四阶偏微分方程在大气科学、应用科学、流体力学、数学物理、海洋环境流体动力学等一些学科领域均得到了广泛应用, 比较有代表性的方程包括Cahn-Hilliard方程、薄膜方程和半导体方程. 近年来, 国内外对于四阶偏微分方程的研究越来越多，特别是关于Cahn-Hilliard模型和薄膜方程的研究. 在文献[1]中, Liu研究了具有非退化流动性和零质量通量边界条件的Cahn-Hilliard方程:

$ \omega_{t}+\operatorname{div}\left(m(\omega) k \nabla \Delta \omega-|\nabla \omega|^{p-2} \nabla \omega\right)=0, \quad(x, t) \in \Omega \times(0, T), $

其中$ k>0 $, $ p>2 $, $ \Omega $是二维空间上的有界区域. 作者利用Schauder估计方法和Campanato空间理论, 给出了在小初始能量下经典解的全局存在性. Liang在文献[2] 中研究了一维空间中边界条件为正常数的四阶抛物方程:

$ \omega_{t}+\varepsilon\left(\omega^{n} \omega_{x x x}\right)_{x}-\delta \omega_{x x}=0, \quad(x, t) \in \Omega \times(0, T), $

其中$ \Omega=(0,1) $, $ \varepsilon>0 $, $ \delta>0 $. 通过将四阶方程转化为二阶椭圆抛物型方程组, 得到了初值函数接近边值时强正解的存在性和正则性. 此外, Liang在文献[3]中还研究了多维情形下的相关问题:

$ \omega_{t}+\nabla \cdot\left(\omega^{n} \nabla \Delta \omega\right)=0, \quad(x, t) \in \Omega \times(0, T), $

其中初始函数在拉普拉斯方程的正稳态解附近. 对于梯度退化问题, Xu和Zhou在文献[4]中研究了带有齐次边界条件的四阶偏微分方程:

$ \omega_{t}+\operatorname{div}\left(\left.\nabla \Delta \omega\right|^{p-2} \nabla \Delta \omega\right)=f-\operatorname{div} g, \quad(x, t) \in \Omega \times(0, T], $

得到了弱解的存在性与正则性. 在文献[5]中, Guo和Gao研究了相应的变指数问题:

$ \omega_{t}+\operatorname{div}\left(|\nabla \Delta \omega|^{p(x)-2} \nabla \Delta \omega\right)=f(x, u), \quad(x, t) \in \Omega \times(0, T), $

其中非线性源$ f(x, u) $满足$ |f(x, u)| \leq M|u|^{m(x)}+a(x) $, $ p(x)>2 $, $ m(x)>1 $. 在文献[6]中, Zhan研究了如下的对流扩散方程的初边值问题:

$ \omega_{t}=\operatorname{div}\left(\left|\nabla \omega^{n}\right|^{p-2} \nabla \omega^{n}\right)+\sum\limits_{i=1}^{N} \frac{\partial b_{i}\left(\omega^{n}\right)}{\partial x_{i}},\quad(x, t) \in \Omega \times(0, T). $

通过Moser迭代技巧得到了其正则化问题解的$ L^{\infty} $模与其梯度的$ L^{p} $模的局部有界性. 再通过紧致性定理得到了原问题解的存在性. 同时讨论了解的唯一性、正性以及熄灭性. 此外, Zhan在文献[7]中还研究了边界退化的对流扩散方程的初边值问题:

$ \omega_{t}=\operatorname{div}\left(d^{\alpha}|\nabla \omega|^{p-2} \nabla \omega\right)+\sum\limits_{i=1}^{N} \frac{\partial b_{i}(\omega)}{\partial x_{i}},(x, t) \in \Omega \times(0, T), $

其中对流项$ \sum_{i=1}^{N} \frac{\partial b_{i}(\omega)}{\partial x_{i}} $满足$ \left|b_{i}(w)\right| \leq C|w|^{1+\mu}, \left|b_{i}^{\prime}(w)\right| \leq C|w|^{\mu} $, 通过抛物方程正则化理论讨论了该方程初边值问题解的存在性, 并且在$ \frac{p-2}{2}>\alpha>1 $的条件下, 得到了该问题解的存在唯一性. 在此基础上，Zhan在[8]中还研究了上述边界退化方程解的稳定性. 当$ \alpha<p-1 $时, 弱解属于某个$ H^{k}(k>1) $空间, Dirichlet边界条件可以照常施加, 并证明了解的稳定性. 当$ \alpha \geq p-1 $时, 弱解缺乏在边界上定义迹的正则性, 但是仍然可以证明在不施加边界条件的情况下解的稳定性. 也就是说, 此时方程的解可以不受边界条件的任何限制.

受上述文献启发, 本文考虑如下的四阶抛物型$ p- $Laplace方程的初边值问题:

$ \begin{align} & \omega_{t}+\Delta\left(\rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega\right)=\sum\limits_{i=1}^{N} \frac{\partial g_{i}(\omega)}{\partial x_{i}}, \quad(x, t) \in Q_{T}=\Omega \times(0, T), \end{align} $

(1.1)

$ \begin{align} & \omega=\Delta \omega=0, \quad(x, t) \in \Gamma=\partial \Omega \times(0, T), \end{align} $

(1.2)

$ \begin{align} & \omega(x, 0)=\omega_{0}(x), \quad x \in \Omega. \end{align} $

(1.3)

这里假设$ \Omega $是$ R^{N}(N \leq 2) $上的有界开区域, 维数的要求主要是确保相应嵌入定理成立，以及强收敛的极限的进行. 由于几何问题不是研究重点, 要求边界$ \partial \Omega $是充分光滑的, 且具有足够好的拓扑性质. $ \alpha>0 $, $ p>1 $, $ T>0 $都是常数. $ \rho=\operatorname{dist}(x, \partial \Omega) $表示$ x $到边界的距离, 这意味着$ \rho $在边界$ \partial \Omega $上是退化的. 并且, 限制$ g_{i}(\omega) $及其导数的增长阶条件满足: 对于任意的$ i \in\{1,2,3 \cdots N\} $, $ g_{i}(\omega) $是$ C^{1} $函数, 且存在$ C $, 使得$ \left|g_{i}(\omega)\right| \leq C\omega, \left|g_{i}^{\prime}(\omega)\right| \leq C. $

定义1.1   称$ \omega=\omega(x, t) $是问题(1.1)-(1.3)的一个弱解, 如果其满足:

$ {\rm(i)} $ $ \omega \in L^{\infty}\left(0, T ; L^{2}(\Omega)\right) \cap C\left([0, T]; L^{2}(\Omega)\right) $, $ \rho^{\alpha}|\Delta \omega|^{p} \in L^{1}\left(Q_{T}\right) $, $ \Delta \omega \in L^{p}\left(0, T ; L_{\text {loc }}^{p}(\Omega)\right) $, $ \omega_{t} \in L^{2}\left(Q_{T}\right) $;

$ {\rm(ii)} $对于任意的$ \varphi \in C_{0}^{\infty}\left(Q_{T}\right) $, 有

$ \iint_{Q_{T}} \omega_{t} \varphi \mathrm{d}x \mathrm{d} t+\iint_{Q_{T}} \rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega \Delta \varphi \mathrm{d}x \mathrm{d} t+\iint_{Q_{T}} \sum\limits_{i=1}^{N} g_{i}(\omega) \varphi_{x_{i}} \mathrm{d}x \mathrm{d} t=0. $

下面的定理为本文主要结论.

定理1.1   令$ p>2 $, $ \alpha<\frac{p-2}{2} $, 若$ \omega_{0} \in W_{0}^{2}(\Omega) $, 则问题(1.1)-(1.3)存在弱解$ \omega $, 其中$ W_{0}^{2}(\Omega)=W^{2, p}(\Omega) \cap W_{0}^{1, p}(\Omega). $

本文结构分为两个部分. 第一部分考虑非退化情形下的抛物问题, 将利用Galerkin理论证明相应弱解的存在性. 第二部分考虑退化边界情形下的抛物方程, 给出弱解的存在性. 文中约定$ C $表示一般常数, 不同位置值可能不同.

为研究边界退化问题(1.1)-(1.3), 先考虑其正则化问题:

$ \begin{align} & \omega_{\eta t}+\Delta\left(\rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta}\right)=\sum\limits_{i=1}^{N} \frac{\partial g_{i}\left(\omega_{\eta}\right)}{\partial x_{i}},(x, t) \in Q_{T}=\Omega \times(0, T), \end{align} $

(2.1)

$ \begin{align} & \omega_{\eta}=\Delta \omega_{\eta}=0,(x, t) \in \Gamma=\partial \Omega \times(0, T), \end{align} $

(2.2)

$ \begin{align} & \omega_{\eta}(x, 0)=\omega_{0 \eta}(x), \quad x \in \Omega, \end{align} $

(2.3)

其中$ \rho_{\eta}=\rho+\eta, \eta>0 $. 并假设当$ \eta \rightarrow 0 $时, $ \omega_{0 \eta} \rightarrow \omega_{0} $于$ W^{2, p}(\Omega) $. 若初边值问题(2.1)-(2.3)是可解的, 那么边界退化问题(1.1)-(1.3)的解的存在性可以由$ \eta $的渐近极限过程得到.

问题(2.1)-(2.3)的存在性由下面的命题给出.

命题2.1   假定$ \omega_{0 \eta} \in W_{0}^{2}(\Omega) $, 则(2.1)-(2.3)在$ Q_{T}=\Omega \times(0, T) $上存在弱解$ \omega_{\eta}=\omega_{\eta}(x, t) $, 且$ \omega_{\eta} $满足下列条件:

$ {\rm(i)} $ $ \omega_{\eta} \in L^{\infty}\left(0, T ; W_{0}^{2}(\Omega)\right) \cap C\left([0, T] ; W^{1, p}(\Omega)\right) $, 且$ \omega_{\eta t} \in L^{2}\left(Q_{T}\right) $;

$ {\rm(ii)} $对于任意的$ \varphi \in C_{0}^{\infty}\left(Q_{T}\right) $, 有

$ \iint_{Q_{T}} \omega_{\eta} \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta} \Delta \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{T}} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}\right) \varphi_{x_{i}} \mathrm{d} x \mathrm{d} t=0. $

证   下面将采用Galerkin方法给出弱解的存在性. 令$ \left\{\phi_{i}(x)\right\}_{i=1}^{\infty} $是$ W_{0}^{2}(\Omega) $中的标准正交基底, 构造问题(2.1)–(2.3)的近似解为

$ \omega_{\eta}^{n}(x, t)=\sum\limits_{i=1}^{n} c_{i}^{n}(t) \phi_{i}(x), n=1,2, \cdots $

使其满足如下问题:

$ \begin{align} & \left(\omega_{\eta t}^{n}, \phi_{i}\right)+\left(\rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p-2} \Delta \omega_{\eta}^{n}, \Delta \phi_{i}\right)+\left(\sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}^{n}\right),\left(\phi_{i}\right)_{x_{i}}\right)=0, \end{align} $

(2.4)

$ \begin{align} & \omega_{\eta}^{n}(x, 0)=\sum\limits_{i=1}^{n} d_{i}^{n}(t) \phi_{i}(x) \rightarrow \omega_{0 \eta} \text { 于 } W_{0}^{2}(\Omega),\text { 当 } n \rightarrow +\infty \text { 时. } \end{align} $

(2.5)

问题(2.4)-(2.5)解的局部存在性可以由常微分方程理论中的Peano定理来保证, 为证明解的整体性, 需要对近似解做相应的能量估计. 为此, 将(2.4)的左右两端同时乘以$ c_{i}^{n}(t) $, 对$ i $从1到$ n $求和得

$ \begin{aligned} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t}\left\|\omega_{\eta}^{n}\right\|_{L^{2}(\Omega)}^{2}+\int_{\Omega} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p} \mathrm{ d} x = & -\int_{\Omega} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}^{n}\right) \omega_{\eta x_{i}}^{n} \mathrm{ d} x \\ = & -\sum\limits_{i=1}^{N} \int_{\Omega} \frac{\partial}{\partial x_{i}}\left(\int_{0}^{\omega_{\eta}^{n}} g_{i}(s) \mathrm{d} s\right) \mathrm{d} x \\ = & 0. \end{aligned} $

对其在$ (0, T) $上积分, 得

$ \left\|\omega_{\eta}^{n}(\cdot, T)\right\|_{L^{2}(\Omega)}^{2}+2 \iint_{Q_{T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p} \mathrm{d} x \mathrm{d} t \leq\left\|\omega_{0 \eta}^{n}(x)\right\|_{L^{2}(\Omega)}^{2}. $

由此可知

$ \begin{align} \iint_{Q_{T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p} \mathrm{d} x \mathrm{d} t \leq C . \end{align} $

(2.6)

接下来, 将(2.4)的左右两端同时乘以$ \frac{\mathrm{d}}{\mathrm{d} t} c_{i}^{n}(t) $, 对$ i $从1到$ n $求和, 然后对其在$ (0, t) $上积分, 得

$ \begin{align} \iint_{Q_{t}}\left(\omega_{\eta \tau}^{n}\right)^{2} \mathrm{d} x \mathrm{d} \tau+\iint_{Q_{t}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p-2} \Delta \omega_{\eta}^{n} \Delta \omega_{\eta \tau}^{n} \mathrm{d} x \mathrm{d} \tau +\iint_{Q_{t}} \sum\limits_{i=1}^{N} \frac{\partial g_{i}\left(\omega_{\eta}^{n}\right)}{\partial x_{i}} \omega_{\eta \tau}^{n} \mathrm{d} x \mathrm{d} \tau=0, \end{align} $

(2.7)

其中

$ \begin{align} \iint_{Q_{t}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p-2} \Delta \omega_{\eta}^{n} \Delta \omega_{\eta \tau}^{n} \mathrm{d} x \mathrm{d} \tau =& \frac{1}{2} \iint_{Q_{t}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p-2} \frac{\mathrm{d}}{\mathrm{d} \tau}\left(\Delta \omega_{\eta}^{n}\right)^{2} \mathrm{d} x \mathrm{d} \tau \\ =& \frac{1}{2} \iint_{Q_{t}} \rho_{\eta}^{\alpha} \frac{\mathrm{d}}{\mathrm{d} \tau}\left(\int_{0}^{\left|\Delta \omega_{\eta}^{n}\right|^{2}} s^{\frac{p-2}{2}} \mathrm{d} s\right) \mathrm{d} x \mathrm{d} \tau \\ =& \frac{1}{p} \int_{0}^{t} \frac{\mathrm{d}}{\mathrm{d} \tau} \int_{\Omega} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p} \mathrm{d} x \mathrm{d} \tau \\ =&\frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{\eta}^{n}(\cdot, t)\right\|_{L^{p}(\Omega)}^{p}-\frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{\eta}^{n}(\cdot, 0)\right\|_{L^{p}(\Omega)}^{p}. \end{align} $

(2.8)

由$ g_i $的假定条件可得

$ \begin{align} \iint_{Q_{t}} \sum\limits_{i=1}^{N} \frac{\partial g_{i}\left(\omega_{\eta}^{n}\right)}{\partial x_{i}} \omega_{\eta t}^{n} \mathrm{d} x \mathrm{d} \tau \leq & \sum\limits_{i=1}^{N} \iint_{Q_{t}}\left|g^{\prime}\left(\omega_{\eta}^{n}\right)\right| \omega_{\eta x_{i}}^{n}|| \omega_{\eta t}^{n} \mid \mathrm{d} x \mathrm{d} \tau \\ \leq & \frac{1}{2} \iint_{Q_{t}}\left(\omega_{\eta \tau}^{n}\right)^{2} \mathrm{d} x \mathrm{d} \tau+C \iint_{Q_{t}}\left|\nabla \omega_{\eta}^{n}\right|^{2} \mathrm{d} x \mathrm{d} \tau. \end{align} $

(2.9)

由Sobolev嵌入定理(见文献[9])可得$ \left\|\omega_{\eta}^{n}\right\|_{W^{1, p}(\Omega)} \leq C\left\|\omega_{\eta}^{n}\right\|_{W^{2, p}(\Omega)}. $因此, 结合(2.6)可以得到

$ \begin{align} \iint_{Q_{t}}\left|\nabla \omega_{\eta}^{n}\right|^{2} \mathrm{d} x \mathrm{d} \tau \leq C. \end{align} $

(2.10)

结合(2.7)–(2.10)可以得到

$ \frac{1}{2} \iint_{Q_{t}}\left(\omega_{\eta \tau}^{n}\right)^{2} \mathrm{d} x \mathrm{d} \tau+\frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{\eta}^{n}(\cdot, t)\right\|_{L^{p}(\Omega)}^{p} \leq \frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{\eta}^{n}(\cdot, 0)\right\|_{L^{p}(\Omega)}^{p}+C, $

然后可以得到

$ \begin{align} \frac{1}{2} \iint_{Q_{T}}\left(\omega_{\eta t}^{n}\right)^{2} \mathrm{ d} x \mathrm{d} t+\sup _{t \in[0, T]} \frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{\eta}^{n}(\cdot, t)\right\|_{L^{p}(\Omega)}^{p} \leq C. \end{align} $

(2.11)

进而有

$ \begin{align} & \left\|\omega_{\eta t}^{n}\right\|_{L^{2}\left(0, T ; L^{2}(\Omega)\right)} \leq C, \end{align} $

(2.12)

$ \begin{align} & \left\|\omega_{\eta}^{n}\right\|_{L^{\infty}\left(0, T ; W_{0}^{2}(\Omega)\right)} \leq C. \end{align} $

(2.13)

此外, 结合假定条件和(2.13)可以得到

$ \begin{align} \left\|g_{i}\left(\omega_{\eta}^{n}\right)\right\|_{L^{\infty}\left(0, T ; L^{p}(\Omega)\right)} \leq C. \end{align} $

(2.14)

利用(2.12)-(2.13), Aubin-Lions紧致性定理(见[10]), 以及经典的单调算子理论(见[11], 这里不再阐述细节)可知存在函数$ \omega_{\eta} $和$ \left\{\omega_{\eta}^{n}\right\}_{n=1}^{\infty} $的一个子列(仍记为其本身)使得

$ \begin{align} & \omega_{\eta t}^{n} \rightharpoonup \omega_{\eta t} \text { 于 } L^{2}\left(0, T ; L^{2}(\Omega)\right), \end{align} $

(2.15)

$ \begin{align} & \omega_{\eta}^{n} \stackrel{*}{\rightharpoonup} \omega_{\eta} \text { 于 } L^{\infty}\left(0, T ; W_{0}^{2}(\Omega)\right), \end{align} $

(2.16)

$ \begin{align} & \omega_{\eta}^{n} \rightarrow \omega_{\eta} \text { 于 } L^{2}\left(0, T ; W_{0}^{1, p}(\Omega)\right), \end{align} $

(2.17)

$ \begin{align} & \omega_{\eta}^{n} \rightarrow \omega_{\eta} \text { 几乎处处于 } \Omega \times(0, T), \end{align} $

(2.18)

$ \begin{align} & \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p-2} \Delta \omega_{\eta}^{n}\rightharpoonup \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta} \text { 于 } L^{\frac{p}{p-1}}(Q_T), \end{align} $

(2.19)

$ \begin{align} & g_{i}\left(\omega_{\eta}^{n}\right) \rightarrow g_{i}\left(\omega_{\eta}\right) \text { 几乎处处于 } \Omega \times(0, T), \end{align} $

(2.20)

$ \begin{align} & g_{i}\left(\omega_{\eta}^{n}\right) \stackrel{*}{\rightharpoonup} g_{i}\left(\omega_{\eta}\right) \text { 于 } L^{\infty}\left(0, T ; L^{p}(\Omega)\right), \end{align} $

(2.21)

其中(2.17)成立意味着(2.18)成立, 结合$ g_{i}(\omega)\in C^{1}\left(Q_{T}\right) $和(2.18)可以得到(2.20), 结合(2.14) 和(2.20)可以得到(2.21). (2.4)两侧关于$ t $在$ (0, T) $上积分可得

$ \begin{align} \iint_{Q_{T}} \omega_{\eta t}^{n} \varphi \mathrm{d} x \mathrm{d} t +\iint_{Q_{T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}^{n}\right|^{p-2} \Delta \omega_{\eta}^{n} \Delta \varphi \mathrm{d} x \mathrm{d} t + \iint_{Q_{T}} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}^{n}\right) \varphi_{x_{i}} \mathrm{d} x \mathrm{d} t=0. \end{align} $

(2.22)

结合(2.15)–(2.21), 在(2.22)中令$ n \rightarrow +\infty $可得

$ \iint_{Q_{T}} \omega_{\eta t} \varphi \mathrm{d} x \mathrm{d} t +\iint_{Q_{T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta} \Delta \varphi \mathrm{d} x \mathrm{d} t +\iint_{Q_{T}} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}\right) \varphi_{x_{i}} \mathrm{d} x \mathrm{d} t=0, $

其中$ \varphi \in C_{0}^{\infty}\left(Q_{T}\right) $. 此时则得出弱解的存在性. 此外, 结合$ \omega_{\eta} \in L^{\infty}\left(0, T ; W_{0}^{2}(\Omega)\right) $和$ \omega_{\eta t} \in L^{2}\left(0, T ; L^{2}(\Omega)\right) $, 应用Aubin-Lions紧致性定理(见[10])可得$ \omega_{\eta} \in C\left([0, T] ; W^{1, p}(\Omega)\right) $.

在本节中, 用$ \omega_{\eta} $来表示初边值问题(2.1)-(2.3)的解. 本节的主要内容是建立不依赖于$ \eta $的一致估计以及讨论$ \eta \rightarrow 0 $的极限. 一致估计的结果由下面的引理给出.

引理3.1   逼近解$ \omega_{\eta} $具有如下估计:

$ \begin{align} & \left\|\rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p}\right\|_{L^{1}\left(Q_{T}\right)} \leq C, \end{align} $

(3.1)

$ \begin{align} & \left\|\rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta}\right\|_{L^{\frac{p}{p-1}}\left(Q_{T}\right)} \leq C, \end{align} $

(3.2)

$ \begin{align} & \left\|\omega_{\eta}\right\|_{L^{\infty}\left(0, T ; W^{2,\beta}(\Omega)\right)} \leq C, \end{align} $

(3.3)

$ \begin{align} & \left\|\omega_{\eta t}\right\|_{L^{2}\left(0, T ; L^{2}(\Omega)\right)} \leq C, \end{align} $

(3.4)

$ \begin{align} & \left\|g_{i}\left(\omega_{\eta}\right)\right\|_{L^{\infty}\left(0, T ; L^{2}(\Omega)\right)} \leq C, \end{align} $

(3.5)

其中$ \beta>2 $是一个常数, $ C $不依赖于$ \eta $.

证   令$ \xi_{[0, t]}(t) $为区间$ [0, t] $上的特征函数, 对于任意的$ t \in(0, T] $, 在命题2.1-(ii)中取$ \varphi=\omega_{\eta} \xi_{[0, t]}(t) $. 于是有

$ \begin{aligned} & \frac{1}{2}\left\|\omega_{\eta}(\cdot, t)\right\|_{L^{2}(\Omega)}^{2}-\frac{1}{2}\left\|\omega_{\eta}(\cdot, 0)\right\|_{L^{2}(\Omega)}^{2}+\iint_{Q_{t}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p} \mathrm{d} x \mathrm{d} \tau \\ = & -\iint_{Q_{t}} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}^{n}\right) \omega_{\eta x_{i}}^{n} \mathrm{d} x \mathrm{d} t \\ = & -\sum\limits_{i=1}^{N} \iint_{Q_{t}} \frac{\partial}{\partial x_{i}}\left(\int_{0}^{\omega_{\eta}^{n}} g_{i}(s) \mathrm{d} s\right) \mathrm{d} x \mathrm{d} t \\ = & 0. \end{aligned} $

进而可得

$ \begin{align} \sup _{t \in[0, T]}\left\|\omega_{\eta}(\cdot, t)\right\|_{L^{2}(\Omega)}^{2}+2 \iint_{Q_{T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p} \mathrm{d} x \mathrm{d} t \leq\left\|\omega_{0 \eta}(x)\right\|_{L^{2}(\Omega)}^{2}. \end{align} $

(3.6)

由(3.6)可得估计(3.1)成立. 然后再由$ \rho_{\eta} $的有界性, 可以得到估计(3.2)也成立. 此外, 结合$ g_i $的条件和(3.6)可得$ \int_{\Omega}\left|g_{i}(\omega_{\eta})\right|^{2} \mathrm{d} x \leq C \int_{\Omega}\left| \omega_{\eta}\right|^{2} \mathrm{d} x \leq C $, 即可得到估计(3.5). 为得到(3.3)和(3.4), 不妨设命题2.1-(ii)中的检验函数为$ \varphi=\omega_{\eta t}\xi_{[0, t]}(t) $, 再对等式右侧进行放大可得

$ \begin{align} & \frac{1}{2} \iint_{Q_{T}}\left(\omega_{\eta t}\right)^{2} \mathrm{d} x \mathrm{d} t+\sup _{t \in[0, T]}\frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{\eta}(\cdot, t)\right\|_{L^{p}(\Omega)}^{p} \\ \leq& \frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{0 \eta}(x)\right\|_{L^{p}(\Omega)}^{p}+C \iint_{Q_{T}}\left|\nabla \omega_{\eta}\right|^{2} \mathrm{d} x \mathrm{d} t . \end{align} $

(3.7)

此外, 由$ p>2 $, $ \alpha<\frac{p-2}{2} $, 可得$ \frac{\alpha}{p-2}<\frac{1}{2} $. 因此, 存在一个常数$ \alpha_{0} \in\left(\frac{\alpha}{p-2}, \frac{1}{2}\right) $, 这意味着$ p-\frac{\alpha}{\alpha_{0}}>2 $. 由此可得, 存在一个常数$ \beta $满足$ 2<\beta<\min \left\{p-\frac{\alpha}{\alpha_{0}}, \frac{1}{\alpha_{0}}\right\} $. 所以有$ \alpha_{0} \beta<1, \frac{\alpha}{\alpha_{0}}+\beta<p $. 利用上述这些指标关系, 可得

$ \begin{align} \iint_{Q_{T}}\left|\Delta \omega_{\eta}\right|^{\beta} \mathrm{d} x \mathrm{d} t \leq &\notag \iint_{\left\{\rho_{n}^{\alpha_{0}}\left|\Delta \omega_{\eta}\right| \leq 1\right\}}\left|\Delta \omega_{\eta}\right|^{\beta} \mathrm{d} x \mathrm{d} t+\iint_{\left\{\rho_{\eta}^{\alpha_{0}}\left|\Delta \omega_{\eta}\right|>1\right\}}\left|\Delta \omega_{\eta}\right|^{\beta} \mathrm{d} x \mathrm{d} t \\ \leq & \iint_{Q_{T}} \rho_{\eta}^{-\alpha_{0} \beta} \mathrm{d} x \mathrm{d} t+\iint_{Q_{T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{\frac{\alpha}{\alpha_{0}}+\beta} \mathrm{d} x \mathrm{d} t \\ \leq & \iint_{Q_{T}} \rho_{\eta}^{-\alpha_{0} \beta} \mathrm{d} x \mathrm{d} t+C\iint_{Q_{T}} \rho_{\eta}^{\alpha}\left(1+\left|\Delta \omega_{\eta}\right|^{\mathrm{p}}\right) \mathrm{d} x \mathrm{d} t \\ \leq & C \text {, } \end{align} $

(3.8)

其中$ \alpha_{0} \beta<1 $确保了$ \rho^{-\alpha_{0} \beta} $在$ \Omega $上的可积性, 即$ \int_{\Omega} \rho^{-d} \mathrm{ d} x<C $(对于$ d<1 $). 再结合(3.6), 即可得到(3.8)最后一个不等号成立. 由Sobolev嵌入定理(见文献[9])可得

$ \begin{align} \left\|\omega_{\eta}(\cdot, t)\right\|_{W^{1,p}(\Omega)} \leq C\left\|\omega_{\eta}(\cdot, t)\right\|_{W^{2, \beta}(\Omega)}. \end{align} $

(3.9)

结合(3.7)-(3.9)可以得到估计(3.4)和

$ \begin{align} \sup _{t \in[0, T]}\frac{1}{p}\left\|\rho_{\eta}^{\frac{\alpha}{p}} \Delta \omega_{\eta}(\cdot, t)\right\|_{L^{p}(\Omega)}^{p} \leq C. \end{align} $

(3.10)

再经过一个类似于(3.8)的过程, 结合(3.10)可得

$ \begin{align} \int_{\Omega}\left|\Delta \omega_{\eta}(\cdot, t) \right|^{\beta}\mathrm{d} x \leq &\notag \int_{\left\{\rho_{n}^{\alpha_{0}}\left|\Delta \omega_{\eta}(\cdot, t) \right| \leq 1\right\}}\left|\Delta \omega_{\eta}(\cdot, t) \right|^{\beta} \mathrm{d} x +\int_{\left\{\rho_{\eta}^{\alpha_{0}}\left|\Delta \omega_{\eta}(\cdot, t)\right|>1\right\}}\left|\Delta \omega_{\eta}(\cdot, t) \right|^{\beta} \mathrm{d} x \\ \leq & \int_{\Omega} \rho_{\eta}^{-\alpha_{0} \beta} \mathrm{d} x +\int_{\Omega} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}(\cdot, t)\right|^{\frac{\alpha}{\alpha_{0}}+\beta} \mathrm{d} x \\ \leq & \int_{\Omega} \rho_{\eta}^{-\alpha_{0} \beta} \mathrm{d} x +C\int_{\Omega} \rho_{\eta}^{\alpha}\left(1+\left|\Delta \omega_{\eta}(\cdot, t)\right|^{\mathrm{p}}\right) \mathrm{d} x \\ \leq & C , \end{align} $

(3.11)

由(3.11)可得估计(3.3)成立.

下面证明本文的定理, 即证明解在退化情形下的存在性.

证   结合一致估计(3.1)–(3.5), 应用Sobolev空间的的弱紧性, Aubin-Lions紧致性定理(见[10])等结论可以知道存在函数$ \omega, \gamma $和$ \left\{\omega_{\eta}\right\} $的一个子列(仍记为其本身)使得当$ \eta \rightarrow 0 $时有

$ \begin{align} & \omega_{\eta t}\rightharpoonup \omega_{t} \text { 于 } L^{2}\left(Q_{T}\right), \end{align} $

(3.12)

$ \begin{align} & \omega_{\eta} \stackrel{*}{\rightharpoonup} \omega \text { 于 } L^{\infty}\left(0, T ; W^{2,\beta}(\Omega)\right), \end{align} $

(3.13)

$ \begin{align} & \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta}\rightharpoonup \gamma \text { 于 } L^{\frac{p}{p-1}}\left(Q_{T}\right), \end{align} $

(3.14)

$ \begin{align} & \Delta\omega_{\eta} \rightharpoonup \Delta\omega \text { 于 } L^{p}\left(0, T ; L_{loc}^{p}(\Omega)\right), \end{align} $

(3.15)

$ \begin{align} & \omega_{\eta} \rightarrow \omega \text { 于 } L^{p}\left(0, T ; W^{1,p}(\Omega)\right), \end{align} $

(3.16)

$ \begin{align} & \omega_{\eta} \rightarrow \omega \text { 几乎处处于 } \Omega \times(0, T), \end{align} $

(3.17)

$ \begin{align} & g_{i}\left(\omega_{\eta}\right) \rightarrow g_{i}(\omega) \text { 几乎处处于 } \Omega \times(0, T), \end{align} $

(3.19)

$ \begin{align} & g_{i}\left(\omega_{\eta}\right) \stackrel{*}{\rightharpoonup} g_{i}(\omega) \text { 于 } L^{\infty}\left(0, T ; L^{2}(\Omega)\right), \end{align} $

(3.20)

其中(3.16)成立意味着(3.17)成立, 结合$ g_{i}(\omega) \in C^{1}\left(Q_{T}\right) $和(3.17)可以得到(3.19), 结合(3.5)和(3.19)可以得到(3.20).

在命题2.1–(ii)中令$ \eta \rightarrow 0^{+} $可得

$ \begin{align} \iint_{Q_{T}} \omega_{t} \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{T}} \gamma \Delta \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{T}} \sum\limits_{i=1}^{N} g_{i}(\omega) \varphi_{x_{i}} \mathrm{d} x \mathrm{d} t=0, \end{align} $

(3.21)

对于任意的$ \varphi \in C_{0}^{\infty}\left(Q_{T}\right) $都成立. 为了得到问题(1.1)-(1.3)弱解的存在性, 还需证明

$ \begin{align} \iint_{Q_{T}} \gamma \Delta \varphi \mathrm{d} x \mathrm{d} t=\iint_{Q_{T}} \rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega \Delta \varphi \mathrm{d} x \mathrm{d} t. \end{align} $

(3.22)

下面给出其证明. 对任意的$ \varphi \in C_{0}^{\infty}\left(Q_{T}\right) $, 存在一个足够小的正常数$ a $, 使得$ \operatorname{supp} \varphi $, $ \operatorname{supp} \Delta \varphi \subseteq \Omega_{a} \times(a, T-a) $, 其中$ \Omega_{a}=\{x \in \Omega \mid \operatorname{dist}(x, \partial \Omega)>a\} $. 因此, 命题2.1-(ii)和(3.21)局部成立. 即

$ \begin{align} &\iint_{Q_{a T}} \omega_{\eta t} \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta} \Delta \varphi \mathrm{d} x \mathrm{d} t +\iint_{Q_{a T}} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}\right) \varphi_{x_{i}} \mathrm{d} x \mathrm{d} t=0, \end{align} $

(3.23)

$ \begin{align} &\iint_{Q_{a T}} \omega_{t} \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \gamma \Delta \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \sum\limits_{i=1}^{N} g_{i}(\omega) \varphi_{x_{i}} \mathrm{d} x \mathrm{d} t=0, \end{align} $

(3.24)

其中$ Q_{a T}=\Omega_{a} \times(a, T-a) $.

令$ 0 \leq \psi \in C_{0}^{\infty}\left(Q_{T}\right) $, 令$ \psi=1 $, 当$ (x, t) \in \operatorname{supp} \varphi $时. 首先, 选取$ \varphi=\psi \omega_{\eta} $作为(3.23) 中的检验函数. 可以得到

$ \begin{align} & -\frac{1}{2} \iint_{Q_{a T}} \omega_{\eta}^{2} \frac{\mathrm{d} \psi}{\mathrm{d} t} \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \psi \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p} \mathrm{d} x \mathrm{d} t +\notag\iint_{Q_{a T}} \omega_{\eta} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta} \Delta \psi \mathrm{d} x \mathrm{d} t \\ &+ \iint_{Q_{a T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta}\left(\sum\limits_{i=1}^{N} \omega_{\eta x_{i}} \psi_{x_{i}}\right) \mathrm{d} x \mathrm{d} t +\notag\iint_{Q_{a T}} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}\right)\left(\omega_{\eta x_{i}} \psi+\omega_{\eta} \psi_{x_{i}}\right) \mathrm{d} x \mathrm{d} t \\ =&0 . \end{align} $

(3.25)

同样地, 选取$ \varphi=\psi \omega $作为(3.24)中的检验函数. 可以得到

$ \begin{align} & -\frac{1}{2} \iint_{Q_{a T}} \omega^{2} \frac{\mathrm{d} \psi}{\mathrm{d} t} \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \gamma \psi \Delta \omega \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \gamma \omega \Delta \psi \mathrm{d} x \mathrm{d} t \\ & +\iint_{Q_{a T}} \gamma \sum\limits_{i=1}^{N} \omega_{x_{i}} \psi_{x_{i}} \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \sum\limits_{i=1}^{N} g_{i}(\omega)\left(\omega_{x_{i}} \psi+\omega \psi_{x_{i}}\right) \mathrm{d} x \mathrm{d} t \\ =&0 . \end{align} $

(3.26)

又因为当$ k>1 $时, 对于任意的$ m, n \in R^{N}(N \geq 1) $, 都有

$ \begin{align} \left(|m|^{k-2} m-|n|^{k-2} n\right)(m-n) \geq 0. \end{align} $

(3.27)

在(3.27)中, 不妨选取$ m=\Delta \omega_{\eta}, n=\Delta(\omega-\lambda \varphi) $, 其中$ \lambda>0 $. 然后(3.27)两端在$ Q_{a T} $上积分, 结合(3.25)可以得到

$ \begin{align} & -\frac{1}{2} \iint_{Q_{a T}} \omega_{\eta}^{2} \frac{\mathrm{d} \psi}{\mathrm{d} t} \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \omega_{\eta} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta} \Delta \psi \mathrm{d} x \mathrm{d} t \\ &+\iint_{Q_{a T}} \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta}\left(\sum\limits_{i=1}^{N} \omega_{\eta x_{i}} \psi_{x_{i}}\right) \mathrm{d} x \mathrm{d} t +\notag\iint_{Q_{a T}} \sum\limits_{i=1}^{N} g_{i}\left(\omega_{\eta}\right)\left(\omega_{\eta x_{i}} \psi+\omega_{\eta} \psi_{x_{i}}\right) \mathrm{d} x \mathrm{d} t \\ &-\iint_{Q_{a T}} \psi \rho_{\eta}^{\alpha}|\Delta(\omega-\lambda \varphi)|^{p} \mathrm{d} x \mathrm{d} t +\notag\iint_{Q_{a T}} \psi \rho_{\eta}^{\alpha}\left|\Delta \omega_{\eta}\right|^{p-2} \Delta \omega_{\eta} \Delta(\omega-\lambda \varphi) \mathrm{d} x \mathrm{d} t \\ &+\iint_{Q_{a T}} \psi \rho_{\eta}^{\alpha}|\Delta(\omega-\lambda \varphi)|^{p-2} \Delta(\omega-\lambda \varphi) \Delta \omega_{\eta} \mathrm{d} x \mathrm{d} t \\ \leq& 0. \end{align} $

(3.28)

在(3.28)中, 令$ \eta \rightarrow 0 $可以得到

$ \begin{aligned} & -\frac{1}{2} \iint_{Q_{a T}} \omega^{2} \frac{\mathrm{d} \psi}{\mathrm{d} t} \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \gamma \omega \Delta \psi \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \gamma \sum\limits_{i=1}^{N} \omega_{x_{i}} \psi_{x_{i}} \mathrm{d} x \mathrm{d} t \\ & +\iint_{Q_{a T}} \sum\limits_{i=1}^{N} g_{i}(\omega)(\omega_{x_{i}} \psi+\omega \psi_{x_{i}}) \mathrm{d} x \mathrm{d} t-\iint_{Q_{a T}} \psi \rho^{\alpha}|\Delta(\omega-\lambda \varphi)|^{p} \mathrm{d} x \mathrm{d} t \\ & +\iint_{Q_{a T}} \psi \gamma \Delta(\omega-\lambda \varphi) \mathrm{d} x \mathrm{d} t+\iint_{Q_{a T}} \psi \rho^{\alpha}|\Delta(\omega-\lambda \varphi)|^{p-2} \Delta(\omega-\lambda \varphi) \Delta \omega \mathrm{d} x \mathrm{d} t \\ \leq& 0. \end{aligned} $

结合(3.26)可得

$ \begin{align} \iint_{Q_{a T}} \psi\left(\rho^{\alpha}|\Delta(\omega-\lambda \varphi)|^{p-2} \Delta(\omega-\lambda \varphi)-\gamma\right) \Delta \varphi \mathrm{d} x \mathrm{d} t \leq 0. \end{align} $

(3.29)

在(3.29)中, 令$ \lambda \rightarrow 0 $可得

$ \begin{aligned} \iint_{Q_{T}} \psi\left(\rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega-\gamma\right) \Delta \varphi \mathrm{d} x \mathrm{d} t = \iint_{Q_{a T}} \psi\left(\rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega-\gamma\right) \Delta \varphi \mathrm{d} x \mathrm{d} t \leq 0. \end{aligned} $

如果取$ n=\Delta(\omega-\lambda \varphi) $中的$ \lambda<0 $, 通过类似的过程可以得到

$ \iint_{Q_{T}} \psi\left(\rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega-\gamma\right) \Delta \varphi \mathrm{d} x \mathrm{d} t \geq 0. $

从而$ \iint_{Q_{T}} \psi\left(\rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega-\gamma\right) \Delta \varphi \mathrm{d} x \mathrm{d} t=0. $由$ \varphi $和$ \psi $的任意性, 可得(3.22)成立. 从而有$ \iint_{Q_{T}} \omega_{t} \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{T}} \rho^{\alpha}|\Delta \omega|^{p-2} \Delta \omega \Delta \varphi \mathrm{d} x \mathrm{d} t+\iint_{Q_{T}} \sum\limits_{i=1}^{N} g_{i}(\omega) \varphi_{x_{i}} \mathrm{d} x \mathrm{d} t=0, $对于任意的$ \varphi \in C_{0}^{\infty}\left(Q_{T}\right) $都成立. 此外, 由(3.6)可知$ \omega \in L^{\infty}\left(0, T ; L^{2}(\Omega)\right) $, 结合$ \omega \in L^{\infty}\left(0, T ; L^{2}(\Omega)\right) $和$ \omega_{t} \in L^{2}\left(0, T ; L^{2}(\Omega)\right) $, 应用Aubin-Lions紧致性定理(见文献[10])可以得到$ \omega \in C\left(0, T ; L^{2}(\Omega)\right) $. 定理证明完毕.