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A viscous thin-film equation with a singular diffusion


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.


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 [1014] and [15].

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

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

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. 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. 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. (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. (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.

Approximate problem

This section is devoted to studying the following approximate problem:

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

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. 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. 2.

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

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


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:

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

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

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

It gives

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

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

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

Therefore, we have obtained

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

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

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

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 \(M\rightarrow\infty\),

$$\begin{aligned}& u^{M}\rightharpoonup u\quad \mbox{weakly}^{*}\mbox{ in } L^{\infty}\bigl(0,T; H_{\mathrm {per}}^{1}(\varOmega ) \bigr); \end{aligned}$$
$$\begin{aligned}& w^{M}\rightharpoonup w \quad \mbox{weakly in } L^{2} \bigl(0,T;H_{\mathrm {per}}^{1}(\varOmega )\bigr); \end{aligned}$$
$$\begin{aligned}& u^{M}_{t}\rightharpoonup u_{t} \quad \mbox{weakly in } L^{2}(Q_{T}); \end{aligned}$$
$$\begin{aligned}& u^{M}\rightarrow u \quad \mbox{strongly in } L^{2} \bigl(0,T; H_{\mathrm {per}}^{2}(\varOmega )\bigr); \end{aligned}$$
$$\begin{aligned}& \nabla u^{M}\rightarrow\nabla u \quad \mbox{a.e. in } (Q_{T})^{N}; \end{aligned}$$
$$\begin{aligned}& u^{M}\rightarrow u\quad \mbox{a.e. in } Q_{T} \mbox{ and strongly in } C\bigl([0, T]; L^{2}(\varOmega )\bigr). \end{aligned}$$

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

$$\begin{aligned}& u^{M}, u\in C\bigl([0, T]; H_{\mathrm{per}}^{1}(\varOmega ) \bigr), \end{aligned}$$
$$\begin{aligned}& \nabla u^{M}, \nabla u\in L^{4}(Q_{T}). \end{aligned}$$

From Vitali’s theorem, we get

$$\begin{aligned}& \bigl(u^{M}_{+}+\delta\bigr)^{n}\rightarrow \bigl(u^{M}_{+}+\delta\bigr)^{n} \quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$
$$\begin{aligned}& \nabla u^{M}\rightarrow\nabla u \quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$
$$\begin{aligned}& \frac{(u^{M}_{+}+\delta)^{n}\nabla u^{M}}{(u^{M}_{+}+\varepsilon)^{n+\alpha }(1+\varepsilon|\nabla u^{M}|^{2})} \rightarrow\frac{(u_{+}+\delta)^{n}\nabla u}{(u_{+}+\varepsilon )^{n+\alpha}(1+\varepsilon|\nabla u|^{2})}\quad \mbox{strongly in } L^{2}(Q_{T}). \end{aligned}$$

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

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

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

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


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

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

  1. 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. 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. 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. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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


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

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

This implies

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

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. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 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. 7.

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


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

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

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

$$\begin{aligned}& (u_{+}+\delta)^{n}\rightarrow\overline{u}^{n} \quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$
$$\begin{aligned}& \nabla u\rightarrow\nabla\overline{u}\quad \mbox{strongly in } L^{4}(Q_{T}); \end{aligned}$$
$$\begin{aligned}& (u_{+}+\delta)^{n-1}\nabla u\rightarrow\overline{u}^{n-1} \nabla \overline{u}\quad \mbox{strongly in } L^{2}(Q_{T}) \mbox{ if } n\geq1; \end{aligned}$$
$$\begin{aligned}& \frac{(u_{+}+\delta)^{n}\nabla u}{(u_{+}+\varepsilon)^{n+\alpha }(1+\varepsilon|\nabla u|^{2})} \rightarrow\frac{(\overline{u}_{+}+\delta)^{n}\nabla\overline {u}}{(\overline{u}_{+}+\varepsilon)^{n+\alpha}(1+\varepsilon|\nabla \overline{u}|^{2})}\quad \mbox{strongly in } L^{2}(Q_{T}). \end{aligned}$$

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

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

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

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

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

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

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

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

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

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


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

Equations (33) and (35) yield

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

The Lebesgue-dominated theorem yields

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

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

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

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

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

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

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

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


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

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

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

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

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

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

Finally, it is easy to check that

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

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

Lemma 8

If one of the following conditions holds:

  1. (I)

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

  2. (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.


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

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

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

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 \(\varepsilon\rightarrow0\),

$$\begin{aligned}& \overline{u}\rightharpoonup u\quad \mbox{in } L^{2}\bigl(0, T; H_{\mathrm {per}}^{2}(\varOmega )\bigr); \end{aligned}$$
$$\begin{aligned}& \overline{u}_{t}\rightharpoonup u_{t}\quad \mbox{in } L^{2}(Q_{T}); \end{aligned}$$
$$\begin{aligned}& \overline{w}\rightharpoonup w\quad \mbox{in } L^{2}(Q_{T}); \end{aligned}$$
$$\begin{aligned}& \overline{u}\rightarrow u\quad \mbox{in } C\bigl([0, T]; H_{\mathrm {per}}^{1}( \varOmega )\bigr); \end{aligned}$$
$$\begin{aligned}& \overline{u}\rightarrow u\quad \mbox{in } L^{2}\bigl(0, T; H_{\mathrm {per}}^{1}(\varOmega )\bigr); \end{aligned}$$
$$\begin{aligned}& \overline{u}\rightarrow u,\qquad \nabla\overline{u}\rightarrow\nabla u\quad \mbox{a.e. in } Q_{T}. \end{aligned}$$

Furthermore, Lemma 3 yields

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

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

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

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

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

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

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

Hence, we have

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

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

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

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

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

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

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

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 \(u\in C([0, T]; H_{\mathrm{per}}^{1}(\varOmega ))\). □


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This work was supported by the National Natural Science Foundation of China (No. 11201045) and the Education Department Science Foundation of Liaoning Province of China.

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Correspondence to Bo Liang.

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The authors declare that they have no competing interests.

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XP and BL completed the main study. MP and YW verified the calculation. All authors read and approved the final manuscript.

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Peng, X., Liang, B., Pang, M. et al. A viscous thin-film equation with a singular diffusion. Bound Value Probl 2016, 142 (2016).

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  • fourth-order parabolic
  • thin-film equation
  • entropy functional
  • singular diffusion