Mathematical analysis of a fourth-order degenerate equation with a nonlinear second-order term

Abstract

A fourth-order partial differential model in this paper is called the thin-film equation in the field of fluid theory. The existence of the weak solution is obtained by solving two approximation problems. In order to perform the limit for small parameters in the approximation problems, the Galerkin method and the entropy functional method are both used. By means of some classical compactness results we can give the existence of nonnegative weak solutions. Finally, by redefining the entropy functional, the long-time exponential decay is given in the sense of $L^{1}$ -norm.

Keywords

Fourth-order degenerate equation existence long-time behavior entropy functional

1. Introduction

The higher-order degenerated parabolic equations have been used to explain the objective phenomenon problems, such as Blake-Zisserman model in image segmentation, thin-plate problem in rational mechanics, thin-film problem in fluid mechanics, and so on. The thin-film equation, as an essential fourth-order parabolic equation, is widely used in the fields of image processing, biology, fluid mechanics and engineering [1]. In fluid mechanics, this model is often applied to show the movement of thin incompressible viscous fluids along the slope. The thin-film equation has the general form:

$\displaystyle w_{t}+\left({K(w)w_{xxx}+f\left({w,w_{x},w_{xx}}\right)}\right)_% {x}=0$

which is a fourth-order model with mobility $K(w)$ [2]. The mobility function $K(w)$ usually has a form of a power function $w^{n}$ with $n>0$ and so it degenerates if $w=0$ . In fact, different values of $n$ correspond to different models in physics, such as the Hele-Shaw cell flow if $n=1$ , the capillary-driven flow for $n=3$ .

Bernis and Friedman have studied the existence and nonnegativity of solutions for the equation $w_{t}+\left({K(w)w_{xxx}}\right)_{x}=0$ in [3] for the one-dimensional model. In the higher dimensional space, Dal Passo et al. [4] gave the existence and some properties of weak solutions, and Grün [5] studied the existence and regularity of the solution of the Cauchy problem with the obstacle problem. Gao and Yin [6] have studied the existence of radial nonnegative solutions in the two-dimensional space. Bertozzi and Pugh investigated the existence and the large time behavior of weak solutions for a fourth-order partial differential equation with a low order term [7]. For a generalized thin-film equation multidimensional periodic boundary conditions, Boutat et al. [8] have given the existence of solutions. Moreover, the existence of solutions near a constant stationary solution is given in the paper [9]. In this work, we have a different way to show the application for the fourth-order partial differential equation. The diffusion equation

$\displaystyle w_{t}+j_{x}=f(x,t)$

is often used to describe the crystal surface growth, where $w$ denotes the variation of height from the average and $j=w_{xxx}+\frac{w_{x}}{1+w_{x}^{2}}$ denotes the atom current parallel to the surface [10]. The term $\frac{w_{x}}{1+w_{x}^{2}}$ also corresponds to the curvature of a curved substrate from the lubrication model for surface tension driven flow. In the present paper, we study the atom current $j=ww_{xxx}+\frac{w_{x}}{1+w_{x}^{2}}$ that corresponds to the degenerate mobility $K(w)=w$ in the thin-film equation, we are interested in the joint effect of the fourth-order degenerate term and the lower-order term on the solutions. For more information on this topic, we refer to [11, 12, 13, 14, 15, 16, 17]. Thus, the model studied in this paper is as follows:

$\displaystyle\left\{{{\begin{array}[]{l}{w_{t}+\left({ww_{xxx}}\right)_{x}-% \left({\frac{w_{x}}{1+\left|{w_{x}}\right|^{2}}}\right)_{x}=0\ \text{in}\ Q_{% \tau}},\\ {w_{x}(-1,t)=w_{x}(1,t)=0\ \text{on}\ (0,\tau),}\\ {w_{xxx}(-1,t)=w_{xxx}(1,t)=0\ \text{on}\ (0,\tau),}\\ {w(x,0)=w_{0}(x)\ \text{on}\ (-1,1)}\\ \end{array}}}\right.$ (1)

where $Q_{\tau}=(-1,1)\times(0,\tau),I=(-1,1)$ .

The main conclusions are obtained as follows:

Theorem 1. If $w_{0}\in\mathbb{L}^{2}(I),w_{0}\geqslant 0,w_{0}\not{\equiv}0$ , there is at least a weak solution $w$ satisfying

$w\in\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{2}(I)}\right),w_{t}\in\mathbb{L}^{% 1}\left({0,\tau;\mathbb{H}^{-2}(I)}\right),w\in{\cal C}\left({[0,\tau];\mathbb% {L}^{1}(I)-\textit{weak}}\right)$ ;

If $\Phi\in{\cal C}^{1}\left({[0,\tau];{\cal C}_{0}^{2}(I)}\right)$ is an arbitrary test function and $\Phi(x,\tau)=0$ , it has

$\displaystyle-\int\!\!\!\int_{Q_{\tau}}{\Phi_{t}}w\text{d}x\text{d}t-\int\!\!% \!\int_{I}\Phi(x,0)w_{0}(x)\text{d}x+\int\!\!\!\int_{Q_{\tau}}ww_{xx}\Phi_{xx}% \text{d}x\text{d}t$ $\displaystyle\quad+\int\!\!\!\int_{Q_{\tau}}{w_{x}}w_{xx}\Phi_{x}\text{d}x% \text{d}t+\int\!\!\!\int_{Q_{\tau}}{\frac{w_{x}}{1+\left|{w_{x}}\right|^{2}}}% \Phi_{x}\text{d}x\text{d}t=0.$

Theorem 2. For the weak solutions $w$ and any $t>0$ , it has the exponential decay result

$\displaystyle\int_{I}{|w-\frac{1}{2}\int_{I}{w_{0}}(x)\text{d}x|}\,\text{d}x% \leqslant\theta_{1}e^{-\theta_{2}t},$

where $\theta_{1}$ and $\theta_{2}$ are two positive constants independent of $t$ .

For convenience, we make the following description

•

$C$ denotes a general positive constant and it may change from line to line.

•

$\mathbb{H}^{k}(I)$ is Banach space, i.e.,

$\displaystyle\mathbb{H}^{k}(I)=\left\{{w\in\mathbb{W}^{k,2}(I)\mid D^{\alpha}w% \in\mathbb{L}^{2}(I)\ \text{for}\ \alpha\ \text{satisfying}\ |\alpha|\leqslant k% }\right\}.$

•

${\cal C}^{k}(I)$ denotes all $k$ -order continuous differentiable functions in $I$ .

•

$\forall s\in R,\,\,s_{+}=\max\{s,0\},\;\,s_{-}=\min\{s,0\}$ .

•

$<\cdot,\;\cdot>$ denotes the dual product between some Banach space $\mathbb{V}$ and its dual space $\mathbb{V}^{\prime}$

The arrangement of this paper is as follows. First, we consider an approximation problem in Section 2, and then the process of the limits $\sigma\to 0$ and $\varepsilon\to 0$ will be given in the Section 3 and Section 4, Theorems 1 and 2 will be proved in Sections 5 and 6, some conclusions are given in Section 7.

2. Approximate problem

For $0<\sigma<\varepsilon<1$ considering the approximation problem

$\displaystyle\left\{{{\begin{array}[]{l}{w_{\sigma\varepsilon t}+\left({\left(% {w_{\sigma\varepsilon+}+\sigma}\right)w_{\sigma\varepsilon xxx}}\right)_{x}-% \left({\frac{\left({w_{\sigma\varepsilon+}+\sigma}\right)w_{\sigma\varepsilon x% }}{\left({w_{\sigma\varepsilon+}+\varepsilon}\right)\left({1+\left|{w_{\sigma% \varepsilon x}}\right|^{2}}\right)}}\right)_{x}=0\ \text{in}\ Q_{\tau}},\\ {w_{\sigma\varepsilon x}(-1,t)=w_{\sigma\varepsilon x}(1,t)=0\ \text{on}\ (0,% \tau),}\\ {w_{\sigma\varepsilon xxx}(-1,t)=w_{\sigma\varepsilon xxx}(1,t)=0\ \text{on}\ % (0,\tau),}\\ {w_{\sigma\varepsilon}(x,0)=w_{0\sigma\varepsilon}(x)\ \text{on}\ (-1,1),}\\ \end{array}}}\right.$ (2)

we introduce two parameters $\sigma$ and $\varepsilon$ . The parameter $\sigma$ is used to overcome the effect of the degenerate mobility and the parameter $\varepsilon$ is to ensure the positivity of the denominator of the fraction term. To obtain uniform estimations well, we have to perform the limits $\sigma\to 0$ and $\varepsilon\to 0$ for the approximation solutions.

Lemma 1. If $w_{0\sigma\varepsilon}\in\mathbb{H}^{1}(I)$ , then there exists at least one solution $w_{\sigma\varepsilon}$ to Eq. (2) satisfying

$w_{\sigma\varepsilon}\in\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{3}(I)}\right),% w_{\sigma\varepsilon}\in\mathbb{L}^{\infty}\left({0,\tau;\mathbb{H}^{1}(I)}% \right),w_{\sigma\varepsilon t}\in\mathbb{L}^{2}\left({0,\tau;\left({\mathbb{H% }^{-1}(I)}\right)}\right),w_{\sigma\varepsilon}\in{\cal C}\left({[0,\tau];% \mathbb{H}^{1}(I)}\right)$ ;

$\forall\,\Phi\in{\cal C}_{0}^{\infty}\left({Q_{\tau}}\right)$ , one has

$\displaystyle\int_{0}^{\tau}<\,w_{t},\Phi>\text{d}t-\int\!\!\!\int_{Q_{\tau}}{% \left({w_{\sigma\varepsilon+}+\sigma}\right)}\,w_{\sigma\varepsilon xxx}\Phi_{% x}\text{d}x\text{d}t+\int\!\!\!\int_{Q_{\tau}}$ $\displaystyle\quad{\left({\frac{w_{\sigma\varepsilon x}\left({w_{\sigma% \varepsilon+}+\sigma}\right)}{\left({\varepsilon+w_{\sigma\varepsilon+}}\right% )\left({1+\left|{w_{\sigma\varepsilon x}}\right|^{2}}\right)}}\right)}\,\Phi_{% x}\text{d}x\text{d}t=0.$

Proof. For the proof, we will apply the Galerkin method and let $\left\{{y_{j}(x)}\right\}_{j=1,2,\ldots}$ be the orthonormal basis of the spaces $\mathbb{H}^{1}(I)$ and $\mathbb{L}^{2}(I)$ , which satisfies the eigenvalue problems

$\displaystyle-{y}{{}^{\prime\prime}}_{j\sigma\varepsilon}=\lambda_{j}y_{j% \sigma\varepsilon},$ $\displaystyle{{y}^{\prime}_{j\sigma\varepsilon}(-1)={y}^{\prime}_{j\sigma% \varepsilon}(1)=0,}$

where $\lambda_{j}(j=1,2,\ldots)$ are the corresponding eigenvalues.

Let $w_{\sigma\varepsilon}^{N}(x,t)=\sum_{j=1}^{N}{c_{j}}(t)y_{j\sigma\varepsilon}(x)$ be the Galerkin approximate solution with $w_{\sigma\varepsilon}^{N}(x,0)=\sum_{j=1}^{N}{\left({w_{\sigma\varepsilon 0},y% _{j\sigma\varepsilon}}\right)}y_{j\sigma\varepsilon}$ , we consider the following system:

$\displaystyle\frac{d}{dt}\left({w_{\sigma\varepsilon}^{N},y_{j\sigma% \varepsilon}}\right)=\left({\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)w_% {\sigma\varepsilon xxx}^{N},y_{j\sigma\varepsilon x}}\right)-\left({\left({% \frac{w_{\sigma\varepsilon x}^{N}\left({w_{\sigma\varepsilon+}^{N}+\sigma}% \right)}{\left({w_{\sigma\varepsilon+}+\varepsilon}\right)\left({1+\left|{w_{% \sigma\varepsilon x}^{N}}\right|^{2}}\right)}}\right),y_{j\sigma\varepsilon x}% }\right)$ (3)

for $j=1,2,\ldots,N$ . The classic theory of ordinary differential equations can guarantee the existence of local solutions, the existence of global solutions will depend on the following energy estimates. Let $w_{\sigma\varepsilon xx}^{N}$ be the test function of the system Eq. (3) to obtain

$\displaystyle\frac{1}{2}\frac{\text{d}}{\text{d}t}\int_{I}{\left|{w_{\sigma% \varepsilon x}^{N}}\right|^{2}}\text{d}x+\int_{I}{\left({w_{\sigma\varepsilon+% }^{N}+\sigma}\right)}\left|{w_{\sigma\varepsilon xxx}^{N}}\right|^{2}\text{d}x% =\int_{I}{\frac{\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)w_{\sigma% \varepsilon x}^{N}w_{\sigma\varepsilon xxx}^{N}}{\left({w_{\sigma\varepsilon+}% ^{N}+\varepsilon}\right)\left({1+\left|{w_{\sigma\varepsilon x}^{N}}\right|^{2% }}\right)}}\text{d}x.$

The inequality $\sigma<\varepsilon$ and the Cauchy inequality yield

$\displaystyle\left|{\frac{\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)w_{% \sigma\varepsilon x}^{N}w_{\sigma\varepsilon xxx}^{N}}{\left({w_{\sigma% \varepsilon+}^{N}+\varepsilon}\right)\left({1+\left|{w_{\sigma\varepsilon x}^{% N}}\right|^{2}}\right)}}\right|\text{ }\leqslant\frac{1}{2}\left|{w_{\sigma% \varepsilon xxx}^{N}}\right|^{2}\left({w_{\sigma\varepsilon+}^{N}+\sigma}% \right)+C\frac{\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)\left|{w_{% \sigma\varepsilon x}^{N}}\right|^{2}}{\left({w_{\sigma\varepsilon+}^{N}+% \varepsilon}\right)\left({1+\left|{w_{\sigma\varepsilon x}^{N}}\right|^{2}}% \right)}$ $\displaystyle\quad\leqslant\frac{1}{2}\left|{w_{\sigma\varepsilon xxx}^{N}}% \right|^{2}\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)+C.$

So, we can derive that

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{I}{\left|{w_{\sigma\varepsilon x}% ^{N}}\right|^{2}}\text{d}x+\int_{I}{\left|{w_{\sigma\varepsilon xxx}^{N}}% \right|^{2}\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)}\,\text{d}x% \leqslant C.$

The mass conservation property from Eq. (2) gives

$\displaystyle\int_{I}{w_{\sigma\varepsilon}}(x,t)\text{d}x=\int_{I}{w_{0\sigma% \varepsilon}}\text{d}x\buildrel\textstyle.\over{=}C_{0},$

and then the Poincaré’s inequality can yield

$\displaystyle\int_{I}{\left|{w_{\sigma\varepsilon}^{N}-C_{0}}\right|^{2}}\text% {d}x\leqslant C\int_{I}{\left|{w_{\sigma\varepsilon x}^{N}}\right|^{2}}\text{d% }x\leqslant C.$

In virtue of the Sobolev imbedding theorem, we can derive

$\displaystyle w_{\sigma\varepsilon}^{N}\in\mathbb{L}^{\infty}\left({0,\tau;% \mathbb{H}^{1}(I)}\right)\cap\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{3}(I)}% \right)\cap\mathbb{L}^{\infty}\left({Q_{\tau}}\right).$ (4)

For $\Phi\in{\cal C}_{0}^{\infty}\left({Q_{\tau}}\right)$ ,

$\displaystyle\ \ \ \ \left|{\int_{0}^{\tau}<\,w_{\sigma\varepsilon t}^{N},\Phi% >\text{d}t}\right|$ $\displaystyle\leqslant\left|{\int\!\!\!\int_{Q_{\tau}}{\left|{w_{\sigma% \varepsilon xxx}^{N}}\right|^{2}\left({\sigma+w_{\sigma\varepsilon+}^{N}}% \right)}\text{d}x\text{d}t}\right|\,\left|{\int\!\!\!\int_{Q_{\tau}}{\left({% \sigma+w_{\sigma\varepsilon+}^{N}}\right)}\,\Phi_{x}^{2}\text{d}x\text{d}t}\right|$ $\displaystyle\ \ \ \ +\left|{\int\!\!\!\int_{Q_{\tau}}{\left|{w_{\sigma% \varepsilon x}^{N}}\right|^{2}\left({\sigma+w_{\sigma\varepsilon+}^{N}}\right)% }\text{d}x\text{d}t}\right|\left|{\int\!\!\!\int_{Q_{\tau}}{\Phi_{x}^{2}}\text% {d}x\text{d}t}\right|$ $\displaystyle\ \ \ \ \leqslant C.$

Thus, we have

$\displaystyle w_{\sigma\varepsilon t}^{N}\in\mathbb{L}^{2}\left({0,\tau;% \mathbb{H}^{-1}(I)}\right).$ (5)

By Eqs (4) and (5), as $N\to\infty$ , there exist $w_{\sigma\varepsilon}$ and $w_{\sigma\varepsilon}^{N}$ satisfying

$\displaystyle w_{\sigma\varepsilon}^{N}\rightharpoonup w_{\sigma\varepsilon}\ % \text{weakly $\ast$ converges in}\ \mathbb{L}^{\infty}\left(0,\tau;\mathbb{H}^% {1}(I)\right);$ (6) $\displaystyle w_{\sigma\varepsilon}^{N}\rightharpoonup w_{\sigma\varepsilon}\ % \text{weakly converges in}\ \mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{3}(I)}% \right);$ (7) $\displaystyle w_{\sigma\varepsilon t}^{N}\rightharpoonup w_{\sigma\varepsilon t% }\ \text{weakly converges in}\ \mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{-1}(I)}% \right).$ (8)

In virtue of the compactness theorem (see [18, Section 8]), and the Sobolev imbedding theorem $\mathbb{H}^{3}(I)\hookrightarrow\mathbb{W}^{1,4}(I)\hookrightarrow\mathbb{H}^{% -1}(I)$ , combing with Eqs (4) and (5), we get

$\displaystyle w_{\sigma\varepsilon}^{N}\to w_{\sigma\varepsilon}\ \text{ % strongly in}\ \mathbb{L}^{2}\left({0,\tau;\mathbb{W}^{1,4}(I)}\right),$ $\displaystyle w_{\sigma\varepsilon x}^{N}\to w_{\sigma\varepsilon x},\,\;w_{% \sigma\varepsilon}^{N}\to w_{\sigma\varepsilon}\ \text{ a.e. in}\ Q_{\tau},$ $\displaystyle\frac{\left({\sigma+w_{\sigma\varepsilon+}^{N}}\right)w_{\sigma% \varepsilon x}^{N}}{\left({\varepsilon+w_{\sigma\varepsilon+}^{N}}\right)\left% ({1+\left|{w_{\sigma\varepsilon x}^{N}}\right|^{2}}\right)}\to\frac{\left({w_{% \sigma\varepsilon+}+\sigma}\right)w_{\sigma\varepsilon x}}{\left({w_{\sigma% \varepsilon+}+\varepsilon}\right)\left({1+\left|{w_{\sigma\varepsilon x}}% \right|}\right)}\ \text{a.e. in}\ Q_{\tau}.$

The estimates (6)–(8), the compactness theorem [18] and the Vitali’s theorem give

$\displaystyle w_{\sigma\varepsilon x}^{N}\to w_{\sigma\varepsilon x},\,\,w_{% \sigma\varepsilon+}^{N}+\sigma\to w_{\sigma\varepsilon+}+\sigma\ \text{% strongly in}\ \mathbb{L}^{4}\left({Q_{\tau}}\right);$ (9) $\displaystyle\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)w_{\sigma% \varepsilon xxx}^{N}\rightharpoonup\left({w_{\sigma\varepsilon+}+\sigma}\right% )w_{\sigma\varepsilon xxx}\ \text{weakly in}\ \mathbb{L}^{2}\left({Q_{\tau}}% \right);$ (10) $\displaystyle\frac{\left({w_{\sigma\varepsilon+}^{N}+\sigma}\right)w_{\sigma% \varepsilon x}^{N}}{\left({w_{\sigma\varepsilon+}^{N}+\varepsilon}\right)\left% ({1+\left|{w_{\sigma\varepsilon x}^{N}}\right|^{2}}\right)}\to\frac{\left({w_{% \sigma\varepsilon+}+\sigma}\right)w_{\sigma\varepsilon x}}{\left({w_{\sigma% \varepsilon+}+\varepsilon}\right)\left({1+\left|{w_{\sigma\varepsilon x}}% \right|}\right)}\ \text{strongly in}\ \mathbb{L}^{2}\left({Q_{\tau}}\right).$ (11)

Equations (9)–(11) ensure the existence of global solutions.

3. The limit

\sigma\to 0

Assume that $w_{0\sigma\varepsilon}\to w_{0\varepsilon}$ in $\mathbb{H}^{1}(I)$ with $w_{0\varepsilon}\geqslant 0$ and we have:

Proposition 1. For the approximate problem (2), there exists at least one function $w_{\varepsilon}$ satisfying

1.
$\;w_{\varepsilon}\in\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{2}(I)}\right)\cap{% \cal C}\left({[0,\tau];\mathbb{H}^{1}(I)}\right),w_{\varepsilon t}\in\mathbb{L% }^{2}\left({0,\tau;\mathbb{H}^{-1}(I)}\right);$
2.
For any function $\Phi\in\mathbb{L}^{2}\left({0,\tau;{\cal C}^{\infty}(\bar{I})}\right)$ , it has

$\displaystyle{\int_{0}^{\tau}<w_{\varepsilon t},\Phi>\text{d}t+\int\!\!\!\int_% {Q_{\tau}}{w_{\varepsilon}}w_{\varepsilon xx}\Phi_{xx}\text{d}x\text{d}t+\int% \!\!\!\int_{Q_{\tau}}{w_{\varepsilon x}}w_{\varepsilon xx}\Phi_{x}\text{d}x% \text{d}t}$ $\displaystyle\quad+\int\!\!\!\int_{Q_{\tau}}{\frac{w_{\varepsilon}w_{% \varepsilon x}\Phi_{x}}{\left({w_{\varepsilon+}+\varepsilon}\right)\left({1+% \left|{w_{\varepsilon x}}\right|^{2}}\right)}\,}\text{d}x\text{d}t=0.$

Here we call $w_{\varepsilon}$ that satisfies Proposition 1 as the weak solution of Eq. (2) with $\sigma=0$ . To get the existence of solutions as $\sigma\to 0$ , we borrow the function $\Phi_{\sigma}(\cdot)$ as follows:

$\displaystyle\Phi_{\sigma}(x)=\left\{{\begin{array}[]{l}(x+\sigma)\ln(x+\sigma% )-(x+\sigma)+1,\;x\geqslant 0;\\ \frac{x^{2}}{2\sigma}+x(\ln\sigma)+\sigma(\ln\sigma)-\sigma+1,\;x<0.\\ \end{array}}\right.$

$\Phi_{\sigma}$ is a nonnegative convex function and $\Phi_{\sigma}\in\mathbb{W}^{2,+\infty}(R)$ , ${\Phi}{{}^{\prime\prime}}_{\sigma}(x)=\frac{1}{\left({x_{+}+\sigma}\right)}$ . In the proof of this proposition, we want to use this function as an essential tool to construct some test functions.

Lemma 2. If $w_{\sigma\varepsilon}$ is a solution of Eq. (2), then

$\displaystyle\int_{I}{\Phi_{\sigma}}\left({w_{\sigma\varepsilon}}\right)\text{% d}x+\int\!\!\!\int_{Q_{\tau}}{w_{\sigma\varepsilon xx}^{2}}\text{d}x\text{d}t% \leqslant C,$

where $C$ is a constant independent of $\sigma$ .

Proof. Taking ${\Phi}^{\prime}_{\sigma}\left({w_{\sigma\varepsilon}}\right)$ as the test function in Eq. (2), then

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{I}{\Phi_{\sigma}}\left({w_{\sigma% \varepsilon}}\right)\text{d}x+\int_{I}{\left|{w_{\sigma\varepsilon xx}}\right|% ^{2}}\text{d}x+\int_{I}{\frac{w_{\sigma\varepsilon x}^{2}}{\left({\varepsilon+% w_{\sigma\varepsilon+}}\right)\left({1+\left|{w_{\sigma\varepsilon x}}\right|^% {2}}\right)}}\,\text{d}x=0.$ (12)

For $t\in(0,\tau)$ , by integrating Eq. (12), one has

$\displaystyle\int_{I}{\Phi_{\sigma}}\left({w_{\sigma\varepsilon}(x,t)}\right)% \text{d}x+\int\!\!\!\int_{Q_{\tau}}{\left|{w_{\sigma\varepsilon xx}}\right|^{2% }}\text{d}x\text{d}t+\int\!\!\!\int_{Q_{\tau}}{\frac{w_{\sigma\varepsilon x}^{% 2}}{\left({w_{\sigma\varepsilon+}+\varepsilon}\right)\left({1+\left|{w_{\sigma% \varepsilon x}}\right|^{2}}\right)}\,}\text{d}x\text{d}t$ $\displaystyle\quad=\int_{I}{\Phi_{\sigma}}\left({w_{0\sigma\varepsilon}}\right% )\text{d}x$

and then

$\displaystyle\int_{I}{\Phi_{\sigma}}\left({w_{\sigma\varepsilon}(x,t)}\right)% \text{d}x+\int\!\!\!\int_{Q_{\tau}}{\left|{w_{\sigma\varepsilon xx}}\right|^{2% }}\text{d}x\text{d}t\leqslant C.$ (13)

By applying the estimates from Lemma 2, we have the following limits as $\sigma\to 0$ .

Lemma 3. There exists a function $w_{\varepsilon}\geqslant 0$ such that, as $\sigma\to 0$ ,

1.
$w_{\sigma\varepsilon}\rightharpoonup w_{\varepsilon}$ weakly converges in $\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{2}(I)}\right)$ ;
2.
$w_{\sigma\varepsilon t}\rightharpoonup w_{\varepsilon t}$ weakly converges in $\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{-1}(I)}\right)$ ;
3.
$w_{\sigma\varepsilon}\to w_{\varepsilon}$ strongly converges in ${\cal C}\left({[0,\tau];\mathbb{H}^{1}(I)}\right)$

and

$\displaystyle w_{\varepsilon}\in\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{2}(I)}% \right),w_{\varepsilon}\in\mathbb{L}^{\infty}\left({0,\tau;\mathbb{H}^{1}(I)}% \right),w_{\varepsilon t}\in\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{-1}(I)}% \right).$

Proof. By Lemma 1 and Lemma 2, the uniform estimates

$\displaystyle w_{\sigma\varepsilon}\in\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{% 2}(I)}\right),w_{\sigma\varepsilon}\in\mathbb{L}^{\infty}\left({0,\tau;\mathbb% {H}^{1}(I)}\right),w_{\sigma\varepsilon t}\in\mathbb{L}^{2}\left({0,\tau;% \mathbb{H}^{-1}(I)}\right)$

can be obtained, which gives the weak convergence results:

$\displaystyle w_{\sigma\varepsilon}\rightharpoonup w_{\varepsilon}\ \text{in}% \ \mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{2}(I)}\right),\ \text{as}\ \sigma\to 0,$ $\displaystyle w_{\sigma\varepsilon t}\rightharpoonup w_{\varepsilon t}\ \text{% in}\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{-1}(I)}\right),\ \text{as}\ \sigma% \to 0.$

Moreover, the compactness theorem [18] can give the strong convergence results:

$\displaystyle w_{\sigma\varepsilon}\to w_{\varepsilon}\ \text{in}\ \mathbb{L}^% {2}\left({0,\tau;\mathbb{H}^{1}(I)}\right),\ \text{as}\ \sigma\to 0,$ $\displaystyle w_{\sigma\varepsilon}\to w_{\varepsilon}\ \text{in}\ {\cal C}% \left({[0,\tau];\mathbb{H}^{-1}(I)}\right),\ \text{as}\ \sigma\to 0.$

Next, we show that $w_{\varepsilon}$ is nonnegative. By applying Eq. (13), we get $0\leqslant\int_{I}{\Phi_{\sigma}}\left({w_{\sigma\varepsilon}}\right)\text{d}x% \leqslant C,$ and then

$\displaystyle 0\leqslant\int_{w_{\sigma\varepsilon}<0}{\Phi_{\sigma}}\left({w_% {\sigma\varepsilon}}\right)\text{d}x+\int_{w_{\sigma\varepsilon}\geqslant 0}{% \Phi_{\sigma}}\left({w_{\sigma\varepsilon}}\right)\text{d}x\leqslant C.$

The definition of $\Phi_{\sigma}$ yields

$\displaystyle\int_{w_{\sigma\varepsilon}<0}\left[\frac{w_{\sigma\varepsilon}^{% 2}}{2\sigma}+w_{\sigma\varepsilon}(\ln\sigma)+\sigma(\ln\sigma)-\sigma+1\right% ]\text{d}x\leqslant C.$

Since the terms $w_{\sigma\varepsilon}(\ln\sigma)>0$ and $\sigma(\ln\sigma)-\sigma+1\geqslant 0$ are nonnegative and $0<\sigma<\varepsilon<1$ , we have

$\displaystyle 0\leqslant\int_{I}{w_{\sigma\varepsilon^{-}}^{2}}\text{d}x% \leqslant C\sigma.$

Letting $\sigma\to 0,$ we obtain $w_{\varepsilon}\geqslant 0$ a.e. in $Q_{\tau}$ .

Proof of proposition 1. We choose $\Phi\in\mathbb{L}^{2}\left({0,\tau;{\cal C}_{0}^{\infty}(I)}\right)$ as a multiplier of Lemma 1, and we apply the integrating by parts formulas to get

$\displaystyle\int_{0}^{\tau}{<\,}w_{\sigma\varepsilon t},\Phi>\text{d}t+\int\!% \!\!\int_{Q_{\tau}}{\left({w_{\sigma\varepsilon+}+\sigma}\right)}\,w_{\sigma% \varepsilon xx}\Phi_{xx}\text{d}x\text{d}t+\int\!\!\!\int_{Q_{\tau}}{w_{\sigma% \varepsilon x}}w_{\sigma\varepsilon xx}\Phi_{x}\text{d}x\text{d}t$ (14) $\displaystyle\quad+\int\!\!\!\int_{Q_{\tau}}{\frac{\left({w_{\sigma\varepsilon% +}+\sigma}\right)w_{\sigma\varepsilon x}\Phi_{x}}{\left({w_{\sigma\varepsilon+% }+\varepsilon}\right)\left({1+\left|{w_{\sigma\varepsilon x}}\right|^{2}}% \right)}\,}\text{d}x\text{d}t=0.$

By following the process of obtaining the results Eqs (9) and (11), we can deduce

$\displaystyle w_{\sigma\varepsilon x}\to w_{\varepsilon x},\;\;\left({w_{% \sigma\varepsilon+}+\sigma}\right)\to w_{\varepsilon}\ \text{strongly in}\ % \mathbb{L}^{4}\left({Q_{\tau}}\right);$ (15) $\displaystyle\frac{\left({w_{\sigma\varepsilon+}+\sigma}\right)w_{\sigma% \varepsilon x}}{\left({w_{\sigma\varepsilon+}+\varepsilon}\right)\left({1+% \left|{w_{\sigma\varepsilon x}}\right|^{2}}\right)}\to\frac{w_{\varepsilon}w_{% \varepsilon x}}{\left({w_{\varepsilon+}+\varepsilon}\right)\left({1+\left|{w_{% \varepsilon x}}\right|^{2}}\right)}\ \text{strongly in}\ \mathbb{L}^{2}\left({% Q_{\tau}}\right)$ (16)

using Vitali’s theorem and Lemmas 2–3, we can perform that $\sigma\to 0$ in Eq. (3) due to Eqs (15)–(16) .
4. The limit $\varepsilon\to 0$

Assume $w_{0\varepsilon}\to w_{0}$ strongly in $\mathbb{L}^{2}(I)$ . From the idea of the definition of $\Phi_{\sigma}(\cdot)$ , we define $\Phi_{0}(\cdot)$ as

$\displaystyle\Phi_{0}(x)=x\ln x-x+1,$

which will be used as an essential test function.

Lemma 4. In the sense of ${{\cal D}}^{\prime}(0,\tau)$ , it has

$\displaystyle\int_{I}{\left|{w_{\varepsilon xx}}\right|^{2}}\text{d}x+\frac{% \text{d}}{\text{d}t}\int_{I}{\Phi_{0}}\left({w_{\varepsilon}}\right)\text{d}x% \leqslant-\int_{I}{\frac{\left|{w_{\varepsilon x}}\right|^{2}}{\left({w_{% \varepsilon}+\varepsilon}\right)\left({1+\left|{w_{\varepsilon x}}\right|^{2}}% \right)}}\text{d}x.$

Proof. Similar to Eq. (12), we have

$\displaystyle\int_{I}{\ \ \left|{w_{\sigma\varepsilon xx}}\right|^{2}}\text{d}% x+\frac{\text{d}}{\text{d}t}\int_{I}{\Phi_{\sigma}}\left({w_{\sigma\varepsilon% }}\right)\text{d}x=-\int_{I}{\frac{\left|{w_{\sigma\varepsilon x}}\right|^{2}}% {\left({w_{\sigma\varepsilon+}+\varepsilon}\right)\left({1+\left|{w_{\sigma% \varepsilon x}}\right|^{2}}\right)}}\,\text{d}x.$ (17)

For any $\Phi\in{{\cal D}}^{\prime}(0,\tau)$ with $\Phi\geqslant 0$ , we apply the limit $w_{\sigma\varepsilon}\to w_{\varepsilon}$ in ${\cal C}\left({\bar{Q}_{\tau}}\right)$ to get

$\displaystyle-\int_{0}^{\tau}{\Phi}^{\prime}(t)\int_{I}\Phi\left({w_{\sigma% \varepsilon}}\right)\text{d}x\text{d}t\to-\int_{0}^{\tau}{\Phi}^{\prime}(t)% \int_{I}{\Phi_{0}}\left({w_{\varepsilon}}\right)\text{d}x\text{d}t.$ (18)

The weak convergence $w_{\sigma\varepsilon}\rightharpoonup w_{\varepsilon}$ in $\mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{2}(I)}\right)$ yields

$\displaystyle\mathop{\liminf}\limits_{\sigma\to 0}\int_{0}^{\tau}\Phi(t)\int_{% I}{\left|{w_{\sigma\varepsilon xx}}\right|^{2}}\text{d}x\text{d}t\geqslant\int% _{0}^{\tau}\Phi(t)\int_{I}{\left|{w_{\varepsilon xx}}\right|^{2}}\text{d}x% \text{d}t.$ (19)

Moreover, it has

$\displaystyle-\int\!\!\!\int_{Q_{\tau}}{\frac{\left|{w_{\sigma\varepsilon x}}% \right|^{2}\Phi(t)}{\left({1+\left|{w_{\sigma\varepsilon x}}\right|^{2}}\right% )\left({\varepsilon+w_{\sigma\varepsilon+}}\right)}}\,\text{d}x\text{d}t\to-% \int\!\!\!\int_{Q_{\tau}}{\frac{\left|{w_{\varepsilon x}}\right|^{2}\Phi(t)}{% \left({1+\left|{w_{\varepsilon x}}\right|^{2}}\right)\left({\varepsilon+w_{% \varepsilon+}}\right)}}\,\text{d}x\text{d}t.$ (20)

Thus, Eqs (17)–(20) gives the conclusion.

Lemma 5. It has $w_{\varepsilon}\in\mathbb{L}^{\infty}\left({0,\tau;\mathbb{L}^{1}(I)}\right)$ and $w_{\varepsilon t}\in\mathbb{L}^{1}\left({0,\tau;\left({\mathbb{H}^{-2}(I)}% \right)}\right)$ with the estimates of norms independent of $\varepsilon$ .

Proof. Lemma 4, the mass conservation property and the nonnegativity of $w_{\varepsilon}$ give

$\displaystyle w_{\varepsilon}\in\mathbb{L}^{\infty}\left({0,\tau;\mathbb{L}^{1% }(I)}\right).$

We choose $\Phi\in\mathbb{L}^{\infty}\left({0,\tau;\mathbb{H}^{2}(I)}\right)$ and apply the proposition 1 to get

$\displaystyle\left|{\int_{0}^{\tau}<\,w_{\varepsilon t},\Phi>\text{d}t}\right|% \leqslant\text{ }\left|{\int\!\!\!\int_{Q_{\tau}}{w_{\varepsilon}}w_{% \varepsilon xx}\Phi_{xx}\text{d}x\text{d}t}\right|+\left|{\int\!\!\!\int_{Q_{% \tau}}{w_{\varepsilon x}}w_{\varepsilon xx}\Phi_{x}\text{d}x\text{d}t}\right|$ $\displaystyle\quad+\left|{\int\!\!\!\int_{Q_{\tau}}{\frac{w_{\varepsilon}w_{% \varepsilon x}\Phi_{x}}{\left({w_{\varepsilon+}+\varepsilon}\right)\left({1+% \left|{w_{\varepsilon x}}\right|^{2}}\right)}}\,\text{d}x\text{d}t}\right|$ $\displaystyle\quad=I_{1}+I_{2}+I_{3}.$

By Lemma 4 and Hölder’s inequality, we have

$\displaystyle I_{1}\leqslant C\left\|{\Phi_{xx}}\right\|_{\mathbb{L}^{\infty}% \left({Q_{\tau}}\right)}\left\|{w_{\varepsilon}}\right\|_{\mathbb{L}^{2}\left(% {Q_{\tau}}\right)}\left\|{w_{\varepsilon xx}}\right\|_{{\cal L}^{2}\left({Q_{% \tau}}\right)}\leqslant C,$ $\displaystyle I_{2}\leqslant\left\|{\Phi_{x}}\right\|_{{\cal L}^{\infty}\left(% {Q_{\tau}}\right)}\left\|{w_{\varepsilon xx}}\right\|_{{\cal L}^{2}\left({Q_{% \tau}}\right)}\left\|{w_{\varepsilon x}}\right\|_{\mathbb{{\bf L}}^{2}\left({Q% _{\tau}}\right)}\leqslant C,$ $\displaystyle I_{3}\leqslant\left\|{\Phi_{x}}\right\|_{\text{L}^{2}\left({Q_{% \tau}}\right)}\left\|{w_{\varepsilon x}}\right\|_{\text{L}^{2}\left({Q_{\tau}}% \right)}\leqslant C,$

and we can obtain

$\displaystyle w_{\varepsilon t}\in\mathbb{L}^{1}\left({0,\tau;\mathbb{H}^{-2}(% I)}\right)$

with the estimates of norms independent of $\varepsilon$ .

5. The proof of Theorem 1

Using the compactness theorem [18], Lemmas 4 and 5, there exists $w\geqslant 0$ as $\varepsilon\to 0$ such that

$\displaystyle w_{\varepsilon}\rightharpoonup w\ \text{weakly converges in}\ % \mathbb{L}^{2}\left({0,\tau;\mathbb{H}^{2}(I)}\right);$ (21) $\displaystyle w_{\varepsilon}\to w\ \text{converge in}\ \mathbb{L}^{2}\left({0% ,\tau;\mathbb{H}^{1}(I)}\right).$ (22)

By Eq. (22), there exists a subsequence of $w_{\varepsilon}$ satisfying

$\displaystyle w_{\varepsilon}\to w,w_{\varepsilon x}\to w_{x}\ \text{a.e. in}% \ Q_{\tau}.$ (23)

Define $F_{\varepsilon}=\frac{w_{\varepsilon}w_{\varepsilon x}}{\left({\varepsilon+w_{% \varepsilon}}\right)\left({1+\left|{w_{\varepsilon x}}\right|^{2}}\right)}$ , we have

$\displaystyle\int\!\!\!\int_{Q_{\tau}}{\left|{F_{\varepsilon}}\right|^{2}}% \text{d}x\text{d}t\leqslant\int\!\!\!\int_{Q_{\tau}}{\left|{w_{\varepsilon x}}% \right|^{2}}\text{d}x\text{d}t\leqslant C.$

Applying Eq. (23) and Vitali’s theorem, we have

$\displaystyle F_{\varepsilon}\to\frac{w_{x}}{1+w_{x}^{2}}\ \text{strongly in}% \ \mathbb{L}^{2}\left({Q_{\tau}}\right).$ (24)

From the limits Eqs (21)–(24), as $\varepsilon\to 0$ , we get

$\displaystyle\int\!\!\!\int_{Q_{\tau}}{w_{\varepsilon}}w_{\varepsilon xx}\Phi_% {xx}\text{d}x\text{d}t\to\int\!\!\!\int_{Q_{\tau}}{w{\kern 1.0pt}}w_{xx}\Phi_{% xx}\text{d}x\text{d}t,$ $\displaystyle\int\!\!\!\int_{Q_{\tau}}{w_{\varepsilon x}}w_{\varepsilon xx}% \Phi_{x}\text{d}x\text{d}t\to\int\!\!\!\int_{Q_{\tau}}{w_{x}}w_{xx}\Phi_{x}% \text{d}x\text{d}t,$ $\displaystyle\int\!\!\!\int_{Q_{\tau}}{\frac{w_{\varepsilon}w_{\varepsilon x}% \Phi_{x}}{\left({w_{\varepsilon}+\varepsilon}\right)\left({1+\left|{w_{% \varepsilon x}}\right|^{2}}\right)}}\,\text{d}x\text{d}t\to\int\!\!\!\int_{Q_{% \tau}}{\frac{w_{x}\Phi_{x}}{1+\left|{w_{x}}\right|^{2}}}\,\text{d}x\text{d}t.$

By integration by parts, we can derive

$\displaystyle\int_{0}^{\tau}<{\kern 1.0pt}w_{\varepsilon t},\Phi>\text{d}t=-% \int_{I}{w_{0\varepsilon}}\Phi(x,0)\text{d}x-\int\!\!\!\int_{Q_{\tau}}{w_{% \varepsilon}}\Phi_{t}\text{d}x\text{d}t$

for any test function $\Phi\in{\cal C}^{1}\left({[0,\tau];\;{\cal C}_{0}^{2}(I)}\right)$ with $\Phi(x,\tau)=0$ . The strong convergence Eq. (22) and $w_{0\varepsilon}\to w_{0}$ strongly in $\mathbb{L}^{2}(I)$ allow us to give the limits

$\displaystyle\int\!\!\!\int_{Q_{\tau}}{w_{\varepsilon}}\Phi_{t}\text{d}x\text{% d}t\to\int\!\!\!\int_{Q_{\tau}}{\Phi_{t}}w\text{d}x\text{d}t,\int_{I}{\Phi(x,0% )w_{0\varepsilon}}\text{d}x\to\int_{I}\Phi(x,0)w_{0}(x)\text{d}x.$

Therefore, by performing the limit $\varepsilon\to 0$ , we have

$\displaystyle-\int\!\!\!\int_{Q_{\tau}}{\Phi_{t}}w\text{d}x\text{d}t-\int\!\!% \!\int_{I}\Phi(x,0)w_{0}(x)\text{d}x+\int\!\!\!\int_{Q_{\tau}}ww_{xx}\Phi_{xx}% \text{d}x\text{d}t+\int\!\!\!\int_{Q_{\tau}}{w_{x}}w_{xx}\Phi_{x}\text{d}x% \text{d}t$ $\displaystyle\quad+\int\!\!\!\int_{Q_{\tau}}{\frac{w_{x}}{1+\left|{w_{x}}% \right|^{2}}}{\kern 1.0pt}\Phi_{x}\text{d}x\text{d}t=0.$

The initial condition can be defined in the sense of ${\cal C}\left({[0,\tau];\mathbb{L}^{1}(I)-\textit{weak}}\right)$ ([8] or [19]).

6. Long-time behavior

To obtain the exponential decay property, we redefine the entropy functional as the form

$\displaystyle\Phi_{\varepsilon}(x)=x\ln x-x+C_{\varepsilon}+e$

with $C_{\varepsilon}=\frac{1}{2}\int_{I}{w_{0\varepsilon}}(x)\text{d}x$ . For this functional, we have the following natural inequality.

Lemma 6. For $x>0$ , it has $\left|{x-C_{\varepsilon}}\right|\leqslant\Phi_{\varepsilon}(x)$ .

Proof. When $0<x<C_{\varepsilon}$ , we define the function

$\displaystyle f_{1}(x)=\Phi_{\varepsilon}(x)-\left({C_{\varepsilon}-x}\right)=% x\ln x+e.$

then $f_{1}^{\prime}(x)=\ln x+1$ , we can find the function $f_{1}(x)$ has the minimum value $f_{1}(\frac{1}{e})=0$ at the point $x=\frac{1}{e}$ and so $f_{1}(x)\geqslant 0$ . When $x>C_{\varepsilon}$ , we define

$\displaystyle f_{2}(x)=\Phi_{\varepsilon}(x)-\left({x-C_{\varepsilon}}\right)=% x\ln x-2x+2C_{\varepsilon}+e$

Then $f_{2}^{\prime}(x)=\ln x-1$ and the function $f_{2}(x)$ has the minimum value $f_{2}(e)=2C_{\varepsilon}>0$ at the point $x=e$ and then $f_{2}(x)\geqslant 0$ . Therefore, the result of the lemma is true.

The proof of Theorem 2. From Lemma 4, for every $t>$ 0, one has

$\displaystyle\int_{I}{\Phi_{0}}\left({w_{\varepsilon}}\right)\text{d}x+\int_{0% }^{t}{\int_{I}{\left|{w_{\varepsilon xx}}\right|^{2}}}\text{d}x\text{d}t% \leqslant\int_{I}{\Phi_{0}}\left({w_{0\varepsilon}}\right)\text{d}x\leqslant C% _{1},$ (25)

where $C_{1}$ is independent of $t$ and $\varepsilon$ . Lemma 6 yields

$\displaystyle\int_{I}{\left|{w_{\varepsilon}-C_{\varepsilon}}\right|}{\kern 1.% 0pt}\text{d}x+\int_{0}^{t}{\int_{I}{\left|{w_{\varepsilon xx}}\right|^{2}}}% \text{d}x\text{d}t\leqslant C_{1}.$ (26)

Furthermore, the Poincaré’s inequality gives

$\displaystyle\int_{0}^{t}{\left({\int_{I}{\left|{w_{\varepsilon}-C_{% \varepsilon}}\right|}{\kern 1.0pt}\text{d}x}\right)^{2}}\text{d}t\leqslant C_{% 2}\int_{0}^{t}{\int_{I}{\left|{w_{\varepsilon xx}}\right|^{2}}}\text{d}x\text{% d}t\leqslant C_{3},$ (27)

where $C_{2}$ and $C_{3}$ are independent of $t$ and $\varepsilon$ .

Let $H(t)=\left({\int_{I}{\left|{w_{\varepsilon}-C_{\varepsilon}}\right|}\text{d}x}% \right)^{2}$ and apply Eqs (26) and (27) to get

$\displaystyle H(t)\leqslant 2C_{1}^{2}-2C_{1}\int_{0}^{t}{\int_{I}{\left|{w_{% \varepsilon xx}}\right|^{2}}}\text{d}x\text{d}t\leqslant C_{4}-C_{5}\int_{0}^{% t}H(s)\text{d}s,$

where $C_{4}$ and $C_{5}$ are independent of $t$ and $\varepsilon$ . Apply the Gronwall’s inequality to have $H(t)\leqslant C_{4}e^{-C_{5}t}$ , for $t>0$ , i.e.

$\displaystyle\int_{I}{\left|{w_{\varepsilon}-C_{\varepsilon}}\right|}\ \text{d% }x\leqslant C_{6}e^{-C_{7}t}.$

Let the limit $\varepsilon\to 0$ , then

$\displaystyle\int_{I}{\left|{w-C_{0}}\right|}\,\text{d}x\leqslant C_{6}e^{-C_{% 7}t}$

with $C_{0}=\frac{1}{2}\int_{I}{w_{0}}(x)\text{d}x$ .

7. Conclusion

A fourth-order degenerate parabolic equation with a nonlinear second-order diffusion was introduced in the paper, and the model could be used to describe the thin-film problem in fluid mechanics. We overcame the difficulty of the invalidity of the maximum principle and the singularity effect caused by the nonlinear second-order term. Using the Galerkin method and the entropy functional method, we gave the existence of nonnegative weak solutions with the assistance of some classical compactness arguments. The long-time exponential decay behavior was also given in the sense of $L^{1}$ -norm via redefining the entropy function. In the future work, we look forward to extending the conclusions to high dimensional spaces.

Footnotes

Acknowledgments

The paper is supported by the Education Department Science Foundation of Liaoning Province of China (No. LJKMZ20220832).

References

Oron

Davis

Bankoff

. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 1997; 69(3): 931-980. doi: 10.1103/RevModPhys.69.931.

Myers

. Thin films with high surface tension. SIAM Reviews. 1998; 40(3): 441-462. doi: 10.1137/S003614459529284X.

Bernis

Friedman

. Higher order nonlinear degenerate parabolic equations. J. Differential Equations. 1990; 83(1): 179-206. doi: 10.1016/0022-0396(90)90074-Y.

Passo

Garcke

Grün

. On a fourth order degenerate parabolic equation: Global entropy estimates, existence and qualitative behaviour of solutions. SIAM J Math Anal. 1998; 29(2): 321-342. doi: 10.1137/S0036141096306170.

Grün

. Droplet Spreading Under Slippage-Existence for the Cauchy Problem. Commun Partial Differ Equ. 2005; 29(11-12): 1697-1744. doi: 10.1081/PDE-200040193.

Gao

Yin

. Existence of physical solutions to the boundary value problem of an equation in modeling oil film spreading over a solid surface. J Partial Differ Equ. 1998; 11(4): 335-346.

Bertozzi

Pugh

. The lubrication approximation for thin viscous films: The moving contact line with a porous media cut-off of van der Waals interactions. Nonlinearity. 1994; 6: 1535-1564. doi: 10.1088/0951-7715/7/6/002.

Boutat

Hilout

Rakotoson

. A generalized thin-film equation in multidimensional space. Nonlinear Analysis. 2008; 69(4): 1268-1286. doi: 10.1016/j.na.2007.06.028.

Liang

. Existence and asymptotic behavior of solutions to a thin film equation with Dirichlet boundary. Nonlinear Anal-Real. 2011; 12(3): 1828-1840. doi: 10.1016/j.nonrwa.2010.11.014.

10.

Zhao

Liu

Duan

. Time-periodic solution for a fourth-order parabolic equation describing crystal surface growth. Electron J Qual Theory Differ. 2013; 66(7): 1-15. doi: 10.14232/ejqtde.2013.1.7.

11.

Ansini

Giacomelli

. Doubly nonlinear thin-film equations in one space dimension. Arch. Rational Mech. Anal. 2004; 173: 89-131. doi: 10.1007/s00205-004-0313-x.

12.

Bernis

Peletier

Williams

. Source type solutions of a fourth order nonlinear degenerate parabolic equation. J Nonlinear Analysis: Theory, Methods and Applications. 1992; 18(3): 217-234. doi: 10.1016/0362-546X(92)90060-R.

13.

Bertozzi

Pugh

. The lubrication approximation for thin viscous films: regularity and long time behavior of weak solutions. Commun Pure Appl Math. 1996; 49(2): 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2.

14.

Bleher

Lebowitz

Speer

. Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Commun Pure Appl Math. 1994; 47(7): 923-942. doi: 10.1002/cpa.3160470702.

15.

Ferreira

Bernis

. Source-type solutions to thin-film equations in higher dimensions. Eur J Appl Math. 1997; 8(5): 507-524. doi: 10.1017/S0956792597003197.

16.

Khan

Chen

Baleanu

Khan

. Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator. Bound Value Probl. 2017; 1: 1211-1226. doi: 10.1186/s13661-017-0878-6.

17.

Zeidler

. Nonlinear Functional Analysis and Its Applications. Heidelberg: Springer, 2013.

18.

Simon

. Compact sets in the space 𝐿𝑝⁢(0,T;B). Ann Math Pura Appl. 1987; 146: 65-96. doi: 10.1007/BF01762360.

19.

Temam

. Infinite Dimensonal Dynamical Systems in Mechanics and Physics. Heidelberg: Springer, 2012.