ} (s) ‖}_{H_{Γ}}^{2} \\ = (σ - 1) \int_{Γ}^{} \partial_{ν} \partial_{t}^{h} u (s) \partial_{t}^{h} μ_{Γ} (s) d Γ + \int_{Ω} Δ \partial_{t}^{h} u (s) (\partial_{t}^{h} β (u (s)) + \partial_{t}^{h} π (u (s))) d x \\ + σ \int_{Γ}^{} \partial_{t}^{h} (β_{Γ} (u_{Γ} (s)) + π_{Γ} (u_{Γ} (s))) \partial_{t}^{h} μ_{Γ} (s) d Γ \\ ⩽ \frac{1}{2} {‖ \partial_{t}^{h} Δ u (s) ‖}_{H}^{2} + \frac{σ}{2} {‖ \partial_{t}^{h} μ_{Γ} (s) ‖}_{H_{Γ}}^{2} + \frac{3 {(σ - 1)}^{2}}{2 σ} {‖ \partial_{ν} \partial_{t}^{h} u (s) ‖}_{H_{Γ}}^{2} + {‖ \partial_{t}^{h} β (u (s)) ‖}_{H}^{2} \\ (4.28) & + {‖ \partial_{t}^{h} π (u (s)) ‖}_{H}^{2} + \frac{3 σ}{2} {‖ \partial_{t}^{h} β_{Γ} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2} + \frac{3 σ}{2} {‖ \partial_{t}^{h} π_{Γ} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2}, \end{array}$ for a.a. $s ⩾ η$ . Besides, we see from (4.27) that $\begin{array}{l} {‖ \partial_{t}^{h} μ_{Γ} (s) ‖}_{H_{Γ}}^{2} ⩾ & \frac{1}{3} {‖ \partial_{ν} \partial_{t}^{h} u (s) - Δ_{Γ} \partial_{t}^{h} u_{Γ} (s) ‖}_{H_{Γ}}^{2} - {‖ \partial_{t}^{h} β_{Γ} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2} \\ (4.29) & - {‖ \partial_{t}^{h} π_{Γ} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2} . \end{array}$ Now we proceed to estimate the right hand side of (4.28). First, from the Lipschitz continuity of π and $π_{Γ}$ , we have $\begin{matrix} (4.30) & {‖ \partial_{t}^{h} π (u (s)) ‖}_{H}^{2} ⩽ L^{2} {‖ \partial_{t}^{h} u (s) ‖}_{H}^{2}, {‖ \partial_{t}^{h} π_{Γ} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2} ⩽ L^{2} {‖ \partial_{t}^{h} u_{Γ} (s) ‖}_{H_{Γ}}^{2} . \end{matrix}$ Next, from the convexity of $β^{'}$ (see (A1)), we get $\begin{array}{l} {‖ \partial_{t}^{h} β (u (s)) ‖}_{H}^{2} & = \int_{Ω}^{} {| \int_{0}^{1} β^{'} (τ u (s + h) + (1 - τ) u (s)) \partial_{t}^{h} u (s) d τ |}^{2} d x \\ ⩽ {‖ \int_{0}^{1} (τ β^{'} (u (s + h)) + (1 - τ) β^{'} (u (s))) d τ ‖}_{H}^{2} {‖ \partial_{t}^{h} u (s) ‖}_{L^{\infty} (Ω)}^{2} \\ ⩽ {[\int_{0}^{1} (τ {‖ β^{'} (u (s + h)) ‖}_{H} + (1 - τ) {‖ β^{'} (u (s)) ‖}_{H}) d τ]}^{2} {‖ \partial_{t}^{h} u (s) ‖}_{L^{\infty} (Ω)}^{2} \\ ⩽ {(\frac{1}{2} {‖ β^{'} (u (s + h)) ‖}_{H} + \frac{1}{2} {‖ β^{'} (u (s)) ‖}_{H})}^{2} {‖ \partial_{t}^{h} u (s) ‖}_{L^{\infty} (Ω)}^{2} \\ (4.31) & ⩽ ({‖ β^{'} (u (s + h)) ‖}_{H}^{2} + {‖ β^{'} (u (s)) ‖}_{H}^{2}) {‖ \partial_{t}^{h} u (s) ‖}_{L^{\infty} (Ω)}^{2} . \end{array}$ Analogously, it follows that $\begin{matrix} (4.32) & {‖ \partial_{t}^{h} β_{Γ} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2} ⩽ ({‖ β_{Γ}^{'} (u_{Γ} (s + h)) ‖}_{H_{Γ}}^{2} + {‖ β_{Γ}^{'} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2}) {‖ \partial_{t}^{h} u_{Γ} (s) ‖}_{L^{\infty} (Γ)}^{2} . \end{matrix}$ Finally, by the trace theorem and Lemma A.1, we see that for some $r \in (3 / 2, 2)$ , it holds $\begin{array}{l} (4.33) & {‖ \partial_{ν} \partial_{t}^{h} u (s) ‖}_{H_{Γ}}^{2} ⩽ C {‖ \partial_{t}^{h} u (s) ‖}_{H^{r} (Ω)}^{2} ⩽ ζ {‖ \partial_{t}^{h} u (s) ‖}_{H^{2} (Ω)}^{2} + C_{ζ} {‖ \partial_{t}^{h} u (s) ‖}_{H}^{2}, \end{array}$ for any $ζ > 0$ .

For each $z \in W \cap V_{0}$ , it holds $\begin{matrix} {(\partial φ (z), z)}_{H_{0}} = a_{1} (z, z) = ‖ z ‖_{V_{0}}^{2} : = ‖ z ‖_{V_{1, 0}}^{2} . \end{matrix}$ From the generalized Poincaré inequality given by Lemma A.3 and the elliptic regularity theory (e.g., Lemma A.6), we have $\begin{array}{l} (4.34) & ‖ z ‖_{W} ⩽ C {‖ \partial φ (z) ‖}_{H_{0}}, for all z \in W \cap V_{0} . \end{array}$ Thus, by the two dimensional Agmon inequality (see [57, Chapter II, (1.40)]) we infer that for any $z \in W \cap V_{0}$ , $\begin{array}{l} (4.35) & ‖ z ‖_{L^{\infty} (Ω)}^{2} ⩽ C ‖ z ‖_{W} ‖ z ‖_{H} ⩽ {‖ \partial φ (z) ‖}_{H_{0}} ‖ z ‖_{H_{0}}, \end{array}$ and by the Sobolev embedding theorem $\begin{array}{l} (4.36) & ‖ z_{Γ} ‖_{L^{\infty} (Γ)}^{2} ⩽ C ‖ z_{Γ} ‖_{V_{Γ}}^{2} ⩽ C ‖ z ‖_{V}^{2} ⩽ C {‖ \partial φ (z) ‖}_{H_{0}} ‖ z ‖_{H_{0}} . \end{array}$ Now combining (4.28)–(4.36) and taking the constant $ζ > 0$ to be sufficiently small, we obtain that $\begin{array}{l} \frac{d}{d s} {‖ \partial_{t}^{h} u (s) ‖}_{H_{0}}^{2} + min {\frac{1}{2}, \frac{σ}{6}} {‖ \partial φ (\partial_{t}^{h} u (s)) ‖}_{H_{0}}^{2} \\ ⩽ C B (s) {‖ \partial φ (\partial_{t}^{h} u (s)) ‖}_{H_{0}} {‖ \partial_{t}^{h} u (s) ‖}_{H_{0}} + C {‖ \partial_{t}^{h} u (s) ‖}_{H_{0}}^{2} \\ (4.37) & ⩽ min {\frac{1}{4}, \frac{σ}{12}} {‖ \partial φ (\partial_{t}^{h} u (s)) ‖}_{H_{0}}^{2} + C (B {(s)}^{2} + 1) {‖ \partial_{t}^{h} u (s) ‖}_{H_{0}}^{2}, \end{array}$ for a.a. $s ⩾ η$ , where $\begin{array}{l} B (s) : = & {‖ β^{'} (u (s + h)) ‖}_{H}^{2} + {‖ β^{'} (u (s)) ‖}_{H}^{2} + {‖ β_{Γ}^{'} (u_{Γ} (s + h)) ‖}_{H_{Γ}}^{2} \\ + {‖ β_{Γ}^{'} (u_{Γ} (s)) ‖}_{H_{Γ}}^{2} . \end{array}$ Thanks to (3.10) and Lemma 4.3 (with the choice $p = 4$ ), for all $η_{1} > 0$ there exists a positive constant $\tilde{C} (4, η_{1})$ depending on $M_{2}$ and $C (4, η_{1})$ such that $\begin{matrix} \int_{t}^{t + η_{1}} {‖ \partial_{t}^{h} u (s) ‖}_{H_{0}}^{2} d s ⩽ \tilde{C} (4, η_{1}), \int_{t}^{t + η_{1}} B {(s)}^{2} d s ⩽ \tilde{C} (4, η_{1}), \end{matrix}$ for all $t ⩾ η_{1}$ , indeed, we can apply the same way of the proof of Lemma 3.3. Hence applying the uniform Gronwall inequality given by Lemma A.2, we deduce that $\begin{array}{l} {‖ \partial_{t}^{h} u (t + η_{1}) ‖}_{H_{0}}^{2} & ⩽ (\frac{1}{η_{1}} \int_{t}^{t + η_{1}} {‖ \partial_{t}^{h} u (s) ‖}_{H_{0}}^{2} d s) e^{\int_{t}^{t + η_{1}} C (B {(s)}^{2} + 1) d s} \\ (4.38) & ⩽ \frac{1}{η_{1}} \tilde{C} (4, η_{1}) e^{C (\tilde{C} (4, η_{1}) + η_{1})}, for all t ⩾ η_{1} . \end{array}$ Moreover, integrating (4.37) with respect to time and using (4.38), we have $\begin{array}{l} min {\frac{1}{4}, \frac{σ}{12}} \int_{t}^{t + 1} {‖ \partial φ (\partial_{t}^{h} u (s)) ‖}_{H_{0}}^{2} d s \\ ⩽ {‖ \partial_{t}^{h} u (t) ‖}_{H_{0}}^{2} + C \int_{t}^{t + 1} (B {(s)}^{2} + 1) {‖ \partial_{t}^{h} u (s) ‖}_{H_{0}}^{2} d s \\ (4.39) & ⩽ \frac{1}{η_{1}} \tilde{C} (4, η_{1}) e^{C (\tilde{C} (4, η_{1}) + η_{1})} [1 + C (\tilde{C} (4, 1) + 1)], \end{array}$ for all $t ⩾ η + η_{1}$ .

For simplicity, we can just take $η_{1} = η$ . Letting $h \to 0$ in (4.38), we obtain the estimate (4.20). Moreover, taking $h \to 0$ in (4.39), we conclude from (4.34) the second estimate (4.21). The proof is complete. □

We are now in a position to finish the proof of Theorem 2.1. Proof of Theorem 2.1. Part (2).
Consider (3.25)–(3.26) as an elliptic problem for $μ$ . Recalling that now we assume $σ > 0$ , then by a similar reasoning for (4.34), we infer from (4.20) that $\begin{array}{l} {‖ μ - m (μ) ‖}_{L^{\infty} (2 η, \infty; W)} ⩽ C {‖ u^{'} ‖}_{L^{\infty} (2 η, \infty; H_{0})} ⩽ C M_{6} . \end{array}$ This together with (3.24) yields $\begin{array}{l} (4.40) & ‖ μ ‖_{L^{\infty} (2 η, \infty; W)} ⩽ {\tilde{M}}_{7}, \end{array}$ where the constant ${\tilde{M}}_{7}$ depends on $M_{3}^{'}$ , $M_{6}$ , Ω and Γ. Next, set $\tilde{μ} : = μ - π (u)$ and ${\tilde{μ}}_{Γ} : = μ_{Γ} - π_{Γ} (u_{Γ})$ . From (3.4), (4.40), (A3) and the Sobolev embedding theorem, we see that there exists a positive constant $M_{7}$ such that $\begin{matrix} (4.41) & ‖ \tilde{μ} ‖_{L^{\infty} ((t, t + 1) \times Ω)} ⩽ M_{7}, ‖ {\tilde{μ}}_{Γ} ‖_{L^{\infty} ((t, t + 1) \times Γ)} ⩽ M_{7}, \end{matrix}$ for all $t ⩾ 2 η$ .

Thanks to (4.41), we are able to obtain further estimates for the singular terms $β (u)$ and $β (u_{Γ})$ . This is an essence of the proof for the separation property. To this end, for each $p \in [2, \infty)$ , testing the equation $\begin{matrix} - Δ u + β (u) = \tilde{μ} \in L^{\infty} ((t, t + 1) \times Ω) \end{matrix}$ by $| β (u) |^{p - 2} β (u) \in L^{1} ((t, t + 1) \times Ω)$ (recall (3.27)), and testing the equation $\begin{matrix} \partial_{ν} u - Δ_{Γ} u_{Γ} + β_{Γ} (u_{Γ}) = μ_{Γ} - π (u_{Γ}) = : {\tilde{μ}}_{Γ} \in L^{\infty} ((t, t + 1) \times Γ) \end{matrix}$ by $| β (u_{Γ}) |^{p - 2} β (u_{Γ})$ , adding the resultants together and by a similar argument like in Lemma 3.5, we obtain $\begin{array}{l} \int_{Ω}^{} {| β (u (s)) |}^{p} d x + \int_{Γ}^{} {| β (u_{Γ} (s)) |}^{p - 2} β (u_{Γ} (s)) β_{Γ} (u_{Γ} (s)) d Γ \\ ⩽ \int_{Ω}^{} \tilde{μ} (s) {| β (u (s)) |}^{p - 2} β (u (s)) d x + \int_{Ω}^{} {\tilde{μ}}_{Γ} (s) {| β (u_{Γ} (s)) |}^{p - 2} β (u_{Γ} (s)) d Γ \\ ⩽ {‖ \tilde{μ} (s) ‖}_{L^{p} (Ω)} {(\int_{Ω}^{} {| β (u (s)) |}^{p} d x)}^{\frac{p - 1}{p}} + {‖ {\tilde{μ}}_{Γ} (s) ‖}_{L^{p} (Γ)} {(\int_{Γ}^{} {| β (u_{Γ} (s)) |}^{p} d Γ)}^{\frac{p - 1}{p}} \end{array}$ and $\begin{array}{l} \int_{Γ} {| β (u_{Γ} (s)) |}^{p} d Γ & ⩽ 2 c_{1} \int_{Γ} {| β (u_{Γ} (s)) |}^{p - 2} β (u_{Γ} (s)) β_{Γ} (u_{Γ} (s)) d Γ + C, for a.a. s ⩾ 2 η . \end{array}$ It follows from Young’s inequality that $\begin{array}{l} \int_{Ω}^{} {| β (u (s)) |}^{p} d x + \frac{1}{2 c_{1}} \int_{Γ}^{} {| β (u_{Γ} (s)) |}^{p} d Γ \\ ⩽ \frac{1}{2} \int_{Ω}^{} {| β (u (s)) |}^{p} d x + \frac{1}{4 c_{1}} \int_{Γ}^{} {| β (u_{Γ} (s)) |}^{p} d Γ + \frac{1}{p} {(\frac{2 (p - 1)}{p})}^{p - 1} {‖ \tilde{μ} (s) ‖}_{L^{p} (Ω)}^{p} \\ + \frac{1}{2 c_{1} p} {(\frac{2 (p - 1)}{p})}^{p - 1} {‖ {\tilde{μ}}_{Γ} (s) ‖}_{L^{p} (Γ)}^{p} + C, for a.a. s ⩾ 2 η, \end{array}$ where C is independent of p. We note that $\begin{matrix} ‖ \tilde{μ} ‖_{L^{p} ((t, t + 1) \times Ω)} \to ‖ \tilde{μ} ‖_{L^{\infty} ((t, t + 1) \times Ω)}, ‖ {\tilde{μ}}_{Γ} ‖_{L^{p} ((t, t + 1) \times Γ)} \to ‖ {\tilde{μ}}_{Γ} ‖_{L^{\infty} ((t, t + 1) \times Γ)} \end{matrix}$ as $p \to \infty$ (see, e.g., [2, Theorem 2.14]). Therefore, for sufficiently large p, it holds $\begin{array}{l} \int_{t}^{t + 1} \int_{Ω}^{} {| β (u (s)) |}^{p} d x d s + \frac{1}{2 c_{1}} \int_{t}^{t + 1} \int_{Γ}^{} {| β (u_{Γ} (s)) |}^{p} d Γ d s \\ ⩽ (1 + \frac{1}{2 c_{1} p}) {(\frac{2 (p - 1)}{p})}^{p - 1} {(M_{7})}^{p} + C, for all t ⩾ 2 η . \end{array}$ Letting $p \to \infty$ , we deduce that $\begin{matrix} {‖ β (u) ‖}_{L^{\infty} ((t, t + 1) \times Ω)} ⩽ 2 M_{7} + 1, {‖ β (u_{Γ}) ‖}_{L^{\infty} ((t, t + 1) \times Γ)} ⩽ 2 M_{7} + 1, \end{matrix}$ for all $t ⩾ 2 η$ . From assumption (A1) on the singular term β, we see that there exists a constant $δ_{2} \in (0, 1)$ such that $\begin{matrix} {‖ u (t) ‖}_{L^{\infty} ((t, t + 1) \times Ω)} ⩽ 1 - δ_{2}, {‖ u_{Γ} (t) ‖}_{L^{\infty} ((t, t + 1) \times Γ)} ⩽ 1 - δ_{2}, \end{matrix}$ for all $t ⩾ 2 η$ . Besides, it holds that $u \in L^{\infty} (η, \infty; C (\overline{Ω}) \times C (Γ))$ , due to the Sobolev embedding theorem and (3.29). As a consequence, we can conclude the separation property (2.16) (replacing $2 η$ by η since $η > 0$ can be chosen arbitrary).

The proof of Theorem 2.1 is complete. □

5. Long-time behavior

In this section we prove Theorem 2.2 on the long-time behavior of problem (2.2).

5.1. Compactness of the orbit $u (t)$ in $W$

The following lemma implies the compactness of the weak solution $u$ in $W$ for large time.

Lemma 5.1.
Under the assumptions of Proposition 2.1 , there exists a positive constant $M_{8}$ such that $\begin{matrix} (5.1) & ‖ u ‖_{L^{\infty} (T_{1}, \infty; H^{3} (Ω))} + ‖ u_{Γ} ‖_{L^{\infty} (T_{1}, \infty; H^{3} (Γ))} ⩽ M_{8}, for all t ⩾ T_{1}, \end{matrix}$ where $T_{1}$ is the same as in Theorem 2.1 (1).
Proof.
Consider the equation for $u = (u, u_{Γ})$ $\begin{array}{l} - Δ u (t) = μ (t) - β (u (t)) - π (u (t)) = : f (t) a.e. in Ω, \\ u_{|_{Γ}} (t) = u_{Γ} (t) a.e. on Γ, \\ \partial_{ν} u (t) - Δ_{Γ} u_{Γ} (t) + u_{Γ} (t) = μ_{Γ} (t) - β_{Γ} (u_{Γ} (t)) + π_{Γ} (u_{Γ} (t)) + u_{Γ} (t) = : f_{Γ} (t) a.e. on Γ, \end{array}$ for all $t ⩾ T_{1}$ . From the separation property (2.14), (A1) and (3.29), we can obtain $\begin{array}{l} {‖ β (u (t)) ‖}_{V} + {‖ β_{Γ} (u_{Γ} (t)) ‖}_{V_{Γ}} ⩽ C, for a.a. t ⩾ T_{1} . \end{array}$ Next, it follows from (3.19) that $\begin{matrix} {‖ μ (t) ‖}_{V} + {‖ μ_{Γ} (t) ‖}_{V_{Γ}} ⩽ C, for a.a. t ⩾ T_{1} . \end{matrix}$ As a consequence, we have $\begin{matrix} {‖ f (t) ‖}_{V} + {‖ f_{Γ} (t) ‖}_{V_{Γ}} ⩽ C, for a.a. t ⩾ T_{1}, \end{matrix}$ which together with the elliptic regularity theory (see Lemma A.6) yields the uniform estimate (5.1). □
Remark 5.1.
By (3.10), (5.1) and interpolation, we easily see that $u \in C ([t, t + 1]; W)$ for all $t ⩾ T_{1}$ , hence, $\begin{matrix} u \in C ([T_{1}, \infty); W) . \end{matrix}$ The uniform estimate (5.1) and the compact embedding $H^{3} (Ω) \times H^{3} (Γ) ↪ ↪ W \times W_{Γ}$ also imply that the ω-limit set $ω (u_{0})$ is compact in $W$ (an alternative proof for this fact is due to (4.10) and Lemma 4.1).

5.2. Convergence to equilibrium

Since $ω (u_{0})$ is nonempty and compact in $W$ , we immediately have the sequent convergence $\begin{matrix} lim_{t \to \infty} dist (S (t) u_{0}, ω (u_{0})) = 0 in W . \end{matrix}$ Our aim is to prove that for any initial datum $u_{0}$ satisfying (A4), the corresponding ω-limit set reduces to a singleton, namely, there exists an element $u_{\infty} \in S_{m_{0}}$ such that $\begin{matrix} u (t) \to u_{\infty}, in W as t \to \infty . \end{matrix}$ This can be achieved by using the well-known Łojasiewicz–Simon approach, see for instance, [30,32], and for further applications, we refer to [1,3,10,24,26,28,29,52,54,56,58–60].

The main tool is following extended Łojasiewicz–Simon inequality. Let $ψ = (ψ, ψ_{Γ}) \in S_{a}$ for some $a \in (- 1, 1)$ . It is straightforward to verify that $ψ$ is a critical point of the free energy E (see (3.2)). Moreover, we obtain the following lemma:

Lemma 5.2.

Suppose that (A1)–(A3) are satisfied. In addition, we assume that β, $β_{Γ}$ are real analytic on $(- 1, 1)$ and π, $π_{Γ}$ are real analytic on $R$ . Let $ψ = (ψ, ψ_{Γ}) \in S_{a}$ , $a \in (- 1, 1)$ . There exist constants $θ^{*} \in (0, 1 / 2)$ and $b^{*} > 0$ such that $\begin{matrix} (5.2) & {‖P (\begin{matrix} - Δ w + β (w) + π (w) \\ \partial_{ν} w - Δ_{Γ} w_{Γ} + β_{Γ} (w_{Γ}) + π_{Γ} (w_{Γ}) \end{matrix})‖}_{H_{0}} ⩾ {| E (w) - E (ψ) |}^{1 - θ^{*}} \end{matrix}$ for all $w \in W$ satisfying $‖ w - ψ ‖_{W} < b^{*}$ and $m (w) = a$ .

Lemma 4.1 implies that all elements of $S_{a}$ are uniformly separated from $\pm 1$ . Then we can take $b^{*} > 0$ sufficiently small such that any element $w \in W$ satisfying $‖ w - ψ ‖_{W} < b^{*}$ is uniformly separated from $\pm 1$ . In particular, this choice prevents the possible singularity in the nonlinearities β, $β_{Γ}$ . Keeping this fact in mind, we can follow the standard argument like in [32,54] to prove Lemma 5.2. More related to our problem (2.2), we refer to [24] for the case with mass conservation and a linear boundary condition, and to [56,58] for the case with nonlinear boundary condition but without mass conservation. When singular potential is considered, we refer to [29].

Proof of Theorem 2.2.

We now have all the necessary ingredients for the proof:

the characterization of $ω (u_{0})$ ;

the energy identity (3.9);

the Łojasiewicz–Simon type inequality (5.2).

The proof of Theorem 2.2 can be carried out in the same way as for instance, [24, Section 2.4]. We just would like to mention that in Lemma 5.2, if $w$ is taken to be the weak solution $u (t)$ of problem (2.2) that can be shown falling into the small $W$ -neighborhood of a cluster point $u_{\infty} \in ω (u_{0})$ (which is indeed true for sufficiently large time), then by the generalized Poincaré inequality (see Lemma A.3), we have $\begin{matrix} {‖ u^{'} ‖}_{V_{σ, 0}^{*}} = ‖ P μ ‖_{V_{σ, 0}} ⩾ C {| E (u) - E (u_{\infty}) |}^{1 - θ^{*}} . \end{matrix}$ This connects the energy dissipation in (3.9) and the Łojasiewicz–Simon inequality (5.2) that leads to the proof. The rest of details are omitted. □

Footnotes

Auxiliary lemmas

We report some lemmas that have been used in this paper.

Acknowledgements

The authors would like to thank the anonymous referee for his/her careful reading of an initial version of this paper and for many helpful comments that allowed us to improve the presentation. T. Fukao acknowledges the support from the JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Japan Grant Number 17K05321. H. Wu was partially supported by NNSFC Grant No. 11631011 and the Shanghai Center for Mathematical Sciences at Fudan University.

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Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn–Hilliard type with singular potential

Abstract

Keywords

1. Introduction

2. Preliminaries and main results

2.1. Notation

2.2. The initial boundary value problem

2.3. Main results

Theorem 2.1 (Separation from pure states).

3.1. Mass conservation and energy equality

4.1. Eventual separation from pure states

5.1. Compactness of the orbit u ( t ) in W

Footnotes

Auxiliary lemmas

Acknowledgements

References

5.1. Compactness of the orbit $u (t)$ in $W$