本文考虑如下外区域中具有变系数非线性项的半线性波动方程

(1.1) $ \begin{equation} \left\{ \begin{array}{ll} &\partial_{t}^2 u(x, t)-\Delta u(x, t) = \langle x\rangle^{-\alpha}|u(x, t)|^{p}, \quad\; \; \; (x, t)\in\Omega\times(0, T), \\ &u(x, t) = 0, \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (x, t)\in\partial{\Omega}\times (0, T), \\ &u(x, 0) = \varepsilon f(x), \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in\Omega, \\ &\partial_{t}u(x, 0) = \varepsilon g(x), \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in\Omega. \end{array} \right. \end{equation} $

其中$ \Omega = \{x\in{{\Bbb R}} ^{2};|x|>1\}, \varepsilon $为正常数, $ 0<\alpha<2, \langle x\rangle = \sqrt{1+x^{2}} $, $ f(x) $, $ g(x) $为已知光滑函数.

半线性波动方程一直以来都是热点问题. John在文献[1]中第一次提出如下半线性波动方程

(1.2) $ \begin{equation} \left\{ \begin{array}{ll}&\partial_{t}^2 u(x, t)-\Delta u(x, t) = |u(x, t)|^{p}, \quad (x, t)\in{{\Bbb R}} ^{N}\times(0, T), \\ &u(x, 0) = \varepsilon f(x), \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in{{\Bbb R}} ^{N}, \\ &\partial_{t}u(x, 0) = \varepsilon g(x), \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in{{\Bbb R}} ^{N}. \end{array} \right. \end{equation} $

式中$ \varepsilon $为正常数, $ f(x) $, $ g(x) $是已知光滑函数.其证明了方程(1.2)的临界指数$ p_{c}(3) = 1+\sqrt{2} $,并说明了当$ 1<p<1+\sqrt{2} $时,方程(1.2)的解在有限时间内爆破,当$ p>1+\sqrt{2} $时,存在整体解.接着Glassey ^[2]得到了当$ N\leq3 $时,以上结论仍然成立.随后, Strauss ^[3]猜测:当$ 1<p<p_{s}(N) $时,方程(1.2)没有整体解,当$ p>p_{s}(N) $时,方程(1.2)整体解存在,并提出了对于更一般的维数,方程(1.2)存在临界指数$ p_{s}(N) $,且$ p_{s}(N) $由

$ p_{s}(N) = \sup\{p>1:\Upsilon(N, p)>0\}, \; \; \; \Upsilon(N, p) = 2+(N+1)p-(N-1)p^{2} $

给出.随后不久, Sideris ^[4]得出了当$ N\geq4 $时,方程(1.2)在$ 1<p<p_{s}(N) $时没有整体解,对于$ N = 4 $的情况, Zhou ^[5]证实了Strauss猜测. Georgiev, Lindblad和Sogge ^[6]证明了对于所有维数, Strauss猜测都成立.而对于临界情形$ p = p_{s}(N) $, Schaeffer ^[7]证明了$ N = 2, 3 $时没有整体解, Yordanov和Zhang ^[8]及Zhou ^[9]证明了高维情况没有整体解.

对于半线性波动方程的外区域问题,也有很多研究.在文献[10]中, Sobajima和Wakasa考虑了如下外区域中的半线性波动方程

(1.3) $ \begin{equation} \left\{ \begin{array}{ll}&\partial_{t}^2 u(x, t)-\Delta u(x, t) = |u(x, t)|^{p}, \quad (x, t)\in\Omega\times(0, T), \\ &u(x, t) = 0, \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (x, t)\in\partial{\Omega}\times (0, T), \\ &u(x, 0) = \varepsilon f(x), \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in\Omega, \\ &\partial_{t}u(x, 0) = \varepsilon g(x), \quad\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in\Omega, \end{array} \right. \end{equation} $

其中$ \Omega = {{\Bbb R}} ^{N}\backslash\overline{B(0, 1)} $,其提出并证明了当$ N\geq3 $时,解生命跨度的上界.当$ N = 3, 4 $且$ p_{s}(N)<p<\frac{N+3}{N-1} $时,文献[11-12]给出了方程(1.3)解的存在性,当$ N = 3 $且$ 2<p<p_{s}(3) $时, Yu ^[13]得到了方程(1.3)解生命跨度的下界,而对于$ N\geq3 $且$ 1<p<p_{s}(N) $这种情况, Zhou和Han在文献[14]中给出并证明了方程(1.3)解生命跨度的上界.对于临界情形$ p = p_{s}(N) $,可参见文献[15-17].

Zhang^[18]考虑了如下带变系数非线性项的半线性波动方程

(1.4) $ \begin{equation} \left\{ \begin{array}{ll}&\partial_{t}^2 u(x, t)-\Delta u(x, t) = \langle x\rangle^{-\alpha}|u(x, t)|^{2}, \quad (x, t)\in{{\Bbb R}} _{+}\times{{\Bbb R}} ^{3}, \\ &u(x, 0) = \varepsilon f(x), \quad\; \; \; \; \partial_{t}u(x, 0) = \varepsilon g(x). \end{array} \right. \end{equation} $

证明了当$ \alpha\in[0, \frac{1}{2}) $时$ T_{\varepsilon}\leq A\varepsilon^{-\frac{2}{1-2\alpha}} $,当$ \alpha = \frac{1}{2} $时, $ T_{\varepsilon}\leq\exp(B\varepsilon^{-2}) $,这里$ A, B $是一个与$ \varepsilon $无关的正常数, $ T_{\varepsilon} $是解的最大存在区间.当$ \alpha = 0 $时, $ T_{\varepsilon}\leq A\varepsilon^{-2} $与方程(1.2)的结果相同,由此可见,非线性项中变系数对解的生命跨度有很大的影响,受文献[18]的启发,本文研究外区域中具有变系数非线性项的半线性波动方程(1.1).不同于文献[18],我们将对一般$ N $维外区域中半线性波动方程展开讨论,并着重研究变系数对解生命跨度的影响.本文给出$ N\geq2 $的解生命跨度的上界,证明的主要方法是测试函数法,并利用外区域Dirichlet边界条件的调和函数,该想法来自于Ikeda和Sobajima^[19-20].

本文内容安排如下:第2节给出了方程(1.1)解的相关定义,同时给出了本文的主要结果,即$ N = 2 $和$ N\geq3 $时解生命跨度;为了证明本文的主要结果,第3节首先引入截断函数,然后给出可以推导解生命跨度上界的主要特性,利用$ N = 2 $及$ N\geq3 $时外区域Dirichlet边界条件的调和函数以及测试函数方法^[19-20],求出方程(1.1)解生命跨度,从而证明了本文的主要结果.

定义2.1  我们称$ u $为方程(1.1)的弱解,如果

$ u\in C([0, T);H_{0}^{1}(\Omega)\cap L_{loc}^{p}(\overline{\Omega}\times[0, T))), $

其中$ u(x, 0) = \varepsilon f(x) $,以及对于每个$ \psi\in C^{2}(\Omega\times[0, T) $且supp$ \psi\subset\subset\overline{\Omega}\times[0, T)) $及$ \psi|_{\partial\Omega} = 0 $,

$ \begin{eqnarray*} && \varepsilon\int_{\Omega}g(x)\psi(x, 0){\rm d}x+\int_{0}^{T}\int_{\Omega}|u(x, t)|^{p}\langle x\rangle^{-\alpha}\psi(x, t){\rm d}x{\rm d}t\\ & = &\int_{0}^{T}\int_{\Omega}(\nabla u(x, t)\nabla \psi(x, t)-\partial_{t}u(x, t)\partial_{t}\psi(x, t)){\rm d}x{\rm d}t. \end{eqnarray*} $

定义2.2  LifeSpan(u)称为初边值问题(1.1)的解的最大存在时间,即

$\mbox{LifeSpan(u)} = \sup\{T>0;u$是方程$(1.1)$在$[0, T)$中的唯一弱解}.

定理2.1  设$ N = 2 $, $ 0<\alpha<2 $, $ 1<p<\infty $.令$ u $为方程(1.1)的唯一解,如果$ p\leq2-\frac{\alpha}{2} $, $ g(x)\log|x|\in L^{1}(\Omega) $,且设初值满足

(2.1) $ \begin{equation} \int_{\Omega}g(x)\log|x|{\rm d}x>0, \end{equation} $

那么$ \mbox{LifeSpan(u)}<\infty $.并且LifeSpan(u)有如下上界

$ \begin{eqnarray*} \mbox{LifeSpan(u)}\leq \left\{ \begin{array}{ll} C\varepsilon^{-\frac{p-1}{2-\frac{\alpha}{2}-p}}(\log(\varepsilon^{-1}))^{\frac{p-1}{2-\frac{\alpha}{2}-p}}, {\quad} &{ } 1<p<2-\frac{\alpha}{2}, \\ \exp(C\varepsilon^{1-\frac{2}{\alpha}}), &{ } p = 2-\frac{\alpha}{2}, \end{array} \right. \end{eqnarray*} $

这里$ C $是一个与$ \varepsilon $无关的正常数.

定理2.2  设$ N\geq3 $, $ 0<\alpha<2 $, $ 1<p<\infty $.令$ u $为方程(1.1)的唯一解,如果$ p\leq2-\frac{\alpha}{2} $, $ g(x)(1-|x|^{2-N})\in L^{1}(\Omega) $,且设初值满足

(2.2) $ \begin{equation} \int_{\Omega}g(x)(1-|x|^{2-N}){\rm d}x>0, \end{equation} $

那么$ \mbox{LifeSpan(u)}<\infty $.并且$ \mbox{LifeSpan(u)} $有如下上界

$ \begin{eqnarray*} \mbox{LifeSpan(u)}\leq \left\{ \begin{array}{ll} C\varepsilon^{-\frac{2(p-1)}{2-Np+N-\alpha}}, \quad&{ } 1<p<\frac{2+N-\alpha}{N}, \\ \exp(C\varepsilon^{-(p-1)}), &{ } p = \frac{2+N-\alpha}{N}. \end{array} \right. \end{eqnarray*} $

这里$ C $是一个与$ \varepsilon $无关的正常数.

注2.1  由文献[1]可知,当$ N = 3 $时,方程(1.2)生命跨度的上界为

$ T<B\varepsilon^{-2}. $

这里$ B $是一个与$ \varepsilon $无关的正常数.由文献[18]可知,方程(1.4)生命跨度的上界为

$ \begin{eqnarray*} T_{\varepsilon}\leq \left\{ \begin{array}{ll} A\varepsilon^{-\frac{2}{1-2\alpha}}, &{ } \alpha\in[0, \frac{1}{2}), \\ \exp(B\varepsilon^{-2}), {\quad}&{ } \alpha = \frac{1}{2}. \end{array} \right. \end{eqnarray*} $

比较方程(1.2)和(1.4)的结果可发现,方程(1.4)中$ \alpha = 0 $时,生命跨度的上界和方程(1.2)的相同,不同于文献[18],我们将对一般$ N $维外区域中半线性波动方程展开讨论,并着重研究变系数对解生命跨度的影响,且本文$ \alpha $的范围更大.

选取如下截断函数$ \eta\in C^{2}([0, \infty)) $和$ \eta^{*}\in L^{\infty}([0, \infty)) $ (参见文献[19])

$ \begin{eqnarray*} \eta(s) \left\{ \begin{array}{ll} = 1\quad&{ } s\in [0, \frac{1}{2}], \\ \mbox{单调增加} \quad&{ } s\in (\frac{1}{2}, 1), \\ = 0\quad &s\in [1, \infty), \end{array} \right.{\quad} \eta^{*}(s) = \left\{ \begin{array}{ll} 0\quad\quad&{ } s\in [0, \frac{1}{2}), \\ \eta(s)\quad&{ } s\in [\frac{1}{2}, \infty). \end{array} \right. \end{eqnarray*} $

定义3.1  对于$ p>1 $, $ R>0 $,我们引入如下测试函数^[19]

$ \psi_{R}(x, t) = [\eta(s_{R}(x, t))]^{2p'}, \quad(x, t)\in\Omega\times[0, \infty), $

$ \psi_{R}^{*}(x, t) = [\eta^{*}(s_{R}(x, t))]^{2p'}, \quad(x, t)\in\Omega\times[0, \infty), $

其中

$ s_{R}(x, t) = \frac{(|x|-1)^{2}+t}{R}. $

同时令$ P(R) = \{(x, t)\in\Omega\times[0, \infty):(|x|-1)^{2}+t\leq R\} $.

为了完成本节的证明,需要如下引理.

引理3.1^[19]  设维数$ N = 2 $, $ \psi_{R} $和$ \psi_{R}^{*} $如定义3.1,那么$ \psi_{R} $满足如下性质

(ⅰ)如果$ (x, t)\in P(\frac{R}{2}) $,那么$ \psi_{R}(x, t) = 1 $;且如果$ (x, t)\notin P(R) $,那么$ \psi_{R}(x, t) = 0 $.

(ⅱ)存在正常数$ C_{1} $使得对每个$ (x, t)\in P(R) $,有

$ |\partial_{t}^{2}\psi_{R}(x, t)|\leq C_{1}R^{-2}[\psi_{R}^{*}(x, t)]^{\frac{1}{p}}. $

(ⅲ)存在正常数$ C_{2} $使得对每个$ (x, t)\in P(R) $,有

$ |\nabla\psi_{R}(x, t)|\leq C_{2}R^{-1}|x|(\log|x|)[\psi_{R}^{*}(x, t)]^{\frac{1}{p}}. $

(ⅳ)存在正常数$ C_{3} $使得对每个$ (x, t)\in P(R) $,有

$ |\Delta\psi_{R}(x, t)|\leq C_{3}R^{-1}[\psi_{R}^{*}(x, t)]^{\frac{1}{p}}. $

证  该引理由Ikeda和Sobajima ^[19]给出,具体过程可参见文献[19].

下述引理由文献[19]给出,但是论述不够清楚,为了后面讨论方便,我们给出详细的证明.

引理3.2  设维数$ N = 2 $, $ \delta>0 $, $ C_{0}>0 $, $ R_{1}>0 $, $ \theta\geq0 $, $ \kappa\in{{\Bbb R}} $以及对于$ T>R_{1} $, $ 0\leq w\in L_{loc}^{1}([0, T);L^{1}(\Omega)) $.假设对于每个$ R\in[R_{1}, T) $,有

(3.1) $ \begin{equation} \delta+{\int\!\!\!\int} _{P(R)}w(x, t)\psi_{R}(x, t){\rm d}x{\rm d}t\leq C_{0}R^{-\frac{\theta}{p'}}(\log R)^{\frac{\kappa}{P'}} \bigg({\int\!\!\!\int} _{P(R)}w(x, t)\psi_{R}^{*}(x, t){\rm d}x{\rm d}t\bigg)^{\frac{1}{p}}. \end{equation} $

那么$ T $有如下上界

$ \begin{eqnarray*} T\leq \left\{ \begin{array}{ll} { } C\delta^{-\frac{1}{\theta}}(\log(\delta^{-1}))^{\frac{\kappa}{\theta}}, \quad&{ }\theta>0, \kappa\in{{\Bbb R}}, \\ { } \exp(C\delta^{-\frac{p-1}{1-\kappa(p-1)}}), \quad&{ } \theta = 0, \kappa<\frac{1}{p-1}, \\ { } \exp\exp(C\delta^{-(p-1)}), \quad&{ } \theta = 0, \kappa = \frac{1}{p-1}. \end{array} \right. \end{eqnarray*} $

证  令

$ y(r): = {\int\!\!\!\int}_{P(r)}w(x, t)\psi_{r}^{*}(x, t){\rm d}x{\rm d}t, r\in(0, T), $

则由文献[20]的讨论可得

$ \begin{eqnarray*} \int_{0}^{R}y(r)r^{-1}{\rm d}r& = &{\int\!\!\!\int}_{P(R)}w(x, t)\bigg(\int_{((|x|-1)^{2}+t)/R}^{\infty}[\eta^{*}(s)]^{2p'}s^{-1}{\rm d}s\bigg){\rm d}x{\rm d}t\\ &\leq&(\log2){\int\!\!\!\int} _{P(R)}w(x, t)\psi_{R}(x, t){\rm d}x{\rm d}t. \end{eqnarray*} $

令

$ Y(R) = \int_{0}^{R}y(r)r^{-1}{\rm d}r, \; \rho\in(R_{1}, T), $

对于$ R\in(R_{1}, T) $,从(3.1)式可以推断

$ \begin{eqnarray*} (\delta+\frac{1}{\log2}Y(R))^{P}&\leq&C_{0}^{p}R^{-\theta(p-1)}(\log R)^{\kappa(p-1)}{\int\!\!\!\int}_{P(R)}w(x, t)\psi_{R}^{*}(x, t){\rm d}x{\rm d}t\\ & = &C_{0}^{p}R^{1-\theta(p-1)}(\log R)^{\kappa(p-1)}Y'(R). \end{eqnarray*} $

令

$ Y(R) = Z\bigg(\int_{\log R_{1}}^{\log R}e^{\theta(p-1)s}s^{-\kappa(p-1)}{\rm d}s\bigg), \ 0<\rho<\rho_{T} = \int_{\log R_{1}}^{\log T}e^{\theta(p-1)s}s^{-\kappa(p-1)}{\rm d}s, $

于是

(3.2) $ \begin{equation} \frac{{\rm d}}{{\rm d}\rho}((\log2)\delta+Z(\rho))^{1-p}\leq-(p-1)(\log2)^{-p}C_{0}^{-p}, \rho\in(0, \rho_{T}). \end{equation} $

对上式在$ [\rho_{1}, \rho_{2}]\subset(0, \rho_{T}) $上积分,有

$ \rho_{2}<\rho_{1}+(p-1)^{-1}(\log2)C_{0}^{P}\delta^{-(p-1)}. $

令$ \rho_{2}\uparrow\rho_{T} $及$ \rho_{1}\downarrow0 $,有

$ \begin{eqnarray*} \rho_{T} = \int_{\log R_{1}}^{\log T}e^{\theta(p-1)s}s^{-\kappa(p-1)}{\rm d}s\leq(p-1)^{-1}(\log2)C_{0}^{P}\delta^{-(p-1)}. \end{eqnarray*} $

如果函数$ e^{\theta(p-1)s}s^{-\kappa(p-1)} $在无穷远处是不可积的,那么$ T $必须是有限的.更确切地说,对于较大的$ T $, $ \rho_{T} $的渐近性态分别如下:如果$ \theta>0 $,那么$ \rho_{T}\approx\frac{1}{(p-1)\theta}T^{\theta(p-1)}(\log T)^{-\kappa(p-1)} $;如果$ \theta = 0 $且$ \kappa<\frac{1}{p-1} $,那么$ \rho_{T}\approx\frac{1}{1-\kappa(p-1)}(\log T)^{1-\kappa(p-1)} $;如果$ \theta = 0 $且$ \kappa = \frac{1}{p-1} $,那么$ \rho_{T}\approx\log\log T $.由此,引理3.2得到证明.

引理3.3  设维数$ N\geq3 $, $ \psi_{R} $和$ \psi_{R}^{*} $如定义3.1,那么$ \psi_{R} $满足如下性质

(ⅰ)如果$ (x, t)\in P(\frac{R}{2}) $,那么$ \psi_{R}(x, t) = 1 $;且如果$ (x, t)\notin P(R) $,那么$ \psi_{R}(x, t) = 0 $.

(ⅱ)存在正常数$ C_{4} $使得对每个$ (x, t)\in P(R) $,有

$ |\partial_{t}^{2}\psi_{R}(x, t)|\leq C_{4}R^{-2}[\psi_{R}^{*}(x, t)]^{\frac{1}{p}}. $

(ⅲ)存在正常数$ C_{5} $使得对每个$ (x, t)\in P(R) $,有

$ |\nabla\psi_{R}(x, t)|\leq C_{5}R^{-1}|x|(1-|x|^{2-N})[\psi_{R}^{*}(x, t)]^{\frac{1}{p}}. $

(ⅳ)存在正常数$ C_{6} $使得对每个$ (x, t)\in P(R) $,有

$ |\Delta\psi_{R}(x, t)|\leq C_{6}R^{-1}[\psi_{R}^{*}(x, t)]^{\frac{1}{p}}. $

证  该引理由Ikeda和Sobajima^[19]给出,具体过程可参见文献[19].

引理3.4  设维数$ N\geq3 $, $ \delta>0 $, $ C_{0}>0 $, $ R_{1}>0 $, $ \theta\geq0 $, $ \kappa\in{{\Bbb R}} $以及对于$ T>R_{1} $, $ 0\leq w\in L_{loc}^{1}([0, T);L^{1}(\Omega)) $.假设对于每个$ R\in[R_{1}, T) $,有

(3.3) $ \begin{equation} \delta+{\int\!\!\!\int}_{P(R)}w(x, t)\psi_{R}(x, t){\rm d}x{\rm d}t\leq C_{0}R^{-\frac{\theta}{p'}} \bigg({\int\!\!\!\int}_{P(R)}w(x, t)\psi_{R}^{*}(x, t){\rm d}x{\rm d}t\bigg)^{\frac{1}{p}}. \end{equation} $

那么$ T $有如下上界

$ \begin{eqnarray*} T\leq \left\{ \begin{array}{ll} C\delta^{-\frac{1}{\theta}}, &\theta>0, \\ \exp(C\delta^{-(p-1)}), {\quad}& \theta = 0. \end{array} \right. \end{eqnarray*} $

证  令

$ y(r): = {\int\!\!\!\int}_{P(r)}w(x, t)\psi_{r}^{*}(x, t){\rm d}x{\rm d}t, r\in(0, T), $

则由文献[20]的结果,可得

$ \begin{eqnarray*} \int_{0}^{R}y(r)r^{-1}{\rm d}r& = &{\int\!\!\!\int}_{P(R)}w(x, t) \bigg(\int_{((|x|-1)^{2}+t)/R}^{\infty}[\eta^{*}(s)]^{2p'}s^{-1}{\rm d}s\bigg){\rm d}x{\rm d}t\\ &\leq&(\log2){\int\!\!\!\int} _{P(R)}w(x, t)\psi_{R}(x, t){\rm d}x{\rm d}t. \end{eqnarray*} $

令

$ Y(R) = \int_{0}^{R}y(r)r^{-1}{\rm d}r, \; \rho\in(R_{1}, T), $

对于$ R\in(R_{1}, T) $,从(3.3)式我们可以推断

$ \begin{eqnarray*} \bigg(\delta+\frac{1}{\log2}Y(R)\bigg)^{P}&\leq&C_{0}^{p}R^{-\theta(p-1)}{\int\!\!\!\int}_{P(R)}w(x, t)\psi_{R}^{*}(x, t){\rm d}x{\rm d}t \\ & = &C_{0}^{p}R^{1-\theta(p-1)}Y'(R). \end{eqnarray*} $

令

$ Y(R) = Z\bigg(\int_{R_{1}}^{R}r^{(p-1)\theta-1}{\rm d}s\bigg), \ 0<\rho<\rho_{T} = \int_{R_{1}}^{T}r^{(p-1)\theta-1}{\rm d}s, $

上式意味着

(3.4) $ \begin{equation} \frac{{\rm d}}{{\rm d}\rho}((\log2)\delta+Z(\rho))^{1-p}\leq-(p-1)(\log2)^{-p}C_{0}^{-p}, \rho\in(0, \rho_{T}). \end{equation} $

对上式在$ [\rho_{1}, \rho_{2}]\subset(0, \rho_{T}) $上积分,我们有

$ \rho_{2}<\rho_{1}+(p-1)^{-1}(\log2)C_{0}^{P}\delta^{-(p-1)}. $

令$ \rho_{2}\uparrow\rho_{T} $及$ \rho_{1}\downarrow0 $,可得

$ \begin{eqnarray*} \rho_{T} = \int_{R_{1}}^{T}r^{(p-1)\theta-1}{\rm d}s\leq(p-1)^{-1}(\log2)C_{0}^{P}\delta^{-(p-1)}. \end{eqnarray*} $

如果$ \theta>0 $,即$ (p-1)\theta-1>-1 $,那么有

$ T\leq (R_{1}^{(p-1)\theta}+(\log2)C_{0}^{p}\theta\delta^{-(p-1)})^{\frac{1}{(p-1)\theta}}\leq C\delta^{-\frac{1}{\theta}}. $

如果$ \theta = 0 $,即$ (p-1)\theta-1 = -1 $,那么有

$ T\leq \exp(\log R_{1}+(\log2)(p-1)^{-1}C_{0}^{p}\delta^{-(p-1)})\leq\exp(C\delta^{-(p-1)}). $

于是,引理3.4得到证明.

本节我们将给出本文主要结果定理2.1和定理2.2的证明.

定理2.1的证明  当$ N = 2 $时,由定义2.1可知,对每个supp$ \psi\subset\subset\bar{\Omega}\times[0, T) $及$ \psi|_{\partial\Omega} $的$ \psi\in C^{2}(\bar{\Omega}\times[0, T)) $,有

$ \begin{eqnarray*} && \varepsilon\int_{\Omega}(g(x)\psi(0)-f(x)\partial_{t}\psi(0)){\rm d}x+\int_{0}^{T}\int_{\Omega}|u(t)|^{p}\langle x\rangle^{-\alpha}\psi(t){\rm d}x{\rm d}t\\ & = &\int_{0}^{T}\int_{\Omega}u(t)(-\Delta\psi(t)+\partial_{t}^{2}\psi(t)){\rm d}x{\rm d}t. \end{eqnarray*} $

取$ \psi = \Phi(x)\psi_{R}(x, t) $,其中$ \Phi(x) = \log|x| $,由引理3.1有

$ \begin{eqnarray*} |-\Delta(\Phi\psi_{R}(t))+\partial_{t}^{2}(\Phi\psi_{R}(t))|& = &|-2\nabla\Phi\nabla\psi_{R}(t)-\Phi\Delta\psi_{R}(t)+\Phi\partial_{t}^{2}\psi_{R}(t)|\\ &\leq&\frac{2C_{2}}{R}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}+\frac{C_{3}}{R}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}+\frac{C_{1}}{R^{2}}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}\\ &\leq&\frac{C_{7}}{R}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}, \end{eqnarray*} $

其中$ C_{7} = 2C_{2}+C_{3}+\frac{C_{1}}{R_{0}} $.从而有

$ \begin{eqnarray*} && \varepsilon\int_{\Omega}(g(x)\log|x|\psi_{R}(0)-f(x)\log|x|\partial_{t}\psi_{R}(0)){\rm d}x +{\int\!\!\!\int}_{P(R)}|u(t)|^{p}\langle x\rangle^{-\alpha}\Phi\psi_{R}(t){\rm d}x{\rm d}t\\ &\leq&\frac{C_{7}}{R}{\int\!\!\!\int}_{P(R)}|u(t)|\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}{\rm d}x{\rm d}t. \end{eqnarray*} $

由假设(2.1)式,当$ R\rightarrow+\infty $时,有

$ \int_{\Omega}(g(x)\log|x|\psi_{R}(0)-f(x)\log|x|\partial_{t}\psi_{R}(0)){\rm d}x\rightarrow\int_{\Omega}g(x)\log|x|{\rm d}x>0. $

所以存在$ R_{0} $,使得当$ R\geq R_{0} $时, $ \int_{\Omega}g(x)\log|x|{\rm d}x\geq c_{0} $,从而有

$ \begin{eqnarray*} && c_{0}\varepsilon+{\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}(t){\rm d}x{\rm d}t\\ &\leq&\frac{C_{7}}{R}{\int\!\!\!\int}_{P(R)}|u(t)|\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}{\rm d}x{\rm d}t\\ &\leq&\frac{C_{7}}{R}\bigg({\int\!\!\!\int}_{P(R)}\Phi(1+x^{2})^{\frac{\alpha}{2}(p'-1)}{\rm d}x{\rm d}t\bigg)^{\frac{1}{p'}} \bigg({\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}^{*}(t){\rm d}x{\rm d}t\bigg)^{\frac{1}{p}}. \end{eqnarray*} $

而

$ {\int\!\!\!\int}_{P(R)}\Phi(1+x^{2})^{\frac{\alpha}{2}(p'-1)}{\rm d}x{\rm d}t\leq \frac{\pi}{\frac{\alpha}{2}(p'-1)+1}R[1+(\sqrt{R}+1)^{2}]^{\frac{\alpha}{2}(p'-1)+1}\log(\sqrt{R}+1), $

于是

$ \begin{eqnarray*} && c_{0}\varepsilon+{\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}(t){\rm d}x{\rm d}t\\ &\leq& C_{8}R^{1-(1-\frac{\alpha}{4})\frac{2}{p}}(\log R)^{\frac{1}{p'}}\bigg({\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}^{*}(t){\rm d}x{\rm d}t\bigg)^{\frac{1}{p}}. \end{eqnarray*} $

令$ w = |u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi $,则由引理3.2可得定理2.1.证毕.

定理2.2的证明  当$ N\geq3 $时,由定义2.1,对每个有supp$ \psi\subset\subset\bar{\Omega}\times[0, T) $及$ \psi|_{\partial\Omega} $的$ \psi\in C^{2}(\bar{\Omega}\times[0, T)) $,我们有

$ \begin{eqnarray*} && \varepsilon\int_{\Omega}(g(x)\psi(0)-f(x)\partial_{t}\psi(0)){\rm d}x+\int_{0}^{T}\int_{\Omega}|u(t)|^{p}\langle x\rangle^{-\alpha}\psi(t){\rm d}x{\rm d}t\\ & = &\int_{0}^{T}\int_{\Omega}u(t)(-\Delta\psi(t)+\partial_{t}^{2}\psi(t)){\rm d}x{\rm d}t. \end{eqnarray*} $

取$ \psi = \Phi(x)\psi_{R}(x, t) $,其中$ \Phi(x) = 1-|x|^{2-N} $,由引理3.3可得

$ \begin{eqnarray*} |-\Delta(\Phi\psi_{R}(t))+\partial_{t}^{2}(\Phi\psi_{R}(t))|& = &|-2\nabla\Phi\nabla\psi_{R}(t)-\Phi\Delta\psi_{R}(t)+\Phi\partial_{t}^{2}\psi_{R}(t)|\\ &\leq&\frac{2C_{5}}{R}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}+\frac{C_{6}}{R}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}+\frac{C_{4}}{R^{2}}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}\\ &\leq&\frac{C_{9}}{R}\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}, \end{eqnarray*} $

其中$ C_{9} = 2C_{5}+C_{6}+\frac{C_{4}}{R_{1}} $.从而有

$ \begin{eqnarray*} && \varepsilon\int_{\Omega}(g(x)(1-|x|^{2-N})\psi_{R}(0)-f(x)(1-|x|^{2-N})\partial_{t}\psi_{R}(0)){\rm d}x\\ &&+{\int\!\!\!\int}_{P(R)}|u(t)|^{p}\langle x\rangle^{-\alpha}\Phi\psi_{R}(t){\rm d}x{\rm d}t\\ &\leq&\frac{C_{9}}{R}{\int\!\!\!\int}_{P(R)}|u(t)|\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}{\rm d}x{\rm d}t. \end{eqnarray*} $

由假设(2.2)式,当$ R\rightarrow+\infty $时,有

$ \int_{\Omega}(g(x)(1-|x|^{2-N})\psi_{R}(0)-f(x)(1-|x|^{2-N})\partial_{t}\psi_{R}(0)){\rm d}x\rightarrow\int_{\Omega}g(x)(1-|x|^{2-N}){\rm d}x>0, $

所以存在$ R_{1} $,使得当$ R\geq R_{1} $时, $ \int_{\Omega}g(x)\log|x|{\rm d}x\geq c_{1} $,从而有

$ \begin{eqnarray*} && c_{1}\varepsilon+{\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}(t){\rm d}x{\rm d}t\\ &\leq&\frac{C_{9}}{R}{\int\!\!\!\int}_{P(R)}|u(t)|\Phi[\psi_{R}^{*}(t)]^{\frac{1}{p}}{\rm d}x{\rm d}t\\ &\leq&\frac{C_{9}}{R}\bigg({\int\!\!\!\int}_{P(R)}\Phi(1+x^{2})^{\frac{\alpha}{2}(p'-1)}{\rm d}x{\rm d}t\bigg)^{\frac{1}{p'}} \bigg({\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}^{*}(t){\rm d}x{\rm d}t\bigg)^{\frac{1}{p}}. \end{eqnarray*} $

而

$ {\int\!\!\!\int}_{P(R)}\Phi(1+x^{2})^{\frac{\alpha}{2}(p'-1)}{\rm d}x{\rm d}t\leq 2\pi^{N-1}N^{-1}R(\sqrt{R}+1)^{N}[1+(\sqrt{R}+1)^{2}]^{\frac{\alpha}{2}(p'-1)}, $

于是

$ \begin{eqnarray*} && c_{1}\varepsilon+{\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}(t){\rm d}x{\rm d}t\\ &\leq&C_{10}R^\frac{Np-2-N+\alpha}{2p} \bigg({\int\!\!\!\int}_{P(R)}|u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi\psi_{R}^{*}(t){\rm d}x{\rm d}t\bigg)^{\frac{1}{p}}. \end{eqnarray*} $

取$ w = |u(t)|^{p}\frac{1}{(1+x^{2})^{\frac{\alpha}{2}}}\Phi $,则由引理3.4可得定理2.2.证毕.