Processing math: 0%

数学物理学报, 2019, 39(1): 81-94 doi:

论文

记忆型无阻尼抽象发展方程的强时间依赖全局吸引子

胡弟弟, 汪璇,

The Strong Time-Dependent Global Attractors for the Non-Damping Abstract Evolution Equations with Fading Memory

Hu Didi, Wang Xuan,

通讯作者: 汪璇, E-mail: wangxuan@nwnu.edu.cn; 2295423708@qq.com

收稿日期: 2017-03-8  

基金资助: 国家自然科学基金.  11761062
国家自然科学基金.  11561064
国家自然科学基金.  11661071
西北师范大学创新团队基金.  NWNU-LKQN-14-6

Received: 2017-03-8  

Fund supported: the NSFC.  11761062
the NSFC.  11561064
the NSFC.  11661071
the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University.  NWNU-LKQN-14-6

摘要

运用修正的拉回吸引子理论、先验估计技巧和算子分解方法,得到了记忆型无阻尼抽象发展方程强时间依赖全局吸引子的存在性和正则性.

关键词: 衰退记忆 ; 抽象发展方程 ; 强时间依赖全局吸引子 ; 存在性 ; 正则性

Abstract

In this paper, by applying the modified pullback attractors theory, asymptotic a priori estimate method and operator decomposition technique, we obtain the existence as well as regularity of strong time-dependent global attractors for the non-damping abstract evolution equations with fading memory.

Keywords: Fading memory ; Abstract evolution equation ; Strong time-dependent global attractor ; Existence ; Regularity

PDF (398KB) 元数据 多维度评价 相关文章 导出 EndNote| Ris| Bibtex  收藏本文

本文引用格式

胡弟弟, 汪璇. 记忆型无阻尼抽象发展方程的强时间依赖全局吸引子. 数学物理学报[J], 2019, 39(1): 81-94 doi:

Hu Didi, Wang Xuan. The Strong Time-Dependent Global Attractors for the Non-Damping Abstract Evolution Equations with Fading Memory. Acta Mathematica Scientia[J], 2019, 39(1): 81-94 doi:

1 引言

本文研究边界充分光滑的有界域ΩRn (n上的记忆型无阻尼抽象发展方程

\left\{\begin{array}{ll} \displaystyle \varepsilon(t)u_{tt}+k(0)A^{\theta}u+\int^{\infty}_{0}k^{\prime}(s)A^{\theta} u(t-s){\rm d}s+g(u)=f(x),&(x, t) \in \Omega\times{\mathbb{R}} , \\[5pt] u(x, t)=0,&x\in\partial\Omega, t\in{\mathbb{R}} , \\ u(x, t)=u_{\tau}(x, t), u_{t}(x, \tau)=u_{t\tau}(x),&x\in\Omega, t\leqslant\tau, \end{array} \right.
(1.1)

其中A=-\Delta,

\theta\in\left\{\begin{array}{ll} \Big(\frac{2n}{n+4}, \frac{n}{4}\Big), &5\leqslant n<8, \\[8pt] \Big (\frac{n}{6}, \frac{n}{4}\Big), &n\geqslant 8, \end{array} \right.
(1.2)

\varepsilon(t)\in C^{1}({\mathbb{R}} )是单调递减的正函数,且满足

\lim\limits_{t\to +\infty}\varepsilon(t)=0,
(1.3)

特别地,存在常数L>0,使得

\sup\limits_{t\in {\mathbb{R}} }(|\varepsilon(t)|+|\varepsilon^{\prime}(t)|)\leqslant L.
(1.4)

方程(1.1)中函数A^{\theta}u(\cdot)和记忆核函数k(\cdot)的线性时间卷积使得衰退记忆在无阻尼动力系统的能量耗散过程中起到主要作用.设k^{\prime}(s)<0, \forall\, s\in{{\mathbb{R}}^{+}}, k(\infty)=1.进一步,假设\mu(s)=-k^{\prime}(s),且满足以下条件

\label{mus}\mu(s)\in{\rm C^{1}}\left({{\mathbb{R}}^{+}}\right)\cap L^{1}\left({{\mathbb{R}}^{+}}\right), \mu(s)\geqslant0, \mu^{\prime}(s)\leqslant0, \quad \forall\, s\in{{\mathbb{R}}^{+}},
(1.5)

\label{musk}\int ^{\infty}_{0}\mu(s){\rm d}s=k_{0},
(1.6)

\label{musdelta}\mu^{\prime}(s)+\delta\mu(s)\leqslant0, \quad \forall\, s\in {{\mathbb{R}}^{+}},
(1.7)

其中k_{0}, \delta为正常数,显然\mu(s)沿指数衰退到零.

设非线性项 g\in C^{2}({\mathbb{R}} ;{\mathbb{R}} ), g(0)=0,满足

g^{\prime\prime}(s) \in L^{\infty}({\mathbb{R}} ),
(1.8)

且满足以下增长条件

|g'(s)| \leqslant C(1+|s|^{\gamma}), ~\mbox{且}\ 0<\gamma<\frac{4}{n-4}, \quad \forall\, s\in {\mathbb{R}}
(1.9)

和耗散性条件

\liminf\limits_{|s| \to +\infty} \frac{g(s)}{s}>-\lambda_{1}^{\theta},
(1.10)

其中\lambda_{1}为算子-\Delta在空间H_{0}^{1}(\Omega)中的第一特征值.为了证明有界吸收集的存在性,我们假设

2\langle g(u), u\rangle \geqslant 2\langle G(u), 1\rangle-(1-\nu)\|u\|^{2}_{\theta}-C,
(1.11)

其中G(u)=\int^{u}_{0}g(y){\rm d}y,以及

g(s)s\geqslant 0, \quad \forall\, s\in{\mathbb{R}} ,
(1.12)

并且存在常数l>0,使得

g'(s)\leqslant l, \quad \forall\, s\in{\mathbb{R}} .
(1.13)

方程(1.1)来源于等温黏弹性理论(见文献[2, 5-6]),描述了一种各向同性的黏弹性物质的能量耗散过程.当\varepsilon(t)\equiv1时,在文献[16]中,我们证明了当非线性项满足较弱的耗散型条件时强全局吸引子的存在性.在文献[14]中证明了非线性项临界增长时全局吸引子的存在性.当\varepsilon(t)为关于时间t的函数时,通常的动力系统理论已经不再适用于该模型的解在时间依赖函数空间中渐近性态的研究.面对这种情况, Di Plinio和Conti提出了时间依赖吸引子的概念和理论(修正的拉回吸引子理论)来研究解的动力学行为.

关于时间依赖动力系统的渐近性态研究已有以下一些结果.当非线性项满足次临界指数增长条件时,在文献[3-4, 8, 10-11],作者研究了半线性波方程时间依赖渐近性行为. Conti和Pata等人在文献[3]中运用修正的拉回吸引子理论证明了半线性波方程时间依赖吸引子的存在性及正则性,在文献[4]中作者进一步研究了该吸引子的渐近结构.孟和杨等人在文献[10]中用收缩函数的方法证明了半线性波方程在时间依赖空间中吸引子的存在性.孟和刘在文献[11]中通过非紧性测度中提出了时间依赖吸引子存在的充要条件并证明了弱阻尼半线性波方程在强拓扑空间中时间依赖吸引子的存在性.最近,在文献[8]中,我们关于记忆型无阻尼抽象发展方程在弱空间中得到了时间依赖全局吸引子的存在性和正则性结果.当非线性项满足临界指数增长条件时,马和王在文献[9]中证明了非经典反应扩散方程时间依赖吸引子的存在性和正则性.

据我们所知,关于方程(1.1)的解在强时间依赖函数空间的渐近性态尚未有人研究.在研究中,我们发现存在一些难以克服的本质性困难:首先,由于系统中正的递减函数\varepsilon(t)在无穷大处的值趋于零,因此不能运用经典的动力系统理论得到能量耗散估计;其次, A^{\theta}为抽象算子,这给紧性的验证带来了困难;再者,方程中包含记忆项,对应的记忆空间非紧,所以Sobolev紧嵌入不成立;最后,研究的模型不包含阻尼项,系统的能量耗散通过记忆项来实现,这也给紧性的验证造成了干扰.针对上述难点,我们运用了先验估计和算子分解技巧,结合修正的时间依赖吸引子理论,成功地克服了这些困难,并得到了强时间依赖全局吸引子的存在性和正则性结果.

本文结构如下:在第二节,介绍即将用到的函数空间、符号和一般的抽象结果.在第三节,证明了方程(1.1)时间依赖强全局吸引子的存在性和正则性.

为了方便估计,本文中出现的C, C_i\rho均表示正常数,且\rho\in(0, 1).

2 预备知识

H=L^{2}(\Omega), \langle Au, v\rangle=b(u, v), \forall\, u, \, v \in H ,其中b(u, v)H上的双线性型,且为对称和强制的. AH上的线性无界自伴算子,其定义域 D(A)\subset H .\left\{\lambda_{j}\right\}_{j \in {\Bbb N}}, \, \left\{\omega_{j}\right\}_{j \in {\Bbb N}}分别为A的特征值和特征向量,因此\left\{\omega_{j}\right\}_{j\in {\Bbb N}}可构成H的一组正交基,且有

\begin{eqnarray*} \left\{\begin{array}{ll} A \omega_{j}=\lambda_{j} \omega_{j}, \\ 0<\lambda_{1}\leqslant\lambda_{2}\leqslant\cdots\leqslant\lambda_{j}, \lambda_{j} \rightarrow \infty, ~\mbox{当$ j\rightarrow \infty.$} \end{array} \right.\end{eqnarray*}

利用这组基可定义如下与A同构的幂算子族A^{\theta}

D(A^{\theta})=\left\{u \in H, \sum\limits^{\infty}_{j=1}\lambda^{2\theta}_{j}(u, \omega_{j})^2<\infty \right\},

其定义域为 D\left(A^{\theta}\right)\subset H.分别赋予内积和范数

\langle u, v\rangle_{D(A^{\theta})}=\langle A^{\theta}\cdot, A^{\theta}\cdot\rangle, \quad \|u\|_{D(A^{\theta})}^{2}=\|A^{\theta}\cdot\|^{2}.

V_{s}=D\left(A^{\frac{s}{2}}\right),则D\left(A^{0}\right)=H, D\left(A^{\frac{\theta}{2}}\right)=V_{\theta}, D\left(A^{-\frac{\theta}{2}}\right)=V_{\theta}^{*}.分别赋予空间H, V_{\theta}V_{2\theta}以下形式的内积与范数

\langle u, v\rangle =\int_\Omega u(x)v(x){\rm d}x, \quad \|u\|^{2}=\int_\Omega |u(x)|^{2}{\rm d}x, \quad \forall\, u, \, v \in H;
(2.1)

\langle u, v\rangle _{\theta}=\int_\Omega A^{\frac{\theta}{2}}u(x)A^{\frac{\theta}{2}}v(x){\rm d}x, \quad \|u\|^{2}_{\theta}=\int_\Omega |A^{\frac{\theta}{2}}u(x)|^{2}{\rm d}x, \quad \forall\, u, \, v \in V_{\theta};
(2.2)

\langle u, v\rangle _{2\theta}=\int_\Omega A^{\theta}u(x)A^{\theta}v(x){\rm d}x, \quad \|u\|^{2}_{2\theta}=\int_\Omega |A^{\theta}u(x)|^{2}{\rm d}x, \quad \forall\, u, \, v \in V_{2\theta}.
(2.3)

因为算子A是无界自伴的,故A^{\theta}也为无界自伴算子.因此,有

\begin{array}{ll}D(A^{\theta_{1}})\hookrightarrow D(A^{\theta_{2}}), \quad &\forall\, \theta_{1}>\theta_{2}, \\[2mm] D(A^{\theta})\hookrightarrow L^{\frac{2n}{n-4\theta}}, \quad & \forall\, \theta\in(0, \frac{n}{4}). \end{array}
(2.4)

并且有嵌入不等式

\lambda_{1}^{\theta}\int_{\Omega}|A^{\frac{\theta}{2}}v|^{2}{\rm d}x\leqslant\int_{\Omega}|A^{\theta}v|^{2}{\rm d}x, \quad \forall\, v\in V_{\theta}.
(2.5)

L_{\mu}^{2}\left({\mathbb{R}}^{+};V_{2\theta}\right)为定义于{{\mathbb{R}}^{+}}取值于V_{2\theta}的Hilbert空间族,赋予相应的内积和范数

\langle \varphi, \psi\rangle_{\mu, 2\theta}=\int^{\infty}_{0}\mu(s)\int_{\Omega}A^{\theta}\varphi A^{\theta} \psi {\rm d}x{\rm d}s, \quad \|\varphi\|^{2}_{\mu, 2\theta}=\int^{\infty}_{0}\mu(s)\int_{\Omega}|A^{\theta}\varphi|^{2} {\rm d}x{\rm d}s.

定义时间依赖空间

{\cal H}_{t}^{\theta+\sigma}=V_{2\theta+\sigma} \times V_{\theta+\sigma} \times L_{\mu}^{2}\left({\mathbb{R}}^{+};V_{2\theta+\sigma}\right),

并且赋予相应的范数

\|z\|^{2}_{{\cal H}_{t}^{\theta+\sigma}}=\left\|\left(u, u_{t}, \eta^{t}\right)\right\|^{2}_{{\cal H}_{t}^{\theta+\sigma}} =\|u\|^{2}_{2\theta+\sigma}+\varepsilon(t)\|u_{t}\|^{2} _{\theta+\sigma} +\|\eta^{t}\|^{2}_{\mu, 2\theta+\sigma}.

\sigma=0时,我们记{\cal H}_{t}^{\theta}=V_{2\theta} \times V_{\theta} \times L_{\mu}^{2}\left({\mathbb{R}}^{+};V_{2\theta}\right),对应的范数为

\|z\|^{2}_{{\cal H}_{t}^{\theta}}=\left\|\left(u, u_{t}, \eta^{t}\right)\right\|^{2}_{{\cal H}_{t}^{\theta}}=\|u\|^{2}_{2\theta} +\varepsilon(t)\|u_{t}\|_{\theta}^{2}+\|\eta^{t}\|^{2}_{\mu, 2\theta}.

定义变量

\eta^{t}(s)=\eta^{t}(x, s)=u(x, t)-u(x, t-s).

\mu(s)=-k^{\prime}(s)k(\infty)=1, 则方程(1.1)转化为以下形式

\left\{\begin{array}{ll} \displaystyle \varepsilon(t)u_{tt}+A^{\theta}u+\int^{\infty}_{0}\mu(s)A^{\theta}\eta^{t}(s){\rm d}s+g(u)=f, \\[5pt] \eta^{t}_{t}=-\eta^{t}_{s}+u_{t}, \end{array} \right.
(2.6)

相应初-边值条件为

\left\{\begin{array}{ll} u(x, t)=0, \eta^{t}(x, s)=0,&x\in\partial\Omega, \\ u(x, \tau)= u_{\tau}(x), u_{t}(x, \tau)=u_{t\tau}(x),&x\in\Omega, \\ \eta^{\tau}(x, s)=\eta_{\tau}(x, s)=u_{\tau}(x, \tau)-u_{\tau}(x, -s),&(x, s)\in\Omega\times{{\mathbb{R}}^{+}}, \end{array} \right.
(2.7)

这里, u满足\int_{0}^{\infty}e^{-\sigma s}\|\nabla u(-s)\|^{2}{\rm d}s\leqslant N,其中, \sigma\geqslant0, N>0.

引理2.1[15]  设记忆核函数\mu(s)满足条件(1.5)和(1.7),则对\forall\, T>\tau, \forall\, \eta^{t} \in C\big([\, \tau, T\, ];L^{2}_{\mu}\left({\Bbb R^{+}};V_{\theta}\right)\big),有\left\langle\eta^{t}, \eta^{t}_{s}\right\rangle_{\mu, \theta}\geqslant\frac{\delta}{2}\|\eta^{t}\|^{2}_{\mu, \theta}.

定义2.1  设\{X_{t}\}是一个赋范空间.双参数算子族\{U(t, \tau):X_{\tau}\rightarrow X_{t}, t\geqslant\tau, \tau\in{\mathbb{R}} \}满足以下性质

(i)   对任意的\tau\in{\mathbb{R}} , U(\tau, \tau)={\rm Id}X_{\tau}上的恒等算子;

(ii)   对任意的\sigma\in{\mathbb{R}} 和任意的t\geqslant \tau\geqslant \sigma, U(t, \tau)U(\tau, \sigma)=U(t, \sigma).

则称U(t, \tau)为过程.

定义2.2  称有界集C_{t}\subset X_{t}的集合族{\mathfrak C}=\{C_{t}\}_{t\in{\mathbb{R}} }为一致有界.如果存在常数R>0,使得

C_{t}\subset\{z\in X_{t}:\|z\|_{X_{t}}\leqslant R\}={\Bbb B}_{t}(R), \quad \forall t\in{\mathbb{R}} .

定义2.3  称集合族{\mathfrak B}=\{B_{t}\}_{t\in{\mathbb{R}} }为拉回吸收.如果它是一致有界,并且对任意的R>0,存在常数t_{0}(t, R)\leqslant t使得\tau\leqslant t_{0},有 U(t, \tau){\Bbb B}_{\tau}(R)\subset B_{t}成立.

定义2.4  称一致有界集族{\mathfrak B}=\{B_{t}\}_{t\in{\mathbb{R}} }为过程U(t, \tau)的时间依赖吸收集.如果对于任意R>0,存在常数t_{0}(t, R)\leqslant t,使得当\tau\leqslant t-t_{0}时,有 U(t, \tau){\Bbb B}_{\tau}(R)\subset B_{t} 成立.

定义2.5  称满足如下性质

(i)   每个A_{t}X_{t}中是紧的;

(ii)   {\mathfrak A}是拉回吸引的,即对每个一致有界族{\mathfrak C}=\{C_{t}\}_{t\in{\mathbb{R}} },以下极限成立

\lim\limits_{\tau \rightarrow -\infty} {\rm dist}(U(t, \tau)C_{\tau}, A_{t})=0

的最小族{\mathfrak A}=\{A_{t}\}_{t\in{\mathbb{R}} }为过程U(t, \tau)的时间依赖吸引子.其中 {\rm dist}(B, C)=\sup\limits_{x\in B}\inf\limits_{y\in C}\|x-y\|_{X_{t}}表示集合BC的Hausdorff半距离.

定理2.1  时间依赖吸引子{\mathfrak A}存在当且仅当过程是渐近紧的,即集合

{\Bbb K}=\{{\mathfrak K}=\{K_{t}\}_{t\in{\mathbb{R}} }: K_{t}\subset X_{t} \ \mbox{紧, $ {\mathfrak K} $为拉回吸引}\}

非空.

定义2.6  时间依赖吸引子{\mathfrak A}=\{A_{t}\}_{t\in{\mathbb{R}} }是不变的,如果 U(t, \tau)A_{\tau}=A_{t}, \ \forall\, t\geqslant \tau.

3 主要结果与证明

3.1 适定性

根据无穷维动力系统的一般理论[13],利用Galerkin逼近方法,容易得到方程(2.6)-(2.7)强解的存在唯一性,这里时间依赖函数\varepsilon(t)没有增加证明的复杂性,即

定理3.1  假设\Omega{\mathbb{R}}^n上带有光滑边界的有界区域,条件(1.4)-(1.7)成立, f\in Hg\in C^2({\mathbb{R}} ; {\mathbb{R}} )满足条件(1.8)-(1.13).则对任意的\tau, T\in{\mathbb{R}} , T>\tau和任意的初值z(\tau)=(u(\tau), u_{t}(\tau), \eta^{\tau}(s))\in {\cal H}^{\theta}_{\tau},如果

u\in C([\tau, T];V_{2\theta}), u_{t}\in L^{2}([\tau, T];V_{2\theta}), \eta^{t}\in C([\tau, T];L^{2}_{\mu}({{\mathbb{R}}^{+}};V_{2\theta})),

\eta^{t}_{t}+\eta^{t}_{s}\in L^{\infty}([\tau, T];L^{2}_{\mu}({{\mathbb{R}}^{+}};V_{\theta}))\cap L^{2}([\tau, T];L^{2}_{\mu}({{\mathbb{R}}^{+}};V_{2\theta})),

\left\{\begin{array}{ll} \displaystyle \langle\varepsilon(t)u_{tt}, v\rangle+\langle u, v\rangle_{\theta}+\langle\eta^{t}(s), v\rangle_{\mu, \theta}+\langle g(u), v\rangle=\langle f, v\rangle, \\ \langle\eta^{t}_{t}(s)+\eta^{t}_{s}(s), \varphi(s)\rangle_{\mu, \theta}=\langle u_{t}, \varphi(s)\rangle_{\mu, \theta}, \end{array} \right.

对于任意的v\in V_{\theta}, \, \varphi\in L^{2}_{\mu}({{\mathbb{R}}^{+}};V_{\theta}), \, t\in I几乎处处成立.则问题(2.6)-(2.7)存在唯一的解z(t)=(u(t), u_{t}(t), \eta^{t}(s))\in L^{\infty}([\tau, T];{\cal H}_{t}^{\theta})\cap C([\tau, T];{\cal H}_{t}^{\theta}).

根据定理3.1,可定义强弱连续的过程 U(t, \tau):{\cal H}^{\theta}_{\tau}\rightarrow {\cal H}^{\theta}_{t},使得 U(t, \tau)z(\tau)=z(t)=(u(t), u_{t}(t), \eta^{t}(s)), t\geqslant\tau\in{\mathbb{R}} .

3.2 时间依赖吸收集

引理3.1  假设条件(1.3)-(1.13)成立,设z(t)是方程(2.6)对应于初值z(\tau)(\|z(\tau)\|_{{\cal H}^{\theta}_{\tau}}\leqslant R)的解,则存在一个正常数C_{1}=C_{1}(R),使得

\|z(t)\|_{{\cal H}_{t}^{\theta}}=\|U(t, \tau)z(\tau)\|_{{\cal H}_{t}^{\theta}}\leqslant C_{1}, \quad \forall\, t\geqslant \tau.
(3.1)

  用2A^{\theta}(u_{t}+\rho u)与方程(2.6)在H中作内积,有

\begin{eqnarray}&& \langle \varepsilon(t)u_{tt}, 2A^{\theta}(u_{t}+\rho u)\rangle+\langle A^{\theta}u, 2A^{\theta}(u_{t}+\rho u)\rangle+ \langle\int_{0}^{\infty}\mu(s)A^{\theta}\eta^{t}(s){\rm d}s, 2A^{\theta}(u_{t}+\rho u)\rangle \\&& +\langle g(u), 2A^{\theta}(u_{t}+\rho u)\rangle=\langle f, 2A^{\theta}(u_{t}+\rho u)\rangle.\end{eqnarray}
(3.2)

对于记忆项,由引理2.1,有

\begin{eqnarray}\langle\int_{0}^{\infty}\mu(s)A^{\theta}\eta^{t}(s){\rm d}s, 2A^{\theta}u_{t}\rangle&=&\int_{\Omega}\int_{0}^{\infty}2A^{\theta}(\eta^{t}_{t}+\eta^{t}_{s})\mu(s)A^{\theta}\eta^{t}(s){\rm d}s{\rm d}x \\&\geqslant&\frac{{\rm d}}{{\rm d}t}\|\eta^{t}\|_{\mu, 2\theta}^{2}+\delta \|\eta^{t}\|_{\mu, 2\theta}^{2}. \end{eqnarray}
(3.3)

利用Hölder不等式和Young不等式,可得

\begin{eqnarray}\langle\int_{0}^{\infty}\mu(s)A^{\theta}\eta^{t}(s){\rm d}s, 2\rho A^{\theta}u\rangle &\geqslant&-2\rho\int_{\Omega}|A^{\theta}u|\int_{0}^{\infty}\mu(s)|A^{\theta}\eta^{t}(s)|{\rm d}s{\rm d}x \\ &\geqslant&-\frac{\rho\nu}{2}\|u\|_{2\theta}^{2}-\frac{2k_{0}\rho}{\nu}\|\eta^{t}\|_{\mu, 2\theta}^{2}. \end{eqnarray}
(3.4)

将(3.3)和(3.4)式代入(3.2)式,可得

\begin{eqnarray}&& \frac{\rm d}{{\rm d}t}(\|u\|^{2}_{2\theta}+\varepsilon(t)\|u_{t}\|_{\theta}^{2}+\|\eta^{t}\|^{2}_{\mu, 2\theta}+2\rho\varepsilon(t)\langle u_{t}, A^{\theta}u\rangle-2\langle f, A^{\theta}u\rangle)+\frac{3\rho}{2}\|u\|^{2}_{2\theta} \\ && -(\varepsilon^\prime(t)+2\rho\varepsilon(t))\|u_{t}\|_{\theta}^{2}+(\delta-2\rho k_{0})\|\eta^{t}\|^{2}_{\mu, 2\theta}-2\rho\varepsilon^\prime(t)\langle u_{t}, A^{\theta}u\rangle-2\rho\langle f, u\rangle \\ &\leqslant&-2\langle g(u), A^{\theta}(u_{t}+\rho u)\rangle.\end{eqnarray}
(3.5)

根据文献[8,引理2.2],可得

\begin{eqnarray} \|z(t)\|_{{\cal H}_{t}}=\|U(t, \tau)z(\tau)\|_{{\cal H}_{t}}\leqslant R_0, \quad \forall\, t\geqslant \tau.\end{eqnarray}
(3.6)

由(3.1)式, (3.6)式和条件(1.12)-(1.13),我们可对(3.5)式的右边项做以下估计

\begin{eqnarray} -\langle g(u), 2A^{\theta}(u_{t}+\rho u)\rangle&=&-\langle g^\prime(\xi)u, 2A^{\theta}(u_{t}+\rho u)\rangle \\ &\leqslant&l|\langle u, 2A^{\theta}(u_{t}+\rho u)\rangle| \\ &\leqslant&l|\langle A^{\frac{\theta}{2}}u, 2A^{\frac{\theta}{2}}u_{t}\rangle|+l|\langle A^{\frac{\theta}{2}}u, 2\rho A^{\frac{\theta}{2}}u\rangle| \\ &\leqslant&2lR_{0}\|u_{t}\|_{\theta}+2\rho l\|u\|^{2}_{\theta} \\& \leqslant&\rho l\|u_{t}\|^{2}_{\theta}+C, \end{eqnarray}
(3.7)

其中0\leqslant\xi\leqslant u (或u\leqslant\xi\leqslant0),并且0\leqslant g'(\xi)\leqslant l.

对合适的常数C,我们定义泛函

\begin{eqnarray} {\cal M}_{0}(t)=\|u\|^{2}_{2\theta}+\varepsilon(t)\|u_{t}\|_{\theta}^{2}+\|\eta^{t}\|^{2}_{\mu, 2\theta}+2\rho\varepsilon(t)\langle u_{t}, A^{\theta}u\rangle-2\langle f, A^{\theta}u\rangle+C. \end{eqnarray}
(3.8)

利用条件(1.4)和(2.5)式,结合Hölder不等式和Young不等式,可得

2\rho\varepsilon(t)|\langle u, A^{\theta}u_{t}\rangle|\leqslant \rho\|u\|_{2\theta}^{2}+\frac{\rho L}{\lambda_{1}^{\theta}}\varepsilon(t)\|u_{t}\|_{\theta}^{2}.

从而对足够小的\rho,存在一个正常数C, C_2C_3,使得

\begin{eqnarray} C\|z(t)\|^2_{{\cal H}_{t}^{\theta}}-C_2\leqslant {\cal M}_{0}(t)\leqslant C\|z(t)\|^2_{{\cal H}_{t}^{\theta}}+C_3. \end{eqnarray}
(3.9)

将(3.7)-(3.8)式代入(3.5)式,可得

\begin{eqnarray}&&\frac{\rm d}{{\rm d}t}{\cal M}_{0}(t)+\rho {\cal M}_{0}(t)+\frac{\rho}{2}\|u\|_{2\theta}^{2}-(\varepsilon^\prime(t)+3\rho\varepsilon(t)+\rho l)\|u_{t}\|_{\theta}^{2}+(\delta-k_{0}\rho-\rho)\|\eta^{t}\|_{\mu, 2\theta}^{2} \\ &&-2\rho(\varepsilon^\prime(t)+\rho\varepsilon(t))\langle u_{t}, A^{\theta}u\rangle\leqslant C. \end{eqnarray}
(3.10)

由(1.4)和(2.5)式,我们得到

-2\rho(\varepsilon^\prime(t)+\rho\varepsilon(t))\langle u_{t}, A^{\theta}u\rangle\geqslant-\frac{\rho}{2}\|u\|_{2\theta}^{2}-\frac{2\rho L^{2}}{\lambda^{\theta}_{1}}\|u_{t}\|_{\theta}^{2}.
(3.11)

将(3.1)式带入到(3.10)式,并取\rho足够小,使得

\varepsilon'(t)+3\rho\varepsilon(t)+\rho l+\frac{2\rho L^{2}}{\lambda^{\theta}_{1}}\leqslant0, \quad \delta-k_{0}\rho-\rho\geqslant0.

\frac{\rm d}{{\rm d}t}{\cal M}_{0}(t)+\rho {\cal M}_{0}(t) \leqslant C.

应用Gronwall引理,结合(3.9)式可证得(3.1)式成立.证毕.

{\Bbb B}_{t}(R)=\{z(t)\in{\cal H}^{\theta}_{t}:\|z(t)\|_{{\cal H}_{t}^{\theta}}\leqslant R\}.从而,我们可直接得到以下时间依赖吸收集的存在性定理.

定理3.2  假设条件(1.3)-(1.10)及(1.5)-(1.7)成立,则对应于问题(2.6)-(2.7)的过程\{U(t, \tau)\}拥有时间依赖吸收集{\mathfrak B}=\left\{{\Bbb B}_{t}(R)\right\}.

3.3 时间依赖强全局吸引子的存在性

为了验证过程的渐近紧性,我们需要构造非空紧的拉回吸引族.为此,我们将过程分解为一个紧性部分和一个衰减部分的和.

当条件(1.9)-(1.13)成立时,非线性项g可分解为g=g_{0}+g_{1},其中g_{0}\in C^{1}({\mathbb{R}} ;{\mathbb{R}} ), g_{1}\in C^{2}({\mathbb{R}} ;{\mathbb{R}} ), \forall\, s\in {\mathbb{R}} ,存在l_{0}>0满足

g_{0}^\prime(s)\leqslant l_{0},
(3.12)

\label{gzero}g_{0}(0)=g_{1}(0)=0, g_{0}(s)s\geqslant0,
(3.13)

\label{gupghbcnlj} g_{1}^{\prime\prime}(s) \in L^{\infty}({\Bbb R}),
(3.14)

|g_{0}^\prime(s)|\leqslant k(1+|s|^{\gamma}), ~\mbox{且}\ 0<\gamma<\frac{4}{n-4},
(3.15)

|g_{1}^\prime(s)|\leqslant k(1+|s|^{\gamma}), ~\mbox{且}\ 0<\gamma<\frac{4}{n-4}.
(3.16)

{\mathfrak B}=\{{\Bbb B}_{t}(R)\}_{t\in {\mathbb{R}} }是由定理3.2所得到的一个时间依赖吸收集,且\tau\in {\mathbb{R}} 是固定的,那么对任意的z(\tau)\in {\Bbb B}_{\tau}(R),可以将U(t, \tau)z(\tau)分解为

U(t, \tau)z(\tau)=z(t)=(u(t), u_{t}(t), \eta^{t}(s))=U_{1}(t, \tau)z(\tau)+U_{2}(t, \tau)z(\tau),

其中

U_{1}(t, \tau)z(\tau)=(v(t), v_{t}(t), \zeta^{t}(s)), \quad U_{2}(t, \tau)z(\tau)=(w(t), w_{t}(t), \xi^{t}(s)),

分别满足

\left\{\begin{array}{ll} \varepsilon(t) v_{tt}+A^{\theta}v+\int_{0}^{\infty}\mu(s)A^{\theta}\zeta^{t}(s){\rm d}s+g_{0}(v)=0, \\[5pt] \zeta_{t}^{t}=-\zeta_{s}^{t}+v, \\ v(x, t)|_{\partial\Omega}=0, \zeta^{t}(x, t)|_{\partial\Omega}=0, \\ v(x, \tau)=u_{\tau}(x), \zeta^{\tau}(x, s)=\eta_{\tau}(x, s) \end{array} \right.
(3.17)

\left\{\begin{array}{ll} \varepsilon(t) w_{tt}+A^{\theta}w+\int_{0}^{\infty}\mu(s)A^{\theta}\xi^{t}(s){\rm d}s+g(u)-g_{0}(v)=f, \\[5pt] \xi_{t}^{t}=-\xi_{s}^{t}+w, \\ w(x, t)|_{\partial\Omega}=0, \xi^{t}(x, t)|_{\partial\Omega}=0, \\ w(x, \tau)=0, \xi^{\tau}(x, s)=0. \end{array} \right.
(3.18)

由Galerkin逼近方法,可得到方程(3.17)和(3.18)解的存在唯一性.进一步,类似于引理3.1的证明,对于方程(3.17)的解z_1(t),我们可得到

引理3.2  令z_1(t)是方程(3.17)关于初值z_1(\tau) (\|z(\tau)\|_{\mathcal H^{\theta}_\tau}^2\leqslant C_5)的解.假设g_0满足条件(3.12)-(3.15), f\in H,并且条件(1.4)-(1.13)成立.则对于任意的\epsilon>0,存在常数C_6=C_6({\mathfrak B})>0,和单调递增正函数Q(\cdot),使得方程(3.17)的解满足

\|z_1(t)\|_{\mathcal H^{\theta}_t}^2= \|U_{1}(t, \tau)z(\tau)\|_{{\cal H}^{\theta}_{t}}^2\leqslant Ce^{-C_6(t-\tau)}Q(\|z(\tau)\|_{\mathcal H^{\theta}_\tau})+\epsilon, \quad \forall\, t\geqslant\tau.
(3.19)

  用2A^{\theta}(v_{t}(t)+\rho v(t))与方程(3.17)做内积,我们得到

\begin{eqnarray}&&\frac{\rm d}{{\rm d}t}(\|v\|^{2}_{2\theta}+\varepsilon(t)\|v_{t}\|_{\theta}^{2}+\|\zeta^{t}\|^{2}_{\mu, 2\theta}+2\rho\varepsilon(t)\langle A^{\theta}v, v_{t}\rangle)+\frac{3\rho}{2}\|v\|_{2\theta}^{2}-(\varepsilon^\prime(t)+2\rho\varepsilon(t))\|v_{t}\|_{\theta}^{2} \\ &&+(\delta-2\rho k_{0})\|\zeta^{t}\|_{\mu, 2\theta}^{2}-2\rho\varepsilon^\prime(t)\langle A^{\theta}v, v_{t}\rangle\leqslant -2\langle g_{0}(v), A^{\theta}(v_{t}+\rho v)\rangle. \end{eqnarray}
(3.20)

根据文献[8,引理2.4],可知

\|z_1(t)\|_{{\cal H}_{t}}^2=\|U_{1}(t, \tau)z(\tau)\|_{{\cal H}_{t}}^2\leqslant Ce^{-C(t-\tau)}Q(\|z(\tau)\|_{\mathcal H_\tau})+\epsilon, \quad \forall\, t\geqslant \tau.
(3.21)

对于(3.20)式右边的第一项,由条件(3.12)-(3.13)和(3.21)式,有

\begin{eqnarray*} -2\langle g_{0}(v), A^{\theta}(v_{t}+\rho v)\rangle &\leqslant &l_{0}|\langle A^{\frac{\theta}{2}}v, 2A^{\frac{\theta}{2}}v_{t}\rangle|+l_{0}|\langle A^{\frac{\theta}{2}}v, 2\rho A^{\frac{\theta}{2}}v\rangle| \\ &\leqslant& 2l_{0}\epsilon\|v_{t}\|_{\theta}+2\rho l_{0}\epsilon^{2} \\ &\leqslant&\frac{\rho}{4} \|v_{t}\|^{2}_{\theta}+C\epsilon. \end{eqnarray*}

我们定义如下泛函

{\cal M}_{2}(t)= \|v\|^{2}_{2\theta}+\varepsilon(t)\|v_{t}\|_{\theta}^{2}+\|\zeta^{t}\|^{2}_{\mu, 2\theta}+2\rho\varepsilon(t)\langle A^{\theta}v, v_{t}\rangle,

事实上,由条件(1.4)和(2.5),有

2\rho\varepsilon(t)|\langle A^{\theta}v, v_{t}\rangle|\leqslant \frac{1}{2}\|v\|_{2\theta}^{2}+\frac{2\rho^{2} L}{\lambda_{1}^{\theta}}\varepsilon(t)\|v_{t}\|_{\theta}^{2}.

\frac{1}{2}\|U_{1}(t, \tau)z(\tau)\|_{{\cal H}^{\theta}_{t}}^{2}\leqslant{\cal M}_{2}(t)\leqslant C\|U_{1}(t, \tau)z(\tau)\|_{{\cal H}^{\theta}_{t}}^{2}.
(3.22)

类似地,我们可得

-2\rho\varepsilon'(t)\langle A^{\theta}v, v_{t}\rangle\geqslant -\frac{\rho}{4}\|v\|_{2\theta}^{2}-\frac{4\rho L^{2}}{\lambda_{1}^{\theta}}\|v_{t}\|_{\theta}^{2}.

\rho足够小,使得

\varepsilon'(t)+3\rho\varepsilon(t)+\frac{4\rho L^{2}}{\lambda_{1}^{\theta}}\leqslant0, \quad \delta-2\rho k_{0}-\rho\geqslant0.

结合以上估计,我们得到

\frac{\rm d}{{\rm d}t}{\cal M}_{2}(t)+\rho\|U_{1}(t, \tau)z(\tau)\|_{{\cal H}^{\theta}_{t}}^2\leqslant\epsilon.

应用Gronwall引理,结合(3.22)式,即得(3.19)式成立.

由以上证明可知,下面的估计式成立

\sup\limits_{t\geqslant\tau}\{\|U(t, \tau)z(\tau)\|_{{\cal H}^{\theta}_{t}}+\|U_{1}(t, \tau)z(\tau)\|_{{\cal H}^{\theta}_{t}}+ \|U_{2}(t, \tau)z(\tau)\|_{{\cal H}^{\theta}_{t}}\}\leqslant C.
(3.23)

证毕.

引理3.3  假设z_2(t)是方程(3.18)关于初值\|z_2(\tau)\|_{\mathcal H^{\theta}_\tau}的解.若条件(1.4)-(1.13)和(3.12)-(3.16)成立,则存在M=M({\mathfrak B})>0,使得

\sup\limits_{t\geqslant\tau}\|z_2(t)\|_{{\cal H}_{t}^{\theta+\sigma}}=\sup\limits_{t\geqslant\tau}\|U_{2}(t, \tau)z_2(\tau)\|_{{\cal H}_{t}^{\theta+\sigma}}\leqslant M,
(3.24)

其中0<\sigma<\min\{1, \theta-\frac{n-4\theta}{2}\gamma, 2\theta-\frac{n-4\theta+2}{2}\gamma, 3\theta-\frac{n}{2}\}.

  因为

2\langle g(u)-g_{0}(v), A^{\theta+\sigma}(w_{t}+\rho w)\rangle =2\langle g(u)-g(v), A^{\theta+\sigma}(w_{t}+\rho w)\rangle+2\langle g_{1}(v), A^{\theta+\sigma}(w_{t}+\rho w)\rangle.

2A^{\theta+\sigma}(w_{t}(t)+\rho w(t))与(3.18)式做内积,可得

\begin{eqnarray}&& \frac{\rm d}{{\rm d}t}(\|w\|^{2}_{2\theta+\sigma}+\varepsilon(t)\|w_{t}\|^{2}_{\theta+\sigma} +\|\xi^{t}\|^{2}_{\mu, 2\theta+\sigma}+2\rho\varepsilon(t)\langle w_{t} , A^{\theta+\sigma}w\rangle+2\langle g(u)-g_0(v), A^{\theta+\sigma} w\rangle \\ & & -2\langle f, A^{\theta+\sigma}w \rangle)+\frac{3\rho}{2}\|w\|^{2}_{2\theta+\sigma}-(\varepsilon^\prime(t)+2\rho\varepsilon(t))\|w_{t}\|^{2}_{\theta+\sigma}+(\delta-2\rho k_{0})\|\xi^{t}\|_{\mu, 2\theta+\sigma}^{2} \\ && -2\rho\varepsilon^\prime(t)\langle w_{t}, A^{\theta+\sigma}w\rangle+2\rho\langle g(u)-g_0(v), A^{\theta+\sigma} w\rangle-2\rho\langle f, A^{\theta+\sigma}w\rangle \\ &\leqslant &2\langle g^\prime(u)u_{t}-g^\prime(v)v_{t}, A^{\theta+\sigma} w\rangle+2\langle g_{1}^\prime(v)v_{t}, A^{\theta+\sigma} w\rangle.\end{eqnarray}
(3.25)

我们对(3.25)式右边的项逐一做估计.首先,由条件(1.13), (3.14)和(3.23)式,有

\begin{eqnarray}2\langle g^\prime(u)u_{t}, A^{\theta+\sigma} w\rangle % \\ &=&2\langle g^{\prime\prime}(u)\nabla uu_{t}, A^{\theta+\sigma-1/2}w\rangle+2\langle g^\prime(u)\nabla u_{t}, A^{\theta+\sigma-1/2}w\rangle \\ &\leqslant &C\bigg(\int_{\Omega}|A^{1/2}u|^{\frac{n}{\theta-\sigma+1}}{\rm d}x\bigg)^{\frac{\theta-\sigma+1}{n}}\bigg(\int_{\Omega} |u_{t}|^{\frac{2n}{n-2\theta}}{\rm d}x\bigg)^{\frac{n-2\theta}{2n}} \\&&\times\bigg(\int_{\Omega}|A^{\theta+\sigma-1/2}w|^{\frac{2n}{n-2(1-\sigma)}}{\rm d}x\bigg)^{\frac{n-2(1-\sigma)}{2n}} \\ & &+C\bigg(\int_{\Omega}(1+|u|^{\gamma})^{\frac{n}{\theta-\sigma}}{\rm d}x\bigg)^{\frac{\theta-\sigma}{n}}\bigg(\int_{\Omega}|A^{1/2}u_{t}|^{\frac{2n}{n-2(\theta-1)}} {\rm d}x\bigg)^{\frac{n-2(\theta-1)}{2n}} \\&&\times\bigg(\int_{\Omega}|A^{\theta+\sigma-1/2}w|^{\frac{2n}{n-2(1-\sigma)}}{\rm d}x\bigg)^{\frac{n-2(1-\sigma)}{2n}} \\ &\leqslant &C\|u\|_{2\theta}^\gamma\|u_{t}\|_{\theta}\|w\|_{2\theta+\sigma}\leqslant \frac{\rho}{8}\|w\|^{2}_{2\theta+\sigma}+C, \end{eqnarray}
(3.26)

其中\frac{n}{\theta-\sigma+1}\leqslant\frac{2n}{n-2(2\theta-1)}, \frac{n\gamma}{\theta-\sigma}\leqslant\frac{2n}{n-4\theta}.同理,有

-2\langle g^\prime(v)v_{t}, A^{\theta+\sigma} w\rangle\leqslant\frac{\rho}{8}\|w\|^{2}_{2\theta+\sigma}+C.

其次,对于(3.25)式右边的第二项,由条件(3.14), (3.16)和(3.23)式,类似于(3.26)式的计算,有

2\langle g_{1}^\prime(v)v_{t}, A^{\theta+\sigma} w\rangle\leqslant\frac{\rho}{8}\|w\|^{2}_{2\theta+\sigma}+C,

取合适的常数C> 0,定义泛函

\begin{eqnarray*} {\cal M}_{3}(t)&=&\|w\|^{2}_{2\theta+\sigma}+\varepsilon(t)\|w_{t}\|^{2}_{\theta+\sigma} +\|\xi^{t}\|^{2}_{\mu, 2\theta+\sigma}+2\rho\varepsilon(t)\langle w_{t} , A^{\theta+\sigma}w \rangle \\& &+2\langle g(u)-g_0(v), A^{\theta+\sigma} w\rangle-2\langle f, A^{\theta+\sigma}w \rangle+ C. \end{eqnarray*}

事实上,由(1.4)和(2.5)式,我们有

2\rho\varepsilon(t)|\langle w_{t} , A^{\theta+\sigma}w \rangle|\leqslant \frac{1}{4}\|w\|_{2\theta+\sigma}^{2}+\frac{\rho L}{\lambda_{1}^{\theta}}\varepsilon(t)\|w_{t}\|_{\theta+\sigma}^{2}.

由条件(1.13),有

\begin{eqnarray*} &&2|\langle g(u)-g_0(v), A^{\theta+\sigma} w\rangle| \\ &=&2|\langle g^\prime(u)A^{1/2}u-g_0^\prime(v)A^{1/2}v, A^{\theta+\sigma-1/2} w\rangle| \\ &\leqslant &C\bigg(\int_{\Omega}(1+|u|^{\gamma}+|v|^{\gamma})^{\frac{n}{2\theta-\sigma}}{\rm d}x\bigg)^{\frac{2\theta-\sigma}{n}} \bigg(\bigg(\int_{\Omega}|A^{1/2}v|^{\frac{2n}{n-2(2\theta-1)}}{\rm d}x\bigg)^{\frac{n-2(2\theta-1)}{2n}} \\ & &+\bigg(\int_{\Omega}|A^{1/2}u|^{\frac{2n}{n-2(2\theta-1)}}{\rm d}x\bigg)^{\frac{n-2(2\theta-1)}{2n}}\bigg) \cdot \bigg(\int_{\Omega}|A^{\theta+\sigma-1/2}w|^{\frac{2n}{n-2(1-\sigma)}}{\rm d}x\bigg)^{\frac{n-2(1-\sigma)}{2n}} \\ &\leqslant &C\|u\|_{2\theta}^\gamma(\|u\|_{2\theta}+\|v\|_{2\theta})\|w\|_{2\theta+\sigma} \leqslant \frac{1}{4}\|w\|^{2}_{2\theta+\sigma}+C, \end{eqnarray*}

其中\frac{n\gamma}{2\theta-\sigma}\leqslant\frac{2n}{n-4\theta}.因此当\rho足够小时,可得

\frac{1}{2}\|U_{2}(t , \tau)z(\tau)\|^{2}_{{\cal H}^{\theta+\sigma}_{t}}\leqslant{\cal M}_{3}(t)\leqslant C\|U_{2}(t , \tau)z(\tau)\|^{2}_{{\cal H}^{\theta+\sigma}_{t}}+C.
(3.27)

再次应用(1.4)和(2.5)式,有

-2\rho(\varepsilon'(t)+\rho\varepsilon(t))\langle w_{t}, A^{\theta+\sigma}w\rangle \geqslant-\frac{\rho}{8}\|w\|_{2\theta+\sigma}^{2}-\frac{8\rho L^{2}}{\lambda_{1}^{\theta}}\|w_{t}\|_{\theta+\sigma}^{2}.

\rho足够小,使得

\varepsilon'(t)+3\rho\varepsilon(t)+\frac{8\rho L^{2}}{\lambda_{1}^{\theta}}\leqslant0, \quad \delta-\rho-2\rho k_{0}\geqslant0.

结合以上估计,我们可推导出

\frac{\rm d}{{\rm d}t}{\cal M}_{3}(t)+ \rho{\cal M}_{3}(t)\leqslant C.

应用Gronwall引理,结合(3.27)式可得(3.24)式成立.证毕.

为了构造紧集的拉回吸引集,我们还需要以下结果验证记忆项的紧性.

引理3.4[1, 7, 12]  假定\mu(s)\in C^{1}({{\mathbb{R}} }^{+})\cap L^{1}({\mathbb{R}}^{+})是一个非负函数,且满足以下条件:如果存在s_{0}\in {\mathbb{R}}^{+},使得\mu(s_{0})=0,那么对所有的s\geqslant s_{0},有\mu(s_{0})=0.进一步,设B_{0}, B_{1}, B_{2}是Banach空间, B_{0}, B_{1}是自反的,且满足B_{0}\hookrightarrow B_{1}\hookrightarrow B_{2},其中嵌入B_{0}\hookrightarrow B_{1}是紧的.设C\subset L_{\mu}^{2}({\Bbb R}^{+};B_{1})满足

(i)   C\subset L_{\mu}^{2}({\mathbb{R}}^{+};B_{0})\cap H_{\mu}^{1}({\mathbb{R}}^{+};B_{2});

(ii)    \sup\limits_{\eta\in C}\|\eta(s)\|_{B_{1}}^{2}\leqslant h(s), \forall\, s\in {\mathbb{R}}^{+}, h(s)\in L_{\mu}^{1}({\mathbb{R}}^{+}).

那么C在空间L_{\mu}^{2}({\mathbb{R}}^{+};B_{1})中是相对紧的.

另外,对任意的\xi^{t}\in L_{\mu}^{2}({\mathbb{R}}^{+};V_{2\theta}), Cauchy问题

\left\{ \begin{array}{l}\xi _t^t = - \xi _s^t + {w_t},\;\;\,t \ge \tau ,\\{\xi ^\tau } = {\xi _\tau }{\rm{ }}\end{array} \right.
(3.28)

有唯一解\xi^{t}\in C([\tau, \infty);L_{\mu}^{2}({{\mathbb{R}} }^{+};V_{2\theta})),因此对方程(3.18),有

\xi^{t}(x, s)=\left\{\begin{array}{ll} w(x, t)-w(x, t-s), &\tau<s<t, \\ w(x, t)-w(\tau), &s\geqslant t.\end{array} \right.
(3.29)

类似于文献[14,引理9]的证明,应用引理3.4容易得到以下结果.

引理3.5  假定条件(1.4)-(1.13)和(3.12)-(3.16)成立,外力项f\in H,对任意给定的T>\tau和任意的\epsilon>0,令

{\cal K}_{T}=\Pi U_{1}(T, \tau){\mathfrak B},

则存在一个正常数N_{1}=N_{1}(\|{\mathfrak B}\|_{{\cal H}^{\theta}_{t}}),使得

(i)    {\cal K}_{T}在空间L_{\mu}^{2}({\mathbb{R}}^{+};V_{2\theta+\sigma})\cap H_{\mu}^{1}({\mathbb{R}}^{+};V_{2\theta})中有界,

(ii)    \sup\limits_{\xi\in {\cal K}_{T}^{\epsilon}}\|\xi(s)\|_{2\theta}^{2}\leqslant N_{1},

其中\{U_{1}(T, \tau)\}_{t\geqslant\tau}是方程(3.18)的解过程, \Pi: V_{2\theta} \times V_{\theta} \times L_{\mu}^{2}({\mathbb{R}}^{+};V_{2\theta})\rightarrow L_{\mu}^{2}({{\mathbb{R}} }^{+};V_{2\theta})为投影算子.

根据引理3.5可知{\cal K}_{T}在空间L_{\mu}^{2}({\Bbb R}^{+};V_{2\theta})上是相对紧的.进一步,应用嵌入V_{2\theta+\sigma}\hookrightarrow V_{2\theta},有

引理3.6  假定条件(1.4)-(1.13)和(3.12)-(3.16)成立, \{U_{2}(t, \tau)\}_{t\geqslant\tau}是方程(3.18)的对应的解过程,则对任意的T>\tau, U_{2}(t, \tau){\mathfrak B}{\cal H}^{\theta}_{t}中是相对紧的.

定理3.3(3.18)假定条件(1.3)-(1.13)和(3.12)-(3.16)成立,则关于方程(2.6)-(2.7)对应过程U(t, \tau):{\cal H}^{\theta}_{\tau}\rightarrow{\cal H}^{\theta}_{t}{\cal H}^{\theta}_{t}中拥有一个不变的时间依赖全局吸引子{\mathfrak A}=\left\{A_{t}\right\}_{t\in {\mathbb{R}} }.

  根据引理3.3和引理3.5,我们可以考虑{\cal K}=\{K_{t}^{\theta+\sigma}\}_{t\in {\mathbb{R}} },其中

K_{t}^{\theta+\sigma}=\{z(t)\in {\cal H}_{t}^{\theta+\sigma}:\|z(t)\|_{{\cal H}_{t}^{\theta+\sigma}}\leqslant M\}.

K_{t}^{\theta+\sigma}{\cal H}^{\theta}_{t}中是紧的.此外, {\cal K}是一致有界的.最后,根据定理3.2,引理3.2,引理3.3和引理3.6,就可证得{\cal K}是拉回吸引的.事实上

\delta_t(U(t, \tau){\Bbb B}_{\tau}(R_{1}), K_{t}^{\theta+\sigma})\leqslant Ce^{-\alpha(t-\tau)}, \quad \forall\, t\geqslant\tau,

因此,过程U(t, \tau)是渐近紧的,这就证明了U(t, \tau)的时间依赖全局吸引子{\mathfrak A}=\left\{A_{t}\right\}_{t\in {\Bbb R}}的存在性.最后,由过程U(t, \tau)_{t\geqslant\tau}的强连续性,我们就能得到{\mathfrak A}的不变性.证毕.

3.4 时间依赖强全局吸引子的正则性

{\cal K}中,对所有的t\in {\mathbb{R}} , {\mathfrak A}的最小性就保证了A_{t}\subset K_{t},从而A_{t}{\cal H}_{t}^{\theta+\sigma}上是有界的,且其界与t无关.进一步,我们还能得到下面的正则性结果.

对任意给定的\tau\in{\mathbb{R}} z(\tau)\in A_{t},我们将U(t, \tau)z(\tau)分解为

U(t, \tau)z(\tau)=z(t)=\left(u(t), u_{t}(t), \eta^{t}(s)\right)=U_{3}(t, \tau)z(\tau)+U_{4}(t, \tau)z(\tau),

其中

U_{3}(t, \tau)z(\tau)=\left(v(t), v_{t}(t), \zeta^{t}(s)\right), \quad U_{4}(t, \tau)z(\tau)=\left(w(t), w_{t}(t), \xi^{t}(s)\right),

分别满足

\left\{\begin{array}{ll} \varepsilon(t) v_{tt}+A^{\theta}v+\int_{0}^{\infty}\mu(s)A^{\theta}\zeta^{t}(s){\rm d}s=0, \\[5pt] \zeta_{t}^{t}=-\zeta_{s}^{t}+v, \\ v(x, t)|_{\partial\Omega}=0, \zeta^{t}(x, t)|_{\partial\Omega}=0, \\ v(x, \tau)=u_{\tau}(x), v_{t}(x, \tau)=u_{t\tau}(x), \zeta^{\tau}(x, s)=\eta_{\tau}(x, s) \end{array} \right.
(3.30)

\left\{\begin{array}{ll} \varepsilon(t) w_{tt}+A^{\theta}w+\int_{0}^{\infty}\mu(s)A^{\theta}\xi^{t}(s){\rm d}s+g(u)=f, \\[5pt] \xi_{t}^{t}=-\xi_{s}^{t}+w, \\ w(x, t)|_{\partial\Omega=0}, \xi^{t}(x, t)|_{\partial\Omega}=0, \\ w(x, \tau)=0, w_{t}(x, \tau)=0, \xi^{\tau}(x, s)=0. \end{array} \right.
(3.31)

作为引理3.2的一个特例,我们可以得到

\|U_{3}(t, \tau)z(\tau)\|_{{\cal H}_{t}^{\theta}}\leqslant C{\rm e}^{-\alpha(t-\tau)}, \quad \forall\, t\geqslant\tau.
(3.32)

引理3.7  假设条件(1.4)-(1.13)成立, z_i(t)是满足\|z_{i}(\tau)\|_{{\cal H}^{\theta}_{\tau}}\leqslant R\|z_{i}(\tau)\|_{{\cal H}^{\theta}_{\tau}}\leqslant R, i=1, 2关于问题(2.6)-(2.7)的解.则存在N_{1}=N_{1}({\mathfrak A})>0,使得以下估计成立

\sup\limits_{t\geqslant\tau}\|U_{4}(t, \tau)z(\tau)\|_{{\cal H}_{t}^{\theta+\tilde{\sigma}}}\leqslant N_{1} ,

其中\sigma<\tilde{\sigma}<\min\{1, 3\theta+2\sigma-\frac{n}{2}, \theta+\sigma-\frac{n-4\theta-2\sigma}{2}\gamma\}.

  用2A^{\theta+\tilde{\sigma}}(w_{t}(t)+\rho w(t))与方程(3.31)做内积,有

\begin{eqnarray}&&\frac{\rm d}{{\rm d}t}(\|w\|_{2\theta+\tilde{\sigma}}^{2}+\varepsilon(t)\|w_{t}\|_{\theta+\tilde{\sigma}}^{2}+\|\xi\|_{\mu, 2\theta+\tilde{\sigma}}^{2} +2\rho\varepsilon(t)\langle w_{t} , A^{\theta+\tilde{\sigma}}w \rangle+2\langle g(u), A^{\theta+\tilde{\sigma}} w\rangle \\ && -2\langle f, A^{\theta+\tilde{\sigma}}w \rangle)+\frac{3}{2}\rho\|w\|^{2}_{2\theta+\tilde{\sigma}}-(\varepsilon^\prime(t)+2\rho\varepsilon(t))\|w_{t}\|^{2}_{\theta+\tilde{\sigma}}+(\delta-2\rho k_{0})\|\xi^{t}\|^{2}_{\mu, 2\theta+\tilde{\sigma}} \\ &&+2\rho\langle g(u), A^{\theta+\tilde{\sigma}}w\rangle-2\rho\varepsilon^\prime(t)\langle w_{t}, A^{\theta+\tilde{\sigma}}w\rangle-2\rho\langle f, A^{\theta+\tilde{\sigma}}w \rangle\leqslant2\langle g^\prime(u)u_{t}, A^{\theta+\tilde{\sigma}} w\rangle. \end{eqnarray}
(3.33)

取合适的常数C>0,使得

\begin{eqnarray*} 0\leqslant{\cal M}_{4}(t)&=& \|w\|_{2\theta+\tilde{\sigma}}^{2}+\varepsilon(t)\|w_{t}\|_{\theta+\tilde{\sigma}}^{2}+\|\xi\|_{\mu, 2\theta+\tilde{\sigma}}^{2}+2\rho\varepsilon(t)\langle w_{t} , A^{\tilde{\theta+\sigma}}w \rangle \\& &+2\langle g(u), A^{\theta+\tilde{\sigma}} w\rangle-2\langle f, A^{\theta+\tilde{\sigma}}w \rangle+C.\end{eqnarray*}

类似于(3.27)式,对足够小的\rho,我们有

\frac{1}{2}\|U_{4}(t, \tau)z(\tau)\|^{2}_{{\cal H}^{\theta+\tilde{\sigma}}_{t}} \leqslant {\cal M}_{4}(t)\leqslant C\|U_{4}(t, \tau)z(\tau)\|^{2}_{{\cal H}^{\theta+\tilde{\sigma}}_{t}}+C.
(3.34)

由吸引子{\mathfrak A}的不变性,我们有

\|U(t, \tau)z(\tau)\|_{{\cal H}_{t}^{\theta+\sigma}}\leqslant C,

其中C > 0是在 {\cal H}^{\theta+\sigma}_{t}中与A_{t}的界有关的常数.对于(3.33)式的右边,有

\begin{eqnarray*} 2\langle g^\prime(u)u_{t}, A^{\theta+\tilde{\sigma}} w\rangle &=&2\langle g^{\prime\prime}(u)\nabla uu_{t}, A^{\theta+\tilde{\sigma}-1/2}w\rangle+2\langle g^\prime(u)\nabla u_{t}, A^{\theta+\tilde{\sigma}-1/2}w\rangle \\ &\leqslant &C\bigg(\int_{\Omega}|A^{1/2}u|^{\frac{n}{\theta+\sigma-\tilde{\sigma}+1}}{\rm d}x\bigg)^{\frac{\theta+\sigma-\tilde{\sigma}+1}{n}}\bigg(\int_{\Omega} |u_{t}|^{\frac{2n}{n-2(\theta+\sigma)}}{\rm d}x\bigg)^{\frac{n-2(\theta+\sigma)}{2n}}\\&& \times\bigg(\int_{\Omega}|A^{\theta+\tilde{\sigma}-1/2}w|^{\frac{2n}{n-2(1-\tilde{\sigma})}}{\rm d}x\bigg)^{\frac{n-2(1-\tilde{\sigma})}{2n}} \\ & &+C\bigg(\int_{\Omega}(1+|u|^{\gamma})^{\frac{n}{\theta+\sigma-\tilde{\sigma}}}{\rm d}x\bigg)^{\frac{\theta+\sigma-\tilde{\sigma}}{n}}\bigg(\int_{\Omega}|A^{1/2}u_{t}|^{\frac{2n}{n-2(\theta+\sigma-1)}} {\rm d}x\bigg)^{\frac{n-2(\theta+\sigma-1)}{2n}} \\ && \times\bigg(\int_{\Omega}|A^{\theta+\tilde{\sigma}-1/2}w|^{\frac{2n}{n-2(1-\tilde{\sigma})}}{\rm d}x\bigg)^{\frac{n-2(1-\tilde{\sigma})}{2n}} \\ &\leqslant &C(\|u\|_{2\theta+\sigma}+\|u\|_{2\theta+\sigma}^\gamma)\|u_{t}\|_{\theta+\sigma}\|w\|_{2\theta+\tilde{\sigma}} \\ &\leqslant&\frac{\rho}{8}\|w\|^{2}_{2\theta+\tilde{\sigma}}+C, \end{eqnarray*}

其中\frac{n}{\theta+\sigma-\tilde{\sigma}+1}\leqslant\frac{2n}{n-2(2\theta+\sigma-1)}, \frac{n\gamma}{\theta+\sigma-\tilde{\sigma}}\leqslant\frac{2n}{n-2(2\theta+\sigma)}.结合以上估计,我们可推导出

\frac{\rm d}{{\rm d}t}{\cal M}_{3}(t)+\rho{\cal M}_{3}(t)\leqslant C.

应用Gronwall引理,结合(3.34)式,我们证得\|U_{4}(t, \tau)z(\tau)\|_{{\cal H}_{t}^{\theta+\tilde{\sigma}}}是一致有界的.证毕.

定理3.4  如果引理3.1的假设条件成立,则A_{t}{\cal H}_{t}^{\theta+\tilde{\sigma}}中是有界的(且其界与t无关).

  令

K^{\theta+\tilde{\sigma}}_{t}=\{z(t)\in {\cal H}^{\theta+\tilde{\sigma}}_{t}: \|z(t)\|_{{\cal H}^{\theta+\tilde{\sigma}}_{t}} \leqslant N_{1} \},

由不等式(3.32)和引理3.7,对\forall\, t\in {\mathbb{R}} ,有

\lim\limits_{\tau \rightarrow -\infty} {\rm dist}(U(t, \tau)A_{\tau}, K^{\theta+\tilde{\sigma}}_{t})=0,

从而由{\mathfrak A}的不变性,有

{\rm dist}(A_{t}, K^{\theta+\tilde{\sigma}}_{t})=0.

因此, A_{t}\subset \overline{{K^{\theta+\tilde{\sigma}}_{t}}}=K^{\theta+\tilde{\sigma}}_{t},即证明了A_{t}{\cal H}^{\theta+\tilde{\sigma}}_{t}中是有界的,并且其界与t\in {\mathbb{R}} 无关.证毕.

参考文献

Borini S , Pata V .

Uniform attractors for a strongly damped wave equations with linear memory

Asymptot Anal, 1999, 20: 263- 277

URL     [本文引用: 1]

Coleman B D , Noll W .

Foundations of linear viscoelasticity

Rev Mod Phys, 1961, 33: 239- 249

DOI:10.1103/RevModPhys.33.239      [本文引用: 1]

Conti M , Pata V , Temam R .

Attractors for processes on time-dependent spaces and applications to wave equation

J Differ Equations, 2013, 255: 1254- 1277

DOI:10.1016/j.jde.2013.05.013      [本文引用: 2]

Conti M , Pata V .

Asymptotic structure of the attractor for processes on time-dependent spaces

Nonlinear Anal, 2014, 19: 1- 10

DOI:10.1016/j.nonrwa.2014.02.002      [本文引用: 2]

Dafermos C M .

Asymptotic stability in viscoelasticity

Arch Ration Mech Anal, 1970, 37: 297- 308

DOI:10.1007/BF00251609      [本文引用: 1]

Fabrizio M , Morro A . Mathematical Problems in Linear Viscoelasticity. Philadelphia: SAIM Stud Appl Math, 1992

[本文引用: 1]

Gatti C , Miranville A , Pata V , Zelik S V .

Attractors for semilinear equations of viscoelasticity with very low dissipation

R Mountain J Math, 2008, 38: 1117- 1138

DOI:10.1216/RMJ-2008-38-4-1117      [本文引用: 1]

胡弟弟, 汪璇.

记忆型抽象发展方程时间依赖吸引子的存在性

华东师范大学学报(自然科学版), 2018, 2018: 35- 49

DOI:10.3969/j.issn.1000-5641.2018.01.005      [本文引用: 4]

Hu D D , Wang X .

The existence of time-dependent attractors for abstract evolution equations with fading memory

J East China Normal Univ (Nat Sci), 2018, 2018: 35- 49

DOI:10.3969/j.issn.1000-5641.2018.01.005      [本文引用: 4]

Ma Q Z , Wang X P , Xu L .

Existence and regularity of time-dependent global attractos for the nonclassical reaction-diffusion equations with lower forcing term

Bound Value Probl, 2016, 2016: 1- 11

DOI:10.1186/s13661-015-0477-3      [本文引用: 1]

Meng F J , Yang M H , Zhong C K .

Attractors for wave equations with nonlinear damping on time-dependent space

Discrete Contin Dyn Syst, 2017, 21: 205- 225

URL     [本文引用: 2]

Meng F J , Liu C C .

Necessary and sufficient conditions for the existence of time-dependent global attractor and application

J Math Phys, 2017, 58: 032702

DOI:10.1063/1.4978329      [本文引用: 2]

Pata V , Zucchi A .

Attractors for a damped hyperbolic equation with linear memory

Adv Math Sci Appl, 2001, 260: 505- 529

URL     [本文引用: 1]

Temam R . Infinit Dimensional Dynamical System in Mechanichs and Physics. New York: Springer-Verlag, 1997

[本文引用: 1]

Wang X , Duan F X , Hu D D .

Attractors for a class of abstract evolution equations with fading memory

Math Probl Eng, 2017, 2017: 1- 16

URL     [本文引用: 2]

Wang X , Yang L , Zhong C K .

Attractors for the nonclassical diffusion equations with fading memory

J Math Anal Appl, 2010, 362: 327- 337

DOI:10.1016/j.jmaa.2009.09.029      [本文引用: 1]

张玉宝, 汪璇.

无阻尼弱耗散抽象发展方程的强全局吸引子

华东师范大学学报(自然科学版), 2017, 2017: 8- 19

DOI:10.3969/j.issn.1000-5641.2017.02.002      [本文引用: 1]

Zhang Y B , Wang X .

The strong global attractors for the non-damping weak diffusion abstract evolution equations

J East China Normal Univ (Nat Sci), 2017, 2017: 8- 19

DOI:10.3969/j.issn.1000-5641.2017.02.002      [本文引用: 1]

/