近年来, 时滞微分方程由于其在数学、物理、化学、生物和通讯系统等领域的广泛应用而备受人们的广泛关注并成为各领域研究的热点问题, 参见文献[1-2]. 在数学领域, 人们主要关注并研究时滞微分方程解的适定性和长时间行为, 并涌现出一些研究成果, 参见文献[3-26].

基于上述情况, 本文研究下述带有时滞项的弱阻尼波方程解的长时间行为

(1.1) $ \begin{equation} \left\{\begin{array}{ll} \partial_{tt} u+\gamma\partial_{t}u-\Delta u =f(u)+F(u_{t})+g(t), \quad&(x, t)\in \Omega\times(\tau, \infty), \\ u(x, \, t)=0, &(x, t)\in \partial\Omega\times(\tau, \infty), \\ u(x, \, t)=\phi(x, t-\tau), &x\in\Omega, t\in[\tau-h, \tau], \\ \partial_{t}u(x, \, t)=\partial_{t}\phi(x, t-\tau), &x\in\Omega, t\in[\tau-h, \tau], \end{array}\right. \end{equation} $

其中$ \Omega\subset{{\Bbb R}} ^{3} $是有界光滑区域, 系数$ \gamma>0 $, $ F $是作用在带有某种遗传特征的解上的算子, 非自治外力项$ g(\cdot)\in L^{2}_{\rm loc}({{\Bbb R}} ; L^{2}(\Omega)) $, $ \phi $是区间$ [\tau-h, \tau] $上的初值, $ h(>0) $是时滞影响的长度. 对任意的$ t\geq\tau $, $ \tau\in{{\Bbb R}} $, $ u_{t} $表示定义在$ [-h, 0] $上的函数$ u_{t}(\theta)=u(t+\theta) $, $ \theta\in[-h, 0] $.

对于非线性项$ f $, 我们假设$ f\in C^{1}({{\Bbb R}} , {{\Bbb R}} ) $并满足$ f(0)=0 $和下述增长条件

(1.2) $ \begin{equation} |f'(u)|\leq C(1+|u|^{p-1}), {\quad} p\in[1, 5), \quad\forall u\in{{\Bbb R}} . \end{equation} $

记

$ {\cal F}(u):=\int_{0}^{u}f(s){\rm d}s, $

则由条件(1.2)可知, 存在常数$ C>0 $使得

(1.3) $ \begin{eqnarray} |{\cal F}(u)|\leq C(1+|u|^{p+1}), \quad\forall u\in{{\Bbb R}} . \end{eqnarray} $

类似于文献[27], 我们对$ f $和$ {\cal F} $做如下假设:

(1.4) $ \begin{equation} \limsup\limits_{|u|\rightarrow \infty}\frac{{\cal F}(u)}{u^{2}}\leq0, \end{equation} $

(1.5) $ \begin{equation} \limsup\limits_{|u|\rightarrow \infty}\frac{f(u)u-\mu{\cal F}(u)}{u^{2}}\leq0, \end{equation} $

其中常数$ \mu>0 $.

设$ C_{X} $表示Banach空间$ C([-h, 0]; X) $并赋予上确界模, 则对任意的$ u\in C_{X} $, 它的模表示为$ { }\|u\|_{C_{X}}=\max\limits_{t\in [-h, 0]}\|u(t)\|_{X} $. 设$ (X, \|\cdot\|_{X}) $, $ (Y, \|\cdot\|_{Y}) $是两个满足连续嵌入$ X\subset Y $的Banach空间, 用$ C_{X, Y} $表示Banach空间$ C_{X}\cap C^{1}([-h, 0];Y) $并定义其模$ \|\cdot\|_{C_{X, Y}} $为

$ \|\phi\|^{2}_{C_{X, Y}}=\|\phi\|^{2}_{C_{X}}+\|\partial_{t}\phi\|^{2}_{C_{Y}} \quad\forall\phi\in C_{X, Y}. $

对于算子$ F $, 类似于文献[3, 9, 28], 假设$ F: C_{L^{2}(\Omega)}\rightarrow L^{2}(\Omega) $且

(Ⅰ) $ F(0)=0 $;

(Ⅱ) $ \forall\xi, \eta\in C_{L^{2}} $, $ \exists L_{F}>0 $使得

$ \begin{eqnarray*} \|F(\xi)-F(\eta)\|_{2}\leq L_{F}\|\xi-\eta\|_{C_{L^{2}}}; \end{eqnarray*} $

(Ⅲ) $ \exists m_{0}\geq0 $和$ C_{F}>0 $, 使得对$ \forall u, v\in C([\tau-h, t]; L^{2}(\Omega)) $和$ m\in [0, m_{0}] $, 有

$ \begin{eqnarray*} \int_{\tau}^{t}e^{ms}\|F(u_{s})-F(v_{s})\|_{2}^{2}{\rm d}s \leq C^{2}_{F}\int_{\tau-h}^{t}e^{ms}\|u(s)-v(s)\|_{2}^{2}{\rm d}s. \end{eqnarray*} $

对于不含时滞项的弱阻尼波方程, 其解的长时间行为已被许多学者进行研究, 参见文献[29-32]. 当非线性项$ f $满足临界指数(3次)增长时, Arrieta等^[29]和Babin等^[30]利用分解技巧证明了$ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $中全局吸引子的存在性; Ball^[31]利用能量方法, 在不需要对方程做分解的情况下证明了$ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $中全局吸引子的存在性. 当非线性项$ f $满足5次增长时, 基于Strichartz估计在有界域上的延拓(参见文献[33-35]), Kalantarov等^[32]建立了Shatah-Struwe解的全局适定性并证明了$ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $中全局吸引子的存在性.

对于带有时滞项的弱阻尼波方程, 其解的长时间行为也被一些学者进行研究, 参见文献[3, 18, 21, 26]. Caraballo等^[3]和Wang^[18]主要研究线性情形并分别证明了$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中一致向前吸引子和$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $, $ C_{H_{0}^{1}(\Omega)\cap H^{2}(\Omega)}\times C_{H_{0}^{1}(\Omega)} $中拉回吸引子的存在性. 当非线性项满足临界(3次)增长时, Zhu等^[21]建立了弱解的全局适定性并利用收缩函数的思想(参见文献[37-42])证明了$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中拉回吸引子的存在性. 当非线性项的增长次数大于3而小于5时, Zhu等^[36]建立了Shatah-Struwe解的全局适定性并证明了$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中拉回吸引子的存在性.

本文考虑带有时滞项的弱阻尼波方程在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中一致吸引子的存在性, 其中非线性项的增长次数大于3而小于5, 非自治外力项只满足平移有界. 我们通过构造能量泛函并结合收缩函数的思想验证Shatah-Struwe解过程$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中的一致渐近紧性, 从而得到$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中一致吸引子的存在性.

本节首先简要给出非自治动力系统(一致吸引子)的一些相关概念, 参见文献[43-44].

给定$ g_{0}\in L_{b}^{2}({{\Bbb R}} ; V) $, 考虑到空间$ L_{b}^{2}({{\Bbb R}} ; V) $关于时间平移群$ T(s) $, $ s\in{{\Bbb R}} $的不变性, 我们定义$ g_{0} $的符号空间$ {\cal H}(g_{0}) $如下.

定义2.1  设$ g_{0}\in L_{b}^{2}({{\Bbb R}} ; V) $, 定义$ g_{0} $的符号空间$ {\cal H}(g_{0}) $为轨道$ {\cal O}(g_{0}) $在$ L_{{\rm loc}, w}^{2}({{\Bbb R}} ; V) $中的闭包, 即

$ {\cal H}(g_{0}):=[{\cal O}(g_{0})]_{L_{{\rm loc}, w}^{2}({{\Bbb R}} ; V)}, $

其中轨道$ {\cal O}(g_{0}):=\{T(s)g_{0}\mid s\in{{\Bbb R}} \}\subset L_{b}^{2}({{\Bbb R}} ; V) $.

考虑自反Banach空间$ W $上的Cauchy问题

(2.1) $ \begin{equation} \partial_{t}u=A(u)+g_{0}(t), \quad u|_{t=\tau}=u_{\tau}, \end{equation} $

其中$ A(u) $是非线性(无界)算子, 外力项$ g_{0}\in L_{b}^{2}({{\Bbb R}} ; V) $. 假设问题(2.1)在$ W $中是全局适定的, 则我们可得到一族由$ {\cal H}(g_{0}) $参数化的过程$ U_{g}(t, \tau):W\rightarrow W, \, g\in{\cal H}(g_{0}) $, 并且$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $满足平移等式

$ U_{T(s)g}(t, \tau)=U_{g}(t+s, \tau+s), \quad t\geq\tau, \tau, s\in{{\Bbb R}} . $

为了研究一致吸引子的存在性及其结构, Chepyzhov等^[43]构造出扩展相空间$ \Phi:=W\times{\cal H}(g_{0}) $上的斜积流

$ {\Bbb S}(t)(u_{0}, g):=(U_{g}(t, 0)u_{0}, T(t)g), \quad (u_{0}, g)\in\Phi, t\geq0, $

从而$ \{{\Bbb S}(t)\}_{t\geq0} $也构成扩展相空间$ \Phi $上的一个(扩展)算子半群.

定义2.2  如果一个集合$ {\Bbb A}\subset\Phi $满足下述条件:

$ (1) $$ {\Bbb A} $是由嵌入$ \Phi\subset W_{w}\times L_{{\rm loc}, w}^{2}({{\Bbb R}} , V) $$ (\Phi\subset W\times L_{{\rm loc}, w}^{2}({{\Bbb R}} , V)) $诱导的弱(强)拓扑意义下$ \Phi $的一个紧集. 接下来, 我们记这个拓扑为$ \Phi_{w} $$ (\Phi_{s}) $;

$ (2) $$ {\Bbb A} $满足严格不变性: $ {\Bbb S}(t){\Bbb A}={\Bbb A} $;

$ (3) $$ {\Bbb A} $在弱(强)拓扑$ \Phi_{w} $$ (\Phi_{s}) $意义下吸引有界集$ \Phi $的像, 即在弱(强)拓扑$ \Phi_{w} $$ (\Phi_{s}) $意义下对$ \Phi $中的每个有界集$ {\Bbb B} $和集合$ {\Bbb A} $的每个邻域$ {\Bbb O(\Bbb A)} $, 存在时间$ T=T({\Bbb B, \Bbb O}) $使得

$ {\Bbb S}(t){\Bbb B}\subset{\Bbb O(\Bbb A)}, \quad t\geq T, $

则称$ {\Bbb A} $是$ {\Bbb S}(t) $的弱(强)全局吸引子.

$ {\Bbb A} $在$ W $上的投影称为过程族$ U_{g}(t, \tau):W\rightarrow W $, $ t\geq\tau $, $ g\in{\cal H}(g_{0}) $的弱(强)一致吸引子, 即

$ {\cal A}_{w}:=\Pi_{1}{\Bbb A} ({\cal A}_{s}:=\Pi_{1}{\Bbb A}). $

定义2.3  如果对任意给定的$ t>\tau $, 由$ W $中的弱收敛$ u_{\tau}^{n}\rightarrow u_{\tau} $和$ {\cal H}(g_{0}) $中的弱收敛$ g_{n}\rightarrow g $可得到$ W $中的弱收敛$ U_{g_{n}}(t, \tau)u_{\tau}^{n}\rightarrow U_{g}(t, \tau)u_{\tau} $, 则称过程族$ U_{g}(t, \tau):W\rightarrow W $, $ g\in{\cal H}(g_{0}) $是$ (W\times{\cal H}(g_{0}), W) $ -弱连续.

定理2.1  设过程族$ U_{g}(t, \tau):W\rightarrow W $, $ g\in{\cal H}(g_{0}) $一致耗散并且弱连续, 则半群$ {\Bbb S}(t):\Phi\rightarrow \Phi $存在弱一致全局吸引子$ {\Bbb A} $和相应的弱一致吸引子

(2.2) $ \begin{eqnarray} {\cal A}_{w}:=\Pi_{1}{\Bbb A}=\bigcup\limits_{g\in{\cal H}(g_{0})}{\cal K}_{g}|_{t=0}, \end{eqnarray} $

其中$ {\cal K}_{g}\in L^{\infty}({{\Bbb R}} , V) $是由所有有界解轨道$ u(t) $组成的集合.

若过程族$ U_{g}(t, \tau):W\rightarrow W $, $ g\in{\cal H}(g_{0}) $一致渐近紧, 则弱一致吸引子$ {\cal A}_{w} $也是强一致吸引子$ {\cal A}_{s} $.

定义2.4  设$ V $是自反的Banach空间, $ 1<q<\infty $. 若对任给的$ \varepsilon>0 $, 存在子空间$ V_{\varepsilon}\subset V $$ (\dim V_{\varepsilon}<\infty) $和函数$ g_{\varepsilon}\in L^{q}_{b}({{\Bbb R}} ; V_{\varepsilon}) $使得$ \|g-g_{\varepsilon}\|_{L^{q}_{b}({{\Bbb R}} ; V)}\leq\varepsilon $, 则称函数$ g\in L^{q}_{b}({{\Bbb R}} ; V) $具有空间正则性. 类似地, 若对任给的$ \varepsilon>0 $和任意的$ k>0 $, 存在函数$ g_{\varepsilon}\in H^{k}_{b}({{\Bbb R}} ; V) $, 使得$ \|g-g_{\varepsilon}\|_{L^{q}_{b}({{\Bbb R}} ; V)}\leq\varepsilon $, 则称函数$ g\in L^{q}_{b}({{\Bbb R}} ; V) $具有时间正则性.

为了方便起见, 记$ \xi_{u}:=(u, \partial_{t}u) $, $ {\cal E}:=H_{0}^{1}(\Omega)\times L^{2}(\Omega) $, $ {\cal E}^{-1}:=L^{2}(\Omega)\times H^{-1}(\Omega) $, 则

$ \|\xi_{u}\|_{{\cal E}}^{2}:=\|\nabla u\|_{2}^{2}+\|\partial_{t}u\|_{2}^{2}, {\quad} \|\xi_{u}\|_{{\cal E}^{-1}}^{2}:=\|u\|_{2}^{2}+\|\partial_{t}u\|_{H^{-1}}^{2}. $

下面给出弱(能量)解和Shatah-Struwe解的相关定义和结果^{[32, 45, 46]}.

定义2.5  若$ \xi_{u}(t):=(u(t), \partial_{t}u(t))\in L^{\infty}(\tau, T;{\cal E}) $在分布意义下满足方程(1.1), 即

$ \begin{eqnarray*} &&-\int_{\tau}^{T}(\partial_{t}u, \partial_{t}\nu){\rm d}t-\gamma\int_{\tau}^{T}(u, \partial_{t}\nu){\rm d}t +\int_{\tau}^{T}(\nabla u, \nabla\nu){\rm d}t\\ &=&\gamma(u_{\tau}, \nu)+(u'_{\tau}, \nu)+\int_{\tau}^{T}(f(u), \nu){\rm d}t+\int_{\tau}^{T}(F(u_{t}), \nu){\rm d}t +\int_{\tau}^{T}(g(t), \nu){\rm d}t, \quad\forall T>\tau, \end{eqnarray*} $

其中初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in L^{\infty}(\tau, T;{\cal E}) $, 函数$ \nu $属于

$ W_{T}=\{\nu| \nu(x, t)=\varphi(x)\psi(t), \varphi\in C_{0}^{\infty}(\Omega), \psi\in C^{\infty}([\tau, T]), \psi(T)=0\}, $

则称函数$ u(t) $是方程(1.1)的弱(能量)解. 这里$ (\cdot, \cdot) $表示$ L^{2}(\Omega) $中的内积.

定义2.6  若方程(1.1)的弱解$ u(t) $, $ t\in[\tau, T] $满足下述正则性:

$ u\in L^{4}(\tau, T;L^{12}(\Omega)), \quad\forall T>\tau, $

则$ u(t) $, $ t\in[\tau, T] $是方程(1.1)的Shatah-Struwe解.

下面我们给出下述线性波方程的Strichartz估计^[33-35]

(2.3) $ \begin{equation} \left\{\begin{array}{ll} \partial_{tt}\upsilon+\gamma\partial_{t}\upsilon-\Delta\upsilon=G(t), {\quad} &(x, t)\in \Omega\times(\tau, \infty), \\ \upsilon(x, \, t)=0, &(x, t)\in \partial\Omega\times(\tau, \infty), \\ \xi_{\upsilon}(t)|_{t=\tau}=\xi_{\tau}, &x\in\Omega, \end{array}\right. \end{equation} $

其中$ \Omega $是$ {{\Bbb R}} ^{3} $中的有界域, 初值$ \xi_{\upsilon}(\tau)=\xi_{\tau}:=(\upsilon_{\tau}, \upsilon'_{\tau})\in{\cal E} $.

引理2.1  设$ \xi_{\tau}\in{\cal E} $, $ G\in L^{1}(\tau, T;L^{2}(\Omega)) $, $ \upsilon(t) $是方程(2.3)的解且$ \xi_{\upsilon}\in C(\tau, T;{\cal E}) $, 则有下述估计

(2.4) $ \begin{eqnarray} \|\xi_{\upsilon}(t)\|_{{\cal E}}\leq C\left(\|\xi_{\tau}\|_{{\cal E}}e^{-\beta(t-\tau)} +\int_{\tau}^{t}e^{-\beta(t-s)}\|G(s)\|_{2}{\rm d}s\right), \end{eqnarray} $

其中的正常数$ \beta $, $ C $依赖于常数$ \gamma>0 $, 但是不依赖于$ t $, $ \tau $, $ \xi_{\tau} $和$ G $.

注2.1  根据能量估计(2.4)和插值不等式

(2.5) $ \begin{eqnarray} \|\upsilon\|_{L^{5}(\tau, T;L^{10}(\Omega))}\leq C\|\upsilon\|_{L^{4}(\tau, T;L^{12}(\Omega))}^{\frac{4}{5}} \|\upsilon\|^{\frac{1}{5}}_{L^{\infty}(\tau, T;H_{0}^{1}(\Omega))}, \end{eqnarray} $

可得到解$ \upsilon $的$ L^{5}(\tau, T;L^{10}(\Omega)) $ -模的有界性.

下述定义和定理将用来证明一致吸引子的存在性^[32-36].

定义2.7  设$ (X, \|\cdot\|_{X}) $是Banach空间, $ B $是$ X $的有界子集, $ \psi(\cdot, \cdot;\cdot, \cdot) $是定义在$ (X\times X)\times({\cal H}(g_{0})\times{\cal H}(g_{0})) $上的函数. 若对任给的序列$ \{x_{n}\}_{n=1}^{\infty}\subset B $和任意的$ \{g_{n}\}_{n=1}^{\infty}\subset{\cal H}(g_{0}) $, 存在子列$ \{x_{n_{k}}\}_{k=1}^{\infty}\subset\{x_{n}\}_{n=1}^{\infty} $和$ \{g_{n_{k}}\}_{k=1}^{\infty}\subset\{g_{n}\}_{n=1}^{\infty} $使得

$ \begin{eqnarray*} \lim\limits_{k\rightarrow \infty}\lim\limits_{l\rightarrow \infty}\psi(x_{n_{k}}, x_{n_{l}};g_{n_{k}}, g_{n_{l}})=0, \end{eqnarray*} $

则称$ \psi(\cdot, \cdot;\cdot, \cdot) $是$ B\times B $上的收缩函数.

$ B\times B $上的收缩函数组成的集合记为$ Contr(B, {\cal H}(g_{0})) $.

引理2.2  设$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $是Banach空间$ X $中的过程族且存在有界一致吸收集$ B_{0}\subset X $. 若对任给的$ \varepsilon>0 $, 存在$ T_{1}=T_{1}(B_{0}, \varepsilon)=t-\tau $和$ \psi_{T_{1}, \tau}(\cdot, \cdot)\in Contr(B_{0}, {\cal H}(g_{0})) $使得

$ \|U_{g_{1}}(T_{1}+\tau, \tau)x-U_{g_{2}}(T_{1}+\tau, \tau)y\|_{X} \leq\varepsilon+\psi_{T_{1}, \tau}(x, y;g_{1}, g_{2}), \forall x, y\in B_{0}, g_{1}, g_{2}\in{\cal H}(g_{0}), $

则$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ X $中一致渐近紧.

本节我们将证明方程(1.1)在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中弱一致吸引子和强一致吸引子的存在性. 我们首先给出方程(1.1)解的存在性和唯一性, 这可以由Faedo-Galerkin逼近得到^[36].

引理3.1  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{\rm loc}^{2}({{\Bbb R}} ;L^{2}(\Omega)) $, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则对任意的$ \tau\in {{\Bbb R}} $, 存在$ T_{0}=T_{0}(\|\xi_{\tau}\|_{C_{{\cal E}}})\geq \tau $使得方程(1.1)在$ [\tau, T_{0}] $上存在一个Shatah-Struwe解$ u(t) $且满足下述估计

$ \|\xi_{u_{t}}\|_{C_{{\cal E}}}+\|u(t)\|_{L^{4}(\tau, t;L^{12}(\Omega))} \leq Q_{1}(\|\xi_{\tau}\|_{C_{{\cal E}}})+Q_{1}\bigg(\int_{\tau}^{t}\|g(s)\|_{2}^{2}{\rm d}s\bigg), \quad t\in[\tau, T_{0}], $

其中$ Q_{1}(\cdot) $是$ [0, +\infty) $上不依赖于$ T_{0} $和$ u(t) $的单调函数.

引理3.2  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{\rm loc}^{2}({{\Bbb R}} ;L^{2}(\Omega)) $, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则方程(1.1)在$ [\tau, T_{0}] $上的Shatah-Struwe解$ u(t) $具有唯一性. 而且, 对于两个初值不同的解$ u^{1}(t) $和$ u^{2}(t) $有下述李普希茨连续

$ \|\nabla w_{t}\|_{C_{L^{2}(\Omega)}}^{2}+\|w'_{t}\|_{C_{L^{2}(\Omega)}}^{2} \leq C'_{1}(\|\nabla w_{\tau}\|_{C_{L^{2}(\Omega)}}^{2} +\|w'_{\tau}\|_{C_{L^{2}(\Omega)}}^{2})e^{C'_{2}(t-\tau)}, \quad\forall t\geq\tau, $

其中$ w(t)=u^{1}(t)-u^{2}(t) $, 常数

$ C'_{1}=C'_{1}(|\Omega|, \int_{\tau}^{t}(\|u^{1}(s)\|_{L^{12}}^{4}+\|u^{2}(s)\|_{L^{12}}^{4}){\rm d}s), {\quad} C'_{2}=C'_{2}(\lambda_{1}, \gamma, L_{g}, |\Omega|). $

下述定理给出方程(1.1) Shatah-Struwe解$ u(t) $的全局存在性.

引理3.3  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{b}^{2}({{\Bbb R}} ;L^{2}(\Omega)) $, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则方程(1.1)存在一个全局Shatah-Struwe解$ u(t) $, 而且有下述估计

$ \begin{eqnarray*} \|\xi_{u_{t}}\|_{C_{{\cal E}}}+\|u(t)\|_{L^{4}(\max(\tau, t-1), t;L^{12}(\Omega))} \leq Q_{2}(\|\xi_{\tau}\|_{C_{{\cal E}}})+Q_{2}(\|g\|_{L_{b}^{2}({{\Bbb R}} ;L^{2}(\Omega))}^{2}), \end{eqnarray*} $

其中$ Q_{2}(\cdot) $是$ [0, +\infty) $上不依赖于$ t $和$ u(t) $的单调函数.

因而, 我们定义$ C_{{\cal E}} $中的过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $为

(3.1) $ \begin{eqnarray} U_{g}(t, \tau)(\phi, \partial_{t}\phi)=(u_{t}, u'_{t}), \quad\forall t\geq\tau, \end{eqnarray} $

而且$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{{\cal E}} $中满足李普希茨连续.

接下来, 用$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $表示由(3.1)式定义的过程族.

本节我们将证明方程(1.1)在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中弱一致吸引子的存在性. 下述引理表明过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中的一致耗散性.

引理3.4  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{b}^{2}({{\Bbb R}} ;L^{2}(\Omega)) $, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则方程(1.1)的解$ \xi_{u}(t)=(u(t), \partial_{t}u(t)) $满足下述估计

(3.2) $ \begin{eqnarray} \|u_{t}\|_{C_{H_{0}^{1}(\Omega)}}^{2}+\|u'_{t}\|_{C_{L^{2}(\Omega)}}^{2} &\leq&\varrho_{0}^{2}e^{-\alpha(t-h-\tau)}+C_{1}e^{-\alpha(t-h-\tau)}+\frac{C_{4}}{\alpha} \\ & &+2C'_{\delta}|\Omega|+\frac{C_{3}e^{\alpha h}}{1-e^{-\alpha}} \sup\limits_{t\in{{\Bbb R}} }\int_{t}^{t+1}\|g(s)\|_{2}^{2}{\rm d}s, \quad\forall t-h\geq\tau, \end{eqnarray} $

其中$ C_{1}=\frac{2}{\beta}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C^{2}_{F}\|\phi\|_{2}^{2} $, $ C_{2}=\frac{2}{\lambda_{1}}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C'^{2}_{F} $, $ C_{3}=2(\frac{1}{\gamma}+\frac{\epsilon}{\delta}) $, $ C_{4}=\frac{4}{\lambda_{1}}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C'^{2}_{F}C'_{\delta}|\Omega| +2\epsilon C_{\delta}|\Omega| $, $ \alpha=\beta-C_{2}>0 $, $ \beta=\min\{\frac{(2\lambda_{1}-3\delta)\epsilon}{\lambda_{1}+(\gamma+1)\epsilon}, \frac{\gamma-2\epsilon}{1+\epsilon}, \epsilon\mu\} $, $ \varrho_{0}>0 $是一个与$ c_{1} $和$ \|\xi_{\tau}\|^{2}_{C_{{\cal E}}} $相关的常数.

证  让方程(1.1)与$ u $在$ \Omega $上做$ L^{2} $-内积, 可得

$ \begin{eqnarray*} \frac{\rm d}{{\rm d}t}(\partial_{t}u, u)+\frac{\gamma}{2}\frac{\rm d}{{\rm d}t}\|u\|_{2}^{2} +\|\nabla u\|_{2}^{2}-\|\partial_{t}u\|_{2}^{2} =(f(u), u)+(F(u_{t}), u)+(g(t), u). \end{eqnarray*} $

利用Hölder不等式和Young不等式可得

(3.3) $ \begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\left(\gamma\|u\|_{2}^{2}+2(\partial_{t}u, u)\right) +\|\nabla u\|_{2}^{2}-\|\partial_{t}u\|_{2}^{2} \\ &\leq&(f(u), u)+\frac{1}{\delta}\|F(u_{t})\|_{2}^{2} +\frac{1}{\delta}\|g(t)\|_{2}^{2}+\frac{\delta}{2}\|u\|_{2}^{2}, \end{eqnarray} $

这里的常数$ \delta $满足: $ 0<\delta\ll1 $.

再在(3.3)式两边同乘以2$ \epsilon(>0) $, 可得

(3.4) $ \begin{eqnarray} &&\frac{\rm d}{{\rm d}t}\left(\gamma\epsilon\|u\|_{2}^{2}+2\epsilon(\partial_{t}u, u)\right) +(2-\frac{\delta}{\lambda_{1}})\epsilon\|\nabla u\|_{2}^{2}-2\epsilon\|\partial_{t}u\|_{2}^{2} \\ &\leq& 2\epsilon(f(u), u)+\frac{2\epsilon}{\delta}\|F(u_{t})\|_{2}^{2} +\frac{2\epsilon}{\delta}\|g(t)\|_{2}^{2}. \end{eqnarray} $

另一方面, 让方程(1.1)与$ \partial_{t}u $在$ \Omega $上做$ L^{2} $ -内积, 可得

$ \frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\nabla u\|^{2}_{2}+\|\partial_{t}u\|^{2}_{2}) +\frac{\gamma}{2}\|\partial_{t}u\|^{2}_{2}\leq \frac{\rm d}{{\rm d}t}({\cal F}(u), 1) +\frac{1}{\gamma}\|F(u_{t})\|_{2}^{2}+\frac{1}{\gamma}\|g(t)\|_{2}^{2}. $

从而

(3.5) $ \begin{eqnarray} \frac{\rm d}{{\rm d}t}(\|\nabla u\|^{2}_{2}+\|\partial_{t}u\|^{2}_{2}-2({\cal F}(u), 1)) +\gamma\|\partial_{t}u\|^{2}_{2}\leq \frac{2}{\gamma}\|F(u_{t})\|_{2}^{2}+\frac{2}{\gamma}\|g(t)\|_{2}^{2}. \end{eqnarray} $

结合(3.4)和(3.5)式, 可得

(3.6) $ \begin{eqnarray} &&\frac{\rm d}{{\rm d}t}\left(\|\nabla u\|^{2}_{2}+\|\partial_{t}u\|^{2}_{2}-2({\cal F}(u), 1) +\gamma\epsilon\|u\|^{2}_{2}+2\epsilon(\partial_{t}u, u)\right) \\ &&+(2-\frac{\delta}{\lambda_{1}})\epsilon \|\nabla u\|_{2}^{2}+(\gamma-2\epsilon)\|\partial_{t}u\|_{2}^{2} \\ &\leq&2\epsilon(f(u), u)+2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|F(u_{t})\|_{2}^{2} +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|g(t)\|_{2}^{2}. \end{eqnarray} $

由(1.5)式可知存在$ \delta>0 $ (与(3.3)式中相同)和$ C_{\delta}>0 $使得

(3.7) $ \begin{eqnarray} f(u)u-\mu{\cal F}(u)\leq\delta u^{2}+C_{\delta}, \quad\forall u\in{{\Bbb R}} . \end{eqnarray} $

再结合(3.6)式和(3.7)式, 可得

(3.8) $ \begin{eqnarray} &&\frac{\rm d}{{\rm d}t}\left(\|\nabla u\|^{2}_{2}+\|\partial_{t}u\|^{2}_{2}-2({\cal F}(u), 1) +\gamma\epsilon\|u\|^{2}_{2}+2\epsilon(\partial_{t}u, u)\right) +(2-\frac{3\delta}{\lambda_{1}})\epsilon\|\nabla u\|_{2}^{2} \\ &&+(\gamma-2\epsilon)\|\partial_{t}u\|_{2}^{2}-2\epsilon\mu({\cal F}(u), 1) \\ &\leq&2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|F(u_{t})\|_{2}^{2} +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|g(t)\|_{2}^{2} +2\epsilon C_{\delta}|\Omega|. \end{eqnarray} $

记$ \xi_{u}(t)=(u(t), \partial_{t}u(t)) $和

$ E(\xi_{u}(t)):=\|\nabla u\|^{2}_{2}+\|\partial_{t}u\|^{2}_{2} -2({\cal F}(u), 1)+\gamma\epsilon\|u\|^{2}_{2}+2\epsilon(\partial_{t}u, u). $

由(1.4)式可知存在$ \delta>0 $ (与(3.3)式中相同)和$ C'_{\delta}>0 $使得

$ {\cal F}(u)\leq\delta u^{2}+C'_{\delta}, \quad\forall u\in{{\Bbb R}} , $

从而存在常数$ c_{1}>0 $使得

(3.9) $ \begin{eqnarray} c_{1}(\|\nabla u\|^{2}_{2}+\|\partial_{t}u\|^{2}_{2})-2C'_{\delta}|\Omega|\leq E(\xi_{u}(t)) \end{eqnarray} $

以及

(3.10) $ \begin{eqnarray} E(\xi_{u}(t))\leq\left(1+\frac{(\gamma+1)\epsilon}{\lambda_{1}}\right) \|\nabla u\|^{2}_{2}+(1+\epsilon)\|\partial_{t}u\|^{2}_{2}-2({\cal F}(u), 1). \end{eqnarray} $

由(3.8)式得

(3.11) $ \begin{eqnarray} &&\frac{\rm d}{{\rm d}t}E(\xi_{u}(t)) +(2-\frac{3\delta}{\lambda_{1}})\epsilon\|\nabla u\|_{2}^{2} +(\gamma-2\epsilon)\|\partial_{t}u\|_{2}^{2}-2\epsilon\mu({\cal F}(u), 1) \\ &\leq&2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|F(u_{t})\|_{2}^{2} +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|g(t)\|_{2}^{2} +2\epsilon C_{\delta}|\Omega|. \end{eqnarray} $

让$ \gamma-2\epsilon>0 $并设$ \beta=\min\{\frac{(2\lambda_{1}-3\delta)\epsilon}{\lambda_{1}+(\gamma+1)\epsilon}, \frac{\gamma-2\epsilon}{1+\epsilon}, \epsilon\mu\} $, 由(3.10)式和(3.11)式可得

(3.12) $ \begin{eqnarray} \frac{\rm d}{{\rm d}t}E(\xi_{u}(t))+\beta E(\xi_{u}(t)) \leq2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|F(u_{t})\|_{2}^{2} +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|g(t)\|_{2}^{2}+2\epsilon C_{\delta}|\Omega|. \end{eqnarray} $

在(3.12)式两边乘以$ e^{\beta t} $, 可得

(3.13) $ \begin{eqnarray} \frac{\rm d}{{\rm d}t}(E(\xi_{u}(t))e^{\beta t}) \leq2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|F(u_{t})\|_{2}^{2}e^{\beta t} +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\|g(t)\|_{2}^{2}e^{\beta t}+2\epsilon C_{\delta}|\Omega|e^{\beta t}. \end{eqnarray} $

对(3.13)式关于时间$ t $在$ [\tau, t] $上积分, 可得

$ \begin{eqnarray*} E(\xi_{u}(t))e^{\beta t}&\leq&E(\xi_{u}(\tau))e^{\beta\tau} +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\int_{\tau}^{t}\|F(u_{s})\|_{2}^{2}e^{\beta s}{\rm d}s \nonumber\\ &&+2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\int_{\tau}^{t}\|g(s)\|_{2}^{2}e^{\beta s}{\rm d}s +2\epsilon C_{\delta}|\Omega|\int_{\tau}^{t}e^{\beta s}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray*} $

由假设(Ⅲ)可得

$ \begin{eqnarray*} E(\xi_{u}(t))e^{\beta t}&\leq&E(\xi_{u}(\tau))e^{\beta\tau} +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C^{2}_{F}\int_{\tau-h}^{t}\|u(s)\|_{2}^{2}e^{\beta s}{\rm d}s \nonumber\\ && +2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\int_{\tau}^{t}\|g(s)\|_{2}^{2}e^{\beta s}{\rm d}s +2\epsilon C_{\delta}|\Omega|\int_{\tau}^{t}e^{\beta s}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray*} $

再由Poincaré不等式和(3.9)式可得

(3.14) $ \begin{eqnarray} &&E(\xi_{u}(t))e^{\beta t} \\ &\leq& E(\xi_{u}(\tau))e^{\beta\tau} +\frac{2}{\beta}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C^{2}_{F}\|\phi\|_{2}^{2}e^{\beta\tau} +\frac{2}{\lambda_{1}}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C^{2}_{F} \int_{\tau}^{t}\|\nabla u(s)\|_{2}^{2}e^{\beta s}{\rm d}s \\ &&+2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\int_{\tau}^{t}\|g(s)\|_{2}^{2}e^{\beta s}{\rm d}s +2\epsilon C_{\delta}|\Omega|\int_{\tau}^{t}e^{\beta s}{\rm d}s \\ &\leq& E(\xi_{u}(\tau))e^{\beta\tau} +\frac{2}{\beta}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C^{2}_{F}\|\phi\|_{2}^{2}e^{\beta\tau} +\frac{2}{\lambda_{1}}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C'^{2}_{F} \int_{\tau}^{t}(E(\xi_{u}(s))+2C'_{\delta}|\Omega|)e^{\beta s}{\rm d}s \\ &&+2(\frac{1}{\gamma}+\frac{\epsilon}{\delta})\int_{\tau}^{t}\|g(s)\|_{2}^{2}e^{\beta s}{\rm d}s +2\epsilon C_{\delta}|\Omega|\int_{\tau}^{t}e^{\beta s}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray} $

为了简单起见, 记$ C_{1}=\frac{2}{\beta}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C^{2}_{F}\|\phi\|_{2}^{2} $, $ C_{2}=\frac{2}{\lambda_{1}}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C'^{2}_{F} $, $ C_{3}=2(\frac{1}{\gamma}+\frac{\epsilon}{\delta}) $, $ C_{4}=\frac{4}{\lambda_{1}}(\frac{1}{\gamma}+\frac{\epsilon}{\delta})C'^{2}_{F}C'_{\delta}|\Omega| +2\epsilon C_{\delta}|\Omega| $. 再对(3.14)式应用Gronwall引理, 可得

$ \begin{eqnarray*} E(\xi_{u}(t))e^{\beta t}&\leq& E(\xi_{u}(\tau))e^{\beta\tau}e^{C_{2}(t-\tau)} +C_{1}e^{\beta\tau}e^{C_{2}(t-\tau)} \nonumber\\ &&+C_{3}e^{C_{2}t}\int_{\tau}^{t}e^{(\beta-C_{2})s}\|g(s)\|_{2}^{2}{\rm d}s +C_{4}e^{C_{2}t}\int_{\tau}^{t}e^{(\beta-C_{2})s}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray*} $

从而

$ \begin{eqnarray*} E(\xi_{u}(t))&\leq& E(\xi_{u}(\tau))e^{-(\beta-C_{2})(t-\tau)} +C_{1}e^{-(\beta-C_{2})(t-\tau)} \nonumber\\ &&+\frac{C_{4}}{\beta-C_{2}}+C_{3}e^{-(\beta-C_{2})t}\int_{\tau}^{t}e^{(\beta-C_{2})s}\|g(s)\|_{2}^{2}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray*} $

再由(3.9)式并设$ \alpha=\beta-C_{2}>0 $, 然后用$ t+\theta $代替$ t $, 可得

(3.15) $ \begin{eqnarray} \|\xi_{u_{t}}\|^{2}_{C_{V, H}}&=&\|u_{t}\|^{2}_{C_{V}}+\|u'_{t}\|^{2}_{C_{H}} \\ &\leq&\varrho_{0}^{2}e^{-\alpha(t-h-\tau)}+C_{1}e^{-\alpha(t-h-\tau)} +\frac{C_{4}}{\alpha}+2C'_{\delta}|\Omega| \\ &&+C_{3}e^{-\alpha(t-h)}\int_{\tau}^{t}e^{\alpha s}\|g(s)\|_{2}^{2}{\rm d}s, \quad\forall t-h\geq\tau, \end{eqnarray} $

其中$ \varrho_{0}>0 $是依赖于$ c_{1} $和$ \|\xi_{\tau}\|^{2}_{C_{{\cal E}}} $的常数.

而且

(3.16) $ \begin{eqnarray} &&e^{-\alpha t}\int_{\tau}^{t}e^{\alpha s}\|g(s)\|_{2}^{2}{\rm d}s \\ &=&e^{-\alpha t}\left(\int_{t-1}^{t}e^{\alpha s}\|g(s)\|_{2}^{2}{\rm d}s +\int_{t-2}^{t-1}e^{\alpha s}\|g(s)\|_{2}^{2}{\rm d}s+\cdot\cdot\cdot\right) \\ &=&e^{-\alpha t}\left(\int_{0}^{1}e^{\alpha(s+t-1)}\|g(s+t-1)\|_{2}^{2}{\rm d}s +\int_{0}^{1}e^{\alpha(s+t-2)}\|g(s+t-2)\|_{2}^{2}{\rm d}s+\cdot\cdot\cdot\right) \\ &\leq&(e^{-\alpha}+e^{-2\alpha}+\cdot\cdot\cdot) \sup\limits_{t\in{{\Bbb R}} }\int_{0}^{1}e^{\alpha s}\|g(s+t)\|_{2}^{2}{\rm d}s \leq\frac{e^{-\alpha}}{1-e^{-\alpha}} \sup\limits_{t\in{{\Bbb R}} }\int_{0}^{1}e^{\alpha s}\|g(s+t)\|_{2}^{2}{\rm d}s \\ &=&\frac{1}{1-e^{-\alpha}}\sup\limits_{t\in{{\Bbb R}} }\int_{0}^{1}\|g(s+t)\|_{2}^{2}{\rm d}s =\frac{1}{1-e^{-\alpha}}\sup\limits_{t\in{{\Bbb R}} }\int_{t}^{t+1}\|g(s)\|_{2}^{2}{\rm d}s. \end{eqnarray} $

结合(3.15)式和(3.16)式, 可得

$ \begin{eqnarray*} \|u_{t}\|^{2}_{C_{H_{0}^{1}(\Omega)}}+\|u'_{t}\|^{2}_{C_{L^{2}(\Omega)}} &\leq&\varrho_{0}^{2}e^{-\alpha(t-h-\tau)}+C_{1}e^{-\alpha(t-h-\tau)} +\frac{C_{4}}{\alpha}+2C'_{\delta}|\Omega| \nonumber\\ &&+\frac{C_{3}e^{\alpha h}}{1-e^{-\alpha}}\sup\limits_{t\in{{\Bbb R}} }\int_{t}^{t+1}\|g(s)\|_{2}^{2}{\rm d}s, \quad\forall t-h\geq\tau, \end{eqnarray*} $

其中$ \varrho_{0}>0 $是一个与$ c_{1} $和$ \|\xi_{\tau}\|^{2}_{C_{{\cal E}}} $相关的常数.

作为引理3.4的一个直接结果, 下述的推论表明过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{{\cal E}} $中有界一致吸收集的存在性.

推论3.1  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{b}^{2}({{\Bbb R}} ;L^{2}(\Omega)) $, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{{\cal E}} $中存在有界一致吸收集, 即, 存在常数$ \rho_{0}>0 $使得对$ C_{{\cal E}} $中的任意有界子集$ B_{0} $, 存在$ T_{B_{0}}=T(\|B_{0}\|_{C_{{\cal E}}}) $使得

(3.17) $ \begin{eqnarray} \|u_{t}\|_{C_{H_{0}^{1}(\Omega)}}^{2}+\|u'_{t}\|_{C_{L^{2}(\Omega)}}^{2}\leq\rho_{0}^{2}, \quad\forall t-h\geq T_{B_{0}}, \end{eqnarray} $

其中$ \rho_{0}^{2}=1+\frac{C_{4}}{\alpha}+2C'_{\delta}|\Omega| +\frac{C_{3}e^{\alpha h}}{1-e^{-\alpha}}\sup\limits_{t\in{{\Bbb R}} }\int_{t}^{t+1}\|g(s)\|_{2}^{2}{\rm d}s $, $ T_{B_{0}}=\tau+\frac{1}{\alpha}\ln(\varrho_{0}^{2}+C_{1}) $, $ \varrho_{0}>0 $是一个与$ c_{1} $和$ \|\xi_{\tau}\|^{2}_{C_{{\cal E}}} $相关的常数.

下述定理表面过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $的弱连续性.

定理3.1  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{b}^{2}({{\Bbb R}} , L^{2}(\Omega)) $, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则对任意$ t>\tau $, 过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $是$ ({\cal E}\times{\cal H}(g_{0}), {\cal E}) $-弱连续.

证  对任给的$ t>\tau $, $ \tau\in{{\Bbb R}} $, 设$ (\xi_{u_{\tau}^{n}}, g_{n}) $在$ {\cal E}\times{\cal H}(g_{0}) $中弱收敛到$ (\xi_{u_{\tau}}, g) $. 对任给的$ s\in[\tau, t] $, 记$ \xi_{u^{n}}(s)=U_{g_{n}}(s, \tau)\xi_{u_{\tau}^{n}} $, $ \xi_{u}(s)=U_{g}(s, \tau)\xi_{u_{\tau}} $. 由引理3.3和引理3.4可知

$ \{\xi_{u^{n}}(s)\}_{n=1}^{\infty}\ \mbox{在}\ L^{\infty}(\tau, t;{\cal E}) \ \mbox{中有界}, \quad \{u^{n}(s)\}_{n=1}^{\infty}\ \mbox{在}\ L^{4}(\tau, t;L^{12}(\Omega)) \ \mbox{中有界}. $

再由$ \partial_{tt}u^{n}=f(u^{n})+F(u_{t}^{n})+g_{n}(t)-\gamma\partial_{t}u^{n}+\Delta u^{n} $, 可知$ \{\partial_{tt}u^{n}\}_{n=1}^{\infty} $在$ L^{2}(\tau, t;H^{-1}(\Omega)) $中有界.

由Aubin-Lions引理, 通过对$ \{u^{n}\}_{n=1}^{\infty} $的子列(仍记为$ \{u^{n}\}_{n=1}^{\infty} $)所满足的方程(1.1)两边取极限, 可得$ \tilde{u} $是方程(1.1)的对应于初值$ \tilde{u}(\tau)=\phi $的Shatah-Struwe解. 由Shatah-Struwe解的唯一性可知$ \xi_{\tilde{u}}(s)=\xi_{u}(s) $, $ s\in[\tau, t] $, 进而运用反证法可得到全序列$ \{\xi_{u^{n}}\}_{n=1}^{\infty} $在$ L^{2}(\tau, t;{\cal E}) $中弱收敛到$ \xi_{u} $, 全序列$ \{\xi_{u^{n}}\}_{n=1}^{\infty} $在$ L^{2}(\tau, t;{\cal E}^{-1}) $中强收敛到$ \xi_{u} $.

进一步, 对任给的$ \varphi\in{\cal E} $, 全序列$ \{\xi_{u^{n}}(s)\}_{n=1}^{\infty} $在$ {\cal E}^{-1} $中强收敛到$ \xi_{u}(s) $以及$ (\xi_{u^{n}}(s), \varphi)\rightarrow (\xi_{u}(s), \varphi) $, $ s\in[\tau, t] $几乎处处成立.

由引理3.2和引理3.4可知序列$ \{(\xi_{u^{n}}(s), \varphi)\}_{n=1}^{\infty} $满足一致有界且等度连续. 由Arzela–Ascoli定理可知序列$ \{(\xi_{u^{n}}(s), \varphi)\}_{n=1}^{\infty}\in C[\tau, t] $准紧, 从而有

(3.18) $ \begin{eqnarray} (\xi_{u^{n}}(s), \varphi)\rightarrow (\xi_{u}(s), \varphi), \quad\forall s\in[\tau, t], \varphi\in{\cal E}. \end{eqnarray} $

结合考虑方程(1.1)和$ {\cal E} $在它的共轭空间$ H^{-1}(\Omega)\times L^{2}(\Omega) $中的稠密性, 由(3.18)式可知$ \xi_{u^{n}}(s) $在$ {\cal E} $中弱收敛到$ \xi_{u}(s) $.

结合过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $的弱连续性和一致耗散性, 可得下述结果:

定理3.2  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{b}^{2}({{\Bbb R}} , L^{2}(\Omega)) $, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{{\cal E}} $中存在弱一致吸引子$ {\cal A}_{w} $. 而且, 该弱一致吸引子$ {\cal A}_{w} $满足(2.2)式.

本节我们将证明方程(1.1)在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中强一致吸引子的存在性.

下述定理表明过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中的一致渐近紧性.

定理3.3  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{b}^{2}({{\Bbb R}} ;L^{2}(\Omega)) $满足时间正则性, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中一致渐近紧.

证  设$ \xi_{u^{i}}(t)=(u^{i}(t), \partial_{t}u^{i}(t)) $是方程(1.1)的对应于初值$ \xi_{\tau, i}=(\phi^{i}(x, t-\tau), \partial_{t}\phi^{i}(x, t-\tau))\in B_{0}\subset C_{{\cal E}} $的Shatah-Struwe解, $ B_{0} $是推论3.1中的有界吸收集, $ g_{i}\in{\cal H}(g_{0}) $$ (i=1, 2, $$ t\in[\tau-h, \tau]) $, 则$ w(t)=u^{1}(t)-u^{2}(t) $满足方程

(3.19) $ \begin{eqnarray} \partial_{tt}w+\gamma\partial_{t}w-\Delta w=f(u^{1})-f(u^{2})+F(u^{1}_{t})-F(u^{2}_{t})+g_{1}(t)-g_{2}(t), \end{eqnarray} $

其中$ (x, t)\in\Omega\times(\tau, \infty) $, 初值

$ \begin{eqnarray*} &&(w(x, t), \partial_{t}w(x, t)) \nonumber\\ &=&(\phi^{1}(x, t-\tau)-\phi^{2}(x, t-\tau), \partial_{t}\phi^{1}(x, t-\tau)-\partial_{t}\phi^{2}(x, t-\tau)) \quad x\in\Omega, t\in[\tau-h, \tau]. \end{eqnarray*} $

定义能量泛函

$ E_{w}(t)=\frac{1}{2}\int_{\Omega}(|\nabla w(t)|^{2}+|\partial_{t} w(t)|^{2}){\rm d}x. $

在(3.19)式两边同乘以$ \partial_{t}w $并在$ \Omega\times[s, t] $上积分, 再利用Hölder不等式和Young不等式, 可得

(3.20) $ \begin{eqnarray} &&E_{w}(t)+\frac{\gamma}{2}\int_{s}^{t}\|\partial_{t}w(r)\|_{2}^{2}{\rm d}r \\ &\leq &E_{w}(s)+\int_{s}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r \\ &&+\frac{1}{2\gamma}\int_{s}^{t}\|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}dr +\int_{s}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r. \end{eqnarray} $

对(3.20)式关于$ s $在$ [\tau, t] $上积分, 可得

(3.21) $ \begin{eqnarray} &&(t-\tau)E_{w}(t) +\frac{\gamma}{2}\int_{\tau}^{t}\int_{s}^{t}\|\partial_{t}w(r)\|_{2}^{2}{\rm d}r{\rm d}s \\ &\leq&\int_{\tau}^{t}E_{w}(s){\rm d}s+\int_{\tau}^{t}\int_{s}^{t} \int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s \\ &&+\frac{1}{2\gamma}\int_{\tau}^{t}\int_{s}^{t} \|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}{\rm d}r{\rm d}s +\int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s. \end{eqnarray} $

另一方面, 在(3.19)式两边同乘以$ w $并在$ \Omega\times[s, t] $上积分, 再利用Hölder不等式和Young不等式, 可得

(3.22) $ \begin{eqnarray} &&(\partial_{t}w(t), w(t))+\frac{1}{2}\int_{s}^{t}\|\nabla w(r)\|_{2}^{2}{\rm d}r \\ &\leq&(\partial_{t}w(s), w(s))-\gamma\int_{s}^{t}\int_{\Omega}\partial_{t}w(r)w(r){\rm d}x{\rm d}r +\int_{s}^{t}\|\partial_{t}w(r)\|_{2}^{2}{\rm d}r \\ &&+\int_{s}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))w(r){\rm d}x{\rm d}r +\frac{1}{2\lambda_{1}}\int_{s}^{t}\|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}{\rm d}r \\ &&+\int_{s}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))w(r){\rm d}x{\rm d}r. \end{eqnarray} $

进一步, 在(3.22)式中让$ s=\tau $, 可得

(3.23) $ \begin{eqnarray} &&(\partial_{t}w(t), w(t))+\int_{\tau}^{t}E_{w}(r){\rm d}r \\ &\leq&(\partial_{t}w(\tau), w(\tau))-\gamma\int_{\tau}^{t}\int_{\Omega}\partial_{t}w(r)w(r){\rm d}x{\rm d}r +\frac{3}{2}\int_{\tau}^{t}\|\partial_{t}w(r)\|_{2}^{2}{\rm d}r \\ &&+\int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))w(r){\rm d}x{\rm d}r +\frac{1}{2\lambda_{1}}\int_{\tau}^{t}\|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}{\rm d}r \\ &&+\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))w(r){\rm d}x{\rm d}r. \end{eqnarray} $

在(3.20)式两边同乘以$ \frac{3}{\gamma} $并让$ s=\tau $, 可得

(3.24) $ \begin{eqnarray} &&\frac{3}{\gamma}E_{w}(t)+\frac{3}{2}\int_{\tau}^{t}\|\partial_{t}w(r)\|_{2}^{2}{\rm d}r \\ &\leq &\frac{3}{\gamma}E_{w}(\tau)+\frac{3}{\gamma} \int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r \\ &&+\frac{3}{2\gamma^{2}}\int_{\tau}^{t}\|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}{\rm d}r +\frac{3}{\gamma}\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r. \end{eqnarray} $

将(3.24)式代入(3.23)式, 可得

(3.25) $ \begin{eqnarray} &&\int_{\tau}^{t}E_{w}(r){\rm d}r\\ &\leq&-(\partial_{t}w(t), w(t))+(\partial_{t}w(\tau), w(\tau)) -\gamma\int_{\tau}^{t}\int_{\Omega}\partial_{t}w(r) w(r){\rm d}x{\rm d}r \\ &&+\frac{3}{\gamma}E_{w}(\tau)+\frac{3}{\gamma} \int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r \\ &&+\frac{1}{2}(\frac{3}{\gamma^{2}}+\frac{1}{\lambda_{1}})\int_{\tau}^{t}\|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}{\rm d}r +\frac{3}{\gamma}\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r \\ &&+\int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))w(r){\rm d}x{\rm d}r +\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))w(r){\rm d}x{\rm d}r. \end{eqnarray} $

再将(3.25)式代入(3.21)式, 可得

$ \begin{eqnarray*} (t-\tau)E_{w}(t) &\leq&-(\partial_{t}w(t), w(t))+(\partial_{t}w(\tau), w(\tau)) -\gamma\int_{\tau}^{t}\int_{\Omega}\partial_{t}w(r)w(r){\rm d}x{\rm d}r \nonumber\\ &&+\frac{3}{\gamma}E_{w}(\tau)+\frac{3}{\gamma} \int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r \nonumber\\ &&+\frac{1}{2}(\frac{3}{\gamma^{2}}+\frac{1}{\lambda_{1}})\int_{\tau}^{t}\|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}{\rm d}r +\frac{3}{\gamma}\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r \nonumber\\ &&+\int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))w(r){\rm d}x{\rm d}r +\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))w(r){\rm d}x{\rm d}r. \nonumber\\ &&+\int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s \\ &&+\frac{1}{2\gamma}\int_{\tau}^{t}\int_{s}^{t}\|F(u^{1}_{r})-F(u^{2}_{r})\|_{2}^{2}{\rm d}r{\rm d}s +\int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s.\nonumber \end{eqnarray*} $

再由假设(Ⅲ), 可得

(3.26) $ \begin{eqnarray} E_{w}(t)&\leq&-\frac{1}{t-\tau}(\partial_{t}w(t), w(t))+\frac{1}{t-\tau}(\partial_{t}w(\tau), w(\tau)) -\frac{\gamma}{t-\tau}\int_{\tau}^{t}\int_{\Omega}\partial_{t}w(r)w(r){\rm d}x{\rm d}r \\ &&+\frac{3}{\gamma(t-\tau)}E_{w}(\tau)+\frac{3}{\gamma(t-\tau)} \int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r \\ &&+\frac{1}{2}C_{F}^{2}(\frac{3}{\gamma^{2}(t-\tau)}+\frac{1}{\lambda_{1}(t-\tau)}+\frac{1}{\gamma}) \int_{\tau-h}^{t}\|u^{1}(r)-u^{2}(r)\|_{2}^{2}{\rm d}r \\ &&+\frac{1}{t-\tau}\int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))w(r){\rm d}x{\rm d}r \\ &&+\frac{1}{t-\tau}\int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s \\ &&+\frac{1}{t-\tau}\int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s \\ &&+\frac{3}{\gamma(t-\tau)}\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r {}\\ && +\frac{1}{t-\tau}\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))w(r){\rm d}x{\rm d}r. \end{eqnarray} $

下面我们将处理(3.26)式右边的每一项.

首先, 由(1.2)式, Hölder不等式和Sobolev嵌入$ H_{0}^{1}(\Omega)\hookrightarrow L^{6}(\Omega) $, 可得

(3.27) $ \begin{eqnarray} &&\frac{1}{t-\tau}\int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))w(r){\rm d}x{\rm d}r \\ &\leq&\frac{1}{t-\tau}\int_{\tau}^{t} \left(\int_{\Omega}|f(u^{1}(r))-f(u^{2}(r))|^{2}{\rm d}x\right)^{\frac{1}{2}} \left(\int_{\Omega}|w(r)|^{2}{\rm d}x\right)^{\frac{1}{2}}{\rm d}r \\ &\leq&\frac{C}{t-\tau}\left(\int_{\tau}^{t} \left|\int_{\Omega}(1+|u^{1}(r)|^{2p}+|u^{2}(r)|^{2p}){\rm d}x\right|^{\frac{5}{2p}}{\rm d}r\right)^{\frac{p}{5}} \left(\int_{\tau}^{t}\|w(r)\|_{2}^{\frac{5}{5-p}}{\rm d}r\right)^{\frac{5-p}{5}} \\ &\leq&\frac{C}{t-\tau}\Big(|\Omega|^{\frac{5}{2p}}(t-\tau) \\ &&+|\Omega|^{\frac{5-p}{2p}}\int_{\tau}^{t}(\|u^{1}(r)\|_{L^{10}(\Omega)}^{5} +\|u^{2}(r)\|_{L^{10}(\Omega)}^{5}){\rm d}r\Big)^{\frac{1}{2}} \Big(\int_{\tau}^{t}\|w(r)\|_{2}^{\frac{5}{5-p}}{\rm d}r\Big)^{\frac{5-p}{5}}. \end{eqnarray} $

再由引理3.1和(2.5)式, 可得

(3.28) $ \begin{eqnarray} \left(|\Omega|^{\frac{5}{2p}}(t-\tau)+|\Omega|^{\frac{5-p}{2p}}\int_{\tau}^{t}(\|u^{1}(r)\|_{L^{10}(\Omega)}^{5} +\|u^{2}(r)\|_{L^{10}(\Omega)}^{5}){\rm d}r\right)^{\frac{1}{2}}\leq C_{t, \tau, \rho_{0}}<\infty, \end{eqnarray} $

其中$ C_{t, \tau, \rho_{0}} $与$ t-\tau $和$ \rho_{0} $相关, $ B_{0} $和$ \rho_{0} $由推论3.1给出.

另外, 由Hölder不等式和推论3.1中的(3.17)式, 可得

(3.29) $ \begin{eqnarray} -\frac{\gamma}{t-\tau}\int_{\tau}^{t}\int_{\Omega}\partial_{t}w(r)w(r){\rm d}x{\rm d}r &\leq&\frac{\gamma}{t-\tau}\left(\int_{\tau}^{t}\|\partial_{t}w(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}} \left(\int_{\tau}^{t}\|w(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}} \\ &\leq&C_{\rho_{0}}\frac{\gamma}{t-\tau}\left(\int_{\tau}^{t}\|w(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}}, \end{eqnarray} $

以及

(3.30) $ \begin{eqnarray} -\frac{1}{t-\tau}(\partial_{t}w(t), w(t))\leq\frac{1}{t-\tau}\|\partial_{t}w(t)\|_{2}\|w(t)\|_{2} \leq C_{\rho_{0}}\frac{1}{t-\tau}\|w(t)\|_{2}. \end{eqnarray} $

最后

(3.31) $ \begin{eqnarray} &&\frac{1}{t-\tau}\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))w(r){\rm d}x{\rm d}r \\ &\leq&\frac{1}{t-\tau}\left(\int_{\tau}^{t}\|g_{1}(r)-g_{2}(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}} \left(\int_{\tau}^{t}\|w(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}} \\ &\leq&\frac{\sqrt{2}}{t-\tau}\left(\int_{\tau}^{t}\|g_{1}(r)\|_{2}^{2}{\rm d}r +\int_{\tau}^{t}\|g_{2}(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}} \left(\int_{\tau}^{t}\|w(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}}. \end{eqnarray} $

综合(3.26)–(3.31)式, 设$ T_{1}=t-\tau $和

(3.32) $ \begin{eqnarray} &&\psi_{T_{1}, \tau}((\phi^{1}, \partial_{t}\phi^{1}), (\phi^{2}, \partial_{t}\phi^{2});g_{1}, g_{2}) =C_{\rho_{0}}\frac{1}{t-\tau}\|w(t)\|_{2} \\ &&+\left(C\frac{C_{t, \tau, \rho_{0}}}{t-\tau}+C_{\rho_{0}}\frac{\gamma}{t-\tau} +C_{\|g_{1}\|_{L^{2}(\tau, t;L^{2}(\Omega))}, \|g_{2}\|_{L^{2}(\tau, t;L^{2}(\Omega))}} \frac{\sqrt{2}}{t-\tau}\right) \left(\int_{\tau}^{t}\|w(r)\|_{2}^{2}{\rm d}r\right)^{\frac{1}{2}} \\ &&+\frac{1}{2}C_{F}^{2}(\frac{3}{\gamma^{2}(t-\tau)}+\frac{1}{\lambda_{1}(t-\tau)}+\frac{1}{\gamma}) \int_{\tau-h}^{t}\|u^{1}(r)-u^{2}(r)\|_{2}^{2}{\rm d}r \\ &&+\frac{3}{\gamma(t-\tau)}\int_{\tau}^{t}\int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r \\ &&+\frac{1}{t-\tau}\int_{\tau}^{t}\int_{s}^{t} \int_{\Omega}(f(u^{1}(r))-f(u^{2}(r)))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s \\ &&+\frac{1}{t-\tau}\int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r{\rm d}s \\ &&+\frac{3}{\gamma(t-\tau)}\int_{\tau}^{t}\int_{\Omega}(g_{1}(r)-g_{2}(r))\partial_{t}w(r){\rm d}x{\rm d}r. \end{eqnarray} $

因而

$ \begin{eqnarray*} E_{w}(t)\leq\frac{3}{\gamma(t-\tau)}E_{w}(\tau)+\frac{1}{t-\tau}(\partial_{t}w(\tau), w(\tau)) +\psi_{T_{1}, \tau}((\phi^{1}, \partial_{t}\phi^{1}), (\phi^{2}, \partial_{t}\phi^{2});g_{1}, g_{2}). \end{eqnarray*} $

选取足够大的$ t=t(\tau, B_{0}, \varepsilon)>\tau $使得

$ \frac{3}{\gamma(t-\tau)}E_{w}(\tau)+\frac{1}{t-\tau}(\partial_{t}w(\tau), w(\tau))<\varepsilon, $

从而有

(3.33) $ \begin{eqnarray} E_{w}(t)\leq\varepsilon +\psi_{T_{1}, \tau}((\phi^{1}, \partial_{t}\phi^{1}), (\phi^{2}, \partial_{t}\phi^{2});g_{1}, g_{2}), \end{eqnarray} $

其中$ (\phi^{i}, \partial_{t}\phi^{i})\in B_{0} $$ (i=1, 2) $.

下面我们只需验证由(3.32)式所定义的函数$ \psi_{T_{1}, \tau}((\phi^{1}, \partial_{t}\phi^{1}), (\phi^{2}, \partial_{t}\phi^{2});g_{1}, g_{2}) $是$ B_{0}\times B_{0} $上的收缩函数.

设$ (u^{n}, \partial_{t}u^{n}) $是方程(1.1)的与初值$ (\phi^{n}, \partial_{t}\phi^{n})\in B_{0} $和$ g_{n}\in{\cal H}(g_{0}) $, $ n=1, 2, \cdot\cdot\cdot $相关的解. 由于$ B_{0} $在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $有界, 则

$ \begin{eqnarray*} \|(u^{n}(s), \partial_{t}u^{n}(s))\|_{C_{V, H}}\leq M<+\infty, \quad\forall s\in[\tau, t], n\in{\Bbb N}, \end{eqnarray*} $

其中$ M $与(3.2)式中的$ C_{i} $$ (i=1, 2, 3, 4) $和$ \|\xi_{\tau}\|^{2}_{C_{{\cal E}}} $相关.

对任意的$ p\in[1, 5) $, 不失一般性, 有

(3.34) $ \begin{equation} u^{n}\ \mbox{在}\ L^{\infty}(\tau, t; H_{0}^{1}(\Omega))\ \mbox{中弱} \star -\mbox{收敛到}\ u, \end{equation} $

(3.35) $ \begin{equation} u^{n}\ \mbox{在}\ L^{p+1}(\tau, t; L^{p+1}(\Omega)) \ \mbox{中弱收敛到} u, \end{equation} $

(3.36) $ \begin{equation} \partial_{t}u^{n}\ \mbox{在}\ L^{\infty}(\tau, t; L^{2}(\Omega))\ \mbox{中弱} \star -\mbox{收敛到}\ \partial_{t}u, \end{equation} $

(3.37) $ \begin{equation} u^{n}\ \mbox{在} L^{2}(\tau, t; L^{2}(\Omega)) \mbox{和}\ L^{\frac{5}{5-p}}(\tau, t; L^{2}(\Omega)) \ \mbox{中弱收敛到} u, \end{equation} $

(3.38) $ \begin{equation} u^{n}(\tau)\ \mbox{和}\ u^{n}(t)\ \mbox{在}\ L^{p+1}(\Omega) \ \mbox{中分别收敛到}\ u(\tau)\ \mbox{和} \ u(t), \end{equation} $

这里我们用到Sobolev紧嵌入$ H_{0}^{1}(\Omega)\hookrightarrow L^{p+1}(\Omega) $.

下面我们逐项处理(3.32)式中的每一项.

首先, 由(3.36)式可得

(3.39) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\lim\limits_{m\rightarrow \infty} \int_{\tau}^{t}\|u^{n}(r)-u^{m}(r)\|_{2}^{2}{\rm d}r=0, \end{equation} $

(3.40) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\lim\limits_{m\rightarrow \infty} \int_{\tau}^{t}\|u^{n}(r)-u^{m}(r)\|_{2}^{\frac{5}{5-p}}{\rm d}r=0. \end{equation} $

然后

(3.41) $ \begin{eqnarray} &&\int_{\tau}^{t}\int_{\Omega}(f(u^{n}(r))-f(u^{m}(r))) (\partial_{t}u^{n}(r)-\partial_{t}u^{m}(r)){\rm d}x{\rm d}r \\ &=&\int_{\tau}^{t}\int_{\Omega}f(u^{n}(r))\partial_{t}u^{n}(r){\rm d}x{\rm d}r +\int_{\tau}^{t}\int_{\Omega}f(u^{m}(r))\partial_{t}u^{m}(r){\rm d}x{\rm d}r \\ &&-\int_{\tau}^{t}\int_{\Omega}f(u^{n}(r))\partial_{t}u^{m}(r){\rm d}x{\rm d}r -\int_{\tau}^{t}\int_{\Omega}f(u^{m}(r))\partial_{t}u^{n}(r){\rm d}x{\rm d}r \\ &=&\int_{\Omega}{\cal F}(u^{n}(t))-\int_{\Omega}{\cal F}(u^{n}(\tau)) +\int_{\Omega}{\cal F}(u^{m}(t))-\int_{\Omega}{\cal F}(u^{m}(\tau)) \\ &&-\int_{\tau}^{t}\int_{\Omega}f(u^{n}(r))\partial_{t}u^{m}(r){\rm d}x{\rm d}r -\int_{\tau}^{t}\int_{\Omega}f(u^{m}(r))\partial_{t}u^{n}(r){\rm d}x{\rm d}r. \end{eqnarray} $

由(1.2)式, Shatah-Struwe解的定义以及(2.5)式, 可得$ f(u^{m}) $在$ L^{1}(\tau, t;L^{2}(\Omega) $中弱收敛到$ f(u) $. 再综合(3.35)式, (3.37)式和(3.41)式, 可得

(3.42) $ \begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\lim\limits_{m\rightarrow \infty} \int_{\tau}^{t}\int_{\Omega}(f(u^{n}(r))-f(u^{m}(r))) (\partial_{t}u^{n}(r)-\partial_{t}u^{m}(r)){\rm d}x{\rm d}r \\ &=&\int_{\Omega}{\cal F}(u(t))-\int_{\Omega}{\cal F}(u(\tau)) +\int_{\Omega}{\cal F}(u(t))-\int_{\Omega}{\cal F}(u(\tau)) \\ &&-\int_{\tau}^{t}\int_{\Omega}f(u(r))\partial_{t}u(r){\rm d}x{\rm d}r -\int_{\tau}^{t}\int_{\Omega}f(u(r))\partial_{t}u(r){\rm d}x{\rm d}r \\ & =&0. \end{eqnarray} $

类似地, 对任给的$ s\in[\tau, t] $, $ |\int_{s}^{t}\int_{\Omega}(f(u^{n}(r))-f(u^{m}(r))) (\partial_{t}u^{n}(r)-\partial_{t}u^{m}(r)){\rm d}x{\rm d}r| $有界, 再由(3.42)式和勒贝格控制收敛定理, 可得

(3.43) $ \begin{eqnarray} &&\lim\limits_{n\rightarrow \infty}\lim\limits_{m\rightarrow \infty} \int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(f(u^{n}(r))-f(u^{m}(r))) (\partial_{t}u^{n}(r)-\partial_{t}u^{m}(r)){\rm d}x{\rm d}r{\rm d}s \\ &=&\int_{\tau}^{t}\left(\lim\limits_{n\rightarrow \infty}\lim\limits_{m\rightarrow \infty} \int_{s}^{t}\int_{\Omega}(f(u^{n}(r))-f(u^{m}(r))) (\partial_{t}u^{n}(r)-\partial_{t}u^{m}(r)){\rm d}x{\rm d}r\right){\rm d}s \\ &=&\int_{\tau}^{t}0{\rm d}s=0. \end{eqnarray} $

再类似于文献[46]中的命题4.1和定理4.3, 可得

(3.44) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\lim\limits_{m\rightarrow \infty} \int_{\tau}^{t}\int_{s}^{t}\int_{\Omega}(g_{n}(r)-g_{m}(r))(\partial_{t}u_{n}(r)-\partial_{t}u_{m}(r)){\rm d}x{\rm d}r{\rm d}s=0, \end{equation} $

(3.45) $ \begin{equation} \lim\limits_{n\rightarrow \infty}\lim\limits_{m\rightarrow \infty} \int_{\tau}^{t}\int_{\Omega}(g_{n}(r)-g_{m}(r))(\partial_{t}u_{n}(r)-\partial_{t}u_{m}(r)){\rm d}x{\rm d}r=0. \end{equation} $

综合(3.39)–(3.45)式, 可知由(3.32)式所定义的函数$ \psi_{T_{1}, \tau}(\cdot, \cdot;\cdot, \cdot) $是$ B_{0}\times B_{0} $上的收缩函数. 特别地, 在(3.33)式中用$ t+\theta $代替$ t $并根据引理 可知过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $中一致渐近紧.

综合定理3.2和定理3.3, 可得下述结果:

定理3.4  设$ f $满足(1.2)–(1.5)式, $ F(u_{t}) $满足假设(Ⅰ)–(Ⅲ), $ g(\cdot)\in L_{b}^{2}({{\Bbb R}} ;L^{2}(\Omega)) $满足时间正则性, 初值$ \xi_{\tau}=(\phi, \partial_{t}\phi)\in C_{{\cal E}} $, 则过程族$ U_{g}(t, \tau) $, $ g\in{\cal H}(g_{0}) $在$ C_{{\cal E}} $中存在强一致吸引子$ {\cal A}_{s} $. 而且, 该强一致吸引子$ {\cal A}_{s} $满足(2.2)式并等同于定理3.2中的弱一致吸引子$ {\cal A}_{w} $.