On a Cahn–Hilliard–Oono model for image segmentation

Abstract

This paper studies firstly the well-posedness and the asymptotic behavior of a Cahn–Hilliard–Oono type model, with cubic nonlinear terms, which is proposed for image segmentation. In particular, the existences of the global attractor and the exponential attractor have been proved, and it shows that the fractal dimension of the global attractor will tend to infinity as $α \to 0$ . The difficulty here is that we no longer have the conservation of mass. Furthermore, this model with logarithmic nonlinear terms has been studied as well. One difficulty here is to make sure that the logarithmic terms can pass to the limit under the standard Galerkin scheme. Another difficulty is to prove additional regularities on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. It eventually shows that the dimension of the global attractor is finite by proving the existence of the exponential attractor.

Keywords

Image segmentation Cahn–Hilliard–Oono equation well-posedness global attractor exponential attractor strict separation

1. Introduction

Image segmentation is the process of partitioning a image into multiple segments, which plays an significant role in image processing and computer vision. More precisely, it aims to partition a given image into regions in order to recognize and analyze different objects. The practical applications of image segmentation involve medical imaging, machine vision, object detection, video surveillance and so on. In the past few decades, a range of approaches have been proposed to deal with this problem, which include clustering methods, graph partitioning methods, statistics based methods, variational methods and PDEs based methods. We refer the reader to [7,17,22–24] and references therein for more details. And Cahn–Hilliard type models in the context of image processing can be found in [3,6,8–12].

In particular, a Cahn–Hilliard type model for image segmentation has been proposed and studied in [23], the authors prove the existence and uniqueness of solutions for the model, but their conclusion is based on the precondition that the solution u remains in the relevant interval $[0, 1]$ . However, we have presented in [12] that there exist solutions which are unbounded when time goes to infinity, nevertheless, we can not obtain a global in time estimate on u. To overcome this, we consider a modified boundary value problem: $\begin{matrix} (1.1) & \frac{\partial u}{\partial t} + α u + ε_{1} Δ^{2} u - \frac{1}{ε_{1}} Δ f (u) + \frac{ε_{2} h (x)}{ε_{2}^{2} + {(u - \frac{1}{2})}^{2}} = 0, α > 0, \end{matrix}$ where f is a cubic nonlinear term and $\begin{matrix} h (x) = \frac{1}{π} (λ_{1} {(i (x) - c_{1})}^{2} - λ_{2} {(i (x) - c_{2})}^{2}) . \end{matrix}$ Here, $ε_{1}$ , $ε_{2}$ , $λ_{1}$ and $λ_{2}$ are positive constants and i is a given image taking values in $[0, 1]$ . We will set $ε_{i} = 1$ , $i = 1, 2$ in this paper for simplicity. Such a model takes advantage of the favorable features of high-order PDEs and is also easier to handle than Euler’s elastica based models.

Furthermore, if we take $h (x) = 0$ , the equation (1.1) will turn into the well-known Cahn–Hilliard equation when $α = 0$ , and the equation (1.1) will reduce to the Cahn–Hilliard–Oono equation when $α > 0$ . Note that the Cahn–Hilliard–Oono equation is proposed in [19] to account for long-ranged (i.e., nonlocal) interactions in the phase separation processes and also to simplify numerical simulations. We refer the interested reader to [4,5,16,18] for reviews on the Cahn–Hilliard equation and some of its variants, as well as their mathematical analysis.

In this paper, we propose a modification of the model proposed in [23], based on the Cahn–Hilliard–Oono equation, which ensures the global in time boundedness of the solutions. We work on this model in two cases: $f (u)$ is a classic cubic term; $f (u)$ is a logarithmic term. We study the well-posedness and the asymptotic behavior of these models in different cases.

2. Setting of the problem

We consider the following initial and boundary value problem, in a bounded and regular domain $Ω \subset R^{n}$ , $n = 1$ , 2 or 3, with boundary Γ: $\begin{array}{l} (2.1) & \frac{\partial u}{\partial t} + α u + Δ^{2} u - Δ f (u) + \frac{h (x)}{1 + {(u - \frac{1}{2})}^{2}} = 0, \\ (2.2) & \frac{\partial u}{\partial ν} = \frac{\partial Δ u}{\partial ν} = 0 on Γ, \\ (2.3) & u |_{t = 0} = u_{0} . \end{array}$

We assume that $h \in L^{\infty} (Ω)$ , which is consistent with what is considered in [23]. We further take $\begin{matrix} f (s) = s^{3} - s \end{matrix}$ for simplicity, though we can more generally consider any cubic polynomial with positive leading term, or even any regular function with such a cubic growth at infinity. Note in particular that the above nonlinear term satisfies the dissipativity properties $\begin{matrix} (2.4) & f^{'} ⩾ - 1 \end{matrix}$ and $\begin{matrix} F ⩾ - c, c ⩾ 0, \end{matrix}$ where $\begin{matrix} F (s) = \int_{0}^{s} f (ξ) d ξ = \frac{1}{4} s^{4} - \frac{1}{2} s^{2} . \end{matrix}$

We then set $\begin{matrix} g (s) = \frac{1}{1 + {(s - \frac{1}{2})}^{2}} \end{matrix}$ and note that g is nonnegative and bounded by 1.

We finally set, for $v \in L^{1} (Ω)$ , $\begin{matrix} ⟨ v ⟩ = \frac{1}{Vol (Ω)} \int_{Ω} v (x) d x \end{matrix}$ and, for $v \in H^{- 1} (Ω)$ , $\begin{matrix} ⟨ v ⟩ = \frac{1}{Vol (Ω)} {⟨ v, 1 ⟩}_{H^{- 1} (Ω), H^{1} (Ω)} . \end{matrix}$ Furthermore, we set, whenever it makes sense, $\begin{matrix} \overline{v} = v - ⟨ v ⟩ . \end{matrix}$

We denote by $((\cdot, \cdot))$ the usual $L^{2}$ -scalar product, with associated norm $‖ \cdot ‖$ . We also set $‖ \cdot ‖_{- 1} = ‖ {(- Δ)}^{- \frac{1}{2}} \cdot ‖$ , where ${(- Δ)}^{- 1}$ denotes the inverse of the negative Laplace operator associated with Neumann boundary conditions and acting on functions with null spatial average. More generally, we denote by $‖ \cdot ‖_{X}$ the norm on the Banach space X.

We note that $\begin{array}{l} v \mapsto {(‖ \overline{v} ‖_{- 1}^{2} + {⟨ v ⟩}^{2})}^{\frac{1}{2}}, v \mapsto {(‖ \overline{v} ‖^{2} + {⟨ v ⟩}^{2})}^{\frac{1}{2}}, \\ v \mapsto {(‖ \nabla v ‖^{2} + {⟨ v ⟩}^{2})}^{\frac{1}{2}} and v \mapsto {(‖ Δ v ‖^{2} + {⟨ v ⟩}^{2})}^{\frac{1}{2}} \end{array}$ are norms on $H^{- 1} (Ω)$ , $L^{2} (Ω)$ , $H^{1} (Ω)$ and $H^{2} (Ω)$ , respectively, which are equivalent to the usual norms on these spaces. Furthermore, $‖ \cdot ‖_{- 1}$ is a norm on ${v \in H^{- 1} (Ω), ⟨ v ⟩ = 0}$ which is equivalent to the usual $H^{- 1}$ -norm.

At last, we recall several inequalities (see [12,15]), which will be frequently applied below.

∙ The Poincaré–Wirtinger inequality: $\begin{matrix} (2.5) & ‖ \overline{v} ‖ ⩽ c ‖ \nabla v ‖, \forall v \in H^{1} (Ω) . \end{matrix}$

Note that a consequence of this inequality is that $\begin{matrix} (2.6) & ‖ {(- Δ)}^{- 1} v ‖ ⩽ c ‖ v ‖_{- 1}, \forall v \in H^{- 1} (Ω), ⟨ v ⟩ = 0 . \end{matrix}$

∙ An interpolation inequality: $\begin{matrix} (2.7) & ‖ v ‖ ⩽ c ‖ v ‖_{- 1}^{\frac{1}{2}} ‖ \nabla v ‖^{\frac{1}{2}}, \forall v \in H^{1} (Ω), ⟨ v ⟩ = 0 . \end{matrix}$

Throughout this paper, the same coefficients c and $c^{'}$ denote (generally positive) constants which may change from line to line, or even in the same line.

3. The first model with cubic term

3.1. The well-posedness results

We first present the well-posedness result of the problem (2.1)–(2.3), which allows us to construct the dissipative semigroup so that we can prove the existence of the global attractor. The proof of existence is based on a standard Galerkin scheme.

Theorem 3.1.
For everty $u_{0} \in H^{1} (Ω)$ and every $T > 0$ , ( 2.1 )–( 2.3 ) possesses a unique weak solution u such that $\begin{array}{l} u \in L^{\infty} (0, T; H^{1} (Ω)) \cap L^{2} (0, T; H^{2} (Ω)) \cap L^{4} (0, T; L^{4} (Ω)), \\ \frac{\partial u}{\partial t} \in L^{2} (0, T; H^{- 1} (Ω)) . \end{array}$
Proof.
Integrating (2.1) over Ω to have $\begin{matrix} (3.1) & \frac{d ⟨ u ⟩}{d t} + α ⟨ u ⟩ + ⟨ h (x) g (u) ⟩ = 0, \end{matrix}$ then it follows from (2.1) and (3.1) that $\begin{matrix} (3.2) & \frac{\partial \overline{u}}{\partial t} + α \overline{u} + Δ^{2} u - Δ f (u) + \overline{h (x) g (u)} = 0, \end{matrix}$ where $u = \overline{u} + ⟨ u ⟩$ . We can rewrite in the equivalent weaker form of (2.1)–(2.3): $\begin{array}{l} (3.3) & {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t} + α {(- Δ)}^{- 1} \overline{u} - Δ u + \overline{f (u)} + {(- Δ)}^{- 1} \overline{h (x) g (u)} = 0, \\ (3.4) & \frac{\partial \overline{u}}{\partial ν} = 0 on Γ, \\ (3.5) & \overline{u} |_{t = 0} = {\overline{u}}_{0}, ⟨ u ⟩ |_{t = 0} = ⟨ u_{0} ⟩ . \end{array}$

Existence: The proof of existence is based on a standard Galerkin scheme and the a priori estimates. We consider the following approximated problem: for $m \in N$ given, ${ω_{i}}$ ( $i = 1, \dots, m$ ) forms an orthogonal in V basis, we set $V_{m} = span (ω_{1}, \dots, ω_{m})$ and find $u_{m} = \sum_{i = 1}^{m} u_{i, m} ω_{i}$ , such that $\begin{array}{l} (3.6) & \begin{matrix} \frac{d}{d t} (({(- Δ)}^{- 1} {\overline{u}}_{m}, v)) + α (({(Δ)}^{- 1} {\overline{u}}_{m}, v)) + {((u_{m}, v))}_{V} + ((\overline{f (u_{m})}, v)) \\ + (({(- Δ)}^{- 1} \overline{h (x) g (u_{m})}, v)) = 0 in D^{'} (0, T), \forall v \in V_{m}, \end{matrix} \\ (3.7) & \frac{d ⟨ u_{m} ⟩}{d t} + α ⟨ u_{m} ⟩ + ⟨ h (x) g (u_{m}) ⟩ = 0 in D^{'} (0, T), \\ (3.8) & {\overline{u}}_{m} |_{t = 0} = {\overline{u}}_{0}, ⟨ u_{m} ⟩ |_{t = 0} = ⟨ u_{0} ⟩, \end{array}$ where $u_{m} = {\overline{u}}_{m} + ⟨ u_{m} ⟩$ and $D^{'}$ denotes the space of distributions. In view of (3.7), employing the Gronwall lemma to have $\begin{matrix} ⟨ u_{m} (t) ⟩ = ⟨ u_{0} ⟩ e^{- α t} + e^{- α t} \int_{0}^{t} e^{α s} ⟨ h (x) g (u_{m}) ⟩ d s, \end{matrix}$ so that $\begin{matrix} | ⟨ u_{m} (t) ⟩ | ⩽ | ⟨ u_{0} ⟩ | e^{- α t} + \frac{‖ h ‖_{L^{\infty} (Ω)}}{α} (1 - e^{- α t}), t ⩾ 0 . \end{matrix}$ We further assume that $| ⟨ u_{0} ⟩ | ⩽ M$ , $M ⩾ 0$ , in particular, we obtain that $\begin{matrix} (3.9) & | ⟨ u_{m} (t) ⟩ | ⩽ | ⟨ u_{0} ⟩ | + \frac{‖ h ‖_{L^{\infty} (Ω)}}{α} ⩽ M^{'}, \forall t ⩾ 0, \end{matrix}$ which means the spatial average of $u_{m}$ is bounded, uniformly with respect to time.

Taking $v = ω_{i}$ in (3.6), multiplying the resulting equality by ${\overline{u}}_{i, m}$ and summing over $i = 1, \dots, m$ to have $\begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ {\overline{u}}_{m} ‖_{- 1}^{2} + α ‖ {\overline{u}}_{m} ‖_{- 1}^{2} + ‖ \nabla u_{m} ‖^{2} + ((f (u_{m}), {\overline{u}}_{m})) + ((h (x) g (u_{m}), {(- Δ)}^{- 1} {\overline{u}}_{m})) = 0 . \end{matrix}$ Noting that,employing the Young’s inequality, we have $\begin{array}{l} ((f (u_{m}), u_{m})) ⩾ \frac{3}{4} ‖ u_{m} ‖_{L^{4} (Ω)}^{4} - c, \\ | ⟨ u_{m} ⟩ \int_{Ω} f (u_{m}) d x | ⩽ \frac{1}{4} ‖ u_{m} ‖_{L^{4} (Ω)}^{4} + c_{M^{'}}, \end{array}$ and recalling that $h \in L^{\infty} (Ω)$ , g is bounded by 1, we have $\begin{array}{l} | ((h (x) g (u_{m}), {(- Δ)}^{- 1} {\overline{u}}_{m})) | ⩽ c ‖ u_{m} ‖ ⩽ \frac{1}{2} ‖ u_{m} ‖^{2} + c \end{array}$ by employing the Poincaré–Wirtinger inequality, Young’s inequality and the embedding $L^{2} (Ω) ↪ H^{- 1} (Ω)$ is continuous.

We deduce from the above that $\begin{matrix} (3.10) & \frac{d}{d t} ‖ {\overline{u}}_{m} ‖_{- 1}^{2} + 2 α ‖ {\overline{u}}_{m} ‖_{- 1}^{2} + ‖ \nabla u_{m} ‖^{2} + 2 ‖ u_{m} ‖_{L^{4} (Ω)}^{4} ⩽ c, \end{matrix}$ where c is may depend on $M^{'}$ , we omit the subscript $M^{'}$ of constant c here and in the following.

In view of (3.7), we have $\begin{matrix} (3.11) & \frac{d {⟨ u_{m} ⟩}^{2}}{d t} + α {⟨ u_{m} ⟩}^{2} ⩽ c {⟨ u_{m} ⟩}^{2} + 1, \end{matrix}$ recalling (3.9) to deduce that $\begin{matrix} (3.12) & \frac{d}{d t} (‖ {\overline{u}}_{m} ‖_{- 1}^{2} + {⟨ u_{m} ⟩}^{2}) + α (‖ {\overline{u}}_{m} ‖_{- 1}^{2} + {⟨ u_{m} ⟩}^{2}) + ‖ \nabla u_{m} ‖^{2} + 2 ‖ u_{m} ‖_{L^{4} (Ω)}^{4} ⩽ c . \end{matrix}$ Recalling that $V \mapsto {(‖ \overline{u} ‖_{- 1}^{2} + {⟨ v ⟩}^{2})}^{\frac{1}{2}}$ is a norm on $H^{- 1} (Ω)$ , which is equivalent to the usual $H^{- 1} (Ω)$ -norm. It thus follows from (3.12) and the Gronwall lemma that $u_{m}$ is bounded, independent of m, in $L^{\infty} (0, T; H^{- 1} (Ω)) \cap L^{4} (0, T; L^{4} (Ω))$ .

We then take $v = ω_{i}$ in (3.6) and multiply the resulting equation by $λ_{i} u_{i, m}$ to find $\begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ {\overline{u}}_{m} ‖^{2} + α ‖ {\overline{u}}_{m} ‖^{2} + ‖ Δ u_{m} ‖^{2} + ((f^{'} (u) \nabla u_{m}, \nabla u_{m})) + ((h (x) g (u_{m}), u_{m})) = 0, \end{matrix}$ employing the Poincaré–Wirtinger inequality again, we have $\begin{matrix} | ((h (x) g (u_{m}), u_{m})) | ⩽ c ‖ u_{m} ‖ ⩽ c ‖ \nabla u_{m} ‖, \end{matrix}$ it thus follows from (2.4), (3.11), (3.12) to obtain that $\begin{matrix} (3.13) & \frac{1}{2} \frac{d}{d t} ‖ u_{m} ‖^{2} + α ‖ u_{m} ‖^{2} + ‖ Δ u_{m} ‖^{2} ⩽ c ‖ \nabla u_{m} ‖^{2} + c^{'}, \end{matrix}$ it follows from (3.12), (3.13) and the uniform Gronwall lemma that $u_{m}$ is bounded, independent of m, in $L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H^{2} (Ω))$ .

Finally, taking $v = ω_{i}$ in (3.6) and multiply the resulting equation by $\frac{d {\overline{u}}_{i, m}}{d t}$ to have $\begin{array}{l} \frac{d}{d t} (α ‖ {\overline{u}}_{m} ‖_{- 1}^{2} + ‖ \nabla u_{m} ‖^{2} + 2 \int_{Ω} F (u_{m}) d x) + 2 {‖ \frac{\partial {\overline{u}}_{m}}{\partial t} ‖}_{- 1}^{2} - ((f (u_{m}), ⟨ \frac{\partial u_{m}}{\partial t} ⟩)) \\ + ((h (x) g (u_{m}), {(- Δ)}^{- 1} \frac{\partial {\overline{u}}_{m}}{\partial t})) = 0 . \end{array}$ We note that, recalling (3.7), (3.9) and the properties of h and g, we have $\begin{array}{l} - ((f (u_{m}), ⟨ \frac{\partial u_{m}}{\partial t} ⟩)) & = ((f (u_{m}), α ⟨ u_{m} ⟩)) + ((f (u_{m}), ⟨ h (x) g (u_{m}) ⟩)) \\ ⩽ c \int_{Ω} u_{m}^{4} d x + c^{'} \end{array}$ and $\begin{matrix} | ((h (x) g (u_{m}), {(- Δ)}^{- 1} \frac{\partial {\overline{u}}_{m}}{\partial t})) | ⩽ c {‖ \frac{\partial {\overline{u}}_{m}}{\partial t} ‖}_{- 1} ⩽ \frac{1}{2} {‖ \frac{\partial {\overline{u}}_{m}}{\partial t} ‖}_{- 1}^{2} + c . \end{matrix}$ Recalling (3.11), it thus follows from above that $\begin{matrix} (3.14) & \frac{d}{d t} (α ‖ u_{m} ‖_{- 1}^{2} + ‖ \nabla u_{m} ‖^{2} + 2 \int_{Ω} F (u_{m}) d x) + {‖ \frac{\partial u_{m}}{\partial t} ‖}_{- 1}^{2} ⩽ c \int_{Ω} u_{m}^{4} d x + c^{'}, \end{matrix}$ which yields that $u_{m}$ and $\frac{\partial u_{m}}{\partial t}$ are bounded, independent of m, in $L^{\infty} (0, T; H^{1} (Ω))$ and $L^{2} (0, T; H^{- 1} (Ω))$ , respectively.

It follows from the above and the Aubin–Lions compactness results that there exists $u \in L^{\infty} (0, T; H^{1} (Ω)) \cap L^{2} (0, T; H^{2} (Ω)) \cap L^{4} (0, T; L^{4} (Ω))$ , with $\frac{\partial u_{m}}{\partial t} \in L^{2} (0, T; H^{- 1} (Ω))$ such that $\begin{array}{l} u_{m} \overset{*}{⇀} u in L^{\infty} (0, T; H^{1} (Ω)), \\ u_{m} ⇀ u in L^{2} (0, T; H^{2} (Ω)), \\ u_{m} \overset{a . e .}{⟶} u in L^{4} (0, T; L^{4} (Ω)), \\ \frac{\partial u_{m}}{\partial t} ⇀ \frac{\partial u}{\partial t} in L^{2} (0, T; H^{- 1} (Ω)), \end{array}$ passing to the limit in the linear terms is straightforward, the existence results can be deduced by the estimates above.

Uniqueness: Let $u_{1}$ and $u_{2}$ be two solutions with initial data $u_{1, 0}$ and $u_{2, 0}$ , respectively, such that $⟨ u_{1, 0} ⟩ = ⟨ u_{2, 0} ⟩$ . We set $u = u_{1} - u_{2}$ and $u_{0} = u_{1, 0} - u_{2, 0}$ . We then have the following system: $\begin{array}{l} (3.15) & \frac{\partial \overline{u}}{\partial t} + α \overline{u} + Δ^{2} u - Δ (f (u_{1}) - f (u_{2})) + \overline{h (x) (g (u_{1}) - g (u_{2}))} = 0, \\ (3.16) & \frac{d ⟨ u ⟩}{d t} + α ⟨ u ⟩ + ⟨ h (x) (g (u_{1}) - g (u_{2}) ⟩ = 0, \\ (3.17) & \frac{\partial u}{\partial ν} = \frac{\partial Δ u}{\partial ν} = 0 on Γ, \\ (3.18) & u |_{t = 0} = u_{0} . \end{array}$ Proceeding as in the previous subsection, we can rewrite (3.15) in the equivalent weaker form $\begin{array}{l} (3.19) & {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t} + α {(- Δ)}^{- 1} \overline{u} - Δ u + \overline{f (u_{1}) - f (u_{2})} + {(- Δ)}^{- 1} \overline{h (x) (g (u_{1}) - g (u_{2}))} = 0, \end{array}$

Multiplying (3.19) by $\overline{u}$ to have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ \overline{u} ‖_{- 1}^{2} + α ‖ \overline{u} ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + ((f (u_{1}) - f (u_{2}), \overline{u})) \\ + ((h (x) (g (u_{1}) - g (u_{2})), {(- Δ)}^{- 1} \overline{u})) = 0 . \end{array}$ Noting that $\begin{matrix} ((f (u_{1}) - f (u_{2}), \overline{u})) ⩾ - ‖ u ‖^{2} - ⟨ u ⟩ \int_{Ω} f (u_{1}) - f (u_{2}) d x \end{matrix}$ and $\begin{array}{l} | ⟨ u ⟩ \int_{Ω} f (u_{1}) - f (u_{2}) d x | & ⩽ c | ⟨ u ⟩ | \int_{Ω} | u | (u_{1}^{2} + u_{2}^{2} + 1) d x \\ ⩽ c (‖ u ‖^{2} + (‖ u_{1} ‖_{L^{4} (Ω)}^{4} + ‖ u_{2} ‖_{L^{4} (Ω)}^{4} + 1) {⟨ u ⟩}^{2}) \\ ⩽ \frac{1}{4} ‖ \nabla u ‖^{2} + c (‖ u_{1} ‖_{L^{4} (Ω)}^{4} + ‖ u_{2} ‖_{L^{4} (Ω)}^{4} + 1) (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) . \end{array}$ Furthermore, we note that $\begin{matrix} g (u_{1}) - g (u_{2}) = - \frac{(u_{1} + u_{2} - 1) u}{(1 + {(u_{1} - \frac{1}{2})}^{2}) (1 + {(u_{2} - \frac{1}{2})}^{2})}, \end{matrix}$ which yields $\begin{matrix} | h (x) (g (u_{1}) - g (u_{2})) | ⩽ c | u | \end{matrix}$ and $\begin{array}{l} | ((h (x) (g (u_{1}) - g (u_{2})), {(- Δ)}^{- 1} \overline{u})) | & ⩽ c ‖ u ‖ ‖ \overline{u} ‖_{- 1} \\ ⩽ \frac{1}{4} ‖ \nabla u ‖^{2} + c (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) . \end{array}$

We then deduce from above that $\begin{matrix} \frac{d}{d t} ‖ \overline{u} ‖_{- 1}^{2} + 2 α ‖ \overline{u} ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} ⩽ c (‖ u_{1} ‖_{L^{4} (Ω)}^{4} + ‖ u_{2} ‖_{L^{4} (Ω)}^{4} + 1) (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) . \end{matrix}$ In view of (3.1), we have $\begin{matrix} (3.20) & | ⟨ u (t) ⟩ | ⩽ M^{'}, \forall t ⩾ 0, \end{matrix}$ and $\begin{matrix} (3.21) & \frac{d {⟨ u ⟩}^{2}}{d t} + 2 α {⟨ u ⟩}^{2} ⩽ c {⟨ u ⟩}^{2}, \end{matrix}$ to deduce that $\begin{array}{l} \frac{d}{d t} (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) + 2 α (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) + ‖ \nabla u ‖^{2} \\ (3.22) & ⩽ c (‖ u_{1} ‖_{L^{4} (Ω)}^{4} + ‖ u_{2} ‖_{L^{4} (Ω)}^{4} + 1) (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) . \end{array}$

It follows from (3.12), (3.22) and the Gronwall lemma that $\begin{matrix} (3.23) & {‖ u_{1} (t) - u_{2} (t) ‖}_{- 1} ⩽ c e^{c^{'} t} ‖ u_{0, 1} - u_{0, 2} ‖_{- 1}, 0 ⩽ t ⩽ T, \end{matrix}$ which yields the uniqueness, as well as the continuous dependence with respect to the initial data in the $H^{- 1}$ topology. □
Remark 3.2.
It follows from the Lions–Magenes theorem (see [13,15]) that if $u \in L^{2} (0, T; H^{2} (Ω))$ and $\frac{d u}{d t} \in L^{2} (0, T; H^{- 1} (Ω))$ , then $u \in C ([0, T]; H^{\frac{1}{2}} (Ω))$ . It also follows from Strauss’s lemma (see [20]) that if $u \in C ([0, T]; H^{\frac{1}{2}} (Ω)) \cap L^{\infty} (0, T; H^{1} (Ω))$ , then $u \in C ([0, T]; H_{w}^{1} (Ω))$ , where the index w denotes the weak topology.

3.2. Further a priori estimates

We need to derive higher regularities of the solutions and show that absorbing sets exist before we prove the existences of the attractors. We first write $\begin{matrix} (3.24) & \frac{d}{d t} ‖ u ‖_{- 1}^{2} + 2 α ‖ u ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + 2 ‖ u ‖_{L^{4} (Ω)}^{4} ⩽ c, \end{matrix}$ it follows from (3.24) and the Gronwall lemma that $\begin{matrix} (3.25) & ‖ u ‖_{- 1}^{2} ⩽ e^{- α t} ‖ u_{0} ‖_{- 1}^{2} + c, c > 0 . \end{matrix}$ We deduce from (3.25) that exists a bounded subset B of $H^{- 1} (Ω)$ and $t_{0}$ such that $u_{0} \in B$ and $t ⩾ t_{0}$ implies $u (t) \in B_{0}$ , where $B_{0} = {ϕ \in H^{- 1} (Ω), | ⟨ ϕ ⟩ | ⩽ M^{'}, ‖ ϕ ‖_{H^{- 1} (Ω)}^{2} ⩽ c}$ is the bounded absorbing set for the associated dynamical system on $H^{- 1} (Ω)$ , then we can rewrite (3.25) as $\begin{matrix} (3.26) & {‖ u (t) ‖}_{- 1} ⩽ c, \end{matrix}$ and deduce from (3.24) that $\begin{array}{l} (3.27) & \int_{t}^{t + r} ‖ \nabla u ‖^{2} d s ⩽ c_{r}, t ⩾ t_{0}, r > 0, \\ (3.28) & \int_{t}^{t + r} ‖ u ‖_{L^{4} (Ω)}^{4} d s ⩽ c_{r}, t ⩾ t_{0}, r > 0 . \end{array}$

We next write $\begin{matrix} (3.29) & \frac{1}{2} \frac{d}{d t} ‖ u ‖^{2} + α ‖ u ‖^{2} + ‖ Δ u ‖^{2} ⩽ c ‖ \nabla u ‖^{2} + c^{'} . \end{matrix}$ Noting that $\begin{matrix} ‖ u ‖^{2} ⩽ 2 (‖ \overline{u} ‖^{2} + {⟨ u ⟩}^{2}) ⩽ c (‖ \nabla u ‖^{2} + M^{' 2}), \end{matrix}$ we deduce from (3.27), (3.29) and the uniform Gronwall lemma that $\begin{array}{l} (3.30) & ‖ u (t) ‖ ⩽ c, t ⩾ t_{1} (⩾ t_{0}), \\ (3.31) & \int_{t}^{t + r} ‖ Δ u ‖^{2} d s ⩽ c_{r}, t ⩾ t_{1}, \end{array}$ which yields the existence of a bounded absorbing set for the associated dynamical system on $L^{2} (Ω)$ .

Similarly, we write $\begin{matrix} (3.32) & \frac{d}{d t} (α ‖ \overline{u} ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + 2 \int_{Ω} F (u) d x) + {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} ⩽ c \int_{Ω} u^{4} d x + c^{'}, \end{matrix}$ and then deduce from (3.32) and the uniform Gronwall lemma that $\begin{matrix} (3.33) & {‖ u (t) ‖}_{H^{1} (Ω)} ⩽ c, t ⩾ t_{2} (⩾ t_{1}), \end{matrix}$ which yields the existence of a bounded absorbing set for the associated dynamical system in $H^{1} (Ω)$ .

We note that all estimates derived from these differential inequalities are justified within the above Galerkin scheme, passing to the (weak lower) limit.

We finally multiply (2.1) by $Δ^{2} u$ to obtain $\begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ Δ u ‖^{2} + α ‖ Δ u ‖^{2} + {‖ Δ^{2} u ‖}^{2} - ((Δ f (u), Δ^{2} u)) + ((h (x) g (u), Δ^{2} u)) = 0, \end{matrix}$ employing the interpolation inequality and Young’s inequality, we have (see [15]) $\begin{array}{l} | ((Δ f (u), Δ^{2} u)) | & ⩽ ‖ Δ f (u) ‖ ‖ Δ^{2} u ‖ \\ ⩽ c (1 + ‖ u ‖_{H^{1} (Ω)}^{\frac{7}{3}}) {‖ Δ^{2} u ‖}^{\frac{5}{3}} \\ ⩽ \frac{1}{4} {‖ Δ^{2} u ‖}^{2} + c (1 + ‖ u ‖_{H^{1} (Ω)}^{14}) . \end{array}$ We further have $\begin{matrix} | ((h (x) g (u), Δ^{2} u)) | ⩽ c ‖ Δ^{2} u ‖ ⩽ \frac{1}{4} {‖ Δ^{2} u ‖}^{2} + c, \end{matrix}$ hence, owing to (3.21), we have $\begin{matrix} \frac{d}{d t} (‖ Δ u ‖^{2} + {⟨ u ⟩}^{2}) + α (‖ Δ u ‖^{2} + {⟨ u ⟩}^{2}) + {‖ Δ^{2} u ‖}^{2} ⩽ c (1 + ‖ u ‖_{H^{1} (Ω)}^{14} + {⟨ u ⟩}^{2}), \end{matrix}$ and, owing to (3.20) and (3.33), $\exists t_{1}$ such that, $\forall t ⩾ t_{1}$ , we have $\begin{matrix} (3.34) & \frac{d}{d t} (‖ Δ u ‖^{2} + {⟨ u ⟩}^{2}) + α (‖ Δ u ‖^{2} + {⟨ u ⟩}^{2}) + {‖ Δ^{2} u ‖}^{2} ⩽ c, \end{matrix}$

We deduce from (3.34) and the uniform Gronwall lemma that $\begin{array}{l} (3.35) & {‖ u (t) ‖}_{H^{2} (Ω)} ⩽ c, t ⩾ t_{1}, \\ (3.36) & \int_{t}^{t + r} {‖ Δ^{2} u ‖}^{2} d s ⩽ c_{r}, t ⩾ t_{1}, \end{array}$ which yields the existence of a bounded absorbing set for the associated dynamical system in $H^{2} (Ω)$ .

We then obtain the following theorem:

Theorem 3.3.
We assume that $u_{0} \in H^{2} (Ω)$ , with $\frac{\partial u_{0}}{\partial ν} = 0$ on Γ. Then, the solution u given in Theorem 3.1 satisfies $\begin{matrix} u \in C ([0, T]; H^{2} (Ω)) \cap L^{2} (0, T; H^{4} (Ω)) \end{matrix}$ and $\begin{matrix} \frac{\partial u}{\partial t} \in L^{2} (0, T; L^{2} (Ω)), \forall T > 0 . \end{matrix}$
Remark 3.4.
(i) Here we can more generally consider any cubic polynomial with positive leading coefficient or even any regular function with such a cubic growth at infinity, but with restrictions on the degree in two or three space dimensions. In two space dimensions, noting that the embedding $H^{\frac{2}{3}} (Ω) \subset L^{6} (Ω)$ is continuous and employing the interpolation inequality $‖ v ‖_{H^{\frac{2}{3}} (Ω)} ⩽ c ‖ v ‖^{\frac{2}{3}} ‖ v ‖_{H^{2} (Ω)}^{\frac{1}{3}}$ , $\forall v \in H^{\frac{2}{3}} (Ω)$ , we can see that $\overline{f (u)} \in L^{2} (0, T; H)$ , which yields the strong continuity in H; In three space dimensions, noting that the embedding $H^{1} (Ω) \subset L^{6} (Ω)$ and employing the interpolation inequality $‖ v ‖_{H^{1} (Ω)} ⩽ c ‖ v ‖^{\frac{1}{2}} ‖ v ‖_{H^{2} (Ω)}^{\frac{1}{2}}$ , $\forall v \in H^{1} (Ω)$ , it follows that $\overline{f (u)} \in L^{\frac{4}{3}} (0, T; H)$ and $\frac{d \overline{u}}{d t} \in L^{\frac{4}{3}} (0, T; D (A^{- 1}))$ , which yields the weak continuity of $\overline{u}$ in H.

(ii) In particular, we can consider polynomials of degree $2 p + 1$ , $p \in N$ , with positive leading coefficient in three space dimensions. We further note the following assumptions on the nonlinear term f: $\begin{array}{l} f is of class C^{2}, f (0) = 0, \\ f^{'} (s) ⩾ - c_{0}, c_{0} ⩾ 0, s \in R, \\ | f (s) | ⩽ ε F (s) + c_{ε}, \forall ε > 0, s \in R, \\ f (s) (s - κ) ⩾ c_{1} F (s) - c_{2}, F (s) ⩾ - c_{3}, κ = ⟨ s ⟩, c_{1} > 0, c_{2}, c_{3} ⩾ 0, s \in R, \end{array}$ where $F (s) = \int_{0}^{s} f (ξ) d ξ$ . We can differentiate (3.5) with respect to time and obtain the $H^{2}$ -estimate of u, which is globally in time, by steps. And prove the uniqueness, as well as the continuous dependence with respect to the initial data for V-norm, without any restriction on p in three space dimensions. The proof is essentially the same as that for the classical Cahn–Hilliard equation, more details can be found in [15].

3.3. Asymptotic behavior

3.3.1. Basic theories

We first give the preliminary materials and prove the existence of the global attractor.

Proposition 3.5.
We have the continuous (with respect to the $H^{- 1}$ -norm) semigroup $S (t)$ defined as $\begin{matrix} S (t) : L^{2} (Ω) \to L^{2} (Ω), u_{0} \mapsto u (t), t ⩾ 0 . \end{matrix}$

The proposition directly follows from Section 3.1. Furthermore, it follows from Section 3.2 that $S (t)$ possesses a bounded absorbing set $B_{0}^{'}$ which is compact in $L^{2} (Ω)$ and bounded in $H^{2} (Ω)$ , i.e., $\begin{matrix} S (t) : L^{2} (Ω) \to H^{2} (Ω), t > 0 . \end{matrix}$ Setting $\begin{matrix} Φ_{M^{'}} = {v \in L^{2} (Ω), | ⟨ v ⟩ | ⩽ M^{'}}, M^{'} ⩾ 0, \end{matrix}$ it follows from the uniform estimates obtained in Section 3 that we have the dissipative semigroup (still denoted by $S (t)$ ) acting on the phase space $Φ_{M^{'}}$ , $\begin{matrix} S (t) : Φ_{M^{'}} \to Φ_{M^{'}}, t ⩾ 0 . \end{matrix}$ Finally, it is deduced from standard results that we have the following theorem.
Theorem 3.6.
The semigoup $S (t)$ possesses a global attractor $A$ such that $A$ is compact in $L^{2} (Ω)$ and bounded in $H^{2} (Ω)$ .
Remark 3.7.
It is easy to see that we can assume, without loss of generality, that $B_{0}^{'}$ is positively invariant by $S (t)$ , i.e. $S (t) B_{0}^{'} \subset B_{0}^{'}$ , $\forall t ⩾ 0$ .

We next give the definition of the fractal dimension, and prove that the dimension of the global attractor is finite in the next section. Definition 3.8.
Let $X \subset E$ be a (relatively) compact set. For $ε > 0$ , let $N_{ε} (X)$ (if it is necessary to make the topology precise, we will also use the notation $N_{ε} (X, E)$ ) be the minimal number of balls of radius ε which are necessary to cover X. Then, the fractal dimension of X is the quantity (which belongs to $[0, + \infty]$ ) $\begin{matrix} {dim}_{F} X = \underset{ε \to 0^{+}}{lim sup} \frac{{log}_{2} N_{ε} (X)}{{log}_{2} \frac{1}{ε}} (= \underset{ε \to 0^{+}}{lim sup} \frac{ln N_{ε} (X)}{ln \frac{1}{ε}}) . \end{matrix}$ Furthermore, the quantity $H_{ε} (X) (= H_{ε} (X, E)) = {log}_{2} N_{ε} (X)$ is called the Kolmogorov ε-entropy of X.
Theorem 3.9.
Let X be a compact subset of E. We assume that there exist a Banach space $E_{1}$ , with norm $‖ \cdot ‖_{E_{1}}$ , such that $E_{1}$ is compactly embedded in E, and a mapping $L : X \to X$ such that $L (X) = X$ and L satisfies the following smoothing property on the difference of two solutions: $\begin{matrix} ‖ L x_{1} - L x_{2} ‖_{E_{1}} ⩽ η ‖ x_{1} - x_{2} ‖_{E}, \forall x_{1}, x_{2} \in X, c > 0 . \end{matrix}$ Then, the fractal dimension of X is finite and satisfies $\begin{matrix} {dim}_{F} X ⩽ H_{\frac{1}{4 η}} (B_{E_{1}} (0, 1), E), \end{matrix}$ where $B_{E_{1}} (0, 1)$ is the unit ball in $E_{1}$ (note that it is relatively compact in E, so that its $\frac{1}{4 η}$ -entropy is finite).
Proof.
We recommend [15,21] to interested readers for more details about the proof. □

3.3.2. Existence of exponential attractors

We again consider the initial and boundary value problem (3.15)–(3.18), and it is sufficient here to take initial data belonging to the bounded absorbing set $B_{0}^{'}$ defined in the previous section. We first derive a smoothing property on the difference of the two solutions, which is vital to prove the existence of exponential attractors.

Multiplying (3.15) by $t \overline{u}$ to have $\begin{matrix} (3.37) & \frac{d}{d t} (t ‖ \overline{u} ‖^{2}) + α t ‖ \overline{u} ‖^{2} + t ‖ Δ u ‖^{2} ⩽ ‖ \overline{u} ‖^{2} + c t ‖ \nabla u ‖^{2}, \end{matrix}$ noting that $\begin{matrix} | ((\overline{h (x) (g (u_{1}) - g (u_{2}))}, t \overline{u})) | ⩽ c t ‖ \overline{u} ‖^{2} ⩽ c t ‖ \nabla u ‖^{2} \end{matrix}$ and $\begin{matrix} ((\nabla f (u_{1}) - \nabla f (u_{2}), \nabla u)) = ((f^{'} (u_{1}) \nabla u, \nabla u)) + (((f^{'} (u_{1}) - f^{'} (u_{2})) \nabla u_{2}, \nabla u)), \end{matrix}$ where $\begin{matrix} ((f^{'} (u_{1}) \nabla u, \nabla u)) ⩾ - ‖ \nabla u ‖^{2} \end{matrix}$ and $\begin{array}{l} | (((f^{'} (u_{1}) - f^{'} (u_{2})) \nabla u_{2}, \nabla u)) | = 3 \int_{Ω} (u_{1}^{2} - u_{2}^{2}) \nabla u_{2} \cdot \nabla u d x ⩽ c ‖ \nabla u ‖^{2} \end{array}$ by emplying (2.4), the generalised Hölder inequality, interpolation inequality and Theorem 3.1.

Integrating (3.22) between 0 and t, it follows from (3.23) that $\begin{matrix} \int_{0}^{t} ‖ \nabla u ‖^{2} d s ⩽ c e^{c^{'} t} ‖ u_{0, 1} - u_{0, 2} ‖_{- 1}^{2} . \end{matrix}$ Recalling again $\begin{matrix} ‖ u ‖_{H^{1} (Ω)}^{2} ⩽ c (‖ \nabla u ‖^{2} + {⟨ u ⟩}^{2}) \end{matrix}$ so that $\begin{matrix} \int_{0}^{t} ‖ u ‖_{H^{1} (Ω)}^{2} d s ⩽ c e^{c^{'} t} ‖ u_{0, 1} - u_{0, 2} ‖_{- 1}^{2} . \end{matrix}$

Owing to (3.21), we deduce from the above that $\begin{matrix} (3.38) & \frac{d}{d t} (t ‖ \overline{u} ‖^{2} + t {⟨ u ⟩}^{2}) ⩽ ‖ \overline{u} ‖^{2} + {⟨ u ⟩}^{2} + c t (‖ \nabla u ‖^{2} + {⟨ u ⟩}^{2}), \end{matrix}$ Integrating (3.38) between 0 and t, we obtain $\begin{array}{l} (3.39) & {‖ u (t) ‖}^{2} ⩽ c \frac{1 + t}{t} \int_{0}^{t} ‖ u ‖_{H^{1} (Ω)}^{2} d s ⩽ c \frac{1 + t}{t} e^{c^{'} t} ‖ u_{0, 1} - u_{0, 2} ‖_{- 1}^{2}, \end{array}$ where the constants c and $c^{'}$ are depend on $M^{'}$ .

We finally derive a Hölder (both with respect to space and time) estimate. Actually, the Hölder continuity with respect to x follows from (3.23). To prove the Hölder continuity with respect to t, we have $\begin{array}{l} {‖ u (t_{1}) - u (t_{2}) ‖}_{- 1} = {‖ \int_{t_{1}}^{t_{2}} \frac{\partial u}{\partial t} d τ ‖}_{- 1} ⩽ | t_{1} - t_{2} |^{\frac{1}{2}} {| \int_{t_{1}}^{t_{2}} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} d τ |}^{\frac{1}{2}}, \end{array}$ we note that, owing to (3.32), $\begin{matrix} | \int_{t_{1}}^{t_{2}} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} d τ | ⩽ c, \end{matrix}$ where the constant c depends on $B_{0}^{'}$ and T such that $t_{1}, t_{2} \in [0, T]$ and $\begin{matrix} (3.40) & {‖ u (t_{1}) - u (t_{2}) ‖}_{- 1} ⩽ c | t_{1} - t_{2} |^{\frac{1}{2}} . \end{matrix}$

We finally deduce from (3.23), (3.39) and (3.40) the following results.

Theorem 3.10.
The semigroup $S (t)$ possesses an exponential attractor $M \subset B_{0}^{'}$ , i.e.
$M$ is compact in $H^{- 1} (Ω)$ ;

$M$ is positively invariant, $S (t) M \subset M$ , $\forall t ⩾ 0$ ;

$M$ has finite fractal dimension in $H^{- 1} (Ω)$ ;

$M$ attracts exponentially fast the bounded subsets of $L^{2} (Ω)$ , $\forall B \subset L^{2} (Ω)$ bounded, $\begin{matrix} {dist}_{H^{- 1} (Ω)} (S (t) B, M) ⩽ Q (‖ B ‖_{L^{2} (Ω)}) e^{- c t}, c > 0, t ⩾ 0, \end{matrix}$
where the constant c is independent of B and ${dist}_{H^{- 1} (Ω)}$ denotes the Hausdorff semidistance between sets defined by $\begin{matrix} {dist}_{H^{- 1} (Ω)} (A, B) = sup_{a \in A} inf_{b \in B} ‖ a - b ‖_{H^{- 1} (Ω)} . \end{matrix}$
Corollary 3.11.
The semigroup $S (t)$ possesses the finite-dimensional global attractor $A \subset B_{0}^{'}$ .
Proof.
According to Theorem 3.9 and (3.39), the conclusion follows immediately. □
Remark 3.12.
It follows from Theorem 3.9 and (3.39) that η is monotonically increasing with respect to $M^{'}$ . Owing to (3.9), one can easily deduce that $η \to + \infty$ as $α \to 0$ , which means $\frac{1}{4 η} \to 0$ and it is then clear that the estimate on the dimension goes to $+ \infty$ .
Remark 3.13.
A. Miranville has proved in [14,15] that the dynamics of the Cahn–Hilliard–Oono equation is close to that of the original Cahn–Hilliard equation when α is small. In another word, we can construct exponential attractors for the Cahn–Hilliard–Oono equation converging to the exponential attractors for Cahn–Hilliard equation as $α \to 0^{+}$ . However, the conclusion is not valid in our case, we can not obtain the robustness here, since the solutions to the Cahn–Hilliard type model for image segmentation would be unbounded as time goes to infinity, and we can not obtain a global in time estimate on u.

4. The second model with logarithmic term

In this section, we study the Cahn–Hilliard–Oono equation with logarithmic terms as following $\begin{array}{l} (4.1) & \frac{\partial u}{\partial t} + α u + Δ^{2} u - Δ f (u) + h (x) g (u) = 0, α > 0, \\ (4.2) & \frac{\partial u}{\partial ν} = \frac{\partial Δ u}{\partial ν} = 0 on Γ, \\ (4.3) & u |_{t = 0} = u_{0}, \end{array}$ where $\begin{array}{l} f (s) = - c_{0} (s - \frac{1}{2}) + θ ln \frac{s}{1 - s}, s \in (0, 1), 0 < θ < \frac{c_{0}}{4}, \end{array}$ we can note that f is of class $C^{\infty}$ , which satisfies $\begin{matrix} (4.4) & f^{'} ⩾ - c_{0} \end{matrix}$ and $\begin{matrix} (4.5) & f (s) (s - m) ⩾ c_{m} (| f (s) | + F (s)) - c_{m}^{'}, s \in R, m \in (0, 1), c_{m} > 0, c_{m}^{'} ⩾ 0, \end{matrix}$ where $F (s) = \int_{\frac{1}{2}}^{s} f (ξ) d ξ$ and $c_{m}$ and $c_{m}^{'}$ depend continuously on m. Furthermore, we can write $\begin{matrix} f (s) = f_{1} (s) - c_{0} s, f_{1}^{'} ⩾ 0 . \end{matrix}$ The function f also enjoys the above properties.

4.1. The well-posedness results

We again employ a Galerkin scheme. The crucial step, to prove the existence of a solution to the above problem, consists of deriving an a priori estimate independent of N on the approximated logarithmic term $f_{N} (u_{N})$ so that we can pass to the limit $N \to + \infty$ in the following approximated problems.

Theorem 4.1.
We assume that $u_{0}$ is given such that $u_{0} \in H^{1} (Ω)$ , $0 < u_{0} (x) < 1$ and $0 < ⟨ u_{0} ⟩ < 1$ , a.e. $x \in Ω$ . Then there exists $T_{0} = T_{0} (u_{0}) > 0$ , ( 4.1 )–( 4.3 ) possesses a weak solution u on $[0, T_{0}]$ such that $\begin{array}{l} u \in C ([0, T_{0}]; H^{1} {(Ω)}_{W}) \cap L^{\infty} (0, T_{0}; H^{1} (Ω)) \cap L^{2} (0, T_{0}; H^{2} (Ω)), \\ \frac{\partial u}{\partial t} \in L^{2} (0, T_{0}; H^{- 1} (Ω)) . \end{array}$ Furthermore, $0 < u (x, t) < 1$ , a.e $(x, t) \in Ω \times (0, T_{0})$ .
Proof.
We first define, for $N \in N$ , the approximated functions $f_{N} \in C^{1} (R)$ as $\begin{matrix} f_{N} (s) = \{\begin{matrix} f (- 1 + \frac{1}{N}) + f^{'} (- 1 + \frac{1}{N}) (s + 1 - \frac{1}{N}), & s < - 1 + \frac{1}{N}, \\ f (s), & | s | ⩽ 1 - \frac{1}{N}, \\ f (1 - \frac{1}{N}) + f^{'} (1 - \frac{1}{N}) (s - 1 + \frac{1}{N}), & s > 1 - \frac{1}{N} . \end{matrix} \end{matrix}$ We easily see that $f_{N}$ also satisfies $\begin{matrix} (4.6) & f_{N}^{'} ⩾ - c_{0}, \end{matrix}$ and, for $s \in R$ , $m \in (0, 1)$ and N large enough, $f_{N}$ also enjoys the following inequality $\begin{matrix} (4.7) & f_{N} (s) (s - m) ⩾ c_{m} (| f_{N} (s) | + F_{N} (s)) - c_{m}^{'}, c_{m} > 0, c_{m}^{'} ⩾ 0, \end{matrix}$ where $F_{N} (s) = \int_{\frac{1}{2}}^{s} f_{N} (ξ) d ξ$ and the constants are independent of N, we refer the reader to, e.g., [15] for the proof.

We then rewrite (4.1) in the following approximated form, for $N \in N$ : $\begin{array}{l} (4.8) & \frac{\partial u_{N}}{\partial t} + α u_{N} + Δ^{2} u_{N} - Δ f_{N} (u_{N}) + h (x) g (u_{N}) = 0, \\ (4.9) & \frac{\partial u_{N}}{\partial ν} = \frac{\partial Δ u_{N}}{\partial ν} = 0 on Γ, \\ (4.10) & u_{N} |_{t = 0} = u_{0} . \end{array}$

We first integrate (4.8) over Ω and integrate by parts to have $\begin{array}{l} (4.11) & ⟨ \frac{\partial u_{N}}{\partial t} ⟩ + α ⟨ u_{N} ⟩ + ⟨ h (x) g (u_{N}) ⟩ = 0, \end{array}$ which yields that $\begin{array}{l} (4.12) & ⟨ u_{N} ⟩ = ⟨ u_{0} ⟩ e^{- α t} + e^{- α t} \int_{0}^{t} e^{α s} ⟨ h (x) g (u_{N}) ⟩ d s, \end{array}$ so that $\begin{array}{l} (4.13) & | ⟨ u_{N} (t) ⟩ | ⩽ ⟨ u_{0} ⟩ e^{- α t} + \frac{‖ h ‖_{L^{\infty} (Ω)}}{α} (1 - e^{- α t}), \end{array}$ it then follows from the above that there exists $T_{0} = T_{0} (δ, u_{0}) > 0$ such that, for $t \in [0, T_{0}]$ , $\begin{array}{l} (4.14) & | ⟨ u_{N} (t) ⟩ | ⩽ 1 - δ, δ \in (0, \frac{1}{2}) . \end{array}$ Here, we assume that $2 δ < ⟨ u_{0} ⟩ ⩽ 1 - 2 δ$ . Noting that $T_{0}$ can be chosen independent of the approximation parameter, which is essential to pass to the limit.

Then we rewrite (4.8) in the following equivalent form: $\begin{array}{l} (4.15) & \frac{\partial {\overline{u}}_{N}}{\partial t} + α {\overline{u}}_{N} + Δ^{2} {\overline{u}}_{N} - Δ \overline{f_{N} (u_{N})} + \overline{h (x) g (u_{N})} = 0, \\ (4.16) & \frac{\partial {\overline{u}}_{N}}{\partial ν} = \frac{\partial Δ {\overline{u}}_{N}}{\partial ν} = 0 on Γ, \\ (4.17) & u_{N} |_{t = 0} = {\overline{u}}_{0} . \end{array}$

Let us multiply (4.15) by ${(- Δ)}^{- 1} {\overline{u}}_{N}$ to obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ {\overline{u}}_{N} ‖_{- 1}^{2} + α ‖ {\overline{u}}_{N} ‖_{- 1}^{2} + ‖ \nabla u_{N} ‖^{2} + ((\overline{f_{N} (u_{N})}, {\overline{u}}_{N})) + ((\overline{h (x) g (u_{N})}, {(- Δ)}^{- 1} {\overline{u}}_{N})) = 0 . \end{array}$ It follows from (4.7), where we assume that $s = u_{N}$ and $m = ⟨ u_{N} ⟩$ to have $\begin{array}{l} ((\overline{f_{N} (u_{N})}, {\overline{u}}_{N})) = ((f_{N} (u_{N}), {\overline{u}}_{N})) ⩾ c ({‖ f_{N} (u_{N}) ‖}_{L^{1} (Ω)} + \int_{Ω} F_{N} (u_{N}) d x) - c^{'}, c > 0, \end{array}$ and noting that $\begin{array}{l} | ((\overline{h (x) g (u_{N})}, {(- Δ)}^{- 1} {\overline{u}}_{N})) | ⩽ c ‖ {\overline{u}}_{N} ‖_{- 1} ⩽ \frac{α}{2} ‖ \nabla u_{N} ‖_{- 1}^{2} + c . \end{array}$ We deduce from the above that $\begin{array}{l} \frac{d}{d t} ‖ {\overline{u}}_{N} ‖_{- 1}^{2} + α ‖ {\overline{u}}_{N} ‖_{- 1}^{2} + 2 ‖ \nabla u_{N} ‖^{2} + c ({‖ f_{N} (u_{N}) ‖}_{L^{1} (Ω)} + \int_{Ω} F_{N} (u_{N}) d x) ⩽ c^{'}, c > 0 . \end{array}$

In view of (4.11), we have $\begin{array}{l} (4.18) & \frac{d {⟨ u_{N} ⟩}^{2}}{d t} + α {⟨ u_{N} ⟩}^{2} ⩽ c {⟨ u_{N} ⟩}^{2} \end{array}$ and $\begin{array}{l} \frac{d}{d t} (‖ {\overline{u}}_{N} ‖_{- 1}^{2} + {⟨ u_{N} ⟩}^{2}) + α (‖ {\overline{u}}_{N} ‖_{- 1}^{2} + {⟨ u_{N} ⟩}^{2}) + 2 ‖ \nabla u_{N} ‖^{2} \\ (4.19) & + c ({‖ f_{N} (u_{N}) ‖}_{L^{1} (Ω)} + \int_{Ω} F_{N} (u_{N}) d x) ⩽ + {⟨ u_{N} ⟩}^{2} + c^{'}, c > 0 . \end{array}$ It follows from the above and the uniform Gronwall lemma that $u_{N}$ is bounded, independent of N, in $L^{\infty} (0, T; H^{- 1} (Ω)) \cap L^{2} (0, T; H^{1} (Ω))$ .

Let us next multiply (4.15) by ${(- Δ)}^{- 1} \frac{\partial {\overline{u}}_{N}}{\partial t}$ to have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ \nabla u_{N} ‖^{2} + α ‖ {\overline{u}}_{N} ‖_{- 1}^{2}) + {‖ \frac{\partial {\overline{u}}_{N}}{\partial t} ‖}_{- 1}^{2} + ((\overline{f_{N} (u_{N})}, \frac{\partial {\overline{u}}_{N}}{\partial t})) \\ + ((\overline{h (x) g (u_{N})}, {(- Δ)}^{- 1} \frac{\partial {\overline{u}}_{N}}{\partial t})) = 0 . \end{array}$ Noting that $\begin{array}{l} ((\overline{f_{N} (u_{N})}, \frac{\partial {\overline{u}}_{N}}{\partial t})) & = ((f_{N} (u_{N}), \frac{\partial {\overline{u}}_{N}}{\partial t})) \\ = \frac{d}{d t} \int_{Ω} F_{N} (u_{N}) d x - ((f_{N} (u_{N}), \frac{d ⟨ u_{N} ⟩}{d t})) \\ ⩾ \frac{d}{d t} \int_{Ω} F_{N} (u_{N}) d x - c {‖ f_{N} (u_{N}) ‖}_{L^{1} (Ω)} \end{array}$ and $\begin{array}{l} ((\overline{h (x) g (u_{N})}, {(- Δ)}^{- 1} \frac{\partial {\overline{u}}_{N}}{\partial t})) ⩽ c {‖ \frac{\partial {\overline{u}}_{N}}{\partial t} ‖}_{- 1} ⩽ \frac{1}{2} {‖ \frac{\partial {\overline{u}}_{N}}{\partial t} ‖}_{- 1}^{2} + c . \end{array}$ We obtain that $\begin{array}{l} \frac{d}{d t} (‖ \nabla u_{N} ‖^{2} + α ‖ {\overline{u}}_{N} ‖_{- 1}^{2} + 2 \int_{Ω} F_{N} (u_{N}) d x) + {‖ \frac{\partial {\overline{u}}_{N}}{\partial t} ‖}_{- 1}^{2} \\ (4.20) & ⩽ c {‖ f_{N} (u_{N}) ‖}_{L^{1} (Ω)} + c^{'} . \end{array}$ It thus follows from (4.18)–(4.20) and the uniform Gronwall lemma that $u_{N}$ and $\frac{\partial u}{\partial t}$ are bounded, independent of N, in $L^{\infty} (0, T; H^{1} (Ω))$ and $L^{2} (0, T; H^{- 1} (Ω))$ , respectively.

We finally multiply (4.15) by ${\overline{u}}_{N}$ to have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} ‖ {\overline{u}}_{N} ‖^{2} + α ‖ {\overline{u}}_{N} ‖^{2} + ‖ Δ u_{N} ‖^{2} + ((f^{'} (u) \nabla u_{N}, \nabla u_{N})) + ((\overline{h (x) g (u_{N})}, {\overline{u}}_{N})) = 0, \end{array}$ it follows from (4.6) that $\begin{array}{l} ((f_{N}^{'} (u_{N}) \nabla u_{N}, \nabla u_{N})) ⩾ - c_{0} ‖ \nabla u_{N} ‖^{2}, \end{array}$ and noting that $\begin{array}{l} ‖ ((h (x) g (u_{N}), {\overline{u}}_{N})) ‖ ⩽ c ‖ {\overline{u}}_{N} ‖ ⩽ c^{'} ‖ \nabla u_{N} ‖, \end{array}$ and considering the inequality $\begin{array}{l} ‖ \nabla u_{N} ‖^{2} ⩽ \frac{1}{2} ‖ Δ u_{N} ‖^{2} + c ‖ {\overline{u}}_{N} ‖^{2} + c^{'}, \end{array}$ which follows from standard elliptic regularity results and a proper interpolation inequality. Combining the inequalities above to deduce $\begin{array}{l} (4.21) & \frac{d}{d t} ‖ {\overline{u}}_{N} ‖^{2} + ‖ Δ u_{N} ‖^{2} ⩽ c ‖ {\overline{u}}_{N} ‖^{2} + c^{'} . \end{array}$ It follows from (4.18), (4.21) and the uniform Gronwall lemma that $u_{N}$ is bounded, independent of N, in $L^{\infty} (0, T; L^{2} (Ω)) \cap L^{2} (0, T; H^{2} (Ω))$ .

Summing (4.19), (4.20), (4.21) multiplied by $δ > 0$ small enough and (4.18), we finally obtain the energy inequality as following $\begin{array}{l} \frac{d E_{N}}{d t} + c (E_{N} + ‖ {\overline{u}}_{N} ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial {\overline{u}}_{N}}{\partial t} ‖}_{- 1}^{2} + {‖ f_{N} (u_{N}) ‖}_{L^{1} (Ω)}) ⩽ c^{'}, c^{'} > 0, \end{array}$ where $\begin{array}{l} E_{N} = {⟨ u_{N} ⟩}^{2} + ‖ {\overline{u}}_{N} ‖_{- 1}^{2} + ‖ \nabla u_{N} ‖^{2} + 2 \int_{Ω} F_{N} (u_{N}) d x + δ ‖ {\overline{u}}_{N} ‖^{2} \end{array}$ satisfies $\begin{array}{l} E_{N} ⩾ c ‖ u_{N} ‖_{H^{1} (Ω)}^{2} - c^{'}, c > 0 . \end{array}$ We also note that the dissipative estimate follows immediately by employing the Gronwall lemma $\begin{array}{l} (4.22) & E_{N} (t) ⩽ e^{- c t} E_{N} (0) + c^{'}, c > 0, t ⩾ 0, \end{array}$ which yields $\begin{array}{l} (4.23) & {‖ u_{N} (t) ‖}_{H^{1} (Ω)} ⩽ c e^{- c^{'} t} (E_{N} (0) + 1), c^{'} > 0, t ⩾ 0, \end{array}$

Note indeed that (4.15) yields $\begin{array}{l} \overline{f_{N} (u_{N})} = Δ {\overline{u}}_{N} - {(- Δ)}^{- 1} α {\overline{u}}_{N} - {(- Δ)}^{- 1} \frac{\partial {\overline{u}}_{N}}{\partial t} - {(- Δ)}^{- 1} \overline{h (x) g (u_{N})}, \end{array}$ so that $\begin{array}{l} ‖ \overline{f_{N} (u_{N})} ‖ ⩽ c (‖ Δ {\overline{u}}_{N} ‖ + ‖ {\overline{u}}_{N} ‖_{- 1} + {‖ \frac{\partial {\overline{u}}_{N}}{\partial t} ‖}_{- 1} + 1) \end{array}$ and $\begin{array}{l} (4.24) & {‖ \overline{f_{N} (u_{N})} ‖}_{L^{2} (Ω \times (0, T_{0})} ⩽ c E_{N} (0) . \end{array}$ Next, taking $s = u_{N}$ and $⟨ u_{N} ⟩$ in (4.7), it follows from (4.14) that $\begin{array}{l} | ⟨ f_{N} (u_{N}) ⟩ | & ⩽ c ((f_{N} (u_{N}), {\overline{u}}_{N})) + c^{'} \\ = c ((\overline{f_{N} (u_{N})}, u_{N})) + c^{'} \\ (4.25) & ⩽ c ‖ \overline{f_{N} (u_{N})} ‖ ‖ u_{N} ‖ + c^{'}, \end{array}$ where the above constants depend on δ and $⟨ u_{0} ⟩$ . Therefore, $\begin{array}{l} {‖ f_{N} (u_{N}) ‖}_{L^{2} (Ω \times (0, T_{0})}^{2} & ⩽ c ({‖ \overline{f_{N} (u_{N})} ‖}_{L^{2} (Ω \times (0, T_{0})}^{2} + \int_{0}^{T_{0}} {⟨ f_{N} (u_{N}) ⟩}^{2} d t) \\ ⩽ c E_{N}^{2} (0) + c^{'} E_{N} (0) ‖ u_{N} ‖^{2} + c^{″} \\ ⩽ c E_{N}^{2} (0) + c^{'} \end{array}$ so that $\begin{array}{l} (4.26) & {‖ f_{N} (u_{N}) ‖}_{L^{2} (Ω \times (0, T_{0}))} ⩽ c (E (0) + 1) . \end{array}$

As mentioned above, (4.26) is the crucial estimate to pass to the limit in the nonlinear term. Since the above estimates are independent of N, the solution of the approximated problems converges to a limit function u in the sense $\begin{array}{l} u_{N} \overset{*}{⇀} u in L^{\infty} (0, T_{0}; H^{1} (Ω)), \\ u_{N} ⇀ u in L^{2} (0, T_{0}; H^{2} (Ω)), \\ \frac{\partial u_{N}}{\partial t} ⇀ \frac{\partial u}{\partial t} in L^{2} (0, T_{0}; H^{- 1} (Ω)), \\ f_{N} ⇀ f in L^{2} (Ω \times (0, T_{0})), \end{array}$ and $\begin{array}{l} u_{N} \overset{a . e .}{⟶} u in C ([0, T_{0}]; H_{W}^{1} (Ω)) \end{array}$ by using the Aubin–Lions compactness results and W.A. Strauss lemma. We thus finish the proof of the existence.

The uniqueness follows instantly. Let $u_{1}$ and $u_{2}$ be two solutions with initial data $u_{1, 0}$ and $u_{2, 0}$ , respectively, such that $⟨ u_{1, 0} ⟩ = ⟨ u_{2, 0} ⟩$ . Setting $u = u_{1} - u_{2}$ , $u_{0} = u_{1, 0} - u_{2, 0}$ , we then have $\begin{array}{l} (4.27) & \begin{matrix} {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t} + α {(- Δ)}^{- 1} \overline{u} - Δ u + \overline{f (u_{1}) - f (u_{2})} + {(- Δ)}^{- 1} \overline{h (x) (g (u_{1}) - g (u_{2}))} = 0, \end{matrix} \\ (4.28) & \frac{d ⟨ u ⟩}{d t} + α ⟨ u ⟩ + ⟨ h (x) (g (u_{1}) - g (u_{2}) ⟩ = 0, \\ (4.29) & \frac{\partial u}{\partial ν} = \frac{\partial Δ u}{\partial ν} = 0 on Γ, \\ (4.30) & u |_{t = 0} = u_{0} . \end{array}$ Multiplying (4.27) by $\overline{u}$ , we obtain $\begin{matrix} \frac{1}{2} \frac{d}{d t} ‖ \overline{u} ‖_{- 1}^{2} + α ‖ \overline{u} ‖_{- 1}^{2} + ‖ \nabla u ‖^{2} + ((f (u_{1}) - f (u_{2}), \overline{u})) + ((h (x) (g (u_{1}) - g (u_{2})), {(- Δ)}^{- 1} \overline{u})) = 0, \end{matrix}$ in view of (4.4), employing the interpolation inequality, which yields $\begin{matrix} ((f (u_{1}) - f (u_{2}), \overline{u})) ⩾ - c_{0} ‖ u ‖^{2} ⩾ - c_{0} ‖ u ‖_{- 1} ‖ \nabla u ‖, \end{matrix}$ and noting that $\begin{matrix} | h (x) (g (u_{1}) - g (u_{2})) | ⩽ c | u | \end{matrix}$ so that $\begin{array}{l} | ((h (x) (g (u_{1}) - g (u_{2})), {(- Δ)}^{- 1} \overline{u})) | ⩽ c ‖ u ‖ ‖ \overline{u} ‖_{- 1} ⩽ \frac{1}{4} ‖ \nabla u ‖^{2} + c (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) \end{array}$ by using interpolation inequality and Poincaré–Wirtinger inequality. It follows from (4.28) that $\begin{array}{l} (4.31) & \frac{d {⟨ u ⟩}^{2}}{d t} + 2 α {⟨ u ⟩}^{2} ⩽ c {⟨ u ⟩}^{2}, \end{array}$ we then deduce from above that $\begin{matrix} (4.32) & \frac{d}{d t} (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) + 2 α (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) + ‖ \nabla u ‖^{2} ⩽ c (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}), \end{matrix}$ employing the uniform Gronwall lemma, we obtain $\begin{array}{l} (4.33) & {‖ u_{1} (t) - u_{2} (t) ‖}_{- 1} ⩽ c e^{c^{'} t} ‖ u_{1, 0} - u_{2, 0} ‖_{- 1}, 0 ⩽ t ⩽ T, \end{array}$ which yields the uniqueness, as well as the continuous dependence with respect to the initial data in the $H^{- 1}$ topology. □
Remark 4.2.
(i) Note that if $‖ h ‖_{L^{\infty} (Ω)} ⩽ α$ , then it follows from (4.13) that $δ ⩽ ⟨ u_{N} (t) ⟩ ⩽ 1 - δ$ for all times, so that the solution is actually global in time.

(ii) When $‖ h ‖_{L^{\infty} (Ω)} = 0$ , it follows from (4.13) that $⟨ u ⟩ \in (0, 1)$ holds for all times, and the global in time existence of the solution for the Cahn–Hilliard–Oono equation can be obtained easily.

4.2. Regularity and separation from the pure states

We have the following results.

Proposition 4.3.
The solution u to ( 4.1 )–( 4.3 ) given in Theorem 4.1 satisfies $\begin{matrix} \frac{\partial u}{\partial t} \in L^{\infty} (r, + \infty; H^{- 1} (Ω)) \cap L^{2} (r, T; H^{1} (Ω)) \end{matrix}$ $\forall r < T$ , where $r > 0$ and $T > 0$ are given.
Proof.
We recall that we have the equation $\begin{array}{l} (4.34) & {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t} + α {(- Δ)}^{- 1} \overline{u} - Δ u + \overline{f (u)} + {(- Δ)}^{- 1} \overline{h (x) g (u)} = 0 . \end{array}$ Differentiating (4.34) with respect to time, we have $\begin{array}{l} (4.35) & {(- Δ)}^{- 1} \frac{\partial}{\partial t} \frac{\partial \overline{u}}{\partial t} + α {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t} - Δ \frac{\partial \overline{u}}{\partial t} + \overline{f^{'} (u) \frac{\partial u}{\partial t}} + {(- Δ)}^{- 1} \overline{h (x) g^{'} (u) \frac{\partial u}{\partial t}} = 0, \end{array}$ multiplying (4.35) by $\frac{\partial \overline{u}}{\partial t}$ , we obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} + α {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + ((f^{'} (u) \frac{\partial u}{\partial t}, \frac{\partial \overline{u}}{\partial t})) \\ + ((h (x) g^{'} (u) \frac{\partial u}{\partial t}, {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t})) = 0, \end{array}$ recalling the properties of h, g and (4.4), we write instead $\begin{array}{l} | ((h (x) g^{'} (u) \frac{\partial u}{\partial t}, {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t})) | ⩽ c ‖ \frac{\partial u}{\partial t} ‖ {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} \end{array}$ and $\begin{array}{l} ((f^{'} (u) \frac{\partial u}{\partial t}, \frac{\partial \overline{u}}{\partial t})) & = ((f^{'} (u) \frac{\partial u}{\partial t}, \frac{\partial u}{\partial t})) - \frac{d ⟨ u ⟩}{d t} ((f^{'} (u) \frac{\partial u}{\partial t}, 1)) \\ ⩾ - c_{0} {‖ \frac{\partial u}{\partial t} ‖}^{2} - \frac{d ⟨ u ⟩}{d t} \frac{d}{d t} \int_{Ω} F (u) d x \\ = - c_{0} {‖ \frac{\partial u}{\partial t} ‖}^{2} - \frac{d}{d t} (\frac{d ⟨ u ⟩}{d t} \int_{Ω} F (u) d x) + \frac{d^{2} ⟨ u ⟩}{d t^{2}} \int_{Ω} F (u) d x . \end{array}$ Setting $\begin{array}{l} Λ = \frac{1}{2} {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} - \frac{d ⟨ u ⟩}{d t} \int_{Ω} F (u) d x, \end{array}$ we deduce from the above that $\begin{array}{l} (4.36) & \frac{d Λ}{d t} + α {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} + \frac{1}{2} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} ⩽ c ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {| \frac{d ⟨ u ⟩}{d t} |}^{2}) - \frac{d^{2} ⟨ u ⟩}{d t^{2}} \int_{Ω} F (u) d x \end{array}$ by employing proper interpolation inequalities.

Recalling (4.12) and (4.22), we can see that Λ is bounded from below: $\begin{array}{l} Λ ⩾ \frac{1}{2} {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} - c . \end{array}$ Similarly, it is easy to prove that the last two terms on the right-hand side of (4.36) are bounded from above. We finally obtain the differential inequality of the form $\begin{array}{l} (4.37) & \frac{d Λ}{d t} + α {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} + \frac{1}{2} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} ⩽ c ({‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} + {⟨ u_{0} ⟩}^{2} + E {(0)}^{2} + 1), \end{array}$ where E is the equivalent of $E_{N}$ for u, the result follows from (4.18) and (4.37) by using the uniform Gronwall lemma. □
Proposition 4.4.
We assume that $2 ⩽ p < + \infty$ when $n = 2$ and $2 ⩽ p ⩽ 6$ when $n = 3$ . Then, the solution u further satisfies $\begin{matrix} {‖ f (u) ‖}_{L^{\infty} (r, t; L^{p} (Ω))} ⩽ c, {‖ u (t) ‖}_{W^{2, p} (Ω)} ⩽ c \end{matrix}$ $\forall t ⩾ r$ , $r > 0$ given, where the constant c depends on the $H^{1} (Ω)$ -norm of $u_{0}$ .
Proof.
We rewrite (4.34) as an elliptic equation: $\begin{array}{l} (4.38) & - Δ u + f_{1} (u) = - {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t} + ⟨ f (u) ⟩ + c_{0} u - α {(- Δ)}^{- 1} \overline{u} - {(- Δ)}^{- 1} \overline{h (x) g (u)} \equiv h_{u} . \end{array}$ Recall that $f (s) = f_{1} (s) - c_{0} s$ , $f_{1}^{'} ⩾ 0$ , (4.24) and (4.25) for u and f and note that, owing to Proposition 4.3, $\overline{f (u)} \in L^{\infty} (r, t; L^{2} (Ω))$ , so that $\begin{matrix} ⟨ f (u) ⟩ \in L^{\infty} (r, t) . \end{matrix}$ Then, it follows from the above that $\begin{array}{l} ‖ h_{u} ‖_{H^{1} (Ω)} & ⩽ {‖ {(- Δ)}^{- 1} \frac{\partial \overline{u}}{\partial t} ‖}_{H^{1} (Ω)} + | ⟨ f (u) ⟩ | + c_{0} ‖ u ‖_{H^{1} (Ω)} + α {‖ {(- Δ)}^{- 1} \overline{u} ‖}_{H^{1} (Ω)} \\ + | {(- Δ)}^{- 1} \overline{h (x) g (u)} | \\ ⩽ c ({‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} + | ⟨ f (u) ⟩ | + ‖ u ‖_{H^{1} (Ω)} + ‖ \overline{u} ‖_{- 1}) \\ ⩽ c, t ⩾ r, \end{array}$ so that $\begin{array}{l} (4.39) & ‖ h_{u} ‖_{L^{\infty} (r, t; H^{1} (Ω))} ⩽ c, t ⩾ r, \end{array}$ where the constant c depends on the $H^{1} (Ω)$ -norm of $u_{0}$ .

We then multiply (4.38) by $| f_{1} (u) |^{p - 2} f_{1} (u)$ to obtain $\begin{array}{l} - \int_{Ω} Δ u {| f_{1} (u) |}^{p - 2} f_{1} (u) d x + \int_{Ω} {| f_{1} (u) |}^{p} d x = \int_{Ω} h_{u} {| f_{1} (u) |}^{p - 2} f_{1} (u) d x . \end{array}$ We note that $\begin{array}{l} - \int_{Ω} Δ u {| f_{1} (u) |}^{p - 2} f_{1} (u) d x = (p - 1) \int_{Ω} {| f_{1} (u) |}^{p - 2} f_{1}^{'} (u) | \nabla u |^{2} d x ⩾ 0, \end{array}$ since $f_{1}^{'} ⩾ 0$ . Furthermore, we obtain, employing Young’s inequality and proper Sobolev embedding theorems, $\begin{array}{l} | \int_{Ω} h_{u} {| f_{1} (u) |}^{p - 2} f_{1} (u) d x | & ⩽ \frac{1}{2} {‖ f_{1} (u) ‖}_{L^{p} (Ω)}^{p} + ‖ h_{u} ‖_{L^{p} (Ω)}^{p} \\ ⩽ \frac{1}{2} {‖ f_{1} (u) ‖}_{L^{p} (Ω)}^{p} + ‖ h_{u} ‖_{H^{1} (Ω)}^{p} . \end{array}$ It thus follows that $\begin{array}{l} {‖ f_{1} (u) ‖}_{L^{p} (Ω)}^{p} ⩽ c ‖ h_{u} ‖_{H^{1} (Ω)}^{p}, \end{array}$ so that $\begin{array}{l} (4.40) & {‖ f (u) ‖}_{L^{p} (Ω)} ⩽ c (‖ h_{u} ‖_{H^{1} (Ω)} + ‖ u ‖_{H^{1} (Ω)}) . \end{array}$ We finally deduce from (4.39)–(4.40) (and (4.22) again) that $\begin{array}{l} (4.41) & {‖ f (u) ‖}_{L^{\infty} (r, t; L^{p} (Ω)}) ⩽ c, t ⩾ r . \end{array}$ Having this, it follows from standard elliptic regularity results applied to (4.38) and (4.39) that $\begin{array}{l} ‖ u ‖_{L^{\infty} (r, t; W^{2, p} (Ω))} ⩽ c, t ⩾ r, \end{array}$ which finishes the proof, recalling that $u \in C ([0, T]; H^{1} (Ω))$ , $\forall T > 0$ . □

As a consequence of (4.41) we have the following result.
Theorem 4.5.
We assume that $n = 1$ . Then, there exists $δ \in (0, 1)$ depending on the $H^{1} (Ω)$ -norm of $u_{0}$ such that $\begin{matrix} {‖ u (t) ‖}_{L^{\infty} (Ω)} ⩽ 1 - δ, t ⩾ r, \end{matrix}$ where $r > 0$ is given.
Proof.
We can pass to the limit $p \to + \infty$ in (4.41) (see [1,2]) and conclude the proof, owing to the continuity of u with respect to time. □
Remark 4.6.
The separation property from the pure states given in Theorem 4.1 means, roughly speaking, that we never completely reach the pure states during the phase separation process: there always remains at least some trace of the other component. The above strict separation property says that not only can we never completely reach the pure states, but also there remains some given quantity of the other component.

Proving the strict separation property in two space dimensions is more involved and is based on the following result.
Proposition 4.7.
We assume that $n = 2$ . Then, the following holds for every $t ⩾ r$ , $r > 0$ given, and for every $p \in N$ : $\begin{matrix} {‖ f^{'} (u) ‖}_{L^{p} (Ω \times (r, t))} ⩽ c, \end{matrix}$ where the constant c depends on p.
Proof.
We also need to deal with the elliptic equation (4.38), the tricks we use here are similar to those in Proposition 4.4, we recommend to readers [15] for more details. □

Then we can prove additional regularity on the time derivative of u.
Proposition 4.8.
We assume that $n = 2$ . Then, the weak solution u to ( 4.1 )–( 4.3 ) further satisfies $\begin{matrix} \frac{\partial u}{\partial t} \in L^{\infty} (r, + \infty; L^{2} (Ω)) \cap L^{2} (r, T; H^{2} (Ω)) \end{matrix}$ $\forall r < T$ , where $r > 0$ and $T > 0$ are given.
Proof.
We multiply (4.35) by $- Δ \frac{\partial \overline{u}}{\partial t}$ to obtain $\begin{array}{l} \frac{1}{2} \frac{d}{d t} {‖ \frac{\partial \overline{u}}{\partial t} ‖}^{2} + α {‖ \frac{\partial \overline{u}}{\partial t} ‖}^{2} + {‖ Δ \frac{\partial \overline{u}}{\partial t} ‖}^{2} + ((f^{'} (u) \frac{\partial u}{\partial t}, Δ \frac{\partial \overline{u}}{\partial t})) \\ + (({(- Δ)}^{- 1} \overline{h (x) g^{'} (u) \frac{\partial u}{\partial t}}, - Δ \frac{\partial \overline{u}}{\partial t})) = 0, \end{array}$ in view of the properties of h and g again, we have $\begin{array}{l} (({(- Δ)}^{- 1} \overline{h (x) g^{'} (u) \frac{\partial u}{\partial t}}, - Δ \frac{\partial \overline{u}}{\partial t})) ⩽ c {‖ \frac{\partial \overline{u}}{\partial t} ‖}^{2} \end{array}$ and $\begin{array}{l} | ((f^{'} (u) \frac{\partial u}{\partial t}, Δ \frac{\partial \overline{u}}{\partial t})) | & ⩽ {‖ f^{'} (u) ‖}_{L^{4} (Ω)} {‖ \frac{\partial u}{\partial t} ‖}_{L^{4} (Ω)} ‖ Δ \frac{\partial \overline{u}}{\partial t} ‖ \\ ⩽ c {‖ f^{'} (u) ‖}_{L^{4} (Ω)} {‖ \frac{\partial u}{\partial t} ‖}^{\frac{1}{2}} {‖ Δ \frac{\partial \overline{u}}{\partial t} ‖}^{\frac{3}{2}} \\ ⩽ \frac{1}{2} {‖ Δ \frac{\partial \overline{u}}{\partial t} ‖}^{2} + c {‖ f^{'} (u) ‖}_{L^{4} (Ω)}^{4} {‖ \frac{\partial u}{\partial t} ‖}^{2}, \end{array}$ by employing the Hölder, Ladyzhenskaya, and Young inequalities. In view of (4.18), it is deduced from the above that $\begin{array}{l} \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ Δ \frac{\partial u}{\partial t} ‖}^{2} ⩽ (c {‖ f^{'} (u) ‖}_{L^{4} (Ω)}^{4} + c^{'}) {‖ \frac{\partial u}{\partial t} ‖}^{2}, \end{array}$ we can conclude by applying the uniform Gronwall lemma, with (4.36) and Proposition 4.7 (for $p = 4$ ). □
Remark 4.9.
This regularity also holds in one space dimension, owing to the strict separation property given in Theorem 4.5.

We then obtain the following result. Theorem 4.10.
We assume that $n = 2$ . Then, there exists $δ \in (0, 1)$ depending on the $H^{1} (Ω)$ -norm of $u_{0}$ such that $\begin{matrix} {‖ u (t) ‖}_{L^{\infty} (Ω)} ⩽ 1 - δ, t ⩾ r, \end{matrix}$ where $r > 0$ is given.
Proof.
We note that, owing to the regularity given in Proposition 4.8, the right-hand side $h_{u}$ in (4.38) satisfies $\begin{array}{l} ‖ h_{u} ‖_{L^{\infty} (Ω \times (r, t))} ⩽ c, t ⩾ r . \end{array}$ Having this, we can proceed as in the proof of Proposition 4.8 □
Remark 4.11.
(i) The strict separation property is still open in the three space dimension.

(ii) Having the strict separation property also allows us to prove the finite fractal dimensionality of the global attractor.

4.3. Asymptotic behavior

Based on the dissipative estimations in Section 4.1, we set $\begin{matrix} Φ = {v \in H^{1} (Ω) \cap L^{\infty} (Ω), 0 < v (x) < 1 a.e., | ⟨ v ⟩ | < 1}, \end{matrix}$ we can define the continuous (for the $H^{- 1} (Ω)$ -norm) semigroup $\begin{matrix} S_{α} (t) : Φ \to Φ, u_{0}, t ⩾ 0 . \end{matrix}$ It follows from (4.23) that $S_{α} (t)$ is dissipative in Φ, i.e., it possesses a bounded absorbing set $B_{1} \subset Φ$ . We then obtain the following result.

Theorem 4.12.
The semigoup $S_{α} (t)$ possesses a global attractor $A_{α}$ such that $A_{α}$ is compact in Φ.

Furthermore, we have the existence of a uniform (with respect to α) absorbing set $B_{1}^{'} \subset Φ \cap H^{2} (Ω)$ , i.e., $\forall B \subset Φ$ bounded, $\exists t_{0} = t_{0} (B)$ independent of α such that $\begin{matrix} S_{α} (t) B \subset B_{1}^{'}, t ⩾ t_{0}, \end{matrix}$ it thus sufficient to construct the exponential attractors $M_{α}$ . Theorem 4.13.
The semigroup $S_{α} (t)$ possesses an exponential attractor $M_{α}$ on $B_{1}^{'}$ .
Proof.
We again consider the initial and boundary value problem (4.27)–(4.30), and derive a smoothing property on the difference of the two solutions, which is the crucial step to prove the existence of exponential attractors.

We next multiply (4.27) by $t \frac{\partial \overline{u}}{\partial t}$ to have $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (α t ‖ \overline{u} ‖_{- 1}^{2} + t ‖ \nabla u ‖^{2}) + t {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} + t ((f (u_{1}) - f (u_{2}), \frac{\partial \overline{u}}{\partial t})) \\ + t (({(- Δ)}^{- 1} \overline{h (x) (g (u_{1}) - g (u_{2}))}, \frac{\partial \overline{u}}{\partial t})) = \frac{α}{2} ‖ \overline{u} ‖_{- 1}^{2} + \frac{1}{2} ‖ \nabla u ‖^{2} . \end{array}$ We note that $\begin{array}{l} ((f (u_{1}) - f (u_{2}), \frac{\partial \overline{u}}{\partial t})) & ⩽ c ‖ \nabla (f (u_{1}) - f (u_{2})) ‖ {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} \\ ⩽ c ‖ \nabla (\int_{0}^{1} f^{'} (u_{1} + s (u_{2} - u_{1})) d s u) ‖ {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} \\ ⩽ c ‖ \int_{0}^{1} f^{'} (u_{1} + s (u_{2} - u_{1})) d s \nabla u ‖ {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} \\ + ‖ u \int_{0}^{1} f^{″} (u_{1} + s (u_{2} - u_{1})) (\nabla u_{1} + s \nabla (u_{2} - u_{1})) d s ‖ {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} \\ ⩽ c (‖ \nabla u ‖ + ‖ | u | | \nabla u_{1} | ‖ + ‖ | u | | \nabla u_{2} | ‖) {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} \\ ⩽ c ‖ u ‖_{H^{1} (Ω)}^{2} + \frac{1}{4} {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} \end{array}$ and $\begin{array}{l} | (({(- Δ)}^{- 1} \overline{h (x) (g (u_{1}) - g (u_{2}))}, \frac{\partial \overline{u}}{\partial t})) | ⩽ c ‖ u ‖ {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1} ⩽ c ‖ \nabla u ‖^{2} + \frac{1}{4} {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} \end{array}$ by employing Hölder, Young, Poincaré inequality and continuous embedding, where the constant c only depends on $B_{1}^{'}$ . We have $\begin{array}{l} \frac{d}{d t} (α t ‖ \overline{u} ‖_{- 1}^{2} + t ‖ \nabla u ‖^{2}) + t {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} ⩽ α ‖ \overline{u} ‖_{- 1}^{2} + c t ‖ u ‖_{H^{1} (Ω)}^{2} + c^{'} t ‖ \nabla u ‖^{2} . \end{array}$ Owing to (4.31), we further obtain $\begin{array}{l} \frac{d}{d t} (α t ‖ \overline{u} ‖_{- 1}^{2} + t ‖ \nabla u ‖^{2} + t {⟨ u ⟩}^{2}) + t {‖ \frac{\partial \overline{u}}{\partial t} ‖}_{- 1}^{2} + α {⟨ u ⟩}^{2} \\ (4.42) & ⩽ α (‖ \overline{u} ‖_{- 1}^{2} + {⟨ u ⟩}^{2}) + c t (‖ u ‖_{H^{1} (Ω)}^{2} + {⟨ u ⟩}^{2}) + c^{'} t (‖ \nabla u ‖^{2} + {⟨ u ⟩}^{2}) . \end{array}$

Integrating (4.32) over $(0, t)$ , we get $\begin{matrix} (4.43) & \int_{0}^{t} ‖ \nabla u ‖^{2} d s ⩽ c e^{c^{'} t} ‖ u_{0, 1} - u_{0, 2} ‖_{- 1}^{2}, \end{matrix}$ where the constant $c^{'}$ only depends on $B_{1}^{'}$ , hence $\begin{matrix} (4.44) & \int_{0}^{t} ‖ u ‖_{H^{1} (Ω)}^{2} d s ⩽ c e^{c^{'} t} ‖ u_{0, 1} - u_{0, 2} ‖_{- 1}^{2} . \end{matrix}$ Owing to (4.33), (4.42)–(4.44) and the uniform Gronwall lemma, we have $\begin{matrix} (4.45) & ‖ u_{1} - u_{2} ‖_{H^{1} (Ω)}^{2} ⩽ c e^{c^{'} t} ‖ u_{0, 1} - u_{0, 2} ‖_{- 1}^{2}, \forall t > 0 . \end{matrix}$

We finally derive a Hölder (both with respect to space and time) estimate. Actually, the Hölder continuity with respect to x follows from (4.33). To prove the Hölder continuity with respect to t, we have $\begin{array}{l} {‖ u (t_{1}) - u (t_{2}) ‖}_{- 1} = {‖ \int_{t_{1}}^{t_{2}} \frac{\partial u}{\partial t} d τ ‖}_{- 1} ⩽ | t_{1} - t_{2} |^{\frac{1}{2}} {| \int_{t_{1}}^{t_{2}} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} d τ |}^{\frac{1}{2}}, \end{array}$ we note, owing to Proposition 4.3, that $\begin{matrix} | \int_{t_{1}}^{t_{2}} {‖ \frac{\partial u}{\partial t} ‖}_{- 1}^{2} d τ | ⩽ c, \end{matrix}$ where the constant c depends on $B_{1}^{'}$ and T such that $t_{1}, t_{2} \in [0, T]$ and $\begin{matrix} (4.46) & {‖ u (t_{1}) - u (t_{2}) ‖}_{- 1} ⩽ c | t_{1} - t_{2} |^{\frac{1}{2}} . \end{matrix}$

The conclusion is finally deduced from (4.33), (4.45) and (4.46). □
Corollary 4.14.
The semigoup $S_{α} (t)$ possesses the finite-dimensional global attractor $A_{α}$ .
Proof.
It follows immediately in view of Theorem 3.9 and Theorem 4.13. □
Remark 4.15.
Because of the same reason we explained in Remark 3.13, we do not have the robustness of exponential attractor in this case.

5. Conclusion and perspectives

In this paper, we prove the global in time well-posedness of solution to a Cahn–Hilliard–Oono type model proposed for image segmentation. We also prove the existences of the global attractor and the exponential attractor, and show that the fractal dimension of the global attractor will tend to infinity as $α \to 0$ . Next, we study this model with logarithmic nonlinear terms instead. We prove the wellposedness of the weak solution, then we obtain the strict separation results in one and two space dimension, which make sure of proving the finite fractal dimensionality of the global attractor. The robustness of exponential attractor of this model, like A. Miranville presented in [14], can not be derived owing to the limitations of the Cahn–Hilliard type model for image segmentation, which we have studied in [12]. We also have interest in this model with dynamic boundary conditions, which will be addressed in forthcoming papers.

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