期权定价是金融数学的重要研究内容^{[1, 2]}, 变分不等式则在美式期权定价理论中起着至关重要的作用, 美式期权定价问题最终都归结为一个抛物变分不等式问题(见文献[3-6]).在美式期权定价和分期付款模型中构成变分不等式的抛物微分算子完全是线性的(算子的系数是常数), 其解的存在性和唯一性都得到了广泛的研究.后来, 人们在研究Levy模型下的美式期权定价时, 发现构成变分不等式的微分算子不可能是常系数的(见文献[7]).近些年来, 已有文献在拟线性微分算子的基础上开展上述工作.文献[8, 9]利用有限元逼近方法研究了拟线性算子情形下抛物变分不等式解的存在性和唯一性和范数下的误差估计.文献[10]研究了一类混合边界条件下的变分不等式问题, 提出了一种新的离散格式, 得到了解的存在性和唯一性.

到目前有关退化抛物算子情形下的变分不等式问题还未见文献, 本文在推广的${L^{p(x)}}(\Omega)$空间和Sobolev空间${W^{1, p(x)}}(\Omega)$上研究了一类基于退化抛物算子的变分不等式问题.采用惩罚方法给出了其弱解的存在性和唯一性.为了克服退化抛物算子带来的困难, 本文提供了一种新的构造方法.

本文在柱体${\Omega _T}$考虑如下退化抛物变分不等式的初边值问题

$ \min \{ Lu, u - {u_0}\} = 0, \;\;(x, t) \in {\Omega _T}, $

(1)

$ u(x, 0) = {u_0}(x), \;\;x \in \Omega, $

(2)

$ u(x, t) = 0, \;\;(x, t) \in \partial \Omega \times (0, T), $

(3)

其中$\Omega $是${{\rm{R}}^N}$上的有界集, ${\Omega _T} = \Omega \times [0, T]$, $T > 0$, $L$是退化抛物算子满足

$ Lu = {u_t} - {u^\sigma }{\rm{div}}\left( {{{\left| {\nabla u} \right|}^{p(x) - 2}}\nabla u} \right) - \gamma {u^{\sigma {\rm{ - 1}}}}{\left| {\nabla u} \right|^{p(x)}}, $

$p(x)$为$\Omega $上的可测函数, $\gamma \in (0, 1)$, $\sigma \in [1, 1 + \gamma)$.初值条件满足

$ 0 \le {u_0} \in {L^\infty }(\Omega ) \cap W_0^{1, p(x)}(\Omega ), $

令${p^ + } = \mathop {{\rm{esssup}}}\limits_\Omega p(x), \; \; {p^- } = \mathop {{\rm{essinf}}}\limits_\Omega p(x), $本文始终假设$2 + \gamma \le {p^- } \le p(x) \le {p^ + } < \infty, \; \; \forall x \in \Omega.$

为证明主要结论, 需要用到如下有关推广${L^{p(x)}}(\Omega)$空间和${W^{1, p(x)}}(\Omega)$空间方面的理论, 见文献[11,12].设

$ {L^{p(x)}}{\rm{(}}\Omega {\rm{)}} = \left\{ {u\left| {u\text{是}{\Omega _T}\text{上的可测函数}, \int_\Omega {{{\left| {u(x)} \right|}^{p(x)}}{\rm{d}}x} < \infty } \right.} \right\}, $

$ {\left| u \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}} = \inf \left\{ {\lambda \left| {\int_\Omega {{{\left| {\frac{{u(x)}}{\lambda }} \right|}^{p(x)}}{\rm{d}}x \le 1} } \right.} \right\}, $

$ {W^{1, p(x)}}(\Omega ) = \{ u \in {L^{p(x)}}(\Omega )\left| {\nabla u \in {L^{p(x)}}(\Omega )} \right.\}, $

$ {\left| u \right|_{{W^{1, p(x)}}(\Omega )}} = {\left| u \right|_{{L^{p(x)}}(\Omega )}} + {\left| {\nabla u} \right|_{{L^{p(x)}}(\Omega )}}, \;\;u \in {W^{1, p(x)}}(\Omega ), $

$W_0^{1, p(x)}$表示${W^{1.p(x)}}$的支撑, 显然若$u \in {W^{1.p(x)}}$并且$u$在边界$\partial \Omega $上取值为零, 则$u \in W_0^{1, p(x)}$.

引理2.1   (1) ${L^{p(x)}}(\Omega)$和${W^{1, p(x)}}(\Omega)$是自反Banach空间.

(2) 假设${p_1}(x)$和${p_2}(x)$是$\Omega $上的可测函数满足${p_1}{(x)^{ - 1}} + {p_2}{(x)^{ - 1}} = 1$, ${p_1}(x) > 1$, 则对任意的$u \in {L^{{p_1}(x)}}(\Omega)$, $v \in {L^{{p_2}(x)}}(\Omega)$, 有

$ \left| {\int_\Omega {uv{\rm{d}}x} } \right| \le 2{\left| u \right|_{{L^{{p_1}(x)}}(\Omega )}}{\left| v \right|_{{L^{{p_2}(x)}}(\Omega )}}. $

(3) 若${\left| u \right|_{{L^{{p_1}(x)}}(\Omega)}} = 1$, 则$\int_\Omega {{{\left| {u(x)} \right|}^{p(x)}}{\rm{d}}x} = 1, $若${\left| u \right|_{{L^{{p_1}(x)}}(\Omega)}} > 1$, 则

$ \left| u \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}^{{p^ - }} \le \int_\Omega {{{\left| {u(x)} \right|}^{p(x)}}{\rm{d}}x} \le \left| u \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}^{{p^ + }}, $

若${\left| u \right|_{{L^{{p_1}(x)}}(\Omega)}} \le 1$, 则

$ \left| u \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}^{{p^ + }} \le \int_\Omega {{{\left| {u(x)} \right|}^{p(x)}}{\rm{d}}x} \le \left| u \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}^{{p^ - }}. $

(4) 如果${p_1}(x) \le {p_2}(x)$, 则${L^{{p_1}(x)}}(\Omega) \supset {L^{{p_2}(x)}}(\Omega)$.

引理2.2  如果$p(x) \in C(\bar \Omega)$, 则存在正常数$C$使得

$ \left| u \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}^{} \le C\left| {\nabla u} \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}^{}, \forall u \in W_0^{1.p(x)}(\Omega ). $

引理2.2表明$\left| {\nabla u} \right|_{{L^{p(x)}}{\rm{(}}\Omega {\rm{)}}}^{}$和$\left| {\nabla u} \right|_{W_0^{1.p(x)}}^{}$是Banach空间${W^{1, p(x)}}(\Omega)$上的两个等价范数.

引理2.3 设常数$\theta > 0$, $A(\eta) = {({\eta ^2} + \theta)^{\frac{{p(x) - 2}}{2}}}\eta $, 则

$ \left( {A(\eta ) - A(\eta ')} \right) \cdot \left( {\eta - \eta '} \right) \ge C{\left| {\eta - \eta '} \right|^{p(x)}}, \;\forall \eta, \eta ' \in {\rm{R}}, $

其中$C$是仅依赖$p(x)$的正常数.

引理2.4 设${u_1}$和${u_2}$满足

$ L{u_1} \le L{u_2}, \;\;{u_2} \ge c > 0, \;\;\forall (x, t) \in {\Omega _T}. $

若$\forall x \in \Omega $有${u_2}(x, 0) \ge {u_1}(x, 0)$, $\forall(x, t) \in \partial \Omega \times (0, T)$有${u_2}(x, t) \ge {u_1}(x, t)$, 则

$ {u_2}(x, t) \ge {u_1}(x, t), \;\;\forall (x, t) \in {\Omega _T}. $

引理2.5 若$L{u_1} + f(x, t, {u_1}) \le L{u_2} + f(x, t, {u_2}), \; \; \forall (x, t) \in {\Omega _T}$, 则引理2.4中结论依然成立, 其中$f(x, t, u)$关于$u$单调非降.

定义单调极大算子

$ G(\lambda ) = \left\{ \begin{array}{l} 0, \;\lambda > 0, \\ 1, \;\lambda = 0, \end{array} \right. $

且令集合

$ B = \left\{ {u \in {L^\infty }({\Omega _T}) \cap {L^{p(x)}}\left( {0, T;W_0^{1, p(x)}(\Omega )} \right)} \right\}. $

定义2.1称$(u, \xi) \in B \times {L^\infty }({\Omega _T})$为抛物变分不等式(1)-(3)的弱解, 若

(a) $u(x, t) \ge {u_0}(x), \; \; \forall (x, t) \in {\Omega _T}$,

(b) $u(x, 0) = {u_0}(x), \; \; \forall x \in \Omega $,

(c) $\xi \in G(\xi)$,

(d) 对任意的$\varphi \in C_0^\infty ({\Omega _T})$,

$ \int {\int_{{\Omega _T}} { - u{\varphi _t} + {u^\sigma }{{\left| {\nabla u} \right|}^{p(x) - 2}}\nabla u\nabla \varphi + (\sigma - \gamma ){u^{\sigma - 1}}{{\left| {\nabla u} \right|}^{p(x)}}\varphi {\rm{d}}x{\rm{d}}t} } =\\ \int {\int_{{\Omega _T}} {\xi \varphi {\rm{d}}x{\rm{d}}t} }, $

(e) 存在常数$\mu > 0$, $\mathop {\lim }\limits_{t \to \infty } \int_\Omega {\left| {{u^\mu }(x, t) - u_0^\mu (x)} \right|{\rm{d}}x} = 0$.

考虑如下惩罚问题

$ L{u_\varepsilon } + {\beta _\varepsilon }({u_\varepsilon } - {u_0}) = 0, \quad (x, t)\; \in {\Omega _T}, $

(4)

$ {u_\varepsilon }(x, 0) = {u_{0\varepsilon }}(x) = {u_0}(x) + \varepsilon, \quad x \in {\rm{ }}\Omega, $

(5)

$ {u_\varepsilon }(x, t) = \varepsilon, \quad (x, t)\; \in \partial \Omega \times (0, T), $

(6)

其中惩罚函数${\beta _\varepsilon }(\cdot)$满足(见图 3.1)

$ \begin{array}{c} 0 < \varepsilon \le 1, {\beta _\varepsilon }(x) \in {C^2}({\rm{R}}), {\beta _\varepsilon }(x) \le 0, {{\beta '}_\varepsilon }(x) \ge 0, \\ {{\beta ''}_\varepsilon }(x) \le 0, {\beta _\varepsilon }(x) = \left\{ \begin{array}{l} 0, \quad x \ge \varepsilon, \\ - 1, \quad x = 0, \end{array} \right.\mathop {\lim }\limits_{\varepsilon \to 0} {\beta _\varepsilon }(x) = \left\{ \begin{array}{l} 0, \quad x > 0, \\ - \infty, \quad x < 0. \end{array} \right. \end{array} $

(7)

此外由图 3.2, 可得当${\varepsilon _{\rm{1}}} \le {\varepsilon _{\rm{2}}}$时, 对任意的$t \in [0, {\varepsilon _2}]$,

$ {\beta _{{\varepsilon _{\rm{1}}}}}(t) - {\beta _{{\varepsilon _{\rm{2}}}}}(t) \ge 0. $

(8)

根据惩罚函数${\beta _\varepsilon }(\cdot)$的定义

$ L{u_\varepsilon } = 0 \Leftrightarrow {u_\varepsilon } \ge {u_0} + \varepsilon, \ L{u_\varepsilon } > 0 \Leftrightarrow {u_\varepsilon } < {u_0} + \varepsilon . $

(9)

因此当$\varepsilon \to 0$时可以用${\beta _\varepsilon }(\cdot)$控制不等式.下面给出非线性抛物问题(4)-(6)的弱解定义.

定义3.1  称非负函数${u_\varepsilon }$为非线性抛物问题(4)-(6)的弱解, 如果

(a) ${u_\varepsilon } \in {L^\infty }({\Omega _T}) \cap {L^{p(x)}}\left({0, T; W_0^{1, p(x)}(\Omega)} \right), $

(b) 对任意的$\varphi \in C_0^\infty ({\Omega _T})$,

$ \int {\int_{{\Omega _T}} {\left( { - {u_\varepsilon }{\varphi _t} + u_\varepsilon ^\sigma {{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\nabla {u_\varepsilon }\nabla \varphi + (\sigma - \gamma )u_\varepsilon ^{\sigma - 1}{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}\varphi +\\ {\beta _\varepsilon }({u_\varepsilon } - {u_0})\varphi } \right){\rm{d}}x{\rm{d}}t} } = 0, $

(c) 存在常数$\mu > 0$, $\mathop {\lim }\limits_{t \to \infty } \int_\Omega {\left| {{u^\mu }(x, t) - u_0^\mu (x)} \right|{\rm{d}}x} = 0$.

文献[12]利用半离散差分格式证明了非线性抛物方程(4)-(6)存在定义3.1意义下的弱解.本节将在非线性抛物方程(4)-(6)的基础之上, 考察变分不等式的弱解问题.在此之前, 先给出几个有用的引理.

引理3.1 设$\varepsilon $, ${\varepsilon _1}$和${\varepsilon _2}$为正常数满足$\varepsilon \in (0, 1)$, $0 < {\varepsilon _1} \le {\varepsilon _2} < 1$, 则

$ {u_0} \le {u_\varepsilon } \le {\left| {{u_0}} \right|_\infty } + \varepsilon, $

(10)

$ {u_{{\varepsilon _1}}} \le {u_{{\varepsilon _2}}}, \forall (x, t) \in {\Omega _T}. $

(11)

证 首先证明${u_\varepsilon } \ge {u_0}$.考虑公式(4), 即

$ L{u_\varepsilon }{\rm{ = }} - {\beta _\varepsilon }({u_\varepsilon } - {u_0}), $

(12)

令$t = 0$可得

$ L{u_{0\varepsilon }} = - {\beta _\varepsilon }({u_{0\varepsilon }} - {u_0}) = - {\beta _\varepsilon }(\varepsilon ) = 0, \;\;(x, t) \in {\Omega _T}. $

(13)

易见${u_{0\varepsilon }}$和${u_\varepsilon }$在抛物边界上相等, 因此联立公式(12)和(13)并利用引理2.4, 有

${u_\varepsilon } \ge {u_{0\varepsilon }} \ge {u_0}, \;\;\forall (x, t) \in {\Omega _T}. $

其次证明${u_\varepsilon } \le {\left| {{u_0}} \right|_\infty } + \varepsilon $.注意到${\left| {{u_0}} \right|_\infty } + \varepsilon $为常数, 并且

$ {u_\varepsilon } \le {\left| {{u_0}} \right|_\infty } + \varepsilon, \;\forall (x, t) \in \partial \Omega \times (0, T), \ {u_\varepsilon }(x, 0) \le {\left| {{u_0}} \right|_\infty } + \varepsilon, \;\forall x \in \Omega. $

又因为

$ L\left( {{{\left| {{u_0}} \right|}_\infty } + \varepsilon } \right) + {\beta _\varepsilon }({\left| {{u_0}} \right|_\infty } + \varepsilon - {u_0}) \ge {\beta _\varepsilon }(\varepsilon ) = 0, $

所以利用引理2.5可知${u_\varepsilon } \le {\left| {{u_0}} \right|_\infty } + \varepsilon, \; \; \forall (x, t) \in {\Omega _T}.$

最后证明公式(11)成立.因为

$ L{u_{{\varepsilon _1}}} + {\beta _{{\varepsilon _1}}}({u_{{\varepsilon _1}}} - {u_0}) = 0, \ L{u_{{\varepsilon _2}}} + {\beta _{{\varepsilon _2}}}({u_{{\varepsilon _2}}} - {u_0}) = 0, $

进一步利用公式(8)可得

$ L{u_{{\varepsilon _2}}} + {\beta _{{\varepsilon _1}}}({u_{{\varepsilon _2}}} - {u_0}) = {\beta _{{\varepsilon _1}}}({u_{{\varepsilon _2}}} - {u_0}) - {\beta _{{\varepsilon _2}}}({u_{{\varepsilon _2}}} - {u_0}) \ge 0. $

联立初边值条件, 并利用引理2.5可得结论成立.

引理3.2  对任意的$\alpha \in [0, 1- \gamma)$, 非线性抛物方程(4)-(6)的解满足

$ {\left| {{\partial _t}u_\varepsilon ^\mu } \right|_{{L^2}({\Omega _T})}} + {\left| {{\partial _t}{u_\varepsilon }} \right|_{{L^2}({\Omega _T})}} \le C, $

(14)

$ {\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}u_\varepsilon ^{- \alpha }} \right|_{{L^1}({\Omega _T})}} + {\left| {\nabla {u_\varepsilon }} \right|_{{L^{p(x)}}({\Omega _T})}} \le C, $

(15)

其中$C$为不依赖$\varepsilon $的非负常数.

证 注意$2(\mu- 1) = \gamma - \sigma $, 在公式(4)边乘以$u_\varepsilon ^{\gamma - \sigma }{\partial _t}{u_\varepsilon }$并在${\Omega _T}$上的积分, 有

$ \begin{array}{*{20}{l}} {\;\;\;{\mu ^{ - 2}}\int {\int_{{\Omega _T}} {{{\left( {{\partial _t}u_\varepsilon ^\mu } \right)}^2}{\rm{d}}x{\rm{d}}t} } = \int {\int_{{\Omega _T}} {{\partial _t}{u_\varepsilon } \cdot u_\varepsilon ^{\gamma - \sigma }{\partial _t}{u_\varepsilon }{\rm{d}}x{\rm{d}}t} } }\\ { = - \int {\int_{{\Omega _T}} {{{\left( {{{\left| {\nabla {u_\varepsilon }} \right|}^2}} \right)}^{(p(x) - 2)/2}}\nabla {u_\varepsilon }\nabla (u_\varepsilon ^\gamma {\partial _t}{u_\varepsilon }){\rm{d}}x{\rm{d}}t} } }\\ {{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \;\; + \gamma \int {\int_{{\Omega _T}} {{{\left( {{{\left| {\nabla {u_\varepsilon }} \right|}^2}} \right)}^{p(x){\rm{ /}}2}}u_\varepsilon ^{\gamma - 1}{\partial _t}{u_\varepsilon }{\rm{d}}x{\rm{d}}t} } - \int {\int_{{\Omega _T}} {{\beta _\varepsilon }({u_\varepsilon } - {u_0})u_\varepsilon ^{\gamma - \sigma }{\partial _t}{u_\varepsilon }{\rm{d}}x{\rm{d}}t} } }\\ { \le - \int {\int_{{\Omega _T}} {u_\varepsilon ^\gamma {{\left( {{{\left| {\nabla {u_\varepsilon }} \right|}^2}} \right)}^{(p(x) - 2){\rm{ /}}2}}\nabla {u_\varepsilon }\nabla ({\partial _t}{u_\varepsilon }){\rm{d}}x{\rm{d}}t} } {\mkern 1mu} - \int {\int_{{\Omega _T}} {{\beta _\varepsilon }({u_\varepsilon } - {u_0})u_\varepsilon ^{\gamma - \sigma }{\partial _t}{u_\varepsilon }{\rm{d}}x{\rm{d}}t} } .} \end{array} $

由Cauchy不等式, Holder不等式以及公式(10), 可得

$ \begin{array}{l} \left| {\int {\int_{{\Omega _T}} {{\beta _\varepsilon }({u_\varepsilon }-{u_0})u_\varepsilon ^{\gamma-\sigma }{\partial _t}{u_\varepsilon }{\rm{d}}x{\rm{d}}t} } } \right| = \left| {\int {\int_{{\Omega _T}} {{\beta _\varepsilon }({u_\varepsilon }-{u_0})u_\varepsilon ^{\mu - 1}{\mu ^{ - 1}}{\partial _t}u_\varepsilon ^\mu {\rm{d}}x{\rm{d}}t} } } \right|\\ {\kern 150pt} \le \frac{1}{2}\left\| {u_\varepsilon ^{\mu - 1}} \right\|_{{L^2}({\Omega _T})}^2 + \frac{1}{2}{\mu ^{ - 2}}\int {\int_{{\Omega _T}} {{{\left( {{\partial _t}u_\varepsilon ^\mu } \right)}^2}{\rm{d}}x{\rm{d}}t} } \\ {\kern 150pt} \le \frac{1}{2}\left\| {u_{\rm{0}}^{\mu - 1}} \right\|_{{L^2}({\Omega _T})}^2 + \frac{1}{2}{\mu ^{ - 2}}\int {\int_{{\Omega _T}} {{{\left( {{\partial _t}u_\varepsilon ^\mu } \right)}^2}{\rm{d}}x{\rm{d}}t} } {\rm{.}} \end{array} $

从而利用引理2.1 (3)以及引理3.1可得

$ \begin{array}{l} \frac{1}{2}{\mu ^{ - 2}}\int {\int_{{\Omega _T}} {{{\left( {{\partial _t}u_\varepsilon ^\mu } \right)}^2}{\rm{d}}x{\rm{d}}t} } \le \frac{1}{{{p^ - }}}{({\left| {{u_0}} \right|_\infty } + 1)^\gamma }\left( {1 + \left| {\nabla {u_0}} \right|_{{L^{p(x)}}({\Omega _T})}^{{p^ + }/{p^ - }}} \right) + \\ \frac{1}{2}\left\| {u_{\rm{0}}^{\mu - 1}} \right\|_{{L^2}({\Omega _T})}^2. \end{array} $

(16)

又因为$2(\mu - 1) = \gamma - \sigma $, 利用引理3.1可得

$ \int {\int_{{\Omega _T}} {{{\left( {{\partial _t}u_\varepsilon ^{}} \right)}^2}{\rm{d}}x{\rm{d}}t} } = \int {\int_{{\Omega _T}} {{{\left( {{\partial _t}u_\varepsilon ^\mu } \right)}^2}u_\varepsilon ^{\frac{1}{{\gamma - \sigma }}}{\rm{d}}x{\rm{d}}t} } \le {({\left| {{u_0}} \right|_\infty }\\ + 1)^{\frac{1}{{\gamma - \sigma }}}}\int {\int_{{\Omega _T}} {{{\left( {{\partial _t}u_\varepsilon ^\mu } \right)}^2}{\rm{d}}x{\rm{d}}t} } . $

(17)

联立公式(16)和公式(17)可得公式(14)成立.

其次证明公式(15)成立.在等式(4)两端乘以$u_\varepsilon ^{1 - \alpha - \sigma }$并积分可得

$ \begin{array}{l} \int {\int_{{\Omega _T}} {\frac{{\partial {u_\varepsilon }}}{{\partial t}}u_\varepsilon ^{1 - \alpha - \sigma }{\rm{d}}x{\rm{d}}t} } \\ = \int {\int_{{\Omega _T}} {u_\varepsilon ^{1 - \alpha }{\rm{div}}\left\{ {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\nabla {u_\varepsilon }} \right\}{\rm{ + }}\gamma u_\varepsilon ^{ - \alpha }{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}} + u_\varepsilon ^{1 - \alpha - \sigma }{\beta _\varepsilon }({u_\varepsilon } - {u_0}){\rm{d}}x{\rm{d}}t} } \\ = \int_0^T {\int_{\partial \Omega } {\left\{ {u_\varepsilon ^{1 - \alpha }{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\frac{{\partial {u_\varepsilon }}}{{\partial \nu }}} \right\}{\rm{d}}x{\rm{d}}t} } \\ \quad - (1 - \alpha - \gamma )\int {\int_{{\Omega _T}} {u_\varepsilon ^{ - \alpha }{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } + \int {\int_{{\Omega _T}} {u_\varepsilon ^{1 - \alpha - \sigma }{\beta _\varepsilon }({u_\varepsilon } - {u_0}){\rm{d}}x{\rm{d}}t} }, \end{array} $

(18)

其中$\nu $表示曲面$\partial \Omega $的外侧法向量.进一步联立公式(7)和公式(10)有

$\int {\int_{{\Omega _T}} {u_\varepsilon ^{1 - \alpha - \sigma }{\beta _\varepsilon }({u_\varepsilon } - {u_0}){\rm{d}}x{\rm{d}}t} } \le 0. $

(19)

又因为${u_\varepsilon } \ge \varepsilon $, 所以$\frac{{\partial {u_\varepsilon }}}{{\partial \nu }} \le 0, \forall (x, t) \in \Omega \times (0, T), $这意味着

$ \int_0^T {\int_{\partial \Omega } {\left\{ {u_\varepsilon ^{{\rm{1}} - \alpha }{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\frac{{\partial {u_\varepsilon }}}{{\partial \nu }}} \right\}{\rm{d}}x{\rm{d}}t} } \le 0. $

(20)

将公式(19)和公式(20)代入公式(18)可得

$ \int {\int_{{\Omega _T}} {\frac{{\partial {u_\varepsilon }}}{{\partial t}}u_\varepsilon ^{1 - \alpha - \sigma }{\rm{d}}x{\rm{d}}t} } \le - (1 - \alpha - \gamma )\int {\int_{{\Omega _T}} {u_\varepsilon ^{ - \alpha }{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } . $

(21)

进一步利用分步积分, 有

$ \int {\int_{{\Omega _T}} {\frac{{\partial {u_\varepsilon }}}{{\partial t}}u_\varepsilon ^{1 - \alpha - \sigma }{\rm{d}}x{\rm{d}}t} } = \frac{1}{{2 - \alpha - \sigma }}\int_\Omega {u_\varepsilon ^{2 - \alpha - \sigma }(x, T) - u_\varepsilon ^{2 - \alpha - \sigma }(x, 0){\rm{d}}x}.$

注意 $1 - \alpha - \gamma > 0$, $1 - \alpha > 0$.将上式代入公式(21)即可得到

$\int {\int_{{\Omega _T}} {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}u_\varepsilon ^{ - \alpha }{\rm{d}}x{\rm{d}}t} } \le \frac{1}{{(1 - \alpha - \gamma )(2 - \alpha - \sigma )}}\int_\Omega {u_\varepsilon ^{2 - \alpha - \sigma }(x, 0){\rm{d}}x} \le C, $

其中$C$是仅依赖$\alpha $、$\gamma $、$\Omega $和${\left| {{u_0}} \right|_\infty }$的常数.

引理3.1和引理3.2意味着, 对任意的$\varepsilon \in (0, 1)$存在子列$\{ {u_\varepsilon }\} $ (仍记为$\{ {u_\varepsilon }\} $)以及函数$u \in {L^\infty }({\Omega _T})$, 使得

$ {u_\varepsilon } \to u, \;\varepsilon \to 0, $

(22)

$ \nabla {u_\varepsilon }\mathop \to \limits^w \nabla u \in {L^{p(x)}}({\Omega _T}), \;\;\varepsilon \to 0, $

(23)

$ {\partial _t}{u_\varepsilon }\mathop \to \limits^w {\partial _t}u \in {L^2}({\Omega _T}), \;\;\varepsilon \to 0, $

(24)

其中$\mathop \to \limits^w $表示弱收敛, 此外还有

$ {u_0} \le u \le \left| {{u_0}} \right|, \;\;u \le {u_\varepsilon }, \forall (x, t) \in {\Omega _T}. $

(25)

通过下面的引理(引理3.3), 还可以得到

${\left| {\nabla {u_\varepsilon } - \nabla u} \right|_{{L^{p(x)}}({\Omega _T})}} \to 0, \;\;\varepsilon \to 0. $

(26)

引理3.3  设$Q_c^\varepsilon = \left\{ {(x, t) \in {\Omega _T}; {u_\varepsilon } \ge c, c > 0} \right\}$, $Q_c^{} = \left\{ {(x, t) \in {\Omega _T}; u \ge c, c > 0} \right\}$, 则

$ {\left| {\nabla {u_\varepsilon } - \nabla u} \right|_{{L^{p(x)}}(Q_c^\varepsilon )}} \to 0, \;\;\varepsilon \to 0, $

(27)

$ {\left| {\nabla {u_\varepsilon } - \nabla u} \right|_{{L^{p(x)}}(Q_c^{})}} \to 0, \;\;\varepsilon \to 0. $

(28)

证  在公式(4)中选择$\varphi = u_\varepsilon ^{{p^ - } - 2}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2})$可得

$ \begin{array}{l} \int {\int_{{\Omega _T}} {\frac{{\partial {u_\varepsilon }}}{{\partial t}}u_\varepsilon ^{{p^ - } - \sigma - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\rm{d}}x{\rm{d}}t} } \\ = - \int {\int_{{\Omega _T}} {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\nabla {u_\varepsilon }\nabla \{ u_\varepsilon ^{{p^ - } - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2})\} {\rm{d}}x{\rm{d}}t} } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \gamma \int {\int_{{\Omega _T}} {u_\varepsilon ^{{p^ - } - 2}{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\rm{d}}x{\rm{d}}t} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int {\int_{{\Omega _T}} {u_\varepsilon ^{{p^ - } - \sigma - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\beta _\varepsilon }({u_\varepsilon } - {u_0}){\rm{d}}x{\rm{d}}t} } .\\ = (\gamma - {p^ - } + 1)\int {\int_{{\Omega _T}} {u_\varepsilon ^{{p^ - } - 2}{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\rm{d}}x{\rm{d}}t} } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \int {\int_{{\Omega _T}} {u_\varepsilon ^{{p^ - } - 1}{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\nabla {u_\varepsilon }\nabla (u_\varepsilon ^2 - {u^2}){\rm{d}}x{\rm{d}}t} } {\kern 1pt} {\kern 1pt} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int {\int_{{\Omega _T}} {u_\varepsilon ^{{p^ - } - \sigma - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\beta _\varepsilon }({u_\varepsilon } - {u_0}){\rm{d}}x{\rm{d}}t} } . \end{array} $

(29)

利用公式(24)以及罚函数${\beta _\varepsilon }\ (\cdot)$的定义,

$ \int {\int_{{\Omega _T}} {u_\varepsilon ^{{p^ - } - 1}{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\rm{d}}x{\rm{d}}t} } = O({\varepsilon ^2}), $

(30)

$ \int {\int_{{\Omega _T}} {u_\varepsilon ^{{p^ - } - \sigma - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\beta _\varepsilon }({u_\varepsilon } - {u_0}){\rm{d}}x{\rm{d}}t} } = O({\varepsilon ^2}). $

(31)

注意$\gamma - {p^ - } + 1 < 0$, 将公式(30)和公式(31)代入公式(29), 有

$ \begin{array}{l} \int {\int_{{\Omega _T}} {\frac{{\partial {u_\varepsilon }}}{{\partial t}}u_\varepsilon ^{{p^ - } - \sigma - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\rm{d}}x{\rm{d}}t} } \\ \le O({\varepsilon ^2}){\kern 1pt} - {2^{{p^ - } - 1}}\int {\int_{{\Omega _T}} {{{\left| {\nabla u_\varepsilon ^2} \right|}^{p(x) - 2}}\nabla u_\varepsilon ^2\nabla (u_\varepsilon ^2 - {u^2}){\rm{d}}x{\rm{d}}t} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} . \end{array} $

(32)

又因为$\varepsilon \le {u_\varepsilon } \le {\left| {{u_0}} \right|_\infty } + 1$, 所以利用三角不等式可得

$ \begin{array}{l} \int {\int_{{\Omega _T}} {\frac{{\partial {u_\varepsilon }}}{{\partial t}}u_\varepsilon ^{{p^ - } - \sigma - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\rm{d}}x{\rm{d}}t} } \\ + {2^{{p^ - } - 1}}\int {\int_{{\Omega _T}} {\left[{{{\left| {\nabla u_\varepsilon ^2} \right|}^{p(x)-2}}\nabla u_\varepsilon ^2-{{\left| {\nabla u_{}^2} \right|}^{p(x)-2}}\nabla u_{}^2} \right]\nabla (u_\varepsilon ^2 - {u^2}){\rm{d}}x{\rm{d}}t} } \\ \le O({\varepsilon ^2}) - {2^{{p^ - } - 1}}\int {\int_{{\Omega _T}} {{{\left| {\nabla u_{}^2} \right|}^{p(x) - 2}}\nabla u_{}^2\nabla (u_\varepsilon ^2 - {u^2}){\rm{d}}x{\rm{d}}t} }, \end{array} $

(33)

利用公式(10)和公式(14), 并利用Holder不等式可得

$ \int {\int_{{\Omega _T}} {{{\left| {{\partial _t}u_\varepsilon ^{{p^ - } - \sigma }} \right|}^2}{\rm{d}}x{\rm{d}}t} } = {({p^ - } - \sigma )^2}{({\left| {{u_0}} \right|_\infty } + 1)^{{p^ - } - \sigma - 1}}\int {\int_{{\Omega _T}} {{{\left| {{\partial _t}u_\varepsilon ^{}} \right|}^2}{\rm{d}}x{\rm{d}}t} } < \infty, $

这意味着

$ \begin{array}{l} \left| {\int {\int_{{\Omega _T}} {\frac{{\partial {u_\varepsilon }}}{{\partial t}}u_\varepsilon ^{{p^ - } - \sigma - 1}(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2}){\rm{d}}x{\rm{d}}t} } } \right|\\ \le \frac{1}{{{p^ - } - \sigma }}\sqrt {\int {\int_{{\Omega _T}} {{{\left| {{\partial _t}u_\varepsilon ^{{p^ - } - \sigma }} \right|}^2}{\rm{d}}x{\rm{d}}t} } } \cdot \sqrt {\int {\int_{{\Omega _T}} {{{(u_\varepsilon ^2 - {\varepsilon ^2} - {u^2})}^2}{\rm{d}}x{\rm{d}}t} } } = O({\varepsilon ^2}). \end{array} $

(34)

此外由公式(22)和公式(23)有

$ \nabla u_\varepsilon ^2\mathop \to \limits^w \nabla u_{}^2 \in {L^{p(x)}}({\Omega _T}), $

(35)

将公式(34)和公式(35)代入公式(33), 有

$ \mathop {\lim }\limits_{\varepsilon \to 0} \int {\int_{{\Omega _T}} {\left[{{{\left| {\nabla u_\varepsilon ^2} \right|}^{p(x)-2}}\nabla u_\varepsilon ^2-{{\left| {\nabla u_{}^2} \right|}^{p(x)-2}}\nabla u_{}^2} \right]\nabla (u_\varepsilon ^2 - {u^2}){\rm{d}}x{\rm{d}}t} } \le 0. $

进一步利用引理2.3可得

$ \mathop {\lim }\limits_{\varepsilon \to 0} \int {\int_{{\Omega _T}} {{{\left| {\nabla u_\varepsilon ^2-\nabla u_{}^2} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } = 0. $

因此利用公式

$ \nabla u_\varepsilon ^2 - \nabla {u^2} = 2u_\varepsilon ^{}(\nabla {u_\varepsilon } - \nabla u) + 2\nabla u({u_\varepsilon } - u), $

$ \nabla u_\varepsilon ^2 - \nabla {u^2} = 2u(\nabla {u_\varepsilon } - \nabla u) + 2\nabla {u_\varepsilon }({u_\varepsilon } - u), $

从而令$\varepsilon \to 0$可得

$ \begin{array}{l} \;\;\;{2^{{p^- }}}\int {\int_{{\Omega _T}} {u_\varepsilon ^{p(x)}{{\left| {\nabla {u_\varepsilon }- \nabla u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \\ \le {2^{{p^ + }}}\int {\int_{{\Omega _T}} {{{\left| {\nabla u_\varepsilon ^2- \nabla {u^2}} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } + {4^{{p^ + }}}\int {\int_{{\Omega _T}} {{{\left| {\nabla u} \right|}^{p(x)}}{{\left| {{u_\varepsilon } - u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \to 0, \\ \;\;\;{2^{{p^-}}}\int {\int_{{\Omega _T}} {u_\varepsilon ^{p(x)}{{\left| {\nabla {u_\varepsilon }-\nabla u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \\ \le {2^{{p^ + }}}\int {\int_{{\Omega _T}} {{{\left| {\nabla u_\varepsilon ^2-\nabla {u^2}} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } + {4^{{p^ + }}}\int {\int_{{\Omega _T}} {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}{{\left| {{u_\varepsilon } - u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \to 0. \end{array} $

注意$Q_c^\varepsilon \subset {\Omega _T}$, $Q_c^{} \subset {\Omega _T}$, 再令$\varepsilon \to 0$, 有

$ {c^{{p^ - }}}\int {\int_{Q_c^\varepsilon } {{{\left| {\nabla {u_\varepsilon } - \nabla u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \le \int {\int_{{\Omega _T}} {u_\varepsilon ^{p(x)}{{\left| {\nabla {u_\varepsilon } - \nabla u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \to 0, $$ $${c^{{p^ - }}}\int {\int_{Q_c^{}} {{{\left| {\nabla {u_\varepsilon } - \nabla u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \le \int {\int_{{\Omega _T}} {{u^{p(x)}}{{\left| {\nabla {u_\varepsilon } - \nabla u} \right|}^{p(x)}}{\rm{d}}x{\rm{d}}t} } \to 0. $

因此公式(27)和公式(28)成立.进一步利用公式(27)和公式(28), 公式(26)亦是成立的.

引理3.4 当$\varepsilon \to 0$时有

$ {\left| {u_\varepsilon ^{\sigma - 1}{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}} - u_{}^{\sigma - 1}{{\left| {\nabla u} \right|}^{p(x)}}} \right|_{{L^1}({\Omega _T})}} \to 0, $

(36)

$ {\left| {u_\varepsilon ^\sigma {{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\nabla {u_\varepsilon } - {u^\sigma }{{\left| {\nabla u} \right|}^{p(x) - 2}}\nabla u} \right|_{{L^1}({\Omega _T})}} \to 0, $

$ - {\beta _\varepsilon }({u_\varepsilon } - {u_0}) \to \xi \in G(u - {u_0}) . $

证  设${\chi _\eta }$和$\chi _\eta ^{(\varepsilon)}$分别是集合$\left\{ {(x, t) \in {\Omega _T}; u(x, t) < \eta } \right\}$和$\left\{ {(x, t) \in {\Omega _T}; {u_\varepsilon }(x, t) < \eta } \right\}$的特征函数.显然${\chi _\eta } \le \chi _\eta ^{(\varepsilon)}$.利用三角不等式, 有

$ \begin{array}{l} \int {\int_{{\Omega _T}} {\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}} - {{\left| {\nabla u} \right|}^{p(x)}}} \right|{\rm{d}}x{\rm{d}}t} } \\ \le \int {\int_{{\Omega _T}} {\left| {{{\left| {\nabla u} \right|}^{p(x)}}{\chi _\eta } - {{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}\chi _\eta ^{(\varepsilon )}} \right|{\rm{d}}x{\rm{d}}t} } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int {\int_{{\Omega _T}} {\left| {{{\left| {\nabla u} \right|}^{p(x)}}(1 - {\chi _\eta }) - {{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}(1 - \chi _\eta ^{(\varepsilon )})} \right|{\rm{d}}x{\rm{d}}t} } \\ \le \int {\int_{{\Omega _T}} {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}\chi _\eta ^{(\varepsilon )}{\rm{d}}x{\rm{d}}t} } + \int {\int_{{\Omega _T}} {{{\left| {\nabla u} \right|}^{p(x)}}{\chi _\eta }{\rm{d}}x{\rm{d}}t} } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int {\int_{{\Omega _T}} {{{\left| {\nabla u} \right|}^{p(x)}}(\chi _\eta ^{(\varepsilon )} - {\chi _\eta }){\rm{d}}x{\rm{d}}t} } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \int {\int_{{\Omega _T}} {\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}} - {{\left| {\nabla u} \right|}^{p(x)}}} \right|(1 - \chi _\eta ^{(\varepsilon )}){\rm{d}}x{\rm{d}}t} } \\ = {H_1} + {H_2} + {H_3} + {H_4}. \end{array} $

(39)

取$\alpha = (1 - \gamma ){\rm{ }}/2$利用引理3.2可得

$ \begin{array}{l} {H_1} = \int {\int_{{\Omega _T}} {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}\frac{{u_\varepsilon ^\alpha }}{{u_\varepsilon ^\alpha }}\chi _\eta ^{(\varepsilon )}{\rm{d}}x{\rm{d}}t} } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le {\eta ^\alpha }\int {\int_{{\Omega _T}} {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}u_\varepsilon ^{ - \alpha }{\rm{d}}x{\rm{d}}t} } \le C{\eta ^\alpha } \to 0, \;(\eta \to 0). \end{array} $

(40)

利用引理3.3和公式(40), 当$\eta \to 0$时

$ {H_2}{\kern 1pt} {\kern 1pt} \le \int {\int_{{\Omega _T}} {\chi _\eta ^{(\varepsilon )}{{\left| {\nabla {u_\varepsilon }} \right|}^p}{\rm{d}}x{\rm{d}}t} } + \int {\int_{{\Omega _T}} {\chi _\eta ^{(\varepsilon )}{{\left| {\nabla {u_\varepsilon } - \nabla u} \right|}^p}{\rm{d}}x{\rm{d}}t} } \to 0, $

(41)

$ {H_4} \to 0, $

(42)

先固定$\eta > 0$, 易得当$\varepsilon \to 0$时, $\chi _\eta ^{(\varepsilon)} \to {\chi _\eta }$, 故

$ {H_3} \to 0, \;(\eta \to 0). $

(43)

从而将公式(40)-公式(43)代入公式(39), 并令$\eta \to 0$可得

$ \int {\int_{{\Omega _T}} {\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}} - {{\left| {\nabla u} \right|}^{p(x)}}} \right|{\rm{d}}x{\rm{d}}t} } \to 0. $

因此再由公式(22)可得公式(36)成立.

下面证明公式(37)成立.利用三角不等式, 可得

$ \begin{array}{l} \int {\int_{{\Omega _T}} {\left| {u_\varepsilon ^\sigma {{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\nabla {u_\varepsilon } - {u^\sigma }{{\left| {\nabla u} \right|}^{p(x) - 2}}\nabla u} \right|{\rm{d}}x{\rm{d}}t} } \\ \le \int {\int_{{\Omega _T}} {\left| {u_\varepsilon ^\sigma - {u^\sigma }} \right|{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 1}}{\rm{d}}x{\rm{d}}t} } + \int {\int_{{\Omega _T}} {{u^\sigma }{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}\left| {\nabla {u_\varepsilon } - \nabla u} \right|{\rm{d}}x{\rm{d}}t} } \\ \quad + \int {\int_{{\Omega _T}} {{u^\sigma }\left| {\nabla u} \right| \cdot \left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}} - {{\left| {\nabla u} \right|}^{p(x) - 2}}} \right|{\rm{d}}x{\rm{d}}t} } = {H_5} + {H_6} + {H_7}. \end{array} $

(44)

再由Holder不等式和引理2.6, 当$\varepsilon \to 0$时

$ {H_5} \le C\left| {u_\varepsilon ^\sigma - {u^\sigma }} \right|_{{L^{p(x)}}({\Omega _T})}^{{p^ - }} + C\left| {u_\varepsilon ^\sigma - {u^\sigma }} \right|_{{L^{p(x)}}({\Omega _T})}^{{p^ + }} \to 0. $

(45)

再次利用Holder不等式同样有

$ \begin{array}{l} {H_7} = \int {\int_{{\Omega _T}} {{u^\sigma }\left| {\nabla u} \right| \cdot \left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)-2}}-{{\left| {\nabla u} \right|}^{p(x)-2}}} \right|{\rm{d}}x{\rm{d}}t} } \\ {\kern 14pt} \le C\int {\int_{{\Omega _T}} {\left| {\nabla u} \right| \cdot {{\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}} - {{\left| {\nabla u} \right|}^{p(x)}}} \right|}^{\frac{{p(x) - 2}}{{p(x)}}}}{\rm{d}}x{\rm{d}}t} } . \end{array} $

利用引理2.1和公式(26), 当$\varepsilon $足够小时

$ {\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}}-{{\left| {\nabla u} \right|}^{p(x)}}} \right|_{{L^1}({\Omega _T})}} \le 1, $

此时当$\varepsilon \to 0$时

${H_7} \le C\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x)}} - {{\left| {\nabla u} \right|}^{p(x)}}} \right|_{{L^1}({\Omega _T})}^{({p^ - } - 2)/{p^ + }}\left( {\left| {\nabla u} \right|_{{L^{p(x)}}({\Omega _T})}^{{p^ + }} + \left| {\nabla u} \right|_{{L^{p(x)}}({\Omega _T})}^{{p^ - }}} \right) \to 0. $

(46)

下面估计${H_6}$.再次利用三角不等式, 当$\varepsilon $足够小时

$ \begin{array}{l} {H_6} = \int {\int_{{\Omega _T}} {{u^\sigma }{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}} \cdot \left| {\nabla {u_\varepsilon } - \nabla u} \right|{\rm{d}}x{\rm{d}}t} } \\ {\kern 14pt} \le {\kern 1pt} C\int {\int_{{\Omega _T}} {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}} \cdot \left| {\nabla {u_\varepsilon } - \nabla u} \right|{\rm{d}}x{\rm{d}}t} } \\ {\kern 14pt} \le C\left( {\left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}} \right|_{{L^{\frac{{p(x)}}{{p(x) - 2}}}}({\Omega _T})}^{\frac{{{p^ + } - 2}}{{{p^ - }}}} + \left| {{{\left| {\nabla {u_\varepsilon }} \right|}^{p(x) - 2}}} \right|_{{L^{\frac{{p(x)}}{{p(x) - 2}}}}({\Omega _T})}^{\frac{{{p^ - } - 2}}{{{p^ + }}}}} \right)\left| {\nabla {u_\varepsilon } - \nabla u} \right|_{{L^{\frac{{p(x)}}{2}}}({\Omega _T})}^{\frac{{{p^ - }}}{2}}\\ {\kern 14pt} \le C\left( {\left| {\nabla {u_\varepsilon }} \right|_{{L^{p(x)}}({\Omega _T})}^{\frac{{{p^ + }({p^ + } - 2)}}{{{p^ - }}}} + \left| {\nabla {u_\varepsilon }} \right|_{{L^{p(x)}}({\Omega _T})}^{\frac{{{p^ - }({p^ - } - 2)}}{{{p^ + }}}}} \right)\left| {\nabla {u_\varepsilon } - \nabla u} \right|_{{L^{\frac{{p(x)}}{2}}}({\Omega _T})}^{\frac{{{p^ - }}}{2}}. \end{array} $

(47)

因此将公式(14)和公式(25)代入公式(47), 可得

$ {H_6} \to 0 \ ( \varepsilon \to 0 ) . $

(48)

从而, 联立公式(44), 公式(45)和公式(48)可得(37)成立.

最后证明公式(38).利用公式(7)和公式(10), 可见

$ 0 \le - {\beta _\varepsilon }({u_\varepsilon } - {u_0}) \le 1, \ - {\beta _\varepsilon }({u_\varepsilon } - {u_0}) \to \xi \ , \ \varepsilon \to 0. $

(49)

由极限的保号性, 易得

$ 0 \le \xi (x, t) \le 1, \forall (x, t) \in {\Omega _T}. $

(50)

根据$G(\cdot)$的定义, 若证明公式(38)只需证明:当$u({x_0}, {t_0}) > {u_0}({x_0})$时, $\xi ({x_0}, {t_0}) = 0$.事实上当$u({x_0}, {t_0}) > {u_0}(x)$时, 存在常数$\lambda > 0$和邻域${B_\delta }({x_0}, {t_0})$, 当$\varepsilon$足够小时

$ {u_\varepsilon }(x, t) \ge {u_0}(x) + \lambda, \ \forall (x, t) \in {B_\delta }({x_0}, {t_0}). $

当$\varepsilon$足够小时, 对任意的$(x, t) \in {B_\delta }({x_0}, {t_0})$ $0 \ge {\beta _\varepsilon }({u_\varepsilon } - {u_0}) \ge {\beta _\varepsilon }(\lambda) = 0.$因此当$ \varepsilon \to 0 $时

$ \xi (x, t) = 0, \ \forall (x, t) \in {B_\delta }({x_0}, {t_0}). $

公式(38)得证.

引理3.4意味着: $\xi (x, t) = 0$和$u(x, t) > {u_0}(x)$等价, $\xi (x, t) > 0$和$u(x, t) = {u_0}(x)$等价.据此, 我们给出本文的主要结果.

定理3.1  设$\gamma \in (0, 1)$, 则抛物变分不等式(1)-(3)存在唯一的弱解满足

$ {\partial _t}{u^\mu } \in {L^2}({\Omega _T}), \ \mu = \frac{{\gamma - \sigma }}{2} + 1 \in (0, 1). $

证  首先证明解的存在性.由公式(25)和公式(38)可知定义2.1之条件(a)和条件(c)成立, 在公式(5)中令$\varepsilon \to 0$易知定义2.1之条件(b)亦成立.由引理3.4和公式(22)可知定义2.1中等式(d)也成立.最后证明定义2.1中条件(e)成立.定义

$ I = \int_\Omega {\left| {u_\varepsilon ^\mu - u_{0\varepsilon }^\mu } \right|{\rm{d}}x}. $

利用Holder不等式, 有

$ \begin{array}{l} I = \int_\Omega {\left| {u_\varepsilon ^\mu (x, t)-u_{0\varepsilon }^\mu (x)} \right|{\rm{d}}x} \le \int_\Omega {\left| {{\partial _s}u_\varepsilon ^\mu {\rm{d}}x} \right|{\rm{d}}x} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le \sqrt t \int_\Omega {\left| {\sqrt {\int_0^t {{{\left( {{\partial _s}u_\varepsilon ^\mu } \right)}^2}} {\rm{d}}t} } \right|{\rm{d}}x} \le {\left| \Omega \right|^{\frac{1}{2}}}\left| {{\partial _s}u_\varepsilon ^\mu } \right|_{{L^2}({\Omega _T})}^2\sqrt t . \end{array} $

故由公式(14)可得

$ \int_\Omega {\left| {u_\varepsilon ^\mu (x, t) - u_{0\varepsilon }^\mu (x)} \right|{\rm{d}}x} \le C\sqrt t, $

(51)

其中$C$是不依赖$\varepsilon$的正常数.进一步利用三角不等式又有

$\begin{array}{l} \int_\Omega {\left| {u_{}^\mu (x, t)-u_0^\mu (x)} \right|{\rm{d}}x} \\ \le \int_\Omega {\left| {u_{}^\mu (x, t)-u_\varepsilon ^\mu (x, t)} \right|{\rm{d}}x} + |\int_\Omega {\left| {u_\varepsilon ^\mu (x, t)-u_{0\varepsilon }^\mu (x)} \right|{\rm{d}}x + } \int_\Omega {\left| {u_{0\varepsilon }^\mu (x) - u_0^\mu (x)} \right|{\rm{d}}x} . \end{array} $

先令$\varepsilon \to 0$, 可得

$ \int_\Omega {\left| {u_{}^\mu (x, t)-u_0^\mu (x)} \right|{\rm{d}}x} \le |\int_\Omega {\left| {u_\varepsilon ^\mu (x, t)-u_{0\varepsilon }^\mu (x)} \right|{\rm{d}}x}, $

由公式(51)再令$ t \to 0$又有

$ \int_\Omega {\left| {u_{}^\mu (x, t) - u_0^\mu (x)} \right|{\rm{d}}x} \to 0 . $

(52)

因此, 解的存在性得证.

下面证明解的唯一性.假设$({u_1}, {\xi _1})$和$({u_2}, {\xi _2})$是抛物变分不等式(1)-(3)的两个解, 令

$ {\rm{sg}}{{\rm{n}}_\delta }(z) = {\rm{sgn}}(z)\inf \{ \left| z \right|/\delta ,1\} ,\;\delta > 0, $

并定义

${\varphi _{{u_1}}} = {\left( {u_1^{}} \right)^{\gamma - \sigma }}{{\mathop{\rm sgn}} _\delta }\left( {{{\left( {{u_1} - {u_2}} \right)}_ + }} \right), \ {\varphi _{{u_2}}} = {\left( {u_2^{}} \right)^{\gamma - \sigma }}{{\mathop{\rm sgn}} _\delta }\left( {{{\left( {{u_1} - {u_2}} \right)}_ + }} \right), $

则由定义2.1可得

$ \begin{array}{l} \int {\int_{{\Omega _T}} {{{(\gamma + 1 - \sigma )}^{ - 1}}{\partial _t}u_1^{\gamma - \sigma + 1}{{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }) \\+ u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{p(x) - 2}}\nabla {u_1}\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \\ = \int {\int_{{\Omega _T}} {{\xi _1}u_1^{\gamma - \sigma }{{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} }, \end{array} $

$ \begin{array}{l} \int {\int_{{\Omega _T}} {{{(\gamma + 1 - \sigma )}^{ - 1}}{\partial _t}u_2^{\gamma - \sigma + 1}{{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }) \\+ u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{p(x) - 2}}\nabla {u_2}\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \\ = \int {\int_{{\Omega _T}} {{\xi _2}u_2^{\gamma - \sigma }{{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } . \end{array} $

上述两个公式相减可得

$ \begin{array}{l} \int {\int_{{\Omega _T}} {{{(\gamma + 1 - \sigma )}^{ - 1}}{\partial _t}(u_1^{\gamma - \sigma + 1} - u_2^{\gamma - \sigma + 1}){{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \\ +\int {\int_{{\Omega _T}} {\left( {u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{p(x) - 2}}\nabla {u_1} - u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{p(x) - 2}}\nabla {u_2}} \right)\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \\ = \int {\int_{{\Omega _T}} {({\xi _1}u_1^{\gamma - \sigma } - {\xi _2}u_2^{\gamma - \sigma }){{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } . \end{array} $

(53)

下面证明

$ \int {\int_{{\Omega _T}} {({\xi _1}u_1^{\gamma - \sigma } - {\xi _2}u_2^{\gamma - \sigma }){{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \le 0. $

(54)

当${u_1}(x, t) > {u_2}(x, t)$时, ${u_1}(x, t) > {u_0}(x)$, 并且此时${\xi _1} = 0 < {\xi _2}$.故

$ ({\xi _1}u_1^{\gamma - \sigma } - {\xi _2}u_2^{\gamma - \sigma }){{\mathop{\rm sgn}} _\delta }({({u_1} - {u_2})_ + }) \le 0. $

当${u_1}(x, t) \le {u_2}(x, t)$时, ${u_1}(x, t) \le {u_2}(x, t)$, 易得

$ ({\xi _1}u_1^{\gamma - \sigma } - {\xi _2}u_2^{\gamma - \sigma }){{\mathop{\rm sgn}} _\delta }({({u_1} - {u_2})_ + }) = 0. $

因此对任意的$({u_1}, {\xi _1})$和$({u_2}, {\xi _2})$总有

$ ({\xi _1}u_1^{\gamma - \sigma } - {\xi _2}u_2^{\gamma - \sigma }){{\mathop{\rm sgn}} _\delta }({({u_1} - {u_2})_ + }) \le 0. $

故公式(54)成立.

下面证明

$ \int {\int_{{\Omega _T}} {\left( {u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{p(x) - 2}}\nabla {u_1} - u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{p(x) - 2}}\nabla {u_2}} \right)\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \ge 0. $

(55)

注意当$a$, $b$, $c_1$和$c_2$为正常数时, $f(t) = {c_1}{a^t} - {c_2}{b^t}$在$[0, \; \infty)$上是单调函数, 故

$ \left( {u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{p(x)- 2}}\nabla {u_1}- u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{p(x)- 2}}\nabla {u_2}} \right)\nabla {{\mathop{\rm sgn}} _\delta }({({u_1} - {u_2})_ + }) \ge F(x, t), $

(56)

其中

$ \begin{array}{l} F(x, t) = \min \left\{ {(u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{{p^-}-2}}\nabla {u_1}-u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{{p^ - } - 2}}\nabla {u_2})\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }), } \right.\\ \left. { {\kern 70pt}(u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{{p^ + } - 2}}\nabla {u_1} - u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{{p^ + } - 2}}\nabla {u_2})\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + })} \right\}. \end{array} $

因此只需证明$\int {\int_{{\Omega _T}} {F(x, t){\rm{d}}x{\rm{d}}t} } $非负.由引理2.3易得

$ \begin{array}{l} \int {\int_{{\Omega _T}} {(u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{{p^ - } - 2}}\nabla {u_1} - u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{{p^ - } - 2}}\nabla {u_2})\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \\ = \int {\int_{{\Omega _T}} {({{\left| {\nabla g({u_1})} \right|}^{{p^ - } - 2}}\nabla g({u_1}) - {{\left| {\nabla g({u_2})} \right|}^{{p^ - } - 2}}\nabla g({u_2}))\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \ge 0, \end{array} $

其中$g(s) = {s^{1 + \gamma /({p^ - } - 1)}}{(1 + \gamma ({p^ - } - 1))^{ - 1}}$.类推上述证明, 还有

$ \int {\int_{{\Omega _T}} {(u_1^\gamma {{\left| {\nabla {u_1}} \right|}^{{p^ + } - 2}}\nabla {u_1} - u_2^\gamma {{\left| {\nabla {u_2}} \right|}^{{p^ + } - 2}}\nabla {u_2})\nabla {{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \ge 0. $

从而$\int {\int_{{\Omega _T}} {F(x, t){\rm{d}}x{\rm{d}}t} } $非负, 再由公式(56)可知公式(55)成立.进一步将公式(54)和公式(55)代入公式(53), 有

$ \int {\int_{{\Omega _T}} {{{(\gamma + 1 - \sigma )}^{ - 1}}{\partial _t}(u_1^{\gamma - \sigma + 1} - u_2^{\gamma - \sigma + 1}){{{\mathop{\rm sgn}} }_\delta }({{({u_1} - {u_2})}_ + }){\rm{d}}x{\rm{d}}t} } \le 0. $

又因为${{\mathop{\rm sgn}} _\delta }({({u_1} - {u_2})_ + }) = {{\mathop{\rm sgn}} _\delta }({(u_1^{\gamma - \sigma + 1} - u_2^{\gamma - \sigma + 1})_ + })$, 所以

$ \int_\Omega {{{(u_1^{\gamma - \sigma + 1} - u_2^{\gamma - \sigma + 1})}_ + }{\rm{d}}x} \le 0. $

故对任意的$(x, t) \in {\Omega _T}$有${u_1} \le {u_2}$.同理我们亦可证明任意的$(x, t) \in {\Omega _T}$有${u_1} \ge {u_2}$.故解的唯一性成立.