A viscous thin-film equation with a singular diffusion

Abstract

The paper is devoted to studying a viscous thin-film equation with a singular diffusion term and the periodic boundary conditions in multidimensional space, which has a lot of applications in fluids theory such as draining of foams and the movement of contact lenses. In order to obtain the necessary uniform estimates and overcome the difficulty of a singular diffusion term, the entropy functional method is used. Finally, the existence of nonnegative weak solutions is obtained by some compactness arguments.

1 Introduction

The research of the Cahn-Hilliard equation and the thin-film equation has become a hot topic recently. The Cahn-Hilliard equation (see [1]) can describe the evolution of a conserved concentration field during phase separation, which has the form $u_{t}+\nabla\cdot(m\nabla(\varepsilon^{2}\varDelta u+f'(u)))=0$ where m, f, $\varepsilon^{2}$ denote the atomic mobility, the free energy, the parameter proportional to the interface energy, respectively. $-(\varepsilon^{2}\varDelta u+f'(u))$ can be taken as the chemical potential. For the linear or degenerate mobility, Elliott, Zheng, and Garcke [2, 3] have studied its existence and obtained some properties of solutions. Besides, Liang and Zheng [4] obtained the existence and stability results for this model with a gradient mobility by studying the corresponding semi-discrete problems.

The thin-film equation is usually used to describe the motion of a very thin layer of viscous incompressible fluids along an inclined plane such as the draining of foams and the movement of contact lenses. It can be taken as a class of fourth-order degenerate parabolic equations [5]:

$$u_{t}+\bigl(m(u) u_{xxx}+f(u, u_{x}, u_{xx})\bigr)_{x}=0, $$

where the mobility $m(u)$ degenerates at $u=0$. For example, thin-film flows driven by the surface tension can be modeled by the following fourth-order degenerate parabolic equations:

$$ \frac{\partial u}{\partial t} +\frac{\partial}{\partial x} \biggl(\frac {u^{3}}{3}(Ch_{xxx}- \delta Bh_{x}\cos\alpha +B\sin\alpha)+A\frac{u_{x}}{u}+ \frac{M}{2}\sigma_{x} u^{2} \biggr)=0. $$

For the simplified thin-film equation $u_{t}+(u^{n} u_{xxx})_{x}=0$, Bernis and Friedman [6] gave the first result to the existence and nonnegativity of weak solutions. Bertozzi and Pugh [7] have studied the existence in the distributional sense and the long time decay for the model of the thin-film equation with a second-order diffusion term. Boutat et al. [8] studied a generalized thin-film equation with period boundary in multidimensional space. Furthermore, Liang [9] has investigated the existence of the weak solutions and strong solutions with the initial function near a steady state solution. For other results, the reader may refer to [10–14] and [15].

In this paper, we study the following viscous thin-film equation with a singular diffusion:

$$ \left \{ \textstyle\begin{array}{l} u_{t}-\nabla\cdot(u^{n}\nabla w)+A\nabla\cdot (\frac{\nabla u}{u^{\alpha}} )=0\quad \mbox{in } Q_{T}, \\ w=-\varDelta u+\nu u_{t} \quad \mbox{in } Q_{T}, \\ u \mbox{ is } \varOmega \mbox{-periodic}, \\ u(x,0)=u_{0}(x)\quad \mbox{on } \varOmega , \end{array}\displaystyle \right . $$

(1)

where $\varOmega =(-1, 1)^{N}$, $Q_{T}=\varOmega \times(0, T)$. n, A, α, and ν are all constants with $n, \alpha, \nu>0$.

For convenience, we introduce some notations:

• C is denoted as a positive constant and may change from line to line.

• $\varOmega =(-1, 1)^{N}$, $\varGamma _{j}=\partial \varOmega \cap\{x_{j}=-1\}$, $\varGamma _{j+N}=\partial \varOmega \cap\{x_{j}=1\}$.

• $H_{\mathrm{per}}^{m}(\varOmega )$ is the periodic Sobolev space i.e.

  $$H_{\mathrm{per}}^{m}(\varOmega )=\bigl\{ u\in H^{m}(\varOmega )\bigl\vert D^{\xi}u\bigr\vert _{\varGamma _{j}}=D^{\xi}u|_{\varGamma _{j+N}}, j=1, \ldots, N, |\xi|\leq m-1\bigr\} . $$

• The following norms on $H_{\mathrm{per}}^{m}(\varOmega )$ ($m\geq1$) are equivalent:

  $$ \Vert u\Vert _{H^{m}(\varOmega )}, \qquad \Vert u\Vert _{L^{2}(\varOmega )}+\bigl\Vert D^{m} u\bigr\Vert _{L^{2}(\varOmega )}\quad \mbox{and} \quad | \overline{u}|+\bigl\Vert D^{m} u\bigr\Vert _{L^{2}(\varOmega )}, $$

  where $\overline{u}=\frac{1}{2^{N}}\int_{\varOmega }u(x)\,\mathrm{d}x$ (see [8]).

• $C_{\mathrm{per}}^{m}(\overline{\varOmega })=\{u\in C^{m}(\overline {\varOmega })|D^{\xi}u|_{\varGamma _{j}}=D^{\xi}u|_{\varGamma _{j+N}}, j=1, \ldots, N, |\xi|\leq m-1\}$.

• $a_{+}=\max\{a, 0\}$, $a_{-}=\min\{a, 0\}$ for $a\in R$.

Our main result is the following theorem.

Theorem 1

Let $\alpha\in(0, \frac{1}{2}]$, $u_{0}\in H^{1}(\varOmega )$,

$$n\in \left \{ \textstyle\begin{array}{l@{\quad}l} (\frac{6}{7}, 2), & N=1; \\ (\frac{8}{9}, 2), & N=2; \\ (\frac{16}{17}, 2), & N=3. \end{array}\displaystyle \right . $$

Suppose $A\leq0$ or $\alpha\leq1-n$. Then there exist at least one pair of solutions $(u ,w)$ satisfying

• 1. $u\in L^{2}(0, T; H_{\mathrm{per}}^{2}(\varOmega ))\cap C([0, T]; H^{1}(\varOmega ))$, $u^{-\frac{1}{2}}|\nabla u|\in L^{4}(Q_{T})$, $w, u_{t}\in L^{2}(Q_{T})$;

• 2. for any test function $\phi\in C^{1}([0,T]; C_{\mathrm{per}}^{2}(\varOmega ))$, one has

  $$\begin{aligned}& \iint_{Q_{T}}u_{t}\phi\,\mathrm{d}x\,\mathrm{d}t+ \iint_{Q_{T}}u^{n} w\varDelta \phi\,\mathrm{d}x\, \mathrm{d}t \\& \quad {}+n \iint_{Q_{T}}u^{n-1}\nabla uw\nabla\phi\,\mathrm{d}x\, \mathrm{d}t -A \iint_{Q_{T}}\frac{\nabla u\nabla\phi}{u^{\alpha}}\,\mathrm{d}x\,\mathrm{d}t=0, \\& \iint_{Q_{T}}w\phi\,\mathrm{d}x\,\mathrm{d}t=- \iint_{Q_{T}}\varDelta u\phi\, \mathrm{d}x\,\mathrm{d}t+\nu \iint_{Q_{T}}u_{t}\phi\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$

The following lemmas are needed in the paper.

Lemma 1

(Bernis, see [8])

Let $u\in H_{\mathrm{per}}^{2}(\varOmega )$ be a nonnegative function. There exists a constant $\mu>0$ such that the following inequality holds:

$$ \int_{\varOmega }\frac{|\nabla u|^{4}}{u^{2}}\,\mathrm{d}x\leq\mu\|u\| _{H^{2}(\varOmega )}^{2}. $$

Lemma 2

(Aubin-Lions, see [16])

Let X, B, and Y be Banach spaces and assume $X\hookrightarrow B\hookrightarrow Y$ with compact imbedding $X\hookrightarrow B$.

• (1) Let $\mathfrak{F}$ be bounded in $L^{p}(0, T; X)$ where $1\leq p<\infty$, and $\frac{\partial\mathfrak{F}}{\partial t}=\{\frac{\partial f}{\partial t}: f\in\mathfrak{F}\}$ be bounded in $L^{1}(0, T;Y)$. Then $\mathfrak{F}$ is relatively compact in $L^{p}(0, T; B)$.

• (2) Let $\mathfrak{F}$ be bounded in $L^{\infty}(0, T; X)$, and $\frac{\partial\mathfrak{F}}{\partial t}=\{\frac{\partial f}{\partial t}: f\in\mathfrak{F}\}$ be bounded in $L^{r}(0, T;Y)$ where $r>1$. Then $\mathfrak{F}$ is relatively compact in $C([0, T]; B)$.

Lemma 3

(see [17] or [18])

Let V be a real, separable, reflexive Banach space and H is a real, separable, Hilbert space. $V\hookrightarrow H$ is continuous and V is dense in H. Then $\{u\in L^{2}(0, T; V)|u_{t}\in L^{2}(0, T; V')\}$ is continuously imbedded in $C([0, T]; H)$.

The paper is arranged as follows. The existence of solutions to the approximate problem will be proved in Section 2. In Sections 3 and 4, we will take the limit for small parameters $\delta\rightarrow0$ and $\varepsilon\rightarrow0$, respectively.

2 Approximate problem

This section is devoted to studying the following approximate problem:

$$ \left \{ \textstyle\begin{array}{l} u_{t}-\nabla\cdot((u_{+}+\delta)^{n}\nabla w)+A\nabla\cdot (\frac {(u_{+}+\delta)^{n}\nabla u}{(u_{+}+\varepsilon)^{n+\alpha}(1+\varepsilon |\nabla u|^{2})} )=0\quad \mbox{in } Q_{T}, \\ w=-\varDelta u+\nu u_{t}\quad \mbox{in } Q_{T}, \\ u \mbox{ is } \varOmega \mbox{-periodic} , \\ u(x,0)=u_{0\delta\varepsilon}(x)\quad \mbox{on } \varOmega \end{array}\displaystyle \right . $$

(2)

for $0<\delta<\varepsilon<1$ and $u_{+}=\max\{u, 0\}$.

Lemma 4

Let $u_{0\delta\varepsilon}\in H_{\mathrm{per}}^{1}(\varOmega )$, $\alpha>0$, and $0< n<2$. Then there exist at least a pair of solutions $(u, w)$ to (2) satisfying

• 1. $u\in L^{2}(0, T; H_{\mathrm{per}}^{3}(\varOmega ))\cap C([0, T]; H_{\mathrm {per}}^{2}(\varOmega ))$, $u_{t}\in L^{2}(Q_{T})$, $w\in L^{2}(0, T; H_{\mathrm {per}}^{1}(\varOmega ))$, and $u(x, 0)=u_{0}$;

• 2. for any test function $\phi\in L^{2}(0, T; H_{\mathrm{per}}^{1}(\varOmega ))$, one has

  $$\begin{aligned}& \iint_{Q_{T}}u_{t}\phi\,\mathrm{d}x\,\mathrm{d}t+ \iint_{Q_{T}}(u_{+}+\delta )^{n}\nabla w \nabla w\phi\,\mathrm{d}x\,\mathrm{d}t \\& \quad {} -A \iint_{Q_{T}} \biggl(\frac{(u_{+}+\delta)^{n}\nabla u}{(u_{+}+\varepsilon )^{n+\alpha}(1+\varepsilon|\nabla u|^{2})} \biggr)\nabla\phi\,\mathrm {d}x\,\mathrm{d}t=0, \\& \iint_{Q_{T}}w\phi\,\mathrm{d}x\,\mathrm{d}t=- \iint_{Q_{T}}\varDelta u\phi\, \mathrm{d}x\,\mathrm{d}t+\nu \iint_{Q_{T}}u_{t}\phi\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$

Proof

We will apply the Galerkin method to obtain the existence of solutions. Let $\{\phi_{i}\}_{i=1,2,3,\ldots}$ be the eigenfunctions of the Laplace operator $-\varDelta \phi_{i}=\lambda_{i}\phi_{i}$ with periodic boundary value conditions. Moreover, those eigenfunctions are orthogonal in $H^{1}$ and $L^{2}$ spaces and we can normalize $\phi_{i}$ such that $(\phi_{i}, \phi_{j})=\delta_{ij}=\bigl \{\scriptsize{ \begin{array}{l@{\quad}l} 1, & i=j, \\ 0, & i\neq j, \end{array}} \bigr.$ where we define $\lambda_{1}=0$, $\phi_{1}=1$, and $(\cdot, \cdot)$ denotes the scalar product of the $L^{2}$ space.

Let M denote a positive integer and define $w^{M}(x,t)=\sum_{i=1}^{M}d_{i}(t)\phi_{i}(x)$, $u^{M}(x,t)= \sum_{i=1}^{M}c_{i}(t)\phi_{i}(x)$, $u^{M}(x, 0)=\sum_{i=1}^{M}(u_{0}, \phi_{i})\phi_{i}$. For $j=1, \ldots, M$, we consider the following system of ordinary differential equations:

$$\begin{aligned}& \frac{d}{dt}\bigl(u^{M}, \phi_{j} \bigr)=-\bigl(\bigl(u_{+}^{M}+\delta\bigr)^{n} \nabla w^{M}, \nabla\phi _{j}\bigr) \\& \hphantom{\frac{d}{dt}(u^{M}, \phi_{j})={}}{}+A \biggl( \biggl(\frac{(u_{+}^{M}+\delta)^{n}\nabla u^{M}}{(u_{+}+\varepsilon )^{n+\alpha}(1+\varepsilon|\nabla u^{M}|^{2})} \biggr), \nabla \phi_{j} \biggr), \end{aligned}$$

(3)

$$\begin{aligned}& \bigl(w^{M},\phi_{j}\bigr)=-\bigl(\varDelta u^{M}, \phi_{j}\bigr)+\nu\frac{d}{dt}\bigl(u^{M}, \phi_{j}\bigr). \end{aligned}$$

(4)

The ODE existence theorem yields the local unique existence of this initial value problem since the right hand side depends on $c_{i}$ continuously. In order to show the global solvability, we take $-\varDelta u^{M}$ as the test function and apply the Young inequality to get

$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\varOmega }\bigl\vert \nabla u^{M}\bigr\vert ^{2}\,\mathrm{d}x+\nu \int_{\varOmega }\bigl\vert u^{M}_{t}\bigr\vert ^{2}\,\mathrm{d}x + \int_{\varOmega }\bigl(u^{M}_{+}+\delta \bigr)^{n}\bigl\vert \nabla w^{M}\bigr\vert ^{2}\,\mathrm{d}x \\& \quad = A \int_{\varOmega }\frac{(u^{M}_{+}+\delta)^{n}\nabla u^{M} \nabla w^{M}}{(u^{M}_{+}+\varepsilon)^{n+\alpha}(1+\varepsilon \vert \nabla u^{M}\vert ^{2})}\, \mathrm{d}x \\& \quad \leq \frac{1}{2} \int_{\varOmega }\bigl(u^{M}_{+}+\delta \bigr)^{n}\bigl\vert \nabla w^{M}\bigr\vert ^{2}\,\mathrm {d}x+\frac{C}{\varepsilon^{\alpha}} \int_{\varOmega }\bigl\vert \nabla u^{M}\bigr\vert ^{2}\,\mathrm{d}x. \end{aligned}$$

(5)

It gives

$$\begin{aligned}& \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\varOmega }\bigl\vert \nabla u^{M}\bigr\vert ^{2}\,\mathrm{d}x +\nu \int_{\varOmega }\bigl\vert u^{M}_{t}\bigr\vert ^{2}\,\mathrm{d}x + \int_{\varOmega }\bigl(u^{M}_{+}+\delta \bigr)^{n}\bigl\vert \nabla \varDelta u^{M}\bigr\vert ^{2}\,\mathrm {d}x \\& \quad = \frac{C}{\varepsilon^{\alpha}} \int_{\varOmega }\bigl\vert \nabla u^{M}\bigr\vert ^{2}\,\mathrm{d}x. \end{aligned}$$

(6)

The mass conservation property $\int_{\varOmega }u^{M}(x,t)\,\mathrm{d}x=\int _{\varOmega }u^{M}_{0}(x)\,\mathrm{d}x$ (by letting $j=1$) ensures that Poincaré’s inequality can be applied. On the other hand, the Gronwall inequality yields

$$\begin{aligned}& \sup_{t\in(0, T)} \int_{\varOmega }\bigl(\bigl\vert u^{M}\bigr\vert ^{2}+\bigl\vert \nabla u^{M}\bigr\vert ^{2} \bigr) (x, t)\,\mathrm {d}x+ \iint_{Q_{T}}\bigl\vert u^{M}_{t}\bigr\vert ^{2}\,\mathrm{d}x\,\mathrm{d}t+ \iint_{Q_{T}}\bigl\vert \nabla w^{M}\bigr\vert ^{2}\,\mathrm{d}x \\& \quad \leq C. \end{aligned}$$

(7)

Therefore, we have obtained

$$ u^{M}\in L^{\infty}\bigl(0, T; H_{\mathrm{per}}^{1}(\varOmega )\bigr),\qquad w^{M}\in L^{2}\bigl(0, T; H_{\mathrm{per}}^{1}(\varOmega )\bigr),\qquad u^{M}_{t}\in L^{2}(Q_{T}). $$

(8)

The classic $L^{p}$-estimate of the second-order elliptic equations implies

$$ u^{M}\in L^{2}\bigl(0, T; H_{\mathrm{per}}^{3}(\varOmega )\bigr). $$

(9)

By (8), (10), and Lemma 2, we conclude that there exist a pair of functions $(u, w)$ and a subsequence of $(u^{M}, w^{M})$ such that as $M\rightarrow\infty$,

$$\begin{aligned}& u^{M}\rightharpoonup u\quad \mbox{weakly}^{*}\mbox{ in } L^{\infty}\bigl(0,T; H_{\mathrm {per}}^{1}(\varOmega ) \bigr); \end{aligned}$$

(10)

$$\begin{aligned}& w^{M}\rightharpoonup w \quad \mbox{weakly in } L^{2} \bigl(0,T;H_{\mathrm {per}}^{1}(\varOmega )\bigr); \end{aligned}$$

(11)

$$\begin{aligned}& u^{M}_{t}\rightharpoonup u_{t} \quad \mbox{weakly in } L^{2}(Q_{T}); \end{aligned}$$

(12)

$$\begin{aligned}& u^{M}\rightarrow u \quad \mbox{strongly in } L^{2} \bigl(0,T; H_{\mathrm {per}}^{2}(\varOmega )\bigr); \end{aligned}$$

(13)

$$\begin{aligned}& \nabla u^{M}\rightarrow\nabla u \quad \mbox{a.e. in } (Q_{T})^{N}; \end{aligned}$$

(14)

$$\begin{aligned}& u^{M}\rightarrow u\quad \mbox{a.e. in } Q_{T} \mbox{ and strongly in } C\bigl([0, T]; L^{2}(\varOmega )\bigr). \end{aligned}$$

(15)

Moreover, by Lemma 3, (12), (13), and the embedding $H^{2}(\varOmega )\hookrightarrow W^{1, 4}(\varOmega )$, we have

$$\begin{aligned}& u^{M}, u\in C\bigl([0, T]; H_{\mathrm{per}}^{1}(\varOmega ) \bigr), \end{aligned}$$

(16)

$$\begin{aligned}& \nabla u^{M}, \nabla u\in L^{4}(Q_{T}). \end{aligned}$$

(17)

From Vitali’s theorem, we get

$$\begin{aligned}& \bigl(u^{M}_{+}+\delta\bigr)^{n}\rightarrow \bigl(u^{M}_{+}+\delta\bigr)^{n} \quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$

(18)

$$\begin{aligned}& \nabla u^{M}\rightarrow\nabla u \quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$

(19)

$$\begin{aligned}& \frac{(u^{M}_{+}+\delta)^{n}\nabla u^{M}}{(u^{M}_{+}+\varepsilon)^{n+\alpha }(1+\varepsilon|\nabla u^{M}|^{2})} \rightarrow\frac{(u_{+}+\delta)^{n}\nabla u}{(u_{+}+\varepsilon )^{n+\alpha}(1+\varepsilon|\nabla u|^{2})}\quad \mbox{strongly in } L^{2}(Q_{T}). \end{aligned}$$

(20)

Let $T_{M}$ denote the projection from the space $L^{2}(\varOmega )$ to $Span\{ \phi_{1},\ldots,\phi_{M}\}$. By multiplying equation (3) by $T_{M}\phi$ for $\phi\in L^{2}(0, T; H_{\mathrm{per}}^{1}(\varOmega ))$, one has

$$\begin{aligned}& \iint_{Q_{T}} u^{M}_{t} T_{M}\phi \,\mathrm{d}t+ \iint_{Q_{T}}\bigl(u^{M}_{+}+\delta \bigr)^{n}\nabla w^{M}\nabla T_{M}\phi\, \mathrm{d}x\,\mathrm{d}t \\& \quad =A \iint_{Q_{T}} \biggl(\frac{(u^{M}_{+}+\delta)^{n}\nabla u^{M}}{(u^{M}_{+}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla u^{M}|^{2})} \biggr)\nabla T_{M}\phi\,\mathrm{d}x\,\mathrm{d}t, \end{aligned}$$

(21)

$$\begin{aligned}& \iint_{Q_{T}}w^{M}T_{M}\phi\,\mathrm{d}x\, \mathrm{d}t=- \iint_{Q_{T}}\varDelta u^{M}T_{M}\phi\, \mathrm{d}x\,\mathrm{d}t+\nu \iint_{Q_{T}}u^{M}_{t}T_{M}\phi\, \mathrm {d}x\,\mathrm{d}t. \end{aligned}$$

(22)

By (10)-(20), we can perform the limit $M\rightarrow \infty$ in each term of (21)-(22). □

3 The limit $\delta\rightarrow0$

We shall perform the limit $\delta\rightarrow0$ in the section to the solutions obtained by Lemma 4 and we suppose that the initial function $u_{0\delta\varepsilon}\rightarrow u_{0\varepsilon}\in H^{1}(\varOmega )$ as $\delta\rightarrow0$ and $u_{0\varepsilon}\geq0$.

The main result of this section is the following.

Proposition 1

Let

$$n\in \left \{ \textstyle\begin{array}{l@{\quad}l} (\frac{6}{7}, 2), & N=1; \\ (\frac{8}{9}, 2), & N=2; \\ (\frac{16}{17}, 2), & N=3. \end{array}\displaystyle \right . $$

Then there exist at least a pair of functions $(\overline{u}, \overline {w})$ satisfying

• 1. $\overline{w}\in L^{2}(Q_{T})$, $\overline{u}\in L^{2}(0, T; H_{\mathrm {per}}^{2}(\varOmega ))\cap C([0, T]; H_{\mathrm{per}}^{1}(\varOmega ))$, $\overline {u}_{t}\in L^{2}(Q_{T})$, and $\overline{u}(x, 0)=u_{0\varepsilon}$;

• 2. for any test function $\phi\in L^{2}(0, T; C_{\mathrm{per}}^{\infty }(\overline{\varOmega }))$, one has

  $$\begin{aligned}& \iint_{Q_{T}}\overline{u}_{t} \phi\,\mathrm{d}x\, \mathrm{d}t- \iint _{Q_{T}}\overline{u}^{n} \overline{w}\varDelta \phi\, \mathrm{d}x\,\mathrm{d}t -n \iint_{Q_{T}}\overline{u}^{n-1}\overline{w}\nabla \overline{u}\nabla\phi \,\mathrm{d}x\,\mathrm{d}t \\& \quad {}-A \iint_{Q_{T}}\frac{\overline{u}^{n}\nabla\overline{u}\nabla\phi }{(\overline{u}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla\overline {u}|^{2})}\,\mathrm{d}x\,\mathrm{d}t=0, \\& \iint_{Q_{T}}\overline{w}\phi\,\mathrm{d}x\,\mathrm{d}t=- \iint _{Q_{T}}\varDelta \overline{u}\phi\,\mathrm{d}x\,\mathrm{d}t + \nu \iint_{Q_{T}}\overline{u}_{t}\phi\,\mathrm{d}x\, \mathrm{d}t. \end{aligned}$$

In order to prove this proposition, we have to establish some uniform energy estimates independent of δ and thus we introduce a nonnegative convex functional $\varPhi _{\delta}(\cdot )$ (see[8]):

If $0\leq n<2$, $n\neq1$,

$$ \varPhi _{\delta}(\sigma)=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{(1-n)(2-n)}(\sigma+\delta)^{2-n}-\frac{1}{1-n}(\sigma+\delta )+\frac{1}{2-n}, & \sigma\geq0; \\ \frac{(\sigma)^{2}}{2\delta^{n}}+\frac{1}{1-n}(\delta^{1-n}-1)\sigma+\frac {1}{2-n}, & \sigma< 0. \end{array}\displaystyle \right . $$

If $n=1$,

$$ \varPhi _{\delta}(\sigma)=\left \{ \textstyle\begin{array}{l@{\quad}l} (\sigma+\delta)Ln(\sigma+\delta)-(\sigma+\delta)+1, & \sigma\geq0; \\ \frac{(\sigma)^{2}}{2\delta}+\sigma(Ln\delta)+\delta(Ln\delta)-\delta+1, & \sigma< 0. \end{array}\displaystyle \right . $$

It is easy to check that $\varPhi _{\delta}\in{W_{\mathrm{loc}}^{2,+\infty}(R)}$, $\varPhi _{\delta}''(\sigma )=\frac{1}{(\sigma_{+}+\delta)^{n}}$.

By applying this functional, we can get the following estimates.

Lemma 5

There exist some constants C independent of δ (may depend on ε) such that

• 1. $\frac{\mathrm{d}}{\mathrm{d}t}\int_{\varOmega }\varPhi (u(x,t))\,\mathrm {d}x+\int_{\varOmega }|w|^{2}\,\mathrm{d}x+\nu\int_{\varOmega }|u_{t}|^{2}\,\mathrm {d}x\leq C$;

• 2. $\|w\|_{L^{2}(Q_{T})}\leq C$, $\|u\|_{L^{2}(0,T; H_{\mathrm{per}}^{2}(\varOmega ))}\leq C$;

• 3. $\|u\|_{L^{\infty}(0,T;H_{\mathrm{per}}^{1}(\varOmega ))}\leq C$;

• 4. $\iint_{Q_{T}}(u_{+}+\delta)^{n}|\nabla w|^{2}\,\mathrm{d}x\,\mathrm {d}t\leq C$;

• 5. $\|u_{t}\|_{L^{2}(Q_{T})}\leq C$.

Proof

By choosing $\varPhi '(u)$ as the test function in (2), we get

$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t} \int_{\varOmega }\varPhi \bigl(u(x,t)\bigr)\,\mathrm {d}x+ \int_{\varOmega }|w|^{2}\,\mathrm{d}x+\nu \int_{\varOmega }|u_{t}|^{2}\,\mathrm {d}x \\& \quad = A \int_{\varOmega }\frac{|\nabla u|^{2}}{(u_{+}+\varepsilon)^{n+\alpha }(1+\varepsilon|\nabla u|^{2})}\,\mathrm{d}x \leq \frac{|A|}{\varepsilon^{n+\alpha+1}}. \end{aligned}$$

(23)

This implies

$$ \|u\|_{L^{2}(0, T; H_{\mathrm{per}}^{2}(\varOmega ))}\leq C\|w\|_{L^{2}(Q_{T})}\leq C, $$

(24)

which yields the results 1-2. Similar to (5), we conclude that

$$ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \int_{\varOmega }|\nabla u|^{2}\,\mathrm{d}x+\nu \int_{\varOmega }|u_{t}|^{2}\,\mathrm{d}x + \frac{1}{2} \int_{\varOmega }(u_{+}+\delta)^{n}|\nabla w|^{2}\,\mathrm{d}x \leq C(\varepsilon) \int_{\varOmega }|\nabla u|^{2}\,\mathrm{d}x, $$

which gives 3 and 4. □

Lemma 6

There exist a pair of functions $(\overline {u}, \overline{w})$ such that, as $\delta\rightarrow0$,

• 1. $u\rightharpoonup\overline{u}$ weakly in $L^{2}(0,T; H_{\mathrm {per}}^{2}(\varOmega ))$;

• 2. $w\rightharpoonup\overline{w}$ weakly in $L^{2}(Q_{T})$;

• 3. $u_{t}\rightarrow\overline{u}_{t}$ weakly in $L^{2}(Q_{T})$;

• 4. $u\rightarrow\overline{u}$ strongly in $L^{2}(0, T; H_{\mathrm {per}}^{1}(\varOmega ))$ and a.e. in $Q_{T}$;

• 5. $u\rightarrow\overline{u}$ strongly in $C([0, T]; L^{2}(\varOmega ))$;

• 6. if $\sup_{\delta\in(0,1)}\int_{\varOmega }\varPhi (u_{0})\,\mathrm{d}x<\infty$, then $\overline{u}\geq0$ in $\overline{Q}_{T}$ and $\sup_{t\leq T}\|u_{-}(t)\|_{L^{2}(\varOmega )}\leq C\delta^{\frac{n}{2}}$ when $n\leq1$;

• 7. $\overline{u}\in L^{2}(0, T; H_{\mathrm{per}}^{2}(\varOmega ))\cap C([0, T]; L^{2}(\varOmega ))$, $\overline{u}_{t}\in L^{2}(Q_{T})$, $\overline{w}\in L^{2}(Q_{T})$.

Proof

The results 1-3 can be obtained from Lemma 5, and Lemma 2 can give 4 and 5. In order to prove 6-7, we integrate (23) over $(0,T)$ to get

$$ 0\leq \int_{\varOmega }\varPhi \bigl(u(x,t)\bigr)\,\mathrm{d}x \leq \frac{n|A|T}{\varepsilon^{n+\alpha+1}}+\sup_{\delta\in(0,1)} \int _{\varOmega }\varPhi \bigl(u_{0}(x)\bigr)\,\mathrm{d}x \leq C(\varepsilon). $$

If $n<1$, we have

$$ 0\leq\frac{1}{2} \int_{\varOmega }u_{-}^{2}(x,t)\,\mathrm{d}x\leq \frac{\delta ^{n}}{n-1}\bigl(\delta^{1-n}-1\bigr) \int_{\varOmega }u_{-}(x,t)\,\mathrm {d}x+C(\varepsilon) \delta^{n}. $$

If $n=1$, we have

$$ 0\leq\frac{1}{2} \int_{\varOmega }u_{-}^{2}(x,t)\,\mathrm{d}x\leq- \delta\ln \delta \int_{\varOmega }u_{-}(x,t)\,\mathrm{d}x+C(\varepsilon) \delta. $$

By performing the limit $\delta\rightarrow0$, we get $\int_{\varOmega }\overline{u}_{-}^{2}(x,t)\,\mathrm{d}x=0$, which implies 6. Besides, the result 7 can be obtained from 1-3 and Lemma 3. □

Proof of Proposition 1

For any function $\phi\in L^{2}(0, T; C_{\mathrm{per}}^{\infty}(\varOmega ))$, Lemma 4 gives

$$\begin{aligned}& \iint_{Q_{T}}u_{t} \phi\,\mathrm{d}x\,\mathrm{d}t- \iint_{Q_{T}}(u_{+}+\delta )^{n}w\varDelta \phi\, \mathrm{d}x\,\mathrm{d}t -n \iint_{Q_{T}}(u_{+}+\delta)^{n-1}w\nabla u \nabla\phi\,\mathrm{d}x\, \mathrm{d}t \\& \quad {} +A \iint_{Q_{T}} \biggl(\frac{(u_{+}+\delta)^{n}\nabla u\nabla\phi }{(u_{+}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla u|^{2})} \biggr)\, \mathrm{d}x\, \mathrm{d}t=0, \end{aligned}$$

(25)

$$\begin{aligned}& \iint_{Q_{T}}w\phi\,\mathrm{d}x\,\mathrm{d}t=- \iint_{Q_{T}}\varDelta u\phi\, \mathrm{d}x\,\mathrm{d}t +\nu \iint_{Q_{T}}u_{t}\phi\,\mathrm{d}x\, \mathrm{d}t. \end{aligned}$$

(26)

Similar to the proof of (18)-(20) and applying Lemma 5, Lemma 6, and Vitali’s theorem, we can get

$$\begin{aligned}& (u_{+}+\delta)^{n}\rightarrow\overline{u}^{n} \quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$

(27)

$$\begin{aligned}& \nabla u\rightarrow\nabla\overline{u}\quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$

(28)

$$\begin{aligned}& (u_{+}+\delta)^{n-1}\nabla u\rightarrow\overline{u}^{n-1} \nabla \overline{u}\quad \mbox{strongly in } L^{2}(Q_{T}) \mbox{ if } n\geq1; \end{aligned}$$

(29)

$$\begin{aligned}& \frac{(u_{+}+\delta)^{n}\nabla u}{(u_{+}+\varepsilon)^{n+\alpha }(1+\varepsilon|\nabla u|^{2})} \rightarrow\frac{(\overline{u}_{+}+\delta)^{n}\nabla\overline {u}}{(\overline{u}_{+}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla \overline{u}|^{2})}\quad \mbox{strongly in } L^{2}(Q_{T}). \end{aligned}$$

(30)

If (29) holds for $n<1$, (25)-(30) ensure that the limit $\delta\rightarrow0$ can be performed in (25)-(26) and then we can complete the proof of Proposition 1.

Therefore, we only need to prove

$$ (u_{+}+\delta)^{n-1}\nabla u\rightarrow \overline{u}^{n-\frac{1}{2}}\frac{\nabla\overline{u}}{\overline {u}^{\frac{1}{2}}}=\overline{u}^{n-1}\nabla \overline{u}\quad \mbox{strongly in } L^{2}(Q_{T}) $$

(31)

if $n<1$.

From the following three steps, we can prove (29).

Step 1. Define $m(\delta)=\delta+\|u_{-}\|_{C(\overline {Q_{T}})}$ and we have $u+m(\delta)\geq\delta>0$. By applying the Bernis inequality, we get

$$ \iint_{Q_{T}}\frac{|\nabla u|^{4}}{(u+m(\delta))^{2}}\,\mathrm{d}x\,\mathrm{d}t \leq \iint_{Q_{T}}|\varDelta u|^{2}\,\mathrm{d}x\,\mathrm{d}t \leq C, $$

(32)

where C is independent of δ.

Step 2. In this step, we define $U_{\delta}=(u_{+}+\delta )^{n-1}(u+m(\delta))^{\frac{1}{2}}$ and we want to prove that the limit $\lim_{\delta\rightarrow0}\|U_{\delta}-\overline {u}^{n-\frac{1}{2}}\|_{L^{4}(Q_{T})}=0$ holds.

At first, it is obvious that we have

$$ U_{\delta}\geq\bigl(u_{+}+m(\delta) \bigr)^{n-\frac{1}{2}}\quad \mbox{in } Q_{T}. $$

(33)

Now we choose

$$r\left \{ \textstyle\begin{array}{l@{\quad}l} =+\infty, & N=1; \\ < +\infty, & N=2; \\ < 6, & N=3, \end{array}\displaystyle \right . $$

such that $H^{s}(\varOmega )\hookrightarrow W^{1, r}(\varOmega )$ with $\frac{7}{4}< s<2$. By using the Gagliardo-Nirenberg interpolation inequality and Lemma 6, we get

$$\begin{aligned} \bigl\Vert u_{-}(t)\bigr\Vert _{L^{\infty}(\varOmega )} &\leq C\bigl\Vert u_{-}(t)\bigr\Vert _{W^{1, r}(\varOmega )}^{\gamma}\bigl\Vert u_{-}(t)\bigr\Vert _{L^{2}(\varOmega )}^{1-\gamma} \\ &\leq C\bigl\Vert u_{-}(t)\bigr\Vert _{H_{\mathrm{per}}^{s}(\varOmega )}^{\gamma} \delta^{\frac {n}{2}(1-\gamma)} \\ &\leq C(\varepsilon,s)\delta^{\frac{n}{2}(1-\gamma)} \end{aligned}$$

(34)

with $\gamma=\frac{\frac{1}{2}}{\frac{N+2}{2N}-\frac{1}{r}}$. It implies

$$\begin{aligned} U_{\delta}(x,t)&\leq(u_{+}+\delta)^{n-1} \bigl(u_{+}+\delta+2\bigl\Vert u_{-}(t)\bigr\Vert _{L^{\infty}(\varOmega )}\bigr)^{\frac{1}{2}} \\ &\leq(u_{+}+\delta)^{n-\frac{1}{2}}+\delta^{n-1}\bigl(2\bigl\Vert u_{-}(t)\bigr\Vert _{L^{\infty}(\varOmega )}\bigr)^{\frac{1}{2}} \\ &\leq(u_{+}+\delta)^{n-\frac{1}{2}}+C(\varepsilon) \delta^{n-1+\frac {n}{4}(1-\gamma)} \end{aligned}$$

(35)

with

$$n\in \left \{ \textstyle\begin{array}{l@{\quad}l} (\frac{6}{7}, 2), & N=1; \\ (\frac{8}{9}, 2), & N=2; \\ (\frac{16}{17}, 2), & N=3. \end{array}\displaystyle \right . $$

Equations (33) and (35) yield

$$ \lim_{\delta\rightarrow0}U_{\delta}(x, t)= \overline{u}^{n-\frac {1}{2}} \quad \mbox{a.e. in } Q_{T}. $$

(36)

The Lebesgue-dominated theorem yields

$$ \lim_{\delta\rightarrow0} \iint_{Q_{T}}\bigl\vert U_{\delta}-\overline{u}^{n-\frac {1}{2}} \bigr\vert ^{4}\,\mathrm{d}x\,\mathrm{d}t=0. $$

(37)

Step 3. This step is devoted to the proof of (31). For any positive constant η, one has

$$\begin{aligned}& \iint_{Q}\biggl\vert (u_{+}+\delta)^{n-1} \nabla u-\overline{u}^{n-\frac {1}{2}}\frac{\nabla\overline{u}}{\sqrt{\overline{u}}}\biggr\vert ^{2} \,\mathrm {d}x\,\mathrm{d}t \\& \quad \leq \iint_{Q}\bigl\vert U_{\delta}-\overline{u}^{n-\frac{1}{2}} \bigr\vert ^{2}\biggl\vert \frac {\nabla u}{\sqrt{u+m(\delta)}}\biggr\vert ^{2}\,\mathrm{d}x\,\mathrm{d}t + \iint_{Q}\overline{u}^{2n-1}\biggl\vert \frac{\nabla u}{u+m(\delta)}-\frac {\nabla\overline{u}}{\sqrt{\overline{u}}}\biggr\vert ^{2}\,\mathrm{d}x\, \mathrm{d}t \\& \quad \leq \biggl( \iint_{Q}\biggl\vert \frac{\nabla u}{\sqrt{u+m(\delta)}} \biggr\vert ^{4}\,\mathrm{d}x\,\mathrm{d}t \biggr)^{\frac{1}{2}} \biggl( \iint_{Q}\bigl\vert U_{\delta}-\overline{u}^{n-\frac{1}{2}} \bigr\vert ^{4}\,\mathrm {d}x\,\mathrm{d}t \biggr)^{\frac{1}{2}} \\& \qquad {} + \iint_{\{\overline{u}\geq\eta\}}\overline{u}^{2n-1}\biggl\vert \frac {\nabla u}{u+m(\delta)}-\frac{\nabla\overline{u}}{\sqrt{\overline {u}}}\biggr\vert ^{2}\,\mathrm{d}x\, \mathrm{d}t \\& \qquad {} + \iint_{\{\overline{u}< \eta\}}\overline{u}^{2n-1}\biggl\vert \frac{\nabla u}{u+m(\delta)}-\frac{\nabla\overline{u}}{\sqrt{\overline{u}}} \biggr\vert ^{2}\,\mathrm{d}x\, \mathrm{d}t \\& \quad =\mathtt{I}_{1}+\mathtt{I}_{2}+\mathtt{I}_{3}. \end{aligned}$$

(38)

From Step 1 and Step 2, we know $\mathtt{I}_{1}\rightarrow0$ as $\delta \rightarrow0$ and by applying Lemma 6, we have $\mathtt{I}_{2}\rightarrow0$ as $\delta\rightarrow0$. For the last term, we have

$$\begin{aligned} \mathtt{I}_{3}&= \iint_{\{\overline{u}< \eta\}}\overline{u}^{2n-1} \biggl\vert \frac{\nabla u}{u+m(\delta)}-\frac{\nabla\overline{u}}{\sqrt{\overline {u}}}\biggr\vert ^{2}\,\mathrm{d}x\, \mathrm{d}t \\ &\leq\eta^{2n-1} \biggl[ \iint_{\{\overline{u}< \eta\}}\frac{|\nabla u|^{2}}{|u+m(\delta)|}\,\mathrm{d}x\,\mathrm{d}t + \iint_{\{\overline{u}< \eta\}}\frac{|\nabla\overline{u}|^{2}}{|\overline {u}|}\,\mathrm{d}x\,\mathrm{d}t \biggr] \\ &\leq C\eta^{2n-1} \biggl[ \biggl( \iint_{\{\overline{u}< \eta\}}\frac {|\nabla u|^{4}}{|u+m(\delta)|^{2}}\,\mathrm{d}x\,\mathrm{d}t \biggr)^{\frac{1}{2}} + \biggl( \iint_{\{\overline{u}< \eta\}}\frac{|\nabla\overline {u}|^{4}}{|\overline{u}|^{2}}\,\mathrm{d}x\,\mathrm{d}t \biggr)^{\frac {1}{2}} \biggr] \\ &\leq C\eta^{2n-1}. \end{aligned}$$

(39)

Therefore, by performing the limit $\eta\rightarrow0$, we get $\mathtt {I}_{3}\rightarrow0$ and then the estimate (31) holds. □

4 The limit $\varepsilon\rightarrow0$

We will perform the last limit $\varepsilon\rightarrow0$ in this section and assume that the initial function $u_{0\varepsilon}$ converges to $u_{0}$ strongly in $L^{2}(\varOmega )$.

By letting $\delta=0$ in the definition of $\varPhi _{\delta}(\cdot)$, we can define $\varPhi _{0}(\cdot)$ as

$$ \varPhi _{0}(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{(1-n)(2-n)}x^{2-n}-\frac{1}{1-n}x+\frac{1}{2-n} & \mbox{if } n\in[0,2), n\neq1; \\ x\ln x-x+1 & \mbox{if } n=1. \end{array}\displaystyle \right . $$

Lemma 7

In the sense of $\mathcal{D}'(0, T)$, there exists a constant $C_{0}>0$ such that

$$ \frac{\mathrm{d}}{\mathrm{d}t} \int_{\varOmega }\varPhi _{0}(\overline{u})\, \mathrm{d}x+C_{0} \int_{\varOmega }|\varDelta \overline{u}|^{2}\,\mathrm{d}x +\nu \int_{\varOmega }|\overline{u}_{t}|^{2}\, \mathrm{d}x \leq A \int_{\varOmega }\frac{|\nabla\overline{u}|^{2}}{(\overline {u}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla\overline{u}|^{2})}\, \mathrm{d}x. $$

Proof

From the idea of (23) and the $L^{p}$-estimate, we get

$$\begin{aligned}& \frac{\mathrm{d}}{\mathrm{d}t} \int_{\varOmega }\varPhi \bigl(u(x,t)\bigr)\,\mathrm {d}x+C_{0} \int_{\varOmega }|\varDelta u|^{2}\,\mathrm{d}x+\nu \int_{\varOmega }|u_{t}|^{2}\, \mathrm{d}x \\& \quad \leq \frac{\mathrm{d}}{\mathrm{d}t} \int_{\varOmega }\varPhi \bigl(u(x,t)\bigr)\, \mathrm{d}x+ \int_{\varOmega }|w|^{2}\,\mathrm{d}x+\nu \int_{\varOmega }|u_{t}|^{2}\, \mathrm{d}x \\& \quad = A \int_{\varOmega }\frac{|\nabla u|^{2}}{(u_{+}+\varepsilon)^{n+\alpha }(1+\varepsilon|\nabla u|^{2})}\,\mathrm{d}x. \end{aligned}$$

(40)

Since $u\rightarrow\overline{u}$ in $C(\overline{Q}_{T})$ as $\delta \rightarrow0$, we have

$$ - \int_{0}^{T}\phi'(t) \int_{\varOmega }\varPhi (u)\,\mathrm{d}x\,\mathrm{d}t \rightarrow- \int_{0}^{T}\phi'(t) \int_{\varOmega }\varPhi _{0}(\overline{u})\,\mathrm {d}x\, \mathrm{d}t $$

(41)

for any nonnegative function $\phi\in\mathcal{D}'(0, T)$. By applying the limit $\varDelta u\rightharpoonup \varDelta \overline{u}$ in $L^{2}(Q_{T})$ as $\delta\rightarrow0$, one has

$$ \liminf_{\delta\rightarrow0} \int_{0}^{T}\phi(t) \int_{\varOmega }|\varDelta u|^{2}\, \mathrm{d}x\,\mathrm{d}t \geq \int_{0}^{T}\phi(t) \int_{\varOmega }|\varDelta \overline{u}|^{2}\,\mathrm{d}x\, \mathrm{d}t. $$

(42)

Finally, it is easy to check that

$$\begin{aligned}& A \iint_{Q_{T}}\frac{|\nabla u|^{2}\phi(t)}{(u_{+}+\varepsilon)^{n+\alpha }(1+\varepsilon|\nabla u|^{2})}\,\mathrm{d}x\,\mathrm{d}t \\& \quad \rightarrow A \iint_{Q_{T}}\frac{|\nabla\overline{u}|^{2}\phi (t)}{(\overline{u}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla \overline{u}|^{2})}\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$

(43)

Equations (40)-(43) give the result of this lemma. □

Lemma 8

If one of the following conditions holds:

• (I) $\int_{\varOmega }\varPhi _{0}(w_{0})\,\mathrm{d}x<\infty$, $A\leq 0$, and

• (II) $\int_{\varOmega }\varPhi _{0}(w_{0})\,\mathrm{d}x<\infty$, $\alpha \leq1-n$, $n<1$, one has $\overline{u}\in L^{2}(0,T;H_{\mathrm{per}}^{2}(\varOmega ))$, $\overline{w}, \overline{u}_{t}\in L^{2}(Q_{T})$ independent of ε.

Proof

By Lemma 7 and the condition (I), we can prove the result easily.

If the condition (II) holds, Lemma 1 and Lemma 7 give

$$\begin{aligned}& \int_{\varOmega }\varPhi _{0}(\overline{u})\, \mathrm{d}x+C_{0} \int_{\varOmega }|\varDelta \overline{u}|^{2}\,\mathrm{d}x +\nu \iint_{Q_{T}}|\overline{u}_{t}|^{2}\, \mathrm{d}x\,\mathrm{d}t \\& \quad \leq \int_{\varOmega }\varPhi _{0}(\overline{u}_{0})\, \mathrm{d}x+|A| \iint _{Q_{T}}\frac{|\nabla\overline{u}|^{2}}{(\overline{u}+\varepsilon )(1+\varepsilon|\nabla\overline{u}|^{2})}\,\mathrm{d}x\,\mathrm {d}t \\& \quad \leq \int_{\varOmega }\varPhi _{0}(\overline{u}_{0})\, \mathrm{d}x+|A| \iint _{Q_{T}}\frac{|\nabla\overline{u}|^{2}}{(\overline{u}+\varepsilon)^{\alpha +n}}\,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq \int_{\varOmega }\varPhi _{0}(\overline{u}_{0})\, \mathrm{d}x +|A| \biggl( \iint_{Q_{T}}\frac{|\nabla\overline{u}|^{4}}{(\overline {u}+\varepsilon)^{2}}\,\mathrm{d}x\,\mathrm{d}t \biggr)^{\frac{1}{2}} \biggl( \iint_{Q_{T}}(\overline{u}+\varepsilon)^{2(1-(\alpha+n))}\,\mathrm {d}x\,\mathrm{d}t \biggr)^{\frac{1}{2}} \\& \quad \leq \int_{\varOmega }\varPhi _{0}(\overline{u}_{0})\, \mathrm{d}x +C \biggl( \int_{0}^{T}\|\overline{u}\|_{H^{2}(\varOmega )}\, \mathrm{d}t \biggr)^{\frac{1}{2}} \biggl( \iint_{Q_{T}}(\overline{u}+\varepsilon)\,\mathrm{d}x\,\mathrm {d}t \biggr)^{1-(\alpha+n)} \\& \quad \leq \frac{C_{0}}{2} \iint_{Q_{T}}|\varDelta \overline{u}|^{2}\,\mathrm{d}x\, \mathrm{d}t+C, \end{aligned}$$

(44)

which yields $\varDelta \overline{u}\in L^{2}(Q_{T})$. Applying the second equation of Proposition 1, we get $\overline{w}\in L^{2}(Q_{T})$. □

Now we are in the position to prove Theorem 1.

Proof of Theorem 1

By Lemma 8, we can show the existence of two functions $u\geq 0$ and w such that, as $\varepsilon\rightarrow0$,

$$\begin{aligned}& \overline{u}\rightharpoonup u\quad \mbox{in } L^{2}\bigl(0, T; H_{\mathrm {per}}^{2}(\varOmega )\bigr); \end{aligned}$$

(45)

$$\begin{aligned}& \overline{u}_{t}\rightharpoonup u_{t}\quad \mbox{in } L^{2}(Q_{T}); \end{aligned}$$

(46)

$$\begin{aligned}& \overline{w}\rightharpoonup w\quad \mbox{in } L^{2}(Q_{T}); \end{aligned}$$

(47)

$$\begin{aligned}& \overline{u}\rightarrow u\quad \mbox{in } C\bigl([0, T]; H_{\mathrm {per}}^{1}( \varOmega )\bigr); \end{aligned}$$

(48)

$$\begin{aligned}& \overline{u}\rightarrow u\quad \mbox{in } L^{2}\bigl(0, T; H_{\mathrm {per}}^{1}(\varOmega )\bigr); \end{aligned}$$

(49)

$$\begin{aligned}& \overline{u}\rightarrow u,\qquad \nabla\overline{u}\rightarrow\nabla u\quad \mbox{a.e. in } Q_{T}. \end{aligned}$$

(50)

Furthermore, Lemma 3 yields

$$\begin{aligned}& \|\overline{u}\|_{C([0, T]; H_{\mathrm{per}}^{s}(\varOmega )) }\leq C; \end{aligned}$$

(51)

$$\begin{aligned}& \|u\|_{C([0, T]; H_{\mathrm{per}}^{s}(\varOmega )) }\leq C \end{aligned}$$

(52)

for $\frac{3}{2}< s<2$. By the Sobolev embedding theorem with the case $N\leq3$, we have $\|\overline{u}\|_{L^{\infty}(Q_{T})}\leq C$ and $\|u\|_{L^{\infty }(Q_{T})}\leq C$.

Step 1. By using (51)-(52) and Vitali’s theorem, we get $\overline{u}^{n}\rightarrow u^{n}$ in $L^{q}(Q_{T})$ for any $q>0$ and thus one has

$$ \iint_{Q_{T}}\overline{u}^{n}\overline{w}\varDelta \phi\, \mathrm{d}x\,\mathrm {d}t\rightarrow \iint_{Q_{T}}u^{n}w\varDelta \phi\,\mathrm{d}x\,\mathrm{d}t $$

(53)

as $\varepsilon\rightarrow0$ for any test function $\phi\in C^{\infty }([0, T]; C_{\mathrm{per}}^{2}(\overline{\varOmega }))$.

Step 2. In this step, we will prove the limit $\overline {u}^{n-1}\nabla\overline{u}\rightarrow u^{n-1}\nabla u$ in $L^{2}(Q_{T})$.

First of all, the Bernis inequality yields $\iint_{Q_{T}}\vert \frac {\nabla\overline{u}}{\sqrt{\overline{u}}}\vert ^{4}\,\mathrm{d}x\, \mathrm{d}t\leq C$ and then we have

$$\begin{aligned} \iint_{\varDelta _{0}}\overline{u}^{n-1}|\nabla \overline{u}|^{2}\,\mathrm {d}x\,\mathrm{d}t =& \iint_{\varDelta _{0}}\overline{u}^{2n-1}\frac{|\nabla\overline{u}|^{2}}{\sqrt {\overline{u}}}\, \mathrm{d}x\,\mathrm{d}t \\ \leq& C \biggl( \iint_{\varDelta _{0}}\overline{u}^{4n-2}\,\mathrm{d}x\,\mathrm {d}t \biggr)^{\frac{1}{2}} \rightarrow0 \end{aligned}$$

(54)

as $\varepsilon\rightarrow0$ with $\varDelta _{0}=\{(x, t)\in Q_{T}|u(x, t)=0\}$. On the other hand, it is easy to get

$$\frac{\nabla\overline{u}}{\sqrt{\overline{u}}}\rightarrow\frac{\nabla u}{\sqrt{u}}\quad \mbox{a.e. in } Q_{T}\backslash \varDelta _{0} $$

as $\varepsilon\rightarrow0$. By Vitali’s theorem, we have

$$ \overline{u}^{n-1}\nabla\overline{u}\rightarrow u^{n-1}\nabla u \quad \mbox{in } L^{2}(Q_{T} \backslash \varDelta _{0}). $$

(55)

Hence, we have

$$ \overline{u}^{n-1}\nabla\overline{u}\rightarrow u^{n-1}\nabla u \quad \mbox{in } L^{2}(Q_{T}), $$

(56)

where we define $u^{n-1}\nabla u=0$ on $\varDelta _{0}$.

Step 3. In this step, we prove the limit $F_{\varepsilon}=\frac{\overline{u}^{n}\nabla\overline{u}}{(\overline {u}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla\overline {u}|^{2})}\rightarrow u^{-\alpha}\nabla u$ in $L^{2}(Q_{T})$.

If $\alpha\leq\frac{1}{2}$, we have

$$\begin{aligned} \iint_{\varDelta _{0}}|F_{\varepsilon}|^{2}\,\mathrm{d}x\, \mathrm{d}t \leq& \iint_{\varDelta _{0}}\overline{u}^{1-2\alpha}\frac{|\nabla\overline {u}|^{2}}{\overline{u}}\, \mathrm{d}x\,\mathrm{d}t \\ \leq& C \biggl( \iint_{\varDelta _{0}}\overline{u}^{2-4\alpha}\,\mathrm{d}x\, \mathrm{d}t \biggr)^{\frac{1}{2}} \rightarrow0 \end{aligned}$$

(57)

as $\varepsilon\rightarrow0$. Beside, it is easy to show $F_{\varepsilon}\rightarrow u^{-\alpha }\nabla u$ a.e. in $Q_{T}\backslash \varDelta _{0}$ and Vitali’s theorem yields

$$ \iint_{\varDelta _{0}}\bigl\vert F_{\varepsilon}-u^{-\alpha} \nabla u\bigr\vert ^{2}\,\mathrm{d}x\, \mathrm{d}t\rightarrow0 $$

(58)

as $\varepsilon\rightarrow0$. By (57)-(58), we have

$$ F_{\varepsilon}\rightarrow u^{-\alpha}\nabla u \quad \mbox{in } L^{2}(Q_{T}), $$

(59)

where we define $u^{-\alpha}\nabla u=0$ on $\varDelta _{0}$.

As $\varepsilon\rightarrow0$, the convergence (56) and (46)-(47) give $\iint_{Q_{T}}\overline{u}_{t}\phi\,\mathrm{d}x\,\mathrm{d}t\rightarrow \iint_{Q_{T}}u_{t}\phi\,\mathrm{d}x\,\mathrm{d}t$ and $\iint_{Q_{T}}\overline{u}^{n-1}\nabla\overline{u}\overline{w}\nabla\phi \,\mathrm{d}x\,\mathrm{d}t \rightarrow\iint_{Q_{T}}u^{n-1}\nabla u w\nabla\phi\,\mathrm{d}x\, \mathrm{d}t$. Step 3 yields

$$\iint_{Q_{T}}\frac{\overline{u}^{n}\nabla\overline{u}\nabla\phi }{(\overline{u}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla\overline {u}|^{2})}\,\mathrm{d}x\,\mathrm{d}t \rightarrow \iint_{Q_{T}}u^{-\alpha}\nabla u\nabla\phi\,\mathrm{d}x\, \mathrm{d}t. $$

Now we can take the limit $\varepsilon\rightarrow0$ in the equality

$$\begin{aligned}& \iint_{Q_{T}}\overline{u}_{t}\phi\,\mathrm{d}x\, \mathrm{d}t+ \iint _{Q_{T}}\overline{u}^{n} \overline{w}\varDelta \phi\, \mathrm{d}x\,\mathrm{d}t \\& \quad {} +n \iint_{Q_{T}}\overline{u}^{n-1}\nabla\overline{u} \overline{w}\nabla \phi\,\mathrm{d}x\,\mathrm{d}t -A \iint_{Q_{T}}\frac{\overline{u}^{n}\nabla\overline{u}\nabla\phi }{(\overline{u}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla\overline {u}|^{2})}\,\mathrm{d}x\,\mathrm{d}t=0, \\& \iint_{Q_{T}}\overline{w}\phi\,\mathrm{d}x\,\mathrm{d}t=- \iint _{Q_{T}}\varDelta \overline{u}\phi\,\mathrm{d}x\,\mathrm{d}t + \nu \iint_{Q_{T}}\overline{u}_{t}\phi\,\mathrm{d}x\, \mathrm{d}t \end{aligned}$$

for any test function $\phi\in C([0, T]; C_{\mathrm{per}}^{2}(\overline {\varOmega }))$. For the initial value, this holds in the sense of $u\in C([0, T]; H_{\mathrm{per}}^{1}(\varOmega ))$. □