On the degenerate Cahn–Hilliard equation: Global existence and entropy estimates of weak solutions

Abstract

In this paper, we consider a class of the degenerate Cahn–Hilliard equation with a smooth double-well potential. By applying a semi-discretization technique and asymptotic analysis method on non-degenerate Cahn–Hilliard equation, we obtain the existence and regularity of weak solutions to the Cahn–Hilliard equation with degenerate mobility. Moreover, we define a new entropy and obtain the global entropy estimates.

Keywords

Degenerate non-degenerate semi-discretization entropy

1. Introduction

The Cahn–Hilliard (C–H) equation initially describes a process of phase separation, by which two components of a binary system spontaneously separate [5,6]. The equation contains lots of significant prototypes for the evolution of the phase separation dynamics. It also model other phenomena such as the dynamics of populations [8], the tumor growth [14] and the bacterial films [15]. Typically, the fourth-order, nonlinear parabolic C–H equation [9,11] is written as $\begin{array}{l} (1.1) & \partial_{t} u = \nabla \cdot (M (u) \nabla W), (x, t) \in Ω \times (0, \infty), \\ (1.2) & W = - ν Δ u + ψ^{'} (u), (x, t) \in Ω \times (0, \infty), \end{array}$ which is coupled with the zero mass flux, natural boundary conditions and the initial condition $\begin{array}{l} (1.3) & M (u) \nabla W \cdot n = 0, \nabla u \cdot n = 0, (x, t) \in \partial Ω \times (0, \infty), \\ (1.4) & u (x, 0) = u_{0} (x), x \in Ω . \end{array}$ The domain Ω is supposed to be a bounded open domain in $R^{n} (n \in N^{+})$ with sufficiently smooth boundary $\partial Ω$ . We consider the evolution for times $t > 0$ or on a finite time interval $0 < t < T < \infty$ . $Ω_{T}$ stands for a space-time cylinder $Ω \times (0, T)$ . The order parameter $u (x, t)$ is the relative concentration of two phases. The mass flux is defined as $\begin{array}{l} (1.5) & J & = - M (u) \nabla (- ν Δ u + ψ^{'} (u)) . \end{array}$ The phase separation process conserves the total concentration $\begin{array}{l} (1.6) & \frac{d}{d t} \int_{Ω} u (x, t) d x = 0 . \end{array}$ Here, $M (u) > 0$ is a diffusion mobility. $W (u)$ is identified as a generalized chemical potential. $ν > 0$ is a coefficient of gradient energy. $n$ denotes the unit exterior normal on $\partial Ω$ . The homogeneous free energy is denoted as a double-well potential $ψ (u)$ . There are three widely choices of double-well potential functions. The simplest quartic polynomial, the logarithmic potential and the double-barrier potential are presented separately as $\begin{array}{l} (1.7) & ψ (u) = u^{2} {(1 - u)}^{2}, \\ (1.8) & ψ (u) = \frac{θ}{2} (u ln u + (1 - u) ln (1 - u)) + \frac{1}{2} u (1 - u), \\ (1.9) & ψ (u) = \{\begin{matrix} \frac{1}{2} u (1 - u), & if 0 ⩽ u ⩽ 1, \\ \infty, & otherwise . \end{matrix} \end{array}$ Here, $θ > 0$ is sufficiently small. Detailed comparison of (1.7)–(1.9) can refer [3,9,21,26]. The Ginzburg-Landau energy functional is $\begin{array}{l} (1.10) & E (u) = \int_{Ω} (\frac{ν}{2} | \nabla u |^{2} + ψ (u)) d x . \end{array}$ The property of solution to system (1.1)–(1.4) is influenced by both the diffusion mobility $M (u)$ and potential $ψ (u)$ . In the classical C–H equation (the constant mobility $M (u) = M_{0} > 0$ ) theory, various detailed mathematical analysis on the existence, the uniqueness and the regularity of the semi-linear parabolic equation are discussed in [1,4,13,16,19–25]. For the C–H equation with the nonconstant mobility, the asymptotic behavior (the existence of the attractors) can refer [28,29]. The degenerate C–H equation (such as $M (u) = u^{r} {(1 - u)}^{r} (r > 0)$ ) is allowed to degenerate at $u = 0, 1$ . By choosing the suitable degenerate mobility $M (u)$ and potential $ψ (u)$ , the solutions to the degenerate C–H equation remain bounded in $[0, 1]$ if the initial values lie in $[0, 1]$ . Specific analysis on the choice of degenerate mobility $M (u)$ and potential $ψ (u)$ can refer [9,21]. For the 1D degenerate case, J.X. Yin [31] first provided an existence theory of solutions. For the multidimensional space, C.M. Elliot, H. Garcke proved the existence of weak solutions [11] with a degenerate mobility $\begin{matrix} (1.11) & \tilde{M} (u) = \{\begin{matrix} u^{\tilde{m}} {(1 - u)}^{\tilde{m}} \bar{M} (u), & if 0 ⩽ u ⩽ 1, \\ 0, & otherwise \end{matrix} \end{matrix}$ and potential (1.8). Here $\bar{M} (u)$ is a $C^{1}$ -function. $\tilde{m} > 0$ is a real number. Considering the compatibility of physical Gibbs-Thomson effect, S.B. Dai and Q. Du proposed a degenerate mobility $\begin{array}{l} (1.12) & \tilde{\tilde{M}} (u) = | u^{\tilde{m}} {(1 - u)}^{\tilde{m}} |, for all u \in R, \end{array}$ coupled with the quartic polynomial potential (1.7). The significant difference between hypothesis (1.11) and (1.12) is the latter allowing u eventually go outside of $[0, 1]$ , even if the initial value of u lies in $[0, 1]$ . They obtained the coarsening mechanism [10] and the well-posedness of solutions to the C–H equation [9]. Recently, S. Lisini, D. Matthes and G. Savaré proved the weak solutions to (1.1)–(1.4) by applying the variational minimizing movement/JKO scheme under appropriate conditions on a concave function $M (u)$ and the potential $ψ (u)$ subjecting certain regularity assumptions [17].

This paper is devoted to the mathematical analysis of the C–H equation with the degenerate mobility $M (u)$ -(1.11) ( $\bar{M} (u) \equiv 1$ for simplicity) and a smooth double-well potential $ψ (u)$ . The purpose of this paper is to refine the theories of existence and regularity of weak solutions to the C–H equation with degenerate mobility. The main difficulty in technology is the degenerate diffusion mobility $M (u)$ which is allowed to degenerate at 0 at 1. Firstly, inspired by [7,18], we establish the existence of a weak solution $u_{ε} (x, t)$ to the non-degenerate C–H equation by using a semi-discretization technique. This method is based on a discretization in time which is valid even for any number of space dimensions in a Banach space. The ideal of this semi-discretization method consists on approaching $\partial_{t} u_{ε}$ by $\begin{array}{l} (1.13) & \frac{u_{ε} (x, t + h) - u_{ε} (x, t)}{h} . \end{array}$ In order to construct an approximation solution, we introduce a constant interpolation operator and a linear interpolation operator. By applying Schauder fixed-point theorem and Kolmogorov’s compactness criterion, we obtain the existence of weak solutions to the regularized non-degenerate C–H equation. Next, by tending the regularized parameter $ε \to 0$ to the non-degenerate C–H equation, we get the existence and regularity of weak solutions to the degenerate C–H equation (1.1)–(1.4). Next, we consider a basic entropy $G_{ε} (l) = \int_{l}^{\tilde{r}} \int_{\tilde{τ}}^{\tilde{r}} \frac{1}{M_{ε} (τ)} d τ d \tilde{τ}$ satisfying $G_{ε}^{″} (l) = \frac{1}{M_{ε} (l)}$ . Here $\tilde{r}$ is a fixed constant. We obtain the basic entropy dissipation $\begin{matrix} \int_{Ω} G_{ε} (u_{ε} (x, T)) d x + \frac{ν}{2} \int_{Ω_{T}} {| Δ u_{ε} (x, t) |}^{2} d (x, t) ⩽ C, \end{matrix}$ where the constant C is independent of ε. Motivated by the experiments described in [2,27], we also introduce the entropy $\begin{array}{l} G_{α ε} (l) = \int_{l}^{\tilde{r}} \int_{\tilde{τ}}^{\tilde{r}} \frac{1}{τ^{α} M_{ε} (τ)} d τ d \tilde{τ}, 0 ⩽ α < \frac{1}{2}, \end{array}$ such that $G_{ε α}^{″} (l) = \frac{1}{l^{α} M_{ε} (l)}$ . We obtain the additional entropy dissipation $\begin{matrix} \int_{Ω} G_{ε α} (u_{ε} (x, T)) d x + {\tilde{\tilde{C}}}_{1} \int_{Ω_{T}} {| D^{2} u_{ε}^{1 - \frac{α}{2}} |}^{2} d (x, t) + {\tilde{\tilde{C}}}_{1} \int_{Ω_{T}} {| \nabla u_{ε}^{\frac{1}{2} - \frac{α}{4}} |}^{4} d (x, t) ⩽ C, \end{matrix}$ where the constants $C, {\tilde{\tilde{C}}}_{1}$ are independent of ε.

It is worth mentioning that the authors are greatly illuminated by the works of [2,7,9,11,18,23,27].

Main results. For mainly technical reasons, we assume that $M (u) \in C^{2} (R, [0, \infty))$ such that $\begin{matrix} (1.14) & M (u) : = \{\begin{matrix} u^{r} {(1 - u)}^{r}, & if u \in [0, 1], \\ 0, & otherwise . \end{matrix} \end{matrix}$ Here r can be any positive number satisfying $\begin{matrix} \{\begin{matrix} 0 < r < \infty, & if n = 1, 2, \\ 0 < r < \frac{2}{n - 2}, & if n ⩾ 3 . \end{matrix} \end{matrix}$ The exact form of potential $ψ (u)$ is not essential since we only need its smoothness and some growth conditions. There is no loss of generality in assuming that $ψ (u) \in C^{2} (R, R)$ . There exists constants $C_{i} > 0 (i = 1, \dots, 5)$ such that $\begin{array}{l} (1.15) & C_{1} (| u |^{s + 1} - 1) ⩽ ψ (u) ⩽ C_{2} (| u |^{s + 1} + 1), \\ (1.16) & | ψ^{'} (u) | ⩽ C_{3} (| u |^{s} + 1), \\ (1.17) & C_{4} (| u |^{s - 1} - 1) ⩽ ψ^{″} (u) ⩽ C_{5} (| u |^{s - 1} + 1) \end{array}$ for all $u \in R$ . Here s can be any positive number satisfying that $\begin{matrix} \{\begin{matrix} 1 ⩽ s < \infty, & if n = 1, 2, \\ 1 ⩽ s ⩽ \frac{n}{n - 2}, & if n ⩾ 3 . \end{matrix} \end{matrix}$ In the sequel, $C > 0$ represent different constants. We have the following main results.

Theorem 1.1.
Define a set $X : = {(x, t) \in Ω_{T} | M (u) \neq 0}$ . Under the assumptions ( 1.14 )–( 1.17 ), for any $u_{0} \in H^{1} (Ω)$ with $0 ⩽ u_{0} (x) ⩽ 1$ , $T > 0$ , there exists a weak solution $u (x, t) : Ω \to R$ to ( 1.1 )–( 1.4 ) such that $\begin{array}{l} (1) u \in L^{\infty} (Ω_{T}) \cap L^{\infty} (0, T; H^{1} (Ω)) \cap C ([0, T]; L^{2} (Ω)) \cap L^{2} (Ω_{T}) \cap L^{2} (0, T; H^{2} (Ω)); \\ (1.18) & (2) \partial_{t} u \in L^{2} (0, T; {(H^{1} (Ω))}^{'}); \\ (3) 0 ⩽ u (x, t) ⩽ 1 for a.e. (x, t) \in Ω_{T} \end{array}$ in the sense of distributions, $\begin{matrix} (1.19) & \begin{matrix} \int_{Ω_{T}} \partial_{t} u σ d (x, t) = & \int_{Ω_{T} \cap X} M^{'} (u) \nabla u (- ν Δ u + ψ^{'} (u)) \cdot \nabla σ d (x, t) \\ + \int_{Ω_{T} \cap X} M (u) (- ν Δ u + ψ^{'} (u)) Δ σ d (x, t) \end{matrix} \end{matrix}$ for all $σ (x, t) \in C^{2} ({\overline{Ω}}_{T})$ with $\nabla σ \cdot n = 0$ on $\partial Ω \times (0, T)$ . Furthermore, for any real number $0 ⩽ α < \frac{1}{2}$ , a solution $u (x, t)$ to ( 1.1 )–( 1.4 ) satisfies the additional regularity properties $\begin{array}{l} (1.20) & \nabla u^{\frac{1}{2} - \frac{α}{4}} \in L^{4} (Ω_{T}), u^{1 - \frac{α}{2}} \in L^{2} (0, T; H^{2} (Ω)) . \end{array}$

This paper is organized as follows. In Section 2, we proceed with the study of the existence of solutions to the C–H equation with non-degenerate mobility and smooth potential by the technique of semi-discretization and the Schauder fixed-point theory. Section 3 provides the existence and regularity of weak solutions to the C–H equation with degenerate mobility by getting the limits of solutions which proved in Section 2. Section 4 is devoted to obtain the global entropy estimates.
2. Existence of weak solutions to the non-degenerate problem

In this section, we restrict our attention to the existence of weak solutions to the C–H equation with non-degenerate mobility ( $M_{ε} (u) > 0$ ). Since the diffusion degenerate term $M (u)$ which is allowed to vanish at $u = 0, 1$ , it is convenient to choose $\begin{matrix} (2.1) & M_{ε} (u) : = \{\begin{matrix} u^{r} {(1 - u)}^{r}, & if u \in [ε, 1 - ε], \\ ε^{r} {(1 - ε)}^{r}, & otherwise, \end{matrix} \end{matrix}$ for each fixed $0 < ε < 1$ . Throughout this section, we consider the regularized C–H equation as $\begin{array}{l} (2.2) & \partial_{t} u_{ε} = \nabla \cdot (M_{ε} (u_{ε}) \nabla W_{ε}), (x, t) \in Ω \times (0, T), \\ (2.3) & W_{ε} = - ν Δ u_{ε} + ψ^{'} (u_{ε}), (x, t) \in Ω \times (0, T), \\ (2.4) & \nabla W_{ε} \cdot n = 0, \nabla u_{ε} \cdot n = 0, (x, t) \in \partial Ω \times (0, T), \\ (2.5) & u_{ε} (x, 0) = u_{0} (x), x \in Ω . \end{array}$ Here, $W_{ε} : = W (u_{ε})$ . $T > 0$ is a fixed positive number. System (2.2)–(2.5) is rewritten as $\begin{array}{l} (2.6) & \partial_{t} u_{ε} = \nabla \cdot (M_{ε} (u_{ε}) \nabla (- ν Δ u_{ε} + ψ^{'} (u_{ε}))) . \end{array}$

Theorem 2.1.
For any $u_{0} (x) \in H^{1} (Ω)$ and $T > 0$ , the non-degenerate system ( 2.2 )–( 2.5 ) admits a weak solution $u_{ε}$ satisfying $\begin{matrix} (2.7) & \begin{matrix} (1) & u_{ε} \in L^{\infty} (Ω_{T}) \cap L^{2} (Ω_{T}) \cap L^{2} (0, T; H^{3} (Ω)), \\ \partial_{t} u_{ε} \in L^{2} (0, T; {(H^{1} (Ω))}^{'}), \\ (2) & 0 ⩽ u_{ε} (x, t) ⩽ 1 a.e. in Ω_{T} if 0 ⩽ u_{0} (x) ⩽ 1 \end{matrix} \end{matrix}$ such that $\begin{array}{l} (2.8) & \int_{0}^{T} {⟨ \partial_{t} u_{ε}, φ ⟩}_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} d t = - \int_{Ω_{T}} M_{ε} (u_{ε}) (- ν \nabla △ u_{ε} + ψ^{″} (u_{ε}) \nabla u_{ε}) \cdot \nabla φ d (x, t) \end{array}$ for all $φ \in L^{2} (0, T; H^{1} (Ω))$ .

In the following, we prove the existence of weak solutions to non-degenerate problem (2.2)–(2.5). The proof of Theorem 2.1 is summarized in three steps:
We construct an approximating solution by applying the semi-discretization in time.

We get the existence of the approximating solutions by using the Schauder fixed-point theorem.

We obtain the existence and regularity of solutions to the non-degenerate problem (2.2)–(2.5) by considering the limits of conclusions in Step 2.

2.1. Construction of an approximating solution

Similar to [7], we define two interpolation operators: the constant interpolation operator $Π_{N}^{0}$ and the linear interpolation operator $Π_{N}^{1}$ . Let $V$ be a Banach space. We define $h = Δ t = \frac{T}{N} (N \in N_{+}, T > 0)$ as a constant time step. For all $ω = (ω_{0}, ω_{1}, \dots, ω_{N}) \in V^{N + 1}$ , the constant interpolation operators is defined by $Π_{N}^{0} ω : [0, T] \to V$ , $\begin{matrix} (2.9) & \{\begin{matrix} Π_{N}^{0} ω (0) = ω_{0}, & if t = 0, \\ Π_{N}^{0} ω (t) = \sum_{m = 0}^{N - 1} ω_{m + 1} χ_{(m h, (m + 1) h]} (t), & if 0 < t ⩽ T . \end{matrix} \end{matrix}$ We note that $(0, T] = (0, h] \cup (h, 2 h] \cup \dots \cup ((N - 1) h, T]$ . Here $χ_{(m h, (m + 1) h]}$ is the characteristic function in $(m h, (m + 1) h]$ . For example, we define $χ_{(m h, (m + 1) h]} (t) = 1$ if $t \in (m h, (m + 1) h]$ and $χ_{(m h, (m + 1) h]} (t) = 0$ if not.

The linear interpolation operator is defined by $Π_{N}^{1} ω : [0, T] \to V$ , $\begin{matrix} (2.10) & \{\begin{matrix} Π_{N}^{1} ω (0) = ω_{0}, & if t = 0, \\ Π_{N}^{1} ω (t) = \sum_{m = 0}^{N - 1} [(1 + m - \frac{t}{h}) ω_{m} + (\frac{t}{h} - m) ω_{m + 1}] χ_{(m h, (m + 1) h]} (t), & if 0 < t ⩽ T . \end{matrix} \end{matrix}$ Obviously, the function $Π_{N}^{1} ω (t)$ is continuous. For all $t \neq m h$ , the derivative of $Π_{N}^{1} ω (t)$ is $\begin{array}{l} (2.11) & \frac{d}{d t} (Π_{N}^{1} ω (t)) = \sum_{m = 0}^{N - 1} (\frac{ω_{m + 1} - ω_{m}}{h}) χ_{(m h, (m + 1) h]} (t) . \end{array}$ The $L^{p} (1 ⩽ p ⩽ \infty)$ norm of $Π_{N}^{0}$ and $Π_{N}^{1}$ are defined as $\begin{array}{l} (2.12) & {‖ Π_{N}^{0} ω (t) ‖}_{L^{p} (0, T; V)} = {(h \sum_{m = 0}^{N - 1} ‖ ω_{m} ‖_{V}^{p})}^{\frac{1}{p}}, if 1 ⩽ p < \infty . \\ (2.13) & {‖ Π_{N}^{0} ω (t) ‖}_{L^{\infty} (0, T; V)} = {‖ Π_{N}^{1} ω (t) ‖}_{L^{\infty} (0, T; V)} = max_{m = 1, \dots, N} (‖ ω_{m} ‖_{V}), if p = \infty . \end{array}$ In the following proof, we apply above semi-discretization scheme in time to (2.2)–(2.5). We define a family of approximate solutions by the discretization in time. The initial data is given as $\begin{array}{l} (2.14) & u_{0, ε}^{N} = u_{0} (x) . \end{array}$ By applying the mathematical induction, if $u_{0, ε}^{N}, u_{1, ε}^{N}, \dots, u_{m, ε}^{N}$ are known, we define $u_{m + 1, ε}^{N}$ which satisfies the boundary condition (2.4) and $\begin{array}{l} (2.15) & \frac{1}{h} \int_{Ω} (u_{m + 1, ε}^{N} - u_{m, ε}^{N}) φ d x + \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) \nabla (- ν Δ u_{m + 1, ε}^{N} + ψ^{'} (u_{m + 1, ε}^{N})) \cdot \nabla φ d x = 0 \end{array}$ for all $φ \in H^{1} (Ω)$ . In the subsequent section, we devote to the existence of a discrete solution $u_{m + 1, ε}^{N} (x)$ to (2.14)–(2.15).

2.2. The existence of an approximating solution

The aim of this section is to obtain the existence of an approximating solution $u_{m + 1, ε}^{N} (x)$ of (2.14)–(2.15) by using the Schauder fixed-point theorem.

Lemma 2.1.
For all $m = 0, 1, \dots, N - 1$ and $0 ⩽ u_{0} (x) ⩽ 1$ , we have $\begin{array}{l} (2.16) & 0 ⩽ u_{m + 1, ε}^{N} ⩽ 1 for a.e. x \in Ω . \end{array}$
Proof.
For fixed $\tilde{r} > max | u_{m + 1, ε}^{N} | > 0$ and all $0 < ε < 1$ , we consider functions $\begin{aligned} g_{ε} (l) = \int_{\tilde{r}}^{l} \frac{d s}{M_{ε} (s)}, G_{ε} (l) = \int_{\tilde{r}}^{l} g_{ε} (s) d s \end{aligned}$ satisfying that $\begin{matrix} g_{ε} (l) \{\begin{matrix} > 0, & if l > \tilde{r}, \\ = 0, & if l = \tilde{r}, \\ < 0, & if l < \tilde{r}, \end{matrix} G_{ε} (l) \{\begin{matrix} > 0, & if l \neq \tilde{r}, \\ = 0, & if l = \tilde{r} . \end{matrix} \end{matrix}$ Hence, it follows that $\begin{aligned} (1) G_{ε}^{'} (l) = g_{ε} (l), \\ (2) G_{ε}^{″} (l) = g_{ε}^{'} (l) = \frac{1}{M_{ε} (l)}, \\ (3) | g_{ε} (l) | = | \int_{\tilde{r}}^{l} \frac{1}{M_{ε} (s)} d s | ⩽ C | l - \tilde{r} |, \\ (4) | G_{ε} (l) | = | \int_{\tilde{r}}^{l} \int_{\tilde{r}}^{\tilde{τ}} \frac{1}{M_{ε} (s)} d s d \tilde{τ} | ⩽ C | l - \tilde{r} |^{2} . \end{aligned}$ By using the assumption of $M_{ε}$ , it has $\begin{array}{l} G_{ε}^{'} (u_{m + 1, ε}^{N}) = g_{ε} (u_{m + 1, ε}^{N}) \in H^{1} (Ω) . \end{array}$ Choosing $φ = G_{ε}^{'} (u_{m + 1, ε}^{N})$ in (2.15), we obtain $\begin{aligned} \frac{1}{h} \int_{Ω} G_{ε} (u_{m + 1, ε}^{N}) - G_{ε} (u_{m, ε}^{N}) d x \\ = - \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) \nabla (- ν Δ u_{m + 1, ε}^{N} + ψ^{'} (u_{m + 1, ε}^{N})) G_{ε}^{″} (u_{m + 1, ε}^{N}) \cdot \nabla u_{m + 1, ε}^{N} d x \\ \overset{(1)}{=} - \int_{Ω} (- ν \nabla Δ u_{m + 1, ε}^{N} + \nabla ψ^{'} (u_{m + 1, ε}^{N})) \cdot \nabla u_{m + 1, ε}^{N} d x \\ \overset{(2)}{=} - ν \int_{Ω} {| Δ u_{m + 1, ε}^{N} |}^{2} d x + \int_{Ω} ψ^{'} (u_{m + 1, ε}^{N}) Δ u_{m + 1, ε}^{N} d x . \end{aligned}$ It follows that $\begin{array}{l} (2.17) & \begin{matrix} \frac{1}{h} \int_{Ω} G_{ε} (u_{m + 1, ε}^{N}) d x + ν \int_{Ω} {| Δ u_{m + 1, ε}^{N} |}^{2} d x \\ = \frac{1}{h} \int_{Ω} G_{ε} (u_{m, ε}^{N}) d x + \int_{Ω} ψ^{'} (u_{m + 1, ε}^{N}) Δ u_{m + 1, ε}^{N} d x . \end{matrix} \end{array}$ Here we obtain (1) and (2) by applying $\nabla G_{ε}^{'} (u_{m + 1, ε}^{N}) = \frac{\nabla u_{m + 1, ε}^{N}}{M_{ε} (u_{m + 1, ε}^{N})}$ and integration by parts, respectively. Under the assumption of $M_{ε}$ and the uniform bounded of $u_{m, ε}^{N}$ , it gives that $\begin{array}{l} (2.18) & \int_{Ω} G_{ε} (u_{m, ε}^{N}) d x ⩽ C . \end{array}$ Here C is independent of N. Actually, we have $\begin{array}{l} (2.19) & \int_{Ω} ψ^{'} (u_{m + 1, ε}^{N}) Δ u_{m + 1, ε}^{N} d x ⩽ \frac{C^{2}}{2 ν} + \frac{ν}{2} {‖ Δ u_{m + 1, ε}^{N} ‖}_{L^{2} (Ω)}^{2} . \end{array}$ Here we apply the assumptions of ψ, Hölder inequality with $p = 2$ , $q = 2$ and Young’s inequality with ε: $a b ⩽ \frac{ε}{2} a^{2} + \frac{1}{2 ε} b^{2}$ . Associating (2.17) with (2.18)–(2.19), one has $\begin{array}{l} (2.20) & \int_{Ω} G_{ε} (u_{m + 1, ε}^{N}) d x + \frac{ν h}{2} \int_{Ω} {| Δ u_{m + 1, ε}^{N} |}^{2} d x ⩽ C . \end{array}$ Here C is independent of N. Suppose that $l < \tilde{r}$ (other cases can be handled similarly), then it has $\begin{array}{l} g_{ε} (l) = - \int_{l}^{\tilde{r}} \frac{d s}{M_{ε} (s)} < 0, G_{ε} (l) = - \int_{l}^{\tilde{r}} g_{ε} (s) d s > 0 . \end{array}$ In the following, we apply the proof of contradiction to claim that $\begin{array}{l} (2.21) & 0 ⩽ u_{m + 1, ε}^{N} ⩽ 1 a.e. in Ω_{T} . \end{array}$ If it is not true for the left inequality of (2.21), then there exists a point $x_{0} \in Ω$ such that $u_{m + 1, ε}^{N} (x_{0}) < 0$ . There exist $δ > 0, 0 < ε < ε_{0}$ such that $\begin{array}{l} u_{m + 1, ε}^{N} (x) < - δ < 0, | x - x_{0} | < δ, x \in Ω . \end{array}$ We have $\begin{array}{l} G_{ε} (u_{m + 1, ε}^{N} (x)) = - \int_{u_{m + 1, ε}^{N} (x)}^{\tilde{r}} g_{ε} (s) d s ⩾ - \int_{- δ}^{0} g_{ε} (s) d s . \end{array}$ It follows that $\begin{array}{l} lim_{ε \to 0} g_{ε} (δ) = - lim_{ε \to 0} \int_{δ}^{\tilde{r}} \frac{d s}{ε^{r} {(1 - ε)}^{r}} = - \infty . \end{array}$ Under the dominated convergence theorem, it follows that $\begin{array}{l} (2.22) & lim_{ε \to 0} G_{ε} (u_{m + 1, ε}^{N} (x)) ⩾ - lim_{ε \to 0} \int_{- δ}^{0} g_{ε} (s) d s = + \infty, \end{array}$ which contradicting with (2.20). It means that $u_{m + 1, ε}^{N} (x) ⩾ 0$ a.e. in Ω. Similarly, we also obtain $u_{m + 1, ε}^{N} (x) ⩽ 1$ a.e. in Ω. This completes the proof of Lemma 2.1. □

For the purpose of getting the existence of $u_{m + 1, ε}^{N} (x)$ to (2.14)–(2.15), we apply the Schauder fixed-point theorem by introducing the following enclosed convex bounded subset of the Banach space $L^{2} (Ω)$ : $\begin{array}{l} D = {u_{m + 1, ε}^{N} (x) \in L^{2} (Ω); 0 ⩽ u_{m + 1, ε}^{N} (x) ⩽ 1 for a.e. x \in Ω} . \end{array}$ The Lax–Milgram theorem guaranteeing by Lemma 2.1 allows to define the application $\begin{array}{l} L : D \to D \end{array}$ as $\begin{array}{l} (2.23) & L (ω) = u_{m + 1, ε}^{N}, \end{array}$ where $u_{m + 1, ε}^{N}$ is the unique function of $H^{1} (Ω)$ . For $\forall φ \in H^{1} (Ω)$ , it has $\begin{matrix} (2.24) & \{\begin{matrix} \frac{1}{h} \int_{Ω} (u_{m + 1, ε}^{N} - u_{m, ε}^{N}) φ d x \\ + \int_{Ω} M_{ε} (ω) \nabla (- ν Δ u_{m + 1, ε}^{N} + ψ^{'} (ω)) \cdot \nabla φ d x = 0, & \forall x \in Ω, \\ \nabla u_{m + 1, ε}^{N} \cdot n = \nabla Δ u_{m + 1, ε}^{N} \cdot n = 0, & \forall x \in \partial Ω, \\ u_{0, ε}^{N} (x) = u_{0} (x), & \forall x \in Ω . \end{matrix} \end{matrix}$ Each fixed point of $L$ is a solution to (2.14)–(2.15). Let us present that $L (D)$ is relatively compact in $L^{2} (Ω)$ . Choosing $φ = - Δ u_{m + 1, ε}^{N}$ in (2.24)-1, we have $\begin{aligned} \frac{1}{h} \int_{Ω} {| \nabla u_{m + 1, ε}^{N} |}^{2} d x + ν \int_{Ω} M_{ε} (ω) {| \nabla Δ u_{m + 1, ε}^{N} |}^{2} d x \\ = \frac{1}{h} \int_{Ω} \nabla u_{m, ε}^{N} \cdot \nabla u_{m + 1, ε}^{N} d x + \int_{Ω} {(M_{ε} (ω))}^{\frac{1}{2}} \nabla ψ^{'} (ω) {(M_{ε} (ω))}^{\frac{1}{2}} \nabla Δ u_{m + 1, ε}^{N} d x \\ \overset{(3)}{⩽} \frac{1}{h} (\frac{1}{2} {‖ \nabla u_{m, ε}^{N} ‖}_{L^{2} (Ω)}^{2} + \frac{1}{2} {‖ \nabla u_{m + 1, ε}^{N} ‖}_{L^{2} (Ω)}^{2}) + \frac{ν}{2} \int_{Ω} M_{ε} (ω) {| \nabla Δ u_{m + 1, ε}^{N} |}^{2} d x + \frac{C^{2}}{2 ν} . \end{aligned}$ Hence, it follows that $\begin{array}{l} (2.25) & \frac{1}{h} {‖ \nabla u_{m + 1, ε}^{N} ‖}_{L^{2} (Ω)}^{2} + ν \int_{Ω} M_{ε} (ω) {| \nabla Δ u_{m + 1, ε}^{N} |}^{2} d x ⩽ \frac{1}{h} {‖ \nabla u_{m, ε}^{N} ‖}_{L^{2} (Ω)}^{2} + \frac{C^{2}}{ν} \overset{(4)}{⩽} C . \end{array}$ Here C is independent of ω. We get (3) by using the assumptions of $M_{ε}$ and ψ, Hölder inequality with $p = 2, q = 2$ and Young’s inequality $a b ⩽ \frac{ε}{2} a^{2} + \frac{1}{2 ε} b^{2}$ . We obtain (4) by applying the uniform bounded of $\nabla u_{m, ε}^{N}$ . Combining the mass-conserved condition (1.6) with Poincaré’s inequality, we have $\begin{array}{l} {‖ u_{m + 1, ε}^{N} ‖}_{H^{1} (Ω)} ⩽ C, \end{array}$ where the constant $C > 0$ is independent of ω. By applying (2.25), ${(M_{ε} (ω))}^{\frac{1}{2}} \nabla Δ u_{m + 1, ε}^{N}$ is uniform bounded in $L^{2} (Ω)$ . Under the assumption of $M_{ε}$ , we have $\begin{array}{l} (2.26) & \nabla Δ u_{m + 1, ε}^{N} \in L^{2} (Ω) . \end{array}$ Hence, $L (D)$ is bounded in $H^{3} (Ω)$ and $L (D)$ is relatively compact in $L^{2} (Ω)$ .

Now we show that $L$ is a continuous mapping. Let ${ω_{m}}_{m}$ be a sequence in $D$ and $ω \in D$ such that $\begin{array}{l} ω_{m} \to ω strongly in L^{2} (Ω) \end{array}$ as $m \to + \infty$ . Mapping $L$ is continuous to present that $\begin{array}{l} u_{m} = L (ω_{m}) \to L (ω) = u strongly in L^{2} (Ω) \end{array}$ as $m \to + \infty$ . There exists a subsequence of ${ω_{m}}_{m}$ denoted as ${ω_{m_{j}}}_{j}$ converging to ω almost everywhere in Ω. The sequence ${u_{m}}_{m}$ is bounded in $H^{3} (Ω)$ which is a Hilbert space. There exists a subsequence ${u_{m_{j}}}_{j}$ such that $\begin{aligned} u_{(m_{j}) q} ⇀ u weakly in H^{3} (Ω), \\ u_{(m_{j}) q} \to u strongly in L^{2} (Ω) and a.e. in Ω \end{aligned}$ as $q \to + \infty$ . (2.24)-1 yields that $\begin{matrix} \frac{1}{h} \int_{Ω} (u_{(m_{j}) q} - u_{m, ε}^{N}) φ d x + \int_{Ω} M_{ε} (ω_{(m_{j}) q}) \nabla (- ν Δ u_{(m_{j}) q} + ψ^{'} (ω_{(m_{j}) q})) \cdot \nabla φ d x = 0 . \end{matrix}$ Passing to the limit as $q \to + \infty$ , we have $u = L (ω)$ . Accordingly, the subsequence $u_{(m_{j}) q}$ converges to $L (ω) = u$ in $L^{2} (Ω)$ as $q \to + \infty$ . The same argument provided that for every subsequence of ${u_{m}}_{m}$ converging in $L^{2} (Ω)$ . Hence, the sequence ${u_{m}}_{m}$ has a unique accumulation point and since it is included in a relatively compact subset of $L^{2} (Ω)$ . The sequence ${u_{m}}_{m}$ converges to $L (ω)$ in $L^{2} (Ω)$ which shows that $L$ is a continuous mapping. Eventually, the Schauder fixed-point theorem allows to conclude that the operator $L$ has a fixed point $u_{m + 1, ε}^{N}$ such that $\begin{matrix} L (u_{m + 1, ε}^{N}) = u_{m + 1, ε}^{N}, \end{matrix}$ which means that there exists at least one solution $u_{m + 1, ε}^{N}$ to (2.14)–(2.15). This completes the existence of approximating solutions.
2.3. Existence of solutions to the non-degenerate mobility problem (2.2)–(2.5)

This section is intended to obtain the existence of weak solutions to the non-degenerate mobility problem (2.2)–(2.5). Firstly, we obtain a uniform priori estimates on the interpolation of $Π_{N}^{0}$ and $Π_{N}^{1}$ . Then we pass to the limit of above estimates as $N \to \infty$ and get the existence of weak solutions to (2.2)–(2.5) by applying the compactness theorem.

2.3.1. Estimates

We provided the following uniform priori estimates on the interpolation of $u_{ε}^{N}$ with respect to N.

Proposition 2.1.
There exists a positive constant $\tilde{C}$ which is independent of N such that $\begin{array}{l} (2.27) & {‖ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{\infty} (Ω_{T})} = max_{m = 1, \dots, N} {‖ u_{m, ε}^{N} ‖}_{L^{\infty} (Ω)} ⩽ 1, \\ (2.28) & {‖ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T; H^{3} (Ω))} ⩽ \tilde{C}, \\ (2.29) & {‖ τ_{- h^{'}} Π_{N}^{0} u_{ε}^{N} - Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T - h^{'}; L^{2} (Ω))} ⩽ \tilde{C} h^{'}, for \forall h^{'} > 0, \\ (2.30) & {‖ τ_{- h^{'}} Π_{N}^{0} \nabla u_{ε}^{N} - Π_{N}^{0} \nabla u_{ε}^{N} ‖}_{L^{2} (0, T - h^{'}; L^{2} (Ω))} ⩽ \tilde{C} h^{'}, for \forall h^{'} > 0, \\ (2.31) & {‖ \frac{\partial}{\partial t} (Π_{N}^{1} u_{ε}^{N}) ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})} ⩽ \tilde{C}, \\ (2.32) & {‖ Π_{N}^{1} u_{ε}^{N} - Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})} ⩽ \frac{\tilde{C} h^{2}}{3} . \end{array}$
Proof.
From Lemma 2.1 and the Definition (2.13), the estimate (2.27) is obtained immediately.

As for (2.28), we set $φ = - Δ u_{m + 1, ε}^{N}$ in (2.15). $\begin{array}{l} \frac{1}{2 h} \int_{Ω} {| \nabla u_{m + 1, ε}^{N} |}^{2} - {| \nabla u_{m, ε}^{N} |}^{2} d x + ν \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) {| \nabla Δ u_{m + 1, ε}^{N} |}^{2} d x \\ \overset{(4)}{⩽} \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) \nabla ψ^{'} (u_{m + 1, ε}^{N}) (\nabla Δ u_{m + 1, ε}^{N}) d x \\ = \int_{Ω} {(M_{ε} (u_{m + 1, ε}^{N}))}^{\frac{1}{2}} \nabla ψ^{'} (u_{m + 1, ε}^{N}) {(M_{ε} (u_{m + 1, ε}^{N}))}^{\frac{1}{2}} (\nabla Δ u_{m + 1, ε}^{N}) d x \\ ⩽ C^{\frac{1}{2}} {‖ {(M_{ε} (u_{m + 1, ε}^{N}))}^{\frac{1}{2}} (\nabla Δ u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)} {‖ \nabla ψ^{'} (u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)} \\ \overset{(5)}{⩽} \frac{ν}{2} {‖ {(M_{ε} (u_{m + 1, ε}^{N}))}^{\frac{1}{2}} (\nabla Δ u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)}^{2} + \frac{C}{2 ν} {‖ \nabla ψ^{'} (u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)}^{2} . \end{array}$ It gives that $\begin{matrix} (2.33) & \begin{matrix} \frac{1}{h} \int_{Ω} {| \nabla u_{m + 1, ε}^{N} |}^{2} - {| \nabla u_{m, ε}^{N} |}^{2} d x + ν \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) {| \nabla Δ u_{m + 1, ε}^{N} |}^{2} d x \\ ⩽ \frac{C}{ν} {‖ \nabla ψ^{'} (u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)}^{2} \overset{(6)}{⩽} C, \end{matrix} \end{matrix}$ where the contant C is independent of N. Here we obtain (4) by applying the integration by parts and $(a - b) a ⩾ \frac{1}{2} (a^{2} - b^{2})$ : $\begin{array}{l} \frac{1}{h} \int_{Ω} (u_{m + 1, ε}^{N} - u_{m, ε}^{N}) (- Δ u_{m + 1, ε}^{N}) d x & = \frac{1}{h} \int_{Ω} (\nabla u_{m + 1, ε}^{N} - \nabla u_{m, ε}^{N}) (\nabla u_{m + 1, ε}^{N}) d x \\ ⩾ \frac{1}{2 h} \int_{Ω} {| \nabla u_{m + 1, ε}^{N} |}^{2} - {| \nabla u_{m, ε}^{N} |}^{2} d x . \end{array}$ We get (5) by using Hölder inequality with $p = 2, q = 2$ and Young’s inequality with ε: $a b ⩽ \frac{ε}{2} a^{2} + \frac{1}{2 ε} b^{2}$ . $\begin{matrix} (2.34) & \begin{matrix} (6) {‖ \nabla ψ^{'} (u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)}^{2} & = \int_{Ω} {| ψ^{″} (u_{m + 1, ε}^{N}) |}^{2} {| \nabla u_{m + 1, ε}^{N} |}^{2} d x \\ ⩽ {(\int_{Ω} {| ψ^{″} (u_{m + 1, ε}^{N}) |}^{n} d x)}^{\frac{2}{n}} {(\int_{Ω} {| \nabla u_{m + 1, ε}^{N} |}^{\frac{2 n}{n - 2}} d x)}^{\frac{n - 2}{n}} \\ ⩽ C {(1 + \int_{Ω} {| u_{m + 1, ε}^{N} |}^{n (s - 1)} d x)}^{\frac{2}{n}} {‖ \nabla u_{m + 1, ε}^{N} ‖}_{L^{\frac{2 n}{n - 2}} (Ω)}^{2} \\ ⩽ C (1 + {‖ u_{m + 1, ε}^{N} ‖}_{L^{n (s - 1)} (Ω)}^{2 (s - 1)}) {‖ \nabla u_{m + 1, ε}^{N} ‖}_{L^{\frac{2 n}{n - 2}} (Ω)}^{2} \\ ⩽ C (1 + {‖ u_{m + 1, ε}^{N} ‖}_{H^{1} (Ω)}^{2 (s - 1)}) {‖ u_{m + 1, ε}^{N} ‖}_{H^{2} (Ω)}^{2} ⩽ C . \end{matrix} \end{matrix}$ Here we apply Hölder inequality with $p = \frac{n}{2}, q = \frac{n}{n - 2}$ , the assumption (1.17) and Sobolev embedding theorem $H^{1} (Ω) ↪ L^{n (s - 1)} (Ω)$ such that $\begin{matrix} \{\begin{matrix} 0 < (s - 1) n < \infty, & if n = 1, 2, \\ 0 < (s - 1) n < (\frac{n}{n - 2} - 1) n = \frac{2 n}{n - 2}, & if n ⩾ 3, \end{matrix} \end{matrix}$ which is reasonable by using assumption (1.17).

Multiplying estimate (2.33) by h and summing from $m = 0$ to $m = N - 1$ , by using the assumption of $M_{ε}$ , we have $\begin{matrix} (2.35) & \int_{Ω} {| \nabla u_{N, ε}^{N} |}^{2} d x + ν {‖ \nabla Δ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (Ω_{T})}^{2} ⩽ C \cdot T + \int_{Ω} {| \nabla u_{0, ε}^{N} |}^{2} d x = \tilde{C} . \end{matrix}$ In particularly, one has $\begin{matrix} {‖ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T; H^{3} (Ω))} ⩽ \tilde{C} . \end{matrix}$

Now we show estimate (2.29), a time translate estimate of an approximate solution is classical in the analysis of the convergence of implicit finite volumes for non-linear parabolic equation [12]. The method of it is getting compactness property for the sequence $Π_{N}^{0} u_{ε}^{N}$ in $L^{2} (Ω_{T})$ . For $\forall h^{'} > 0$ , it has $\begin{matrix} I : = & {‖ τ_{- h^{'}} Π_{N}^{0} u_{ε}^{N} - Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T - h^{'}; L^{2} (Ω))}^{2} \\ = & \int_{0}^{T - h^{'}} \int_{Ω} {| Π_{N}^{0} u_{ε}^{N} (x, t + h^{'}) - Π_{N}^{0} u_{ε}^{N} (x, t) |}^{2} d (x, t) \\ = & \int_{0}^{T - h^{'}} A (t) d t . \end{matrix}$ For almost every $t \in (0, T - h^{'})$ , we denote $[x]$ (the integer part of a real x), we have $\begin{matrix} A (t) : = & \int_{Ω} {| Π_{N}^{0} u_{ε}^{N} (x, t + h^{'}) - Π_{N}^{0} u_{ε}^{N} (x, t) |}^{2} d x \\ = & \int_{Ω} {| u_{[\frac{t + h^{'}}{h}], ε}^{N} (x) - u_{[\frac{t}{h}], ε}^{N} (x) |}^{2} d x \\ = & \int_{Ω} \sum_{m = [\frac{t}{h}]}^{[\frac{t + h^{'}}{h}] - 1} (u_{m + 1, ε}^{N} (x) - u_{m, ε}^{N} (x)) (u_{[\frac{t + h^{'}}{h}], ε}^{N} (x) - u_{[\frac{t}{h}], ε}^{N} (x)) d x . \end{matrix}$ Setting $m_{0} (t) = [\frac{t}{h}], {\tilde{m}}_{1} (t) = [\frac{t + h^{'}}{h}]$ and choosing $φ = u_{{\tilde{m}}_{1}, ε}^{N} - u_{m_{0}, ε}^{N}$ in (2.15) and summing from $m = m_{0}$ to ${\tilde{m}}_{1} - 1$ , one has $\begin{matrix} \frac{1}{h} \sum_{m = m_{0}}^{{\tilde{m}}_{1} - 1} \int_{Ω} (u_{m + 1, ε}^{N} - u_{m, ε}^{N}) (u_{{\tilde{m}}_{1}, ε}^{N} - u_{m_{0}, ε}^{N}) d x \\ ⩽ - \sum_{m = m_{0}}^{{\tilde{m}}_{1} - 1} \int_{Ω} M (u_{m + 1, ε}^{N}) \nabla (- ν Δ u_{m + 1, ε}^{N} + ψ^{'} (u_{m + 1, ε}^{N})) (\nabla u_{{\tilde{m}}_{1}, ε}^{N} - \nabla u_{m_{0}, ε}^{N}) d x . \end{matrix}$ One defines $\begin{matrix} v_{i, ε} = \int_{Ω} {| \nabla u_{i, ε}^{N} |}^{2} d x, P_{j, ε} = \int_{Ω} ({(m_{2})}^{2} {| \nabla Δ u_{j, ε}^{N} |}^{2} + {| \nabla ψ^{'} (u_{j, ε}^{N}) |}^{2}) d x . \end{matrix}$ Young’s inequality means that $\begin{matrix} I ⩽ h (I_{1} + I_{2} + I_{3}), \end{matrix}$ where $\begin{array}{l} I_{1} : = \int_{0}^{T - h^{'}} \sum_{m = m_{0} (t) + 1}^{{\tilde{m}}_{1} (t)} P_{m, ε} d t, \\ I_{2} : = \int_{0}^{T - h^{'}} \sum_{m = m_{0} (t) + 1}^{{\tilde{m}}_{1} (t)} v_{m_{0} (t), ε} d t, \\ I_{3} : = \int_{0}^{T - h^{'}} \sum_{m = m_{0} (t) + 1}^{{\tilde{m}}_{1} (t)} v_{{\tilde{m}}_{1} (t), ε} d t . \end{array}$ Under the definitions of $χ_{m}$ : $χ_{m} (t, t + h^{'}) = 1$ if $m h \in (t, t + h^{'})$ and $χ_{m} (t, t + h^{'}) = 0$ if not. Hence, $I_{1}$ may be rewritten as $\begin{matrix} I_{1} = \int_{0}^{T - h^{'}} \sum_{m = 1}^{N} P_{m, ε} χ_{m} (t, t + h^{'}) d t = \sum_{m = 1}^{N} P_{m, ε} \int_{0}^{T - h^{'}} χ_{m} (t, t + h^{'}) d t ⩽ h^{'} \sum_{m = 1}^{N} P_{m, ε}, \end{matrix}$ where $\int_{0}^{T - h^{'}} χ_{m} (t, t + h^{'}) d t ⩽ h^{'}$ .

Similarly, we have $\begin{matrix} I_{2} ⩽ h^{'} \sum_{m = 1}^{N} v_{m, ε}, I_{3} ⩽ h^{'} \sum_{m = 1}^{N} v_{m, ε} . \end{matrix}$ We deduce that $\begin{matrix} I ⩽ h^{'} \sum_{i = 1}^{N} (P_{i, ε} + 2 v_{i, ε}) . \end{matrix}$ Under the definitions of $v_{i, ε}, P_{i, ε}$ , estimates (2.25), (2.28) and (2.34), we obtain (2.29).

The proof of estimate (2.30) is similar to (2.29). For $\forall h^{'} > 0$ , it has $\begin{matrix} J : = & {‖ τ_{- h^{'}} Π_{N}^{0} \nabla u_{ε}^{N} - Π_{N}^{0} \nabla u_{ε}^{N} ‖}_{L^{2} (0, T - h^{'}; L^{2} (Ω))}^{2} \\ = & \int_{0}^{T - h^{'}} \int_{Ω} {| Π_{N}^{0} \nabla u_{ε}^{N} (x, t + h^{'}) - Π_{N}^{0} \nabla u_{ε}^{N} (x, t) |}^{2} d (x, t) \\ = & \int_{0}^{T - h^{'}} B (t) d t . \end{matrix}$ For almost every $t \in (0, T - h^{'})$ , we have $\begin{matrix} B (t) : = & \int_{Ω} {| Π_{N}^{0} \nabla u_{ε}^{N} (x, t + h^{'}) - Π_{N}^{0} \nabla u_{ε}^{N} (x, t) |}^{2} d x \\ = & \int_{Ω} {| \nabla u_{[\frac{t + h^{'}}{h}], ε}^{N} (x) - \nabla u_{[\frac{t}{h}], ε}^{N} (x) |}^{2} d x \\ = & \int_{Ω} \sum_{m = [\frac{t}{h}]}^{[\frac{t + h^{'}}{h}] - 1} (\nabla u_{m + 1, ε}^{N} (x) - \nabla u_{m, ε}^{N} (x)) (\nabla u_{[\frac{t + h^{'}}{h}], ε}^{N} (x) - \nabla u_{[\frac{t}{h}], ε}^{N} (x)) d x \end{matrix}$ Choosing $\tilde{φ} = - (Δ u_{{\tilde{m}}_{1}, ε}^{N} - Δ u_{m_{0}, ε}^{N})$ in (2.15) and summing from $m = m_{0}$ to $m_{1} - 1$ , one has $\begin{matrix} \frac{1}{h} \sum_{m = m_{0}}^{{\tilde{m}}_{1} - 1} \int_{Ω} (\nabla u_{m + 1, ε}^{N} - \nabla u_{m, ε}^{N}) (\nabla u_{{\tilde{m}}_{1}, ε}^{N} - \nabla u_{m_{0}, ε}^{N}) d x \\ ⩽ - \sum_{m = m_{0}}^{{\tilde{m}}_{1} - 1} \int_{Ω} M (u_{m + 1, ε}^{N}) \nabla (- ν Δ u_{m + 1, ε}^{N} + ψ^{'} (u_{m + 1, ε}^{N})) \nabla (- (Δ u_{{\tilde{m}}_{1}, ε}^{N} - Δ u_{m_{0}, ε}^{N})) d x . \end{matrix}$ One defines $\begin{matrix} {\tilde{v}}_{i, ε} = \int_{Ω} {| \nabla Δ u_{i, ε}^{N} |}^{2} d x, {\tilde{P}}_{j, ε} = \int_{Ω} m_{2} ({| \nabla Δ u_{j, ε}^{N} |}^{2} + {| \nabla ψ^{'} (u_{j, ε}^{N}) |}^{2}) d x . \end{matrix}$ Young’s inequality means that $\begin{matrix} J ⩽ h (J_{1} + J_{2} + J_{3}), \end{matrix}$ where $\begin{array}{l} J_{1} : = \int_{0}^{T - h^{'}} \sum_{m = m_{0} (t) + 1}^{{\tilde{m}}_{1} (t)} {\tilde{P}}_{m, ε} d t, \\ J_{2} : = \int_{0}^{T - h^{'}} \sum_{m = m_{0} (t) + 1}^{{\tilde{m}}_{1} (t)} {\tilde{v}}_{m_{0} (t), ε} d t, \\ J_{3} : = \int_{0}^{T - h^{'}} \sum_{m = m_{0} (t) + 1}^{{\tilde{m}}_{1} (t)} {\tilde{v}}_{{\tilde{m}}_{1} (t), ε} d t . \end{array}$ $J_{1}, J_{2}$ and $J_{3}$ may be rewritten as $\begin{matrix} J_{1} ⩽ h^{'} \sum_{m = 1}^{N} {\tilde{P}}_{m, ε}, J_{2} ⩽ h^{'} \sum_{m = 1}^{N} {\tilde{v}}_{m, ε}, J_{3} ⩽ h^{'} \sum_{m = 1}^{N} {\tilde{v}}_{m, ε} . \end{matrix}$ We deduce that $\begin{matrix} J ⩽ h^{'} \sum_{i = 1}^{N} ({\tilde{P}}_{i, ε} + 2 {\tilde{v}}_{i, ε}) . \end{matrix}$ Under the definition of ${\tilde{v}}_{i, ε}, {\tilde{P}}_{i, ε}$ , estimates (2.28) and (2.34), we obtain (2.30).

To prove estimate (2.31), from the definition (2.11), we note that $\begin{matrix} {‖ \frac{\partial}{\partial t} (Π_{N}^{1} u_{ε}^{N}) ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})}^{2} = \frac{1}{h} \sum_{m = 0}^{N - 1} {‖ u_{m + 1, ε}^{N} - u_{m, ε}^{N} ‖}_{{(H^{1} (Ω))}^{'}}^{2} . \end{matrix}$ Hence, it suffices to show that $\begin{matrix} (2.36) & \begin{matrix} \frac{1}{h} \sum_{m = 0}^{N - 1} {‖ u_{m + 1, ε}^{N} - u_{m, ε}^{N} ‖}_{{(H^{1} (Ω))}^{'}}^{2} ⩽ \tilde{C} . \end{matrix} \end{matrix}$ Choosing $φ \in H^{1} (Ω)$ in (2.15), we have $\begin{matrix} | \frac{1}{h} {⟨ u_{m + 1, ε}^{N} - u_{m, ε}^{N}, φ ⟩}_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} | \\ = | \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) \nabla (- ν Δ u_{m + 1, ε}^{N} + ψ^{'} (u_{m + 1, ε}^{N})) \cdot \nabla φ d x | \\ ⩽ ν | \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) \nabla Δ u_{m + 1, ε}^{N} \cdot \nabla φ d x | + | \int_{Ω} M_{ε} (u_{m + 1, ε}^{N}) \nabla ψ^{'} (u_{m + 1, ε}^{N}) \cdot \nabla φ d x | \\ \overset{(10)}{⩽} ν {(m_{2})}^{\frac{1}{2}} {‖ {(M_{ε} (u_{m + 1, ε}^{N}))}^{\frac{1}{2}} \nabla Δ u_{m + 1, ε}^{N} ‖}_{L^{2} (Ω)} ‖ φ ‖_{H^{1} (Ω)} + m_{2} {‖ \nabla ψ^{'} (u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)} ‖ φ ‖_{H^{1} (Ω)} . \end{matrix}$ Here we apply (10): Hölder inequality with $p = 2, q = 2$ and the assumption of $M_{ε}$ . Raising the square and using the inequality ${(a + b)}^{2} ⩽ 2 (a^{2} + b^{2})$ for $\forall a, b ⩾ 0$ . For all n, one has $\begin{matrix} (2.37) & \begin{matrix} \frac{1}{h^{2}} {‖ u_{m + 1, ε}^{N} - u_{m, ε}^{N} ‖}_{{(H^{1} (Ω))}^{'}}^{2} \\ ⩽ 2 (ν^{2} m_{2} {‖ {(M_{ε} (u_{m + 1, ε}^{N}))}^{\frac{1}{2}} \nabla Δ u_{m + 1, ε}^{N} ‖}_{L^{2} (Ω)}^{2} + {(m_{2})}^{2} {‖ \nabla ψ^{'} (u_{m + 1, ε}^{N}) ‖}_{L^{2} (Ω)}^{2}) . \end{matrix} \end{matrix}$ Multiplying (2.37) by h and getting a sum from $m = 0$ to $m = N - 1$ , under (2.25) and (2.34), there exists a positive $\tilde{C}$ , not depending on N such that $\begin{matrix} {‖ \frac{\partial}{\partial t} (Π_{N}^{1} u_{m, ε}^{N}) ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})}^{2} ⩽ \tilde{C} . \end{matrix}$ Finally, we prove (2.32). $\begin{matrix} {‖ Π_{N}^{1} u_{m, ε}^{N} - Π_{N}^{0} u_{m, ε}^{N} ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})} \\ = {‖ \sum_{m = 0}^{N - 1} [(1 + m - \frac{t}{h}) u_{m, ε}^{N} + (\frac{t}{h} - m) u_{m + 1, ε}^{N} - u_{m + 1, ε}^{N}] χ_{(m h, (m + 1) h]} (t) ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})} \\ ⩽ \sum_{m = 0}^{N - 1} \int_{m h}^{(m + 1) h} {‖ (1 + m - \frac{t}{h}) (u_{m, ε}^{N} - u_{m + 1, ε}^{N}) ‖}_{{(H^{1} (Ω))}^{'}}^{2} d t \\ \overset{(11, 12)}{⩽} \frac{\tilde{C} h^{2}}{3} . \end{matrix}$ Here we apply $\begin{matrix} (11) \int_{m h}^{(m + 1) h} {(1 + m - \frac{t}{h})}^{2} d t = - h \int_{m h}^{(m + 1) h} - \frac{1}{h} {(1 + m - \frac{t}{h})}^{2} d t \\ = - h {[\frac{1}{3} {(1 + m - \frac{t}{h})}^{3}]}_{m h}^{(m + 1) h} = \frac{h}{3}, \\ (12) \sum_{m = 0}^{N - 1} {‖ u_{m + 1, ε}^{N} - u_{m, ε}^{N} ‖}_{{(H^{1} (Ω))}^{'}}^{2} = h {‖ \frac{\partial}{\partial t} (Π_{N}^{1} u_{ε}^{N}) ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})}^{2} ⩽ \tilde{C} h . \end{matrix}$ □

2.3.2. Passing to the limit

In the following, we devote to the existence of weak solutions to the non-degenerate problem (2.2)–(2.5) as $N \to \infty$ .

Proposition 2.2.
There exist subsequences of $Π_{N}^{0} u_{ε}^{N}$ and $Π_{N}^{1} u_{ε}^{N}$ , for simplicity of notation, still denote by ${(Π_{N}^{0} u_{ε}^{N})}_{N}$ , and function $u_{ε}$ such that $\begin{array}{l} (2.38) & Π_{N}^{0} u_{ε}^{N} ⇀ u_{ε} weakly-* in L^{\infty} (Ω_{T}), \\ (2.39) & Π_{N}^{0} u_{ε}^{N} ⇀ u_{ε} weakly in L^{2} (0, T; H^{3} (Ω)), \\ (2.40) & Π_{N}^{0} u_{ε}^{N} \to u_{ε} strongly in L^{2} (Ω_{T}) and a.e. in Ω_{T}, \\ (2.41) & \nabla Π_{N}^{0} u_{ε}^{N} \to \nabla u_{ε} strongly in L^{2} (Ω_{T}) and a.e. in Ω_{T}, \\ (2.42) & Π_{N}^{1} u_{ε}^{N} ⇀ u_{ε} weakly-* in L^{\infty} (Ω_{T}), \\ (2.43) & \frac{\partial}{\partial t} (Π_{N}^{1} u_{ε}^{N}) ⇀ \frac{\partial}{\partial t} u_{ε} weakly in L^{2} (0, T; {(H^{1} (Ω))}^{'}), \\ (2.44) & Π_{N}^{1} u_{ε}^{N} \to u_{ε} strongly in C ([0, T]; {(H^{1} (Ω))}^{'}) \end{array}$ as $N \to \infty$ . Moreover, $u_{ε}$ verifies that $\begin{array}{l} (2.45) & u_{ε} (x, 0) = u_{0, ε} (x), a.e. x \in Ω, \\ (2.46) & 0 ⩽ u_{ε} (x, t) ⩽ 1, a.e. (x, t) \in Ω_{T} . \end{array}$
Proof.
we note that (2.38), (2.39) and (2.43) are straightforward applications of the estimations of (2.27), (2.28) and (2.31) of Proposition 2.1.

We prove (2.40). Since (2.39) and (2.29) are space and time translate estimates of the approximate solutions $u_{ε}$ , respectively, by applying the Kolmogorov’s compactness criterion, we can deduce that the sequence ${(Π_{N}^{0} u_{ε}^{N})}_{n}$ is relatively compact in $L^{2} (Ω_{T})$ . There exists a subsequence such that (2.40) is hold. Similarly, (2.39) and (2.30) are space and time translate estimates, it follows (2.41).

The assertion (2.42) is similar to the proof of (2.38). Since the interpolation operators $Π_{N}^{0}$ and $Π_{N}^{1}$ have the same $L^{\infty}$ -norm defined in (2.13).

Now we show that (2.44) holds as $N \to \infty$ . Estimates (2.31) and (2.42) permit to prove that there exist ξ and a subsequence of $u_{ε}^{N}$ such that $\begin{matrix} (2.47) & \begin{matrix} Π_{N}^{1} u_{ε}^{N} ⇀ ξ weakly-* in L^{\infty} (Ω_{T}), \\ \frac{\partial}{\partial t} (Π_{N}^{1} u_{ε}^{N}) ⇀ \frac{\partial}{\partial t} ξ weakly in L^{2} (0, T; {(H^{1} (Ω))}^{'}) \end{matrix} \end{matrix}$ as $N \to \infty$ . Since the space $L^{\infty} (Ω)$ is compactly embedded into ${(H^{1} (Ω))}^{'}$ , by J. Simon [30], it has $\begin{matrix} (2.48) & Π_{N}^{1} u_{ε}^{N} \to ξ strongly in C ([0, T]; {(H^{1} (Ω))}^{'}) \end{matrix}$ as $N \to \infty$ . Furthermore, the estimate (2.32) means that $\begin{matrix} {‖ Π_{N}^{1} u_{ε}^{N} - Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})} \to 0 \end{matrix}$ as $N \to \infty$ . Associating (2.40), (2.48) with (2.49), the uniqueness of a limit means that $\begin{matrix} (2.49) & Π_{N}^{1} u_{ε}^{N} ⇀ ξ = u_{ε} strongly in C ([0, T]; {(H^{1} (Ω))}^{'}) . \end{matrix}$ From (2.44), we have $\begin{matrix} Π_{N}^{1} u_{ε}^{N} (x, 0) \to u_{ε} (x, 0) strongly in {(H^{1} (Ω))}^{'} . \end{matrix}$ By using the definition of $Π_{N}^{1}$ -(2.10), for $\forall N$ , it has $\begin{matrix} Π_{N}^{1} u_{ε}^{N} (x, 0) = u_{0, ε} (x) . \end{matrix}$ Hence, it follows that $\begin{matrix} u_{ε} (x, 0) = u_{0, ε} (x) . \end{matrix}$

Finally, let us prove (2.46). Define $\tilde{m} = [t \frac{N}{T}] + 1 = [\frac{t}{h}] + 1$ . $\forall t \in (0, T)$ , we note that $\begin{matrix} Π_{N}^{0} u_{ε}^{N} (x, t) = u_{\tilde{m}, ε}^{N} (x) . \end{matrix}$ We have $\tilde{m} h = ([t \frac{N}{T}] + 1) \frac{T}{N} \to t$ as $N \to \infty$ . Moreover, from (2.40), we have $\begin{matrix} (2.50) & Π_{N}^{0} u_{ε}^{N} (x, t) \to u_{ε} (x, t) a.e. in Ω_{T} \end{matrix}$ as $N \to \infty$ . Hence, if $0 ⩽ u_{\tilde{m}, ε}^{N} (x) ⩽ 1$ , then $0 ⩽ u_{ε} (x, t) ⩽ 1$ which is guarded by dominated convergence theorem. This completes Proposition 2.2. □

It remains to check whether the variational equations (2.8) still hold to prove $u_{ε}$ is a weak solution to (2.2)–(2.5). Under (2.15), for $\forall φ \in L^{2} (0, T; H^{1} (Ω))$ , we have $\begin{matrix} (2.51) & \begin{matrix} \int_{0}^{T} ⟨ \partial_{t} (Π_{N}^{1} u_{ε}^{N}), φ) ⟩_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} d t \\ + \int_{Ω_{T}} M_{ε} (Π_{N}^{0} u_{ε}^{N}) \nabla (- ν Δ Π_{N}^{0} u_{ε}^{N} + ψ^{'} (Π_{N}^{0} u_{ε}^{N})) \cdot \nabla φ d (x, t) = 0 . \end{matrix} \end{matrix}$ Under (2.43), we have $\begin{matrix} (2.52) & \int_{0}^{T} ⟨ \partial_{t} (Π_{N}^{1} u_{ε}^{N}), φ) ⟩_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} d t \to \int_{0}^{T} {⟨ \partial_{t} u_{ε}, φ ⟩}_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} d t \end{matrix}$ as $N \to \infty$ . We first show that $\begin{matrix} (2.53) & \int_{Ω_{T}} M_{ε} (Π_{N}^{0} u_{ε}^{N}) \nabla Δ Π_{N}^{0} u_{ε}^{N} \cdot \nabla φ d (x, t) \to \int_{Ω_{T}} M_{ε} (u_{ε}) \nabla Δ u_{ε}^{N} \cdot \nabla φ d (x, t) \end{matrix}$ as $N \to \infty$ . Actually, from (2.40), we have $\begin{matrix} Π_{N}^{0} u_{ε}^{N} \to u_{ε} a.e. in Ω_{T} . \end{matrix}$ Since $M_{ε} (u_{ε})$ is continuous with respect to $u_{ε}$ , it has $\begin{matrix} (2.54) & M_{ε} (Π_{N}^{0} u_{ε}^{N}) \to M_{ε} (u_{ε}) a.e. in Ω_{T} . \end{matrix}$ By applying the dominated convergence of Lebesgue, it has $\begin{matrix} | \int_{Ω_{T}} [M_{ε} (Π_{N}^{0} u_{ε}^{N}) \nabla Δ Π_{N}^{0} u_{ε}^{N} - M_{ε} (u_{ε}) \nabla Δ (u_{ε}^{N})] \cdot \nabla φ d (x, t) | \\ ⩽ | \int_{Ω_{T}} (M_{ε} (Π_{N}^{0} u_{ε}^{N}) - M_{ε} (u_{ε})) \nabla Δ Π_{N}^{0} u_{ε}^{N} \cdot \nabla φ d (x, t) | \\ + | \int_{Ω_{T}} M_{ε} (u_{ε}) (\nabla Δ Π_{N}^{0} u_{ε}^{N} - \nabla Δ u_{ε}^{N}) \cdot \nabla φ d (x, t) | \\ ⩽ {‖ (M_{ε} (Π_{N}^{0} u_{ε}^{N}) - M_{ε} (u_{ε})) \nabla φ ‖}_{L^{2} (Ω_{T})} {‖ \nabla Δ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (Ω_{T})} \\ + {‖ M_{ε} (u_{ε}) \nabla φ ‖}_{L^{2} (Ω_{T})} {‖ \nabla Δ Π_{N}^{0} u_{ε}^{N} - \nabla Δ u_{ε} ‖}_{L^{2} (Ω_{T})} \to 0 \end{matrix}$ as $N \to \infty$ . Here we apply Hölder inequality with $p = 2, q = 2$ , (2.54) and (2.39). Hence, it follows that (2.53). Finally, we show that $\begin{matrix} (2.55) & \int_{Ω_{T}} M_{ε} (Π_{N}^{0} u_{ε}^{N}) \nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) \cdot \nabla φ d (x, t) \to \int_{Ω_{T}} M_{ε} (u_{ε}) \nabla ψ^{'} (u_{ε}^{N}) \cdot \nabla φ d (x, t) \end{matrix}$ as $N \to \infty$ . Similar to (2.54), we have $\begin{matrix} (2.56) & \begin{matrix} \nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) \to \nabla ψ^{'} (u_{ε}^{N}) a.e. in Ω_{T} . \end{matrix} \end{matrix}$ It suffices to show that $\begin{matrix} | \int_{Ω_{T}} M_{ε} (Π_{N}^{0} u_{ε}^{N}) \nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) \cdot \nabla φ - M_{ε} (u_{ε}) \nabla ψ^{'} (u_{ε}^{N}) \cdot \nabla φ d (x, t) | \\ ⩽ | \int_{Ω_{T}} (M_{ε} (Π_{N}^{0} u_{ε}^{N}) - M_{ε} (u_{ε})) \nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) \cdot \nabla φ d (x, t) | \\ + | \int_{Ω_{T}} M_{ε} (u_{ε}) (\nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) - \nabla ψ^{'} (u_{ε}^{N})) \cdot \nabla φ d (x, t) | \\ ⩽ {‖ (M_{ε} (Π_{N}^{0} u_{ε}^{N}) - M_{ε} (u_{ε})) \nabla φ ‖}_{L^{2} (Ω_{T})} {‖ \nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) ‖}_{L^{2} (Ω_{T})} \\ + m_{2} {‖ (\nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) - \nabla ψ^{'} (u_{ε}^{N})) \nabla φ ‖}_{L^{2} (Ω_{T})} \to 0 . \end{matrix}$ Here we apply Hölder inequality with $p = 2, q = 2$ and (2.54). Actually, similar to the proof of (2.34), by applying the assumption (1.17), Sobolev embedding theorem and (2.28), we have $\begin{array}{l} {‖ \nabla ψ^{'} (Π_{N}^{0} u_{ε}^{N}) ‖}_{L^{2} (Ω_{T})} & = \int_{Ω_{T}} {| ψ^{″} (Π_{N}^{0} u_{ε}^{N}) |}^{2} {| \nabla (Π_{N}^{0} u_{ε}^{N}) |}^{2} d (x, t) \\ ⩽ \int_{0}^{T} {(\int_{Ω} {| ψ^{″} (Π_{N}^{0} u_{ε}^{N}) |}^{n} d x)}^{\frac{2}{n}} {(\int_{Ω} {| \nabla (Π_{N}^{0} u_{ε}^{N}) |}^{\frac{2 n}{n - 2}} d x)}^{\frac{n - 2}{n}} d t \\ ⩽ C \int_{0}^{T} (1 + {‖ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{n (s - 1)} (Ω)}^{2 (s - 1)}) {‖ \nabla (Π_{N}^{0} u_{ε}^{N}) ‖}_{L^{\frac{2 n}{n - 2}} (Ω)}^{2} d t \\ ⩽ C {‖ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T; H^{3} (Ω))}^{2} + C {‖ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{\infty} (Ω_{T})}^{2 (s - 1)} {‖ Π_{N}^{0} u_{ε}^{N} ‖}_{L^{2} (0, T; H^{3} (Ω)}^{2} ⩽ C . \end{array}$ Therefore, we obtain the convergence of (2.15) to the weak formulation of (2.2)–(2.4). This completes the proof of Theorem 2.1.
3. Weak solutions to the degenerate problem

The purpose of this section is to send the regularized parameter $ε \to 0$ in the sequence of solutions ${u_{ε}}$ to the non-degenerate system (2.2)–(2.5) to obtain a weak solution to the original degenerate system (1.1)–(1.4). In Section 2, for each $ε > 0$ , we have presented the existence of a solution $u_{ε}$ to (2.2)–(2.5) such that $0 ⩽ u_{ε} ⩽ 1$ a. e. in $Ω_{T}$ . By applying the compact theorem to uniform priori estimates, we conclude that the existence of weak solutions to (1.1)–(1.4). Since $\nabla Δ u_{ε}$ may not have a limit in $L^{2} (Ω)$ , by using (2.8) and integration by parts, we have $\begin{matrix} (3.1) & \begin{matrix} \int_{Ω_{T}} \partial_{t} u_{ε} σ d (x, t) = & \int_{Ω_{T}} M_{ε}^{'} (u_{ε}) \nabla u_{ε} (- ν Δ u_{ε} + ψ^{'} (u_{ε})) \cdot \nabla σ d (x, t) \\ + \int_{Ω_{T}} M_{ε} (u_{ε}) (- ν Δ u_{ε} + ψ^{'} (u_{ε})) Δ σ d (x, t) \end{matrix} \end{matrix}$ for all $σ (x, t) \in C^{2} ({\overline{Ω}}_{T})$ with $\nabla σ \cdot n = 0$ on $\partial Ω \times (0, T)$ . The goal of the following statement is to show that the existence of solutions to (1.1)–(1.4) and obtain the limit of (3.1).

Lemma 3.1.
Define a set $X : = {(x, t) \in Ω_{T} | M (u) \neq 0}$ . Under the assumptions ( 1.14 )–( 1.17 ), for any $u_{0} \in H^{1} (Ω)$ and $T > 0$ , then there exists at least one solution pair $(u, W)$ to the degenerate mobility system ( 1.1 )–( 1.4 ) such that
$u \in L^{\infty} (Ω_{T}) \cap L^{\infty} (0, T; H^{1} (Ω)) \cap C ([0, T]; L^{2} (Ω)) \cap L^{2} (Ω_{T})$ , $\partial_{t} u \in L^{2} (0, T; {(H^{1} (Ω))}^{'})$ ,

$W \in L^{2} (0, T; H^{1} (Ω))$ ,
satisfying ( 1.1 )–( 1.4 ) in the weak sense $\begin{array}{l} \int_{0}^{T} {⟨ \partial_{t} u, φ ⟩}_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} d t = - \int_{Ω_{T}} M (u) \nabla W \cdot \nabla φ d (x, t), \\ \int_{Ω_{T}} W φ d (x, t) = ν \int_{Ω_{T}} \nabla u \cdot \nabla φ d (x, t) + \int_{Ω_{T}} ψ^{'} (u) φ d (x, t) \end{array}$ for all $φ \in L^{2} (0, T; H^{1} (Ω))$ .
Proof.
Under the free energy (1.10), it has $\begin{array}{l} (3.2) & E (u_{ε}) = \int_{Ω} (\frac{ν}{2} | \nabla u_{ε} |^{2} + ψ (u_{ε})) d x . \end{array}$ Differenting (3.2) by t, one obtains $\begin{matrix} \frac{d}{d t} E (u_{ε}) & = \frac{d}{d t} \int_{Ω} (\frac{ν}{2} | \nabla u_{ε} |^{2} + ψ (u_{ε})) d x = \int_{Ω} (ν (\nabla u_{ε}) \cdot (\nabla \partial_{t} u_{ε}) + ψ^{'} (u_{ε}) \partial_{t} u_{ε}) d x \\ = \int_{Ω} (- ν Δ u_{ε} + ψ^{'} (u_{ε})) \partial_{t} u_{ε} d x = \int_{Ω} W_{ε} \nabla \cdot (M_{ε} (u_{ε}) \nabla W_{ε}) d x \\ = - M_{ε} (u_{ε}) \int_{Ω} | \nabla W_{ε} |^{2} d x ⩽ 0 . \end{matrix}$ Here we apply the integration by parts and (2.2)–(2.3). This implies that $\begin{matrix} (3.3) & \begin{matrix} \int_{Ω} \frac{ν}{2} {| \nabla u_{ε} (x, T) |}^{2} d x + \int_{Ω} ψ (u_{ε} (x, T)) d x + \int_{Ω_{T}} M_{ε} (u_{ε} (x, t)) {| \nabla W_{ε} (x, t) |}^{2} d (x, t) \\ = \int_{Ω} \frac{ν}{2} {| \nabla u_{ε} (x, 0) |}^{2} d x + \int_{Ω} ψ (u_{ε} (x, 0)) d x \\ ⩽ \frac{ν}{2} {‖ \nabla u_{ε} (x, 0) ‖}_{L^{2} (Ω)}^{2} + C \int_{Ω} (1 + {| u_{ε} (x, 0) |}^{s + 1}) d x \\ ⩽ \frac{ν}{2} {‖ u_{0} (x) ‖}_{H^{1} (Ω)}^{2} + C ({‖ u_{0} (x) ‖}_{H^{1} (Ω)}^{s + 1} + | Ω |) ⩽ C . \end{matrix} \end{matrix}$ Here $C > 0$ is a constant which is independ on ε. We apply the assumption (1.15) and Sobolev embedding theorem $H^{1} (Ω) ↪ L^{s + 1} (Ω)$ such that $\begin{matrix} \{\begin{matrix} 0 < s + 1 < \infty, & if n = 1, 2, \\ 0 < s + 1 < \frac{n}{n - 2} + 1 < \frac{2 n}{n - 2}, & if n ⩾ 3, \end{matrix} \end{matrix}$ which is reasonable by applying (1.15). Under the mass-conserved condition , one has $\begin{array}{l} (3.4) & | \int_{Ω} u_{ε} (x, t) d x | = | \int_{Ω} u_{ε} (x, 0) d x | ⩽ C . \end{array}$ Associating (3.3), (3.4) with Poincaré’s inequality, we deduce that $\begin{array}{l} (3.5) & u_{ε} \in L^{\infty} (0, T; H^{1} (Ω)) . \end{array}$ From (3.3) and the assumption of $M_{ε} (u_{ε})$ , we have $\begin{array}{l} ‖ \nabla W_{ε} ‖_{L^{2} (Ω_{T})} ⩽ C . \end{array}$ Poincaré’s inequality implies that $\begin{array}{l} (3.6) & W_{ε} \in L^{2} (0, T; H^{1} (Ω)) . \end{array}$ By applying Proposition 2.2, (3.3), (3.5) and (3.6), we obtain $\begin{matrix} (3.7) & \begin{matrix} {‖ u_{ε} (x, t) ‖}_{L^{\infty} (Ω_{T})} ⩽ C, \\ {‖ u_{ε} (x, t) ‖}_{L^{\infty} (0, T; H^{1} (Ω))} ⩽ C, \\ {‖ \partial_{t} u_{ε} (x, t) ‖}_{L^{2} (0, T; {(H^{1} (Ω))}^{'})} ⩽ C, \\ {‖ W_{ε} (x, t) ‖}_{L^{2} (0, T; H^{1} (Ω))} ⩽ C, \\ {‖ {(M_{ε} (u_{ε} (x, t)))}^{\frac{1}{2}} \nabla W_{ε} (x, t) ‖}_{L^{2} (Ω_{T})} ⩽ C . \end{matrix} \end{matrix}$ Here C represent different positive constants, which is independent of ε. Now we consider the asymptotic behavior as $ε \to 0$ . Under Proposition 2.2, (3.7) and the J. Simon Theorem [30], we can exact subsequences, which we do not rebel, $\begin{matrix} (3.8) & \begin{matrix} u_{ε} ⇀ u weakly- in L^{\infty} (Ω_{T}), \\ u_{ε} ⇀ u weakly- in L^{\infty} (0, T; H^{1} (Ω)), \\ \partial_{t} u_{ε} ⇀ \partial_{t} u weakly in L^{2} (0, T; {(H^{1} (Ω))}^{'}), \\ u_{ε} \to u strongly in L^{2} (Ω_{T}) and a.e. in Ω_{T}, \\ \nabla u_{ε} \to \nabla u strongly in L^{2} (Ω_{T}) and a.e. in Ω_{T}, \\ u_{ε} \to u strongly in C ([0, T]; L^{2} (Ω)), \\ W_{ε} \to W strongly in L^{2} (0, T; H^{1} (Ω)) . \end{matrix} \end{matrix}$ From the hypothesis (1.14), we have $\begin{array}{l} {(M_{ε} (u_{ε}))}^{\frac{1}{2}} ⩽ C (1 + | u_{ε} |^{r}) . \end{array}$ It follows that $\begin{matrix} {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} ‖}_{L^{n} (Ω)} ⩽ C {(\int_{Ω} | u_{ε} |^{n r} + 1 d x)}^{\frac{1}{n}} ⩽ C ‖ u_{ε} ‖_{L^{n r}}^{r} + C \overset{(18)}{⩽} C ‖ u_{ε} ‖_{H^{1} (Ω)}^{r} + C, \end{matrix}$ which means that $\begin{matrix} (3.9) & {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \in L^{\infty} (0, T; L^{n} (Ω)) . \end{matrix}$ Here we apply (18): Sobolev embedding theorem $H^{1} (Ω) ↪ L^{n r} (Ω)$ such that $\begin{matrix} \{\begin{matrix} 0 < r n < \infty, & if n = 1, 2, \\ 0 < r n < \frac{2 n}{n - 2}, & if n ⩾ 3, \end{matrix} \end{matrix}$ We also have $\begin{matrix} {‖ M_{ε} (u_{ε}) ‖}_{L^{\frac{n}{2}} (Ω)} = {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} ‖}_{L^{n} (Ω)}^{2} ⩽ C ‖ u_{ε} ‖_{H^{1} (Ω)}^{2} + C . \end{matrix}$ Here the constant C is independent of ε. This implies that $\begin{matrix} (3.10) & M_{ε} (u_{ε}) \in L^{\infty} (0, T; L^{\frac{n}{2}} (Ω)) . \end{matrix}$ From the assumption (2.1), one has the uniform convergence $\begin{array}{l} (3.11) & M_{ε} \to M, {(M_{ε})}^{\frac{1}{2}} \to M^{\frac{1}{2}} \end{array}$ as $ε \to 0$ . Combining (3.8), (3.9), (3.10) with (3.11), the general dominated convergence theorem indicates that $\begin{matrix} (3.12) & \begin{matrix} M_{ε} (u_{ε}) \to M (u) strongly in C ([0, T]; L^{\frac{n}{2}} (Ω)), \\ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \to {(M (u))}^{\frac{1}{2}} strongly in C ([0, T]; L^{n} (Ω)) \end{matrix} \end{matrix}$ as $ε \to 0$ . Combining (3.7), (3.8) with (3.12), one obtains $\begin{array}{l} (3.13) & {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} ⇀ {(M (u))}^{\frac{1}{2}} \nabla W weakly in L^{2} (Ω_{T}) . \end{array}$

Since $M (u)$ is allowed to degenerate at 0 and 1, we define a set $X : = {(x, t) \in Ω_{T} | M (u) \neq 0}, X = X^{'} \times (0, T), X^{'} \subset Ω$ . In the following asymptotic analysis, we discuss the solutions in set $Ω_{T} ∖ X$ and set $Ω_{T} \cap X$ , respectively. We note that $M_{ε} (u_{ε}) \nabla W_{ε} = {(M_{ε} (u_{ε}))}^{\frac{1}{2}} {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε}$ , for any $ζ \in L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω))$ , one has $\begin{matrix} (3.14) & \begin{matrix} | \int_{Ω_{T}} (M_{ε} (u_{ε}) \nabla W_{ε} - M (u) \nabla W) ζ d (x, t) | \\ ⩽ | \int_{Ω_{T} \cap X} (M_{ε} (u_{ε}) \nabla W_{ε} - M (u) \nabla W) ζ d (x, t) | \\ + | \int_{Ω_{T} ∖ X} (M_{ε} (u_{ε}) \nabla W_{ε} - M (u) \nabla W) ζ d (x, t) | \\ : = K_{1} + K_{2} . \end{matrix} \end{matrix}$ Since $M (u) = 0$ and $\nabla u = 0$ on $Ω_{T} ∖ X$ together with (3.7), it follows that $\begin{array}{l} K_{2} & ⩽ | \int_{Ω_{T} ∖ X} M_{ε} (u_{ε}) \nabla W_{ε} ζ d (x, t) | + | \int_{Ω_{T} ∖ X} M (u) \nabla W ζ d (x, t) | \\ = | \int_{Ω_{T} ∖ X} M_{ε} (u_{ε}) \nabla W_{ε} ζ d (x, t) | \\ = | \int_{Ω_{T} ∖ X} {(M_{ε} (u_{ε}))}^{\frac{1}{2}} {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} ζ d (x, t) | \\ ⩽ {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} ‖}_{C ([0, T]; L^{n} (Ω ∖ X^{'}))} {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} ‖}_{L^{2} (Ω_{T} ∖ X)} ‖ ζ ‖_{L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω ∖ X^{'}))} \\ ⩽ C {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} ‖}_{C ([0, T]; L^{n} (Ω ∖ X^{'}))} ‖ ζ ‖_{L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω ∖ X^{'}))} \\ \to C {‖ {(M (u))}^{\frac{1}{2}} ‖}_{C ([0, T]; L^{n} (Ω ∖ X^{'}))} ‖ ζ ‖_{L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω ∖ X^{'}))} \to 0, \end{array}$ as $ε \to 0$ . Here we apply Hölder inequality with $p = n, q = 2, r = \frac{2 n}{n - 2}$ . Similarly, we have $\begin{matrix} (3.15) & \begin{matrix} K_{1} : = & | \int_{Ω_{T} \cap X} ({(M_{ε} (u_{ε}))}^{\frac{1}{2}} {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} - {(M (u))}^{\frac{1}{2}} {(M (u))}^{\frac{1}{2}} \nabla W) ζ d (x, t) | \\ ⩽ & | \int_{Ω_{T} \cap X} [{(M_{ε} (u_{ε}))}^{\frac{1}{2}} - {(M (u))}^{\frac{1}{2}}] {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} ζ d (x, t) | \\ + | \int_{Ω_{T} \cap X} {(M (u))}^{\frac{1}{2}} [{(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} - {(M (u))}^{\frac{1}{2}} \nabla W] ζ d (x, t) | \\ : = & K_{1 - 1} + K_{1 - 2} . \end{matrix} \end{matrix}$ We note that $\begin{matrix} K_{1 - 1} & ⩽ \int_{0}^{T} {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} - {(M (u))}^{\frac{1}{2}} ‖}_{L^{n} (Ω \cap X^{'})} {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} ‖}_{L^{2} (Ω \cap X^{'})} ‖ ζ ‖_{L^{\frac{2 n}{n - 2}} (Ω \cap X^{'})} d t \\ ⩽ {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} - {(M (u))}^{\frac{1}{2}} ‖}_{C ([0, T]; L^{n} (Ω \cap X^{'}))} {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} ‖}_{L^{2} (Ω_{T} \cap X)} ‖ ζ ‖_{L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω \cap X^{'}))} \\ \to 0 \end{matrix}$ as $ε \to 0$ . Here we apply Hölder inequality with $p = n, q = 2, r = \frac{2 n}{n - 2}$ . From (3.13), it has $\begin{matrix} K_{1 - 2} ⩽ \int_{0}^{T} {‖ {(M_{ε} (u_{ε}))}^{\frac{1}{2}} \nabla W_{ε} - {(M (u))}^{\frac{1}{2}} \nabla W ‖}_{L^{2} (Ω \cap X^{'})} {‖ {(M (u))}^{\frac{1}{2}} ζ ‖}_{L^{2} (Ω \cap X^{'})} d t \to 0 \end{matrix}$ as $ε \to 0$ . Actually, it has $\begin{matrix} {‖ {(M (u))}^{\frac{1}{2}} ζ ‖}_{L^{2} (Ω_{T} \cap X)}^{2} ⩽ & \int_{0}^{T} {‖ M (u) ‖}_{L^{\frac{n}{2}} (Ω \cap X^{'})} ‖ ζ ‖_{L^{\frac{2 n}{n - 2}} (Ω \cap X^{'})}^{2} d t \\ ⩽ & {‖ M (u) ‖}_{C ([0, T]; L^{\frac{n}{2}} (Ω \cap X^{'}))} ‖ ζ ‖_{L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω \cap X^{'}))}^{2} \\ ⩽ & C ‖ ζ ‖_{L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω \cap X^{'}))}^{2} ⩽ C . \end{matrix}$ Here we apply Hölder inequality with $p = \frac{n}{2}, q = \frac{n}{n - 2}$ . From (3.14)–(3.15), it gets for any $ζ \in L^{2} (0, T; L^{\frac{2 n}{n - 2}} (Ω))$ $\begin{matrix} | \int_{0}^{T} \int_{Ω} (M_{ε} (u_{ε}) \nabla W_{ε} - M (u) \nabla W) ζ d (x, t) | \to 0 \end{matrix}$ as $ε \to 0$ . We obtain $\begin{matrix} (3.16) & M_{ε} (u_{ε}) \nabla W_{ε} ⇀ M (u) \nabla W weakly in L^{2} (0, T; L^{\frac{2 n}{n + 2}} (Ω) \end{matrix}$

According to the assumption (1.16), it follows that $\begin{matrix} {‖ ψ^{'} (u_{ε}) ‖}_{L^{2} (Ω)} ⩽ C {(\int_{Ω} {(1 + | u_{ε} |^{s})}^{2} d x)}^{\frac{1}{2}} ⩽ C {‖ u_{ε} (t) ‖}_{L^{2 s} (Ω)}^{s} ⩽ C ‖ u_{ε} ‖_{H^{1} (Ω)}^{s} + C . \end{matrix}$ Here we apply Sobolev embedding theorem $H^{1} (Ω) ↪ L^{2 s} (Ω)$ such that $\begin{matrix} \{\begin{matrix} 2 ⩽ 2 s < \infty, & if n = 1, 2, \\ 2 ⩽ 2 s ⩽ \frac{2 n}{n - 2}, & if n ⩾ 3, \end{matrix} \end{matrix}$ which is reasonable. Together with (3.7), it has $\begin{matrix} (3.17) & ψ^{'} (u_{ε}) \in L^{\infty} (0, T; L^{2} (Ω)) . \end{matrix}$ Accordingly, it has $\begin{matrix} (3.18) & ψ^{'} (u_{ε}) ⇀ ψ^{'} (u) weakly*- in L^{\infty} (0, T; L^{2} (Ω)) . \end{matrix}$ From (2.3), (3.7) and (3.18), by applying the regularity theory, we have $Δ u_{ε} \in L^{2} (Ω_{T})$ . Hence, we have $\begin{matrix} (3.19) & Δ u_{ε} ⇀ Δ u weakly in L^{2} (Ω_{T}) . \end{matrix}$ As for the initial condition, from (3.8), we have $\begin{matrix} u_{ε} (x, 0) \to u (x, 0) a.e. in Ω . \end{matrix}$ Under (2.5) and the uniqueness of limit, we obtain $\begin{matrix} (3.20) & u (x, 0) = u_{0} (x) \end{matrix}$ in $L^{2} (Ω)$ . Synthesizing (3.8), (3.16), (3.18) and (3.20), we completes the proof of Lemma 3.1. □

In the following, we present that $\begin{array}{l} (3.21) & \nabla M_{ε} (u_{ε}) = M_{ε}^{'} (u_{ε}) \nabla u_{ε} \to M^{'} (u) \nabla u strongly in L^{2} (Ω_{T}) . \end{array}$ Actually, it has $\begin{array}{l} (3.22) & \begin{matrix} \int_{Ω_{T}} {| M_{ε}^{'} (u_{ε}) \nabla u_{ε} - M^{'} (u) \nabla u |}^{2} d (x, t) \\ = \int_{Ω_{T} \cap X} {| M_{ε}^{'} (u_{ε}) \nabla u_{ε} - M^{'} (u) \nabla u |}^{2} d (x, t) + \int_{Ω_{T} ∖ X} {| M_{ε}^{'} (u_{ε}) \nabla u_{ε} - M^{'} (u) \nabla u |}^{2} d (x, t) \\ : = L_{1} + L_{2} . \end{matrix} \end{array}$ As for $L_{1}$ , by using (3.11) and assumption of $M_{ε}$ , we have $\begin{array}{l} (3.23) & \begin{aligned} M_{ε}^{'} (u_{ε}) \to M^{'} (u) a.e. in Ω_{T} \cap X . \end{aligned} \end{array}$ Combining (3.8) with (3.23), the generalized Lebesque convergence theorem gives that $\begin{array}{l} (3.24) & \begin{aligned} M_{ε}^{'} (u_{ε}) \nabla u_{ε} \to M^{'} (u) \nabla u a.e. in Ω_{T} \cap X \end{aligned} \end{array}$ as $ε \to 0$ . It follows that $L_{1} \to 0$ as $ε \to 0$ .

As for $L_{2}$ , since $\nabla u = 0$ on $Ω_{T} ∖ X$ , together with (3.8), one obtains $\begin{array}{l} (3.25) & \begin{matrix} L_{2} & ⩽ \int_{Ω_{T} ∖ X} {| M_{ε}^{'} (u_{ε}) \nabla u_{ε} |}^{2} d (x, t) + \int_{Ω_{T} ∖ X} {| M^{'} (u) \nabla u |}^{2} d (x, t) \\ = \int_{Ω_{T} ∖ X} {| M_{ε}^{'} (u_{ε}) \nabla u_{ε} |}^{2} d (x, t) \\ ⩽ C \int_{Ω_{T} ∖ X} | \nabla u_{ε} |^{2} d (x, t) \\ \to C \int_{Ω_{T} ∖ X} | \nabla u |^{2} d (x, t) \to 0 . \end{matrix} \end{array}$ It follows that $L_{2} \to 0$ as $ε \to 0$ . By applying (3.22) and (3.24)–(3.25), we obtain (3.21). Combining Lemma 3.1 with above convergence, we get the limit of (3.1). A solution to (1.1)–(1.4) holds in the following weak sense such that $\begin{matrix} \int_{Ω_{T}} \partial_{t} u σ d (x, t) & = \int_{Ω_{T} \cap X} M^{'} (u) \nabla u (- ν Δ u + ψ^{'} (u)) \cdot \nabla σ d (x, t) \\ + \int_{Ω_{T} \cap X} M (u) (- ν Δ u + ψ^{'} (u)) Δ σ d (x, t) \end{matrix}$ for all $σ (x, t) \in C^{2} ({\overline{Ω}}_{T})$ with $\nabla σ \cdot n = 0$ on $\partial Ω \times (0, T)$ . Thus we obtain (1.19).
4. The global entropy estimates

This section is to establish global entropy estimates under the spirit of previous sections. In Section 2, we choose an entropy $G_{ε} (l) = \int_{l}^{\tilde{r}} \int_{\tilde{τ}}^{\tilde{r}} \frac{1}{M_{ε} (τ)} d τ d \tilde{τ}$ satisfying $G_{ε}^{″} (l) = \frac{1}{M_{ε} (l)}$ . We obtain the basic entropy dissipation in $L^{2} -$ norm. Inspired by [2,27], for the additional regularity, we introduce the entropy $\begin{matrix} (4.1) & \begin{matrix} G_{ε α} (l) = \int_{l}^{\tilde{r}} \int_{\tilde{τ}}^{\tilde{r}} \frac{1}{τ^{α} M_{ε} (τ)} d τ d \tilde{τ} \end{matrix} \end{matrix}$ such that $G_{ε α}^{″} (l) = \frac{1}{l^{α} M_{ε} (l)}$ . We need the following two Lemmas.

Lemma 4.1 ([27]).

If $f (v) \in W^{2, \infty} (R)$ , $v \in H^{2} (Ω)$ and $\frac{\partial v}{\partial n} = 0$ on $\partial Ω \times (0, T)$ , then we have $\begin{array}{rcl} \int_{Ω} f^{'} (v) | \nabla v |^{2} Δ v d x & = & - \frac{1}{3} \int_{Ω} f^{″} (v) | \nabla v |^{4} d x + \frac{2}{3} (\int_{Ω} f (v) {| D^{2} v |}^{2} d x - \int_{Ω} f (v) | Δ v |^{2} d x) \\ (4.2) & + \frac{2}{3} \int_{\partial Ω} f (v) Π (\nabla v) d H^{n - 1} . \end{array}$ Here $Π (\cdot)$ denotes the second fundamental form of $\partial Ω$ . $Π (\nabla v) = - ⟨ \nabla v, D^{2} v, n ⟩$ . $H^{n - 1}$ denotes the $(n - 1)$ -dimensional Hausdorff measure.

Lemma 4.2 ([27]).

Let Ω be of class $C^{0, 1}$ . For each $ε > 0$ , there exists a constant $C_{ε} > 0$ such that $\begin{matrix} (4.3) & ‖ \nabla v ‖_{L^{2} (\partial Ω)} ⩽ ε {‖ D^{2} v ‖}_{L^{2} (Ω)} + C_{ε} ‖ v ‖_{L^{2} (Ω)} \end{matrix}$ for all $v \in H^{2} (Ω)$ .

Proposition 4.1.
Under the assumptions ( 1.14 )–( 1.17 ), let $u_{ε}$ be a solution to the non-degenerate problem ( 2.2 )–( 2.5 ). The real numbers α and β satisfy $0 ⩽ α < \frac{1}{2}$ and $\begin{matrix} (4.4) & \frac{2 - α - \sqrt{(1 - 2 α) (1 + α)}}{3} ⩽ β ⩽ \frac{2 - α + \sqrt{(1 - 2 α) (1 + α)}}{3} . \end{matrix}$ Then we have $\begin{matrix} (4.5) & \begin{matrix} \int_{Ω} G_{ε α} (u_{ε} (x, T)) d x + {\tilde{C}}_{1} \int_{Ω_{T}} u_{ε}^{2 - 2 β - α} (\frac{2}{3} {| D^{2} u_{ε}^{β} |}^{2} + \frac{1}{3} {| Δ u_{ε}^{β} |}^{2}) d (x, t) \\ + {\tilde{C}}_{1} \int_{Ω_{T}} u_{ε}^{- α - 2} | \nabla u_{ε} |^{4} d (x, t) \\ ⩽ \int_{Ω} G_{ε α} (u_{0} (x)) d x + {\tilde{C}}_{2} \int_{Ω_{T}} ({| ψ^{″} (u_{ε}) |}^{2} + 1) u_{ε}^{- α + 2} d (x, t) . \end{matrix} \end{matrix}$ Here the constants ${\tilde{C}}_{1}, {\tilde{C}}_{2}$ are only dependent on the domain Ω.
Proof.
The proof contains two steps. In the first step, we introduce an regularized entropy version of $G_{ε α δ}^{'} (u_{ε} + ξ)$ as a test function in (2.8) and obtain the entropy estimates of $u_{ε}$ . In step 2, we pass to the limit as $ξ \to 0$ and $δ \to 0$ .

Step 1: Consider the following function with the positive parameters $l, δ > 0$ $\begin{matrix} (4.6) & G_{ε α δ} (l) = \int_{l}^{\tilde{r}} \int_{\tilde{τ}}^{\tilde{r}} \frac{1}{{(τ + δ)}^{α} M_{ε} (τ)} d τ d \tilde{τ} > 0 . \end{matrix}$ In Section 2, we have $0 ⩽ u_{ε} (x, t) ⩽ 1$ a.e. in $Ω_{T}$ if $0 ⩽ u_{0} (x) ⩽ 1$ . Hence, we introduce $u_{ε} + ξ$ which denoting the shift of $u_{ε}$ by ξ. Since $u_{ε} \in L^{\infty} (0, T; H^{1} (Ω)) \cap L^{2} (0, T; H^{3} (Ω))$ , the regularity of $u_{ε}$ guarantees the reasonability of these test functions $G_{ε α δ}^{'} (u_{ε} + ξ)$ . We obtain $\begin{matrix} (4.7) & \begin{matrix} \int_{0}^{T} {⟨ \partial_{t} u_{ε}, G_{ε α δ}^{'} (u_{ε} + ξ) ⟩}_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} d t \\ + \int_{Ω_{T}} M_{ε} (u_{ε}) (- ν \nabla Δ u_{ε} + \nabla ψ^{'} (u_{ε})) \cdot \nabla G_{ε α δ}^{'} (u_{ε} + ξ) d (x, t) = 0 . \end{matrix} \end{matrix}$ Obviously, the first term of the left side of (4.7) is $\begin{matrix} (4.8) & \begin{matrix} \int_{0}^{T} {⟨ \partial_{t} u_{ε}, G_{ε α δ}^{'} (u_{ε} + ξ) ⟩}_{{(H^{1} (Ω))}^{'}, H^{1} (Ω)} d t \\ = \int_{Ω} G_{ε α δ} (u_{ε} + ξ) (x, T) d x - \int_{Ω} G_{ε α δ} (u_{0} + ξ) (x) d x . \end{matrix} \end{matrix}$ In the following, we shall make the calculations explicit of test function $G_{ε α δ}^{'} (u_{ε} + ξ)$ . For almost all $t \in (0, T)$ , the second term of the left side of (4.7) is $\begin{matrix} (4.9) & \begin{matrix} \int_{0}^{T} \int_{Ω} M_{ε} (u_{ε}) (- ν \nabla Δ u_{ε} + \nabla ψ^{'} (u_{ε})) G_{ε α δ}^{″} (u_{ε} + ξ) \cdot \nabla u_{ε} d (x, t) \\ = \int_{0}^{T} [- ν \int_{Ω} M_{ε} (u_{ε}) \nabla Δ u_{ε} G_{ε α δ}^{″} (u_{ε} + ξ) \cdot \nabla u_{ε} d x \\ + \int_{Ω} M_{ε} (u_{ε}) \nabla ψ^{'} (u_{ε}) G_{ε α δ}^{″} (u_{ε} + ξ) \cdot \nabla u_{ε} d x] d t \\ : = \int_{0}^{T} ν {\tilde{I}}_{1} + {\tilde{I}}_{2} d t . \end{matrix} \end{matrix}$ By applying the integration by parts and the boundary condition (2.4), we obtain $\begin{array}{l} {\tilde{I}}_{1} = & - \int_{Ω} M_{ε} (u_{ε}) \nabla Δ u_{ε} \frac{1}{{(u_{ε} + ξ + δ)}^{α} M_{ε} (u_{ε} + ξ)} \cdot \nabla u_{ε} d x \\ = & \int_{Ω} \nabla \cdot (M_{ε} (u_{ε}) {(u_{ε} + ξ + δ)}^{- α} \frac{1}{M_{ε} (u_{ε} + ξ)} \nabla u_{ε}) Δ u_{ε} d x \\ = & \int_{Ω} \frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} d x \\ + \int_{Ω} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} {(u_{ε} + ξ + δ)}^{- α})}_{u_{ε}} | \nabla u_{ε} |^{2} Δ u_{ε} d x \\ : = & {\tilde{I}}_{1 - 1} + {\tilde{I}}_{1 - 2} . \end{array}$ The terms ${\tilde{I}}_{1 - 1}$ and ${\tilde{I}}_{1 - 2}$ can be rewritten as $\begin{matrix} {\tilde{I}}_{1 - 1} = \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} d x + \int_{Ω} (\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1) {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} d x \\ : = {\tilde{I}}_{1 - 1 - (1)} + {\tilde{I}}_{1 - 1 - (2)} . \\ {\tilde{I}}_{1 - 2} = \int_{Ω} [{(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)})}_{u_{ε}} {(u_{ε} + ξ + δ)}^{- α} \\ + \frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} {({(u_{ε} + ξ + δ)}^{- α})}_{u_{ε}}] | \nabla u_{ε} |^{2} Δ u_{ε} d x \\ = \int_{Ω} [\frac{M_{ε}^{'} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - \frac{M_{ε} (u_{ε}) M_{ε}^{'} (u_{ε} + ξ)}{{(M_{ε} (u_{ε} + ξ))}^{2}}] {(u_{ε} + ξ + δ)}^{- α} | \nabla u_{ε} |^{2} Δ u_{ε} d x \\ + \int_{Ω} \frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} {({(u_{ε} + ξ + δ)}^{- α})}_{u_{ε}} | \nabla u_{ε} |^{2} Δ u_{ε} d x \\ : = {\tilde{I}}_{1 - 2 - (1)} + {\tilde{I}}_{1 - 2 - (2)} . \end{matrix}$ By using Hölder inequality and Young’s inequality, we get $\begin{matrix} {\tilde{I}}_{1 - 2 - (1)} & ⩽ C \int_{Ω} \frac{ξ}{{(u_{ε} + ξ)}^{2}} {(u_{ε} + ξ + δ)}^{- α} | \nabla u_{ε} |^{2} Δ u_{ε} d x \\ = \int_{Ω} \frac{{(u_{ε} + ξ + δ)}^{- \frac{α}{2}}}{u_{ε} + ξ} | \nabla u_{ε} |^{2} {(u_{ε} + ξ + δ)}^{- \frac{α}{2}} Δ u_{ε} \frac{ξ}{u_{ε} + ξ} d x \\ ⩽ ε_{2} \int_{Ω} \frac{{(u_{ε} + ξ + δ)}^{- α}}{{(u_{ε} + ξ)}^{2}} | \nabla u_{ε} |^{4} d x + C_{ε_{2}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} {(\frac{ξ}{u_{ε} + ξ})}^{2} d x . \end{matrix}$ It follows that $\begin{matrix} {\tilde{I}}_{1 - 2 - (1)} ⩾ - ε_{2} \int_{Ω} \frac{{(u_{ε} + ξ + δ)}^{- α}}{{(u_{ε} + ξ)}^{2}} | \nabla u_{ε} |^{4} d x - C_{ε_{2}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} {(\frac{ξ}{u_{ε} + ξ})}^{2} d x . \end{matrix}$ Here we apply the inequality $| \frac{M_{ε}^{'} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - \frac{M_{ε} (u_{ε}) M_{ε}^{'} (u_{ε} + ξ)}{{(M_{ε} (u_{ε} + ξ))}^{2}} | ⩽ \frac{C ξ}{{(u_{ε} + ξ)}^{2}}$ . Similar to ${\tilde{I}}_{1 - 1}$ , ${\tilde{I}}_{1 - 2 - (2)}$ can be rewritten as $\begin{matrix} {\tilde{I}}_{1 - 2 - (2)} = & \int_{Ω} {({(u_{ε} + ξ + δ)}^{- α})}_{u_{ε}} | \nabla u_{ε} |^{2} Δ u_{ε} d x \\ - α \int_{Ω} {(u_{ε} + ξ + δ)}^{- α - 1} (\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1) | \nabla u_{ε} |^{2} Δ u_{ε} d x \\ : = & {\tilde{I}}_{1 - 2 - (2) a} + {\tilde{I}}_{1 - 2 - (2) b} . \end{matrix}$ From Hölder inequality and Young’s inequality, we obtain $\begin{array}{rcl} {\tilde{I}}_{1 - 2 - (2) b} & = & - α \int_{Ω} {(u_{ε} + ξ + δ)}^{\frac{- α - 2}{2}} | \nabla u_{ε} |^{2} {(u_{ε} + ξ + δ)}^{\frac{- α}{2}} (\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1) Δ u_{ε} d x \\ ⩽ & ε_{1} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} \\ + C_{ε_{1}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} | Δ u_{ε} |^{2} d x . \end{array}$ It has $\begin{array}{rcl} {\tilde{I}}_{1 - 2 - (2) b} & ⩾ & - ε_{1} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} \\ - C_{ε_{1}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} | Δ u_{ε} |^{2} d x . \end{array}$ By using the identity $\begin{matrix} D_{x_{i} x_{j}} v^{β} = β (β - 1) v^{β - 2} D_{x_{i}} v D_{x_{j}} v + β v^{β - 1} D_{x_{i} x_{j}} v, \end{matrix}$ we have $\begin{matrix} D_{x_{i} x_{j}} v = β^{- 1} v^{1 - β} D_{x_{i} x_{j}} v^{β} - (β - 1) v^{- 1} D_{x_{i}} v D_{x_{j}} v . \end{matrix}$ Setting $v = u_{ε} + ξ$ in the above identity, we obtain $\begin{matrix} (4.10) & \begin{matrix} {\tilde{I}}_{1 - 1 - (1)} = & \int_{Ω} β^{- 2} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{2 - 2 β} {| Δ {(u_{ε} + ξ)}^{β} |}^{2} d x \\ - {(β - 1)}^{2} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 2} | \nabla u_{ε} |^{4} d x \\ - 2 (β - 1) \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 1} Δ u_{ε} | \nabla u_{ε} |^{2} d x . \end{matrix} \end{matrix}$ From Lemma 4.1, setting $f (v) = {(u_{ε} + ξ + δ)}^{- α}$ in (4.2), we obtain $\begin{matrix} (4.11) & \begin{matrix} {\tilde{I}}_{1 - 2 - (2) a} = & - \frac{1}{3} \int_{Ω} α (α + 1) {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d x \\ + \frac{2}{3} β^{- 2} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{2 - 2 β} [{| D^{2} {(u_{ε} + ξ)}^{β} |}^{2} - {| Δ {(u_{ε} + ξ)}^{β} |}^{2}] d x \\ + 2 (β - 1) \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 1} Δ u_{ε} | \nabla u_{ε} |^{2} d x \\ + \frac{2}{3} (β - 1) \int_{Ω} {[{(u_{ε} + ξ)}^{- 1} {(u_{ε} + ξ + δ)}^{- α}]}_{u_{ε}} | \nabla u_{ε} |^{4} d x \\ + \frac{2}{3} \int_{\partial Ω} {(u_{ε} + ξ + δ)}^{- α} Π (\nabla u_{ε}) d H^{n - 1} . \end{matrix} \end{matrix}$ Adding (4.10) to (4.11), we have $\begin{matrix} (4.12) & \begin{matrix} {\tilde{I}}_{1 - 1 - (1)} + {\tilde{I}}_{1 - 2 - (2) a} \\ = \int_{Ω} β^{- 2} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{2 - 2 β} [\frac{2}{3} {| D^{2} {(u_{ε} + ξ)}^{β} |}^{2} + \frac{1}{3} {| Δ {(u_{ε} + ξ)}^{β} |}^{2}] d x \\ + \int_{Ω} {- \frac{α (1 + α)}{3} {(u_{ε} + ξ + δ)}^{- α - 2} + \frac{2}{3} (β - 1) {[{(u_{ε} + ξ)}^{- 1} {(u_{ε} + ξ + δ)}^{- α}]}_{u_{ε}} \\ - {(β - 1)}^{2} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 2}} | \nabla u_{ε} |^{4} d x \\ + \frac{2}{3} \int_{\partial Ω} {(u_{ε} + ξ + δ)}^{- α} Π (\nabla u_{ε}) d H^{n - 1} . \end{matrix} \end{matrix}$ If we set $v = \frac{2}{- α + 2} {(u_{ε} + ξ + δ)}^{\frac{- α + 2}{2}}$ in inequality (4.3), then $\nabla v = {(u_{ε} + ξ + δ)}^{- \frac{α}{2}} \nabla u_{ε}$ . From Lemma 4.2, We have $\begin{array}{l} \int_{\partial Ω} {(u_{ε} + ξ + δ)}^{- α} | \nabla u_{ε} |^{2} d H^{n - 1} \\ ⩽ ε_{3} \int_{Ω} {| D^{2} {(u_{ε} + ξ + δ)}^{\frac{- α + 2}{2}} |}^{2} d x + C_{ε_{3}} \int_{Ω} | u_{ε} + ξ + δ |^{- α + 2} d x . \end{array}$ Then it has $\begin{matrix} (4.13) & \begin{matrix} \int_{\partial Ω} {(u_{ε} + ξ + δ)}^{- α} | \nabla u_{ε} |^{2} d H^{n - 1} \\ ⩾ - ε_{3} \int_{Ω} {| D^{2} {(u_{ε} + ξ + δ)}^{\frac{- α + 2}{2}} |}^{2} d x - C_{ε_{3}} \int_{Ω} | u_{ε} + ξ + δ |^{- α + 2} d x . \end{matrix} \end{matrix}$ Combining (4.9) with above estimates, we have $\begin{array}{rcl} \tilde{I_{1}} & = & {\tilde{I}}_{1 - 1 - (1)} + {\tilde{I}}_{1 - 2 - (2) a} + {\tilde{I}}_{1 - 1 - (2)} + {\tilde{I}}_{1 - 2 - (1)} + {\tilde{I}}_{1 - 2 - (2) b} \\ ⩾ & \int_{Ω} β^{- 2} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{2 - 2 β} [\frac{2}{3} {| D^{2} {(u_{ε} + ξ)}^{β} |}^{2} + \frac{1}{3} {| Δ {(u_{ε} + ξ)}^{β} |}^{2}] d x \\ + \int_{Ω} [(1 - β) (β - \frac{1}{3}) {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 2} \\ + \frac{2}{3} α (1 - β) {(u_{ε} + ξ + δ)}^{- α - 1} {(u_{ε} + ξ)}^{- 1} - \frac{1}{3} α (α + 1) {(u_{ε} + ξ + δ)}^{- α - 2} \\ - ε_{2} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 2} - ε_{1} {(u_{ε} + ξ + δ)}^{- α - 2}] | \nabla u_{ε} |^{4} d x \\ - ε_{3} \int_{Ω} {| D^{2} {(u_{ε} + ξ + δ)}^{\frac{- α + 2}{2}} |}^{2} d x + \int_{Ω} (\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1) {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} d x \\ - C_{ε_{3}} \int_{Ω} | u_{ε} + ξ + δ |^{- α + 2} d x - C_{ε_{2}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} {(\frac{ξ}{u_{ε} + ξ})}^{2} d x \\ (4.14) & - C_{ε_{1}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} | Δ u_{ε} |^{2} d x . \end{array}$ In an analogous analysis of term $\tilde{I_{1}}$ , we have $\begin{matrix} (4.15) & \begin{matrix} \tilde{I_{2}} = & \int_{Ω} \frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} ψ^{″} (u_{ε}) {(u_{ε} + ξ + δ)}^{- α} | \nabla u_{ε} |^{2} d x \\ = & \int_{Ω} (\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1) ψ^{″} (u_{ε}) {(u_{ε} + ξ + δ)}^{- α} | \nabla u_{ε} |^{2} d x \\ + \int_{Ω} ψ^{″} (u_{ε}) {(u_{ε} + ξ + δ)}^{- α} | \nabla u_{ε} |^{2} d x \\ = & {\tilde{I}}_{2 - 1} + {\tilde{I}}_{2 - 2} . \end{matrix} \end{matrix}$ Under Hölder inequality and Young’s inequality, we obtain $\begin{matrix} {\tilde{I}}_{2 - 1} = & \int_{Ω} (\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1) | \nabla u_{ε} |^{2} {(u_{ε} + ξ + δ)}^{\frac{- α - 2}{2}} {(u_{ε} + ξ + δ)}^{\frac{- α + 2}{2}} ψ^{″} (u_{ε}) d x \\ ⩽ & ε_{4} \int_{Ω} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d x \\ + C_{ε_{4}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α + 2} {| ψ^{″} (u_{ε}) |}^{2} d x . \end{matrix}$ It follows that $\begin{array}{rcl} {\tilde{I}}_{2 - 1} & ⩾ & - ε_{4} \int_{Ω} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d x \\ (4.16) & - C_{ε_{4}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α + 2} {| ψ^{″} (u_{ε}) |}^{2} d x . \end{array}$ Similar to ${\tilde{I}}_{2 - 1}$ , we have $\begin{array}{rcl} {\tilde{I}}_{2 - 2} & = & \int_{Ω} | \nabla u_{ε} |^{2} {(u_{ε} + ξ + δ)}^{\frac{- α - 2}{2}} {(u_{ε} + ξ + δ)}^{\frac{- α + 2}{2}} ψ^{″} (u_{ε}) d x \\ ⩽ & ε_{5} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d x \\ + C_{ε_{5}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α + 2} {| ψ^{″} (u_{ε}) |}^{2} d x . \end{array}$ It follows that $\begin{matrix} (4.17) & {\tilde{I}}_{2 - 2} ⩾ - ε_{5} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d x - C_{ε_{5}} \int_{Ω} {(u_{ε} + ξ + δ)}^{- α + 2} {| ψ^{″} (u_{ε}) |}^{2} d x . \end{matrix}$ Combining (4.15) with (4.16)–(4.17), we have $\begin{matrix} (4.18) & \begin{matrix} {\tilde{I}}_{2} = & {\tilde{I}}_{2 - 1} + {\tilde{I}}_{2 - 2} \\ ⩾ & - \int_{Ω} [ε_{4} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} {(u_{ε} + ξ + δ)}^{- α - 2} + ε_{5} {(u_{ε} + ξ + δ)}^{- α - 2}] | \nabla u_{ε} |^{4} d x \\ - (C_{ε_{4}} + C_{ε_{5}}) \int_{Ω} {(u_{ε} + ξ + δ)}^{- α + 2} {| ψ^{″} (u_{ε}) |}^{2} d x . \end{matrix} \end{matrix}$ Associating (4.7), (4.14) with (4.18), we have $\begin{array}{l} \int_{Ω} G_{ε α δ} (u_{ε} + ξ) (x, T) d x \\ + \int_{Ω_{T}} ν β^{- 2} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{2 - 2 β} [\frac{2}{3} {| D^{2} {(u_{ε} + ξ)}^{β} |}^{2} + \frac{1}{3} {| Δ {(u_{ε} + ξ)}^{β} |}^{2}] d (x, t) \\ + \int_{Ω_{T}} ν [(1 - β) (β - \frac{1}{3}) {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 2} \\ + \frac{2}{3} α (1 - β) {(u_{ε} + ξ + δ)}^{- α - 1} {(u_{ε} + ξ)}^{- 1} - \frac{1}{3} α (α + 1) {(u_{ε} + ξ + δ)}^{- α - 2}] | \nabla u_{ε} |^{4} d (x, t) \\ + \int_{Ω} ν (\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1) {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} d (x, t) \\ ⩽ \int_{Ω} G_{ε α δ} (u_{0} + ξ) (x) d x \\ + \int_{Ω_{T}} [ν ε_{2} {(u_{ε} + ξ + δ)}^{- α} {(u_{ε} + ξ)}^{- 2} + ν ε_{1} {(u_{ε} + ξ + δ)}^{- α - 2}] | \nabla u_{ε} |^{4} d (x, t) \\ + ε_{4} \int_{Ω_{T}} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d (x, t) \\ + ε_{5} \int_{Ω_{T}} {(u_{ε} + ξ + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d (x, t) \\ + ε_{3} \int_{Ω_{T}} ν {| D^{2} {(u_{ε} + ξ + δ)}^{\frac{- α + 2}{2}} |}^{2} d (x, t) + ν C_{ε_{3}} \int_{Ω_{T}} | u_{ε} + ξ + δ |^{- α + 2} d (x, t) \\ + ν C_{ε_{2}} \int_{Ω_{T}} {(u_{ε} + ξ + δ)}^{- α} | Δ u_{ε} |^{2} {(\frac{ξ}{u_{ε} + ξ})}^{2} d (x, t) \\ + ν C_{ε_{1}} \int_{Ω_{T}} {(u_{ε} + ξ + δ)}^{- α} {(\frac{M_{ε} (u_{ε})}{M_{ε} (u_{ε} + ξ)} - 1)}^{2} | Δ u_{ε} |^{2} d (x, t) \\ (4.19) & + (C_{ε_{4}} + C_{ε_{5}}) \int_{Ω_{T}} {(u_{ε} + ξ + δ)}^{- α + 2} {| ψ^{″} (u_{ε}) |}^{2} d (x, t) : = (\tilde{1}) + (\tilde{2}) + \dots + (\tilde{9}) . \end{array}$ If the real numbers α and β satisfy $0 ⩽ α < \frac{1}{2}$ and $\begin{matrix} (4.20) & \frac{2 - α - \sqrt{(1 - 2 α) (1 + α)}}{3} ⩽ β ⩽ \frac{2 - α + \sqrt{(1 - 2 α) (1 + α)}}{3}, \end{matrix}$ then we obtain the coefficient of the term $| \nabla u_{ε} |^{4}$ of the left-hand side in the inequality (4.19) is positive, i.e., $\begin{array}{rcl} \tilde{C} (α, β) & = & (1 - β) (β - \frac{1}{3}) + \frac{2}{3} α (1 - β) - \frac{1}{3} α (α + 1) \\ = & - β^{2} + \frac{2}{3} (2 - α) β - \frac{1}{3} (α^{2} - α + 1) > 0 . \end{array}$

Step 2. Under the assumption of (4.20), by choosing $ε_{1}$ , $ε_{2}$ , $ε_{3}$ , $ε_{4}$ and $ε_{5}$ sufficiently small, the terms $(\tilde{2}) - (\tilde{5})$ can be absorbed in the terms of the left-hand side of (4.19). We have the following conclusions.
The integrands on the left-hand side of (4.19) are uniformly bounded.

Under the assumptions of $ψ (u_{ε})$ and $u_{ε} \in L^{2} (0, T; H^{2} (Ω))$ , the integrands in the terms $(\tilde{6})$ – $(\tilde{9})$ are uniformly bounded.

Now we pass to the limit $ξ \to 0$ . From $u_{ε} \in L^{2} (0, T; H^{2} (Ω)) \cap L^{\infty} (0, T; H^{1} (Ω))$ and the Lebesgues’s theorem, we obtain $\begin{matrix} (4.21) & \begin{matrix} \int_{Ω} G_{ε α δ} (u_{ε} (x, T)) d x + {\tilde{C}}_{1} \int_{Ω_{T}} {(u_{ε} + δ)}^{- α} u_{ε}^{2 - 2 β} [\frac{2}{3} {| D^{2} u_{ε}^{β} |}^{2} + \frac{1}{3} {| Δ u_{ε}^{β} |}^{2}] d (x, t) \\ + {\tilde{C}}_{1} \int_{Ω_{T}} {(u_{ε} + δ)}^{- α - 2} | \nabla u_{ε} |^{4} d (x, t) \\ ⩽ \int_{Ω} G_{ε α δ} (u_{0} (x)) d x + {\tilde{C}}_{2} \int_{Ω} {(u_{ε} + δ)}^{- α + 2} ({| ψ^{″} (u_{ε}) |}^{2} + 1) d (x, t) . \end{matrix} \end{matrix}$ Here ${\tilde{C}}_{1}, {\tilde{C}}_{2}$ are independent of $ε, δ$ . Passage to the limit $δ \to 0$ in (4.21). By applying Fatou’s lemma and the monotone convergence theorem, we have $\begin{matrix} (4.22) & \begin{matrix} \int_{Ω} G_{ε α} (u_{ε} (x, T)) d x + {\tilde{C}}_{1} \int_{Ω_{T}} u_{ε}^{2 - 2 β - α} [\frac{2}{3} {| D^{2} u_{ε}^{β} |}^{2} + \frac{1}{3} {| Δ u_{ε}^{β} |}^{2}] d (x, t) \\ + {\tilde{C}}_{1} \int_{Ω_{T}} u_{ε}^{- α - 2} | \nabla u_{ε} |^{4} d (x, t) \\ ⩽ \int_{Ω} G_{ε α} (u_{0} (x)) d x + {\tilde{C}}_{2} \int_{Ω} u_{ε}^{- α + 2} | ({| ψ^{″} (u_{ε}) |}^{2} + 1) d (x, t) . \end{matrix} \end{matrix}$ This completes the proof of Proposition 4.1. □

Obviously, under the assumption of α, the inequality (4.22) is rewritten as $\begin{matrix} (4.23) & \begin{matrix} \int_{Ω} G_{ε α} (u_{ε} (x, T)) d x + {\tilde{C}}_{1} \int_{Ω_{T}} u_{ε}^{2 - 2 β - α} {| D^{2} u_{ε}^{β} |}^{2} d (x, t) + {\tilde{C}}_{1} \int_{Ω_{T}} u_{ε}^{- α - 2} | \nabla u_{ε} |^{4} d (x, t) \\ ⩽ \int_{Ω} G_{ε α} (u_{0} (x)) d x + {\tilde{C}}_{2} \int_{Ω} u_{ε}^{- α + 2} ({| ψ^{″} (u_{ε}) |}^{2} + 1) d (x, t) . \end{matrix} \end{matrix}$ By applying the identity $\begin{matrix} D_{x_{i} x_{j}} v^{β} = β (β - 1) v^{β - 2} D_{x_{i}} v D_{x_{j}} v + β v^{β - 1} D_{x_{i} x_{j}} v \end{matrix}$ together with Young’s inequality, we can find a new ${\tilde{\tilde{C}}}_{1}$ is independent of ε and β such that $\begin{matrix} (4.24) & \begin{matrix} sup_{t \in [0, T]} \int_{Ω} G_{ε α} (u_{ε} (x, t)) d x + {\tilde{\tilde{C}}}_{1} \int_{Ω_{T}} {| D^{2} u_{ε}^{1 - \frac{α}{2}} |}^{2} d (x, t) + {\tilde{\tilde{C}}}_{1} \int_{Ω_{T}} {| \nabla u_{ε}^{\frac{1}{2} - \frac{α}{4}} |}^{4} d (x, t) \\ ⩽ \int_{Ω} G_{ε α} (u_{0} (x)) d x + {\tilde{C}}_{2} \int_{Ω} u_{ε}^{- α + 2} ({| ψ^{″} (u_{ε}) |}^{2} + 1) d (x, t) . \end{matrix} \end{matrix}$ Under the hypothesis of $u_{0} (x), M_{ε} (u_{ε}), ψ (u_{ε})$ , $u_{ε} \in L^{\infty} (0, T; H^{1} (Ω))$ and $0 ⩽ α < \frac{1}{2}$ , the right-hand side of (4.24) is uniformly bounded.

Furthermore, from standard parabolic arguments, the uniform bounded in $L^{\infty} (0, T; H^{1} (Ω))$ implies that the sequence ${u_{ε}}$ is uniformly bounded, equi-continuous family of functions in $Ω_{T}$ . Hence, it has a subsequence, which we do not rebel, $\begin{matrix} u_{ε}^{1 - \frac{α}{2}} ⇀ u^{1 - \frac{α}{2}} weakly in L^{2} (0, T; H^{2} (Ω)), \\ \nabla u_{ε}^{\frac{1}{2} - \frac{α}{4}} ⇀ \nabla u^{\frac{1}{2} - \frac{α}{4}} weakly in L^{4} (Ω_{T}), \\ u_{ε}^{1 - \frac{α}{2}} ⇀ u^{1 - \frac{α}{2}} strongly in L^{2} (0, T; H^{1} (Ω)) and a.e. in Ω_{T} \end{matrix}$ as $ε \to 0$ . This completes the proof of Theorem 1.1.
5. Conclusions

This paper presents the mathematical analysis on the degenerate C–H equation with a smooth double-well potential. Firstly, by using the semi-discretization technique, Schauder fixed-point theorem and Kolmogorov’s compactness criterion, we obtain the existence of weak solutions to the non-degenerate C–H equation. Next, passing to the limit $ε \to 0$ , together with the compact theorem, we get the existence and the regularity of weak solutions to the degenerate C–H equation. Furthermore, by introducing a new entropy function $G_{ε α} (l) = \int_{l}^{\tilde{r}} \int_{\tilde{τ}}^{\tilde{r}} \frac{1}{τ^{α} M_{ε} (τ)} d τ d \tilde{τ}$ , we obtain the additional regularity of solution $u (x, t)$ . Regrettably, we still do not establish a uniqueness theory of the degenerate C–H equation.

Footnotes

Acknowledgements

The authors would like to thank Prof. A. Miranville for many valuable suggestions of this paper. This research is supported by NSF of Shanghai University of Engineering Science (Grant No. 0244-E3-0507-19-05155). The work of Wang is supported by the NSFC (11831003, 11771031, 11531010) of China and NSF of Qinghai Province (2017-ZJ-908).

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