数学物理学报, 2020, 40(5): 1341-1353 doi:

论文

带有时滞项的复Ginzburg-Landau方程的拉回吸引子

朱凯旋,1, 谢永钦,2, 周峰,3, 邓习军,1

Pullback Attractors for the Complex Ginzburg-Landau Equations with Delays

Zhu Kaixuan,1, Xie Yongqin,2, Zhou Feng,3, Deng Xijun,1

通讯作者: 朱凯旋, E-mail: zhukx12@163.com

收稿日期: 2019-02-27  

基金资助: 国家自然科学基金.  11601522
中央高校基础研究基金.  17CX02036A
湖南省自然科学基金基金.  2018JJ2416
湖南省自然科学基金基金.  2018JJ2272
湖南文理学院博士科研启动基金.  16BSQD04
湖南文理学院博士科研启动基金.  16BSQD13

Received: 2019-02-27  

Fund supported: the NSFC.  11601522
the Fundamental Research Funds for the Central Universities.  17CX02036A
the NSF of Hunan Province.  2018JJ2416
the NSF of Hunan Province.  2018JJ2272
the Doctoral Research Fund of Hunan University of Arts and Science.  16BSQD04
the Doctoral Research Fund of Hunan University of Arts and Science.  16BSQD13

作者简介 About authors

谢永钦,E-mail:xieyq@csust.edu.cn , E-mail:xieyq@csust.edu.cn

周峰,E-mail:zhoufeng13@upc.edu.cn , E-mail:zhoufeng13@upc.edu.cn

邓习军,E-mail:xijundeng@126.com , E-mail:xijundeng@126.com

摘要

该文考虑带有时滞项的复Ginzburg-Landau方程解的适定性和拉回吸引子的存在性,其中非线性项满足任意$p-1$$p$>2)次多项式增长.利用收缩函数方法验证解过程$\{U(t,\tau)\}_{t\geq\tau}$的紧性,得到$C_{L^{2}(\Omega)}$中拉回吸引子的存在性.

关键词: 复Ginzburg-Landau方程 ; 时滞 ; 拉回吸引子

Abstract

In this paper, we consider the complex Ginzburg-Landau equations with hereditary effects and the nonlinear term satisfying the polynomial growth of arbitrary $p-1$$(p>2)$ order. We analyze the well-posedness of solutions and prove the existence of the pullback attractors in $C_{L^{2}(\Omega)}$ by applying the contractive functions method.

Keywords: Complex Ginzburg-Landau equations ; Delays ; Pullback attractors

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本文引用格式

朱凯旋, 谢永钦, 周峰, 邓习军. 带有时滞项的复Ginzburg-Landau方程的拉回吸引子. 数学物理学报[J], 2020, 40(5): 1341-1353 doi:

Zhu Kaixuan, Xie Yongqin, Zhou Feng, Deng Xijun. Pullback Attractors for the Complex Ginzburg-Landau Equations with Delays. Acta Mathematica Scientia[J], 2020, 40(5): 1341-1353 doi:

1 引言

在物理、化学过程,工程系统,生物系统和通讯系统等领域,时滞微分方程起着重要的作用并受到人们的广泛关注,近年来已成为研究的热点问题,参见文献[18-19].在数学领域,人们主要研究其解的适定性和长时间行为.对于时滞微分方程解的长时间行为的研究,已有一些结果,参见文献[6-11, 13-17, 21-22, 26-27, 32-33]及其参考文献.

基于上述情况,本文研究下述带有时滞项的复Ginburg-Landau方程

$ \begin{equation} \left\{\begin{array}{ll} \partial_{t}u-(\lambda+{\rm i}\alpha)\Delta u+(k+{\rm i}\beta)|u|^{p-2}u-\gamma u = g(t, u_{t})+f(t), \; &x\in\Omega, t\in(\tau, \infty), \\ u(x, \, t) = 0, &x\in\partial\Omega, t\in(\tau, \infty), \\ u(x, \, \tau+\theta) = \phi(x, \, \theta), &x\in\Omega, \; \theta\in[-h, 0], \end{array}\right. \end{equation} $

其中$ \Omega\subset{{\Bbb R}} ^N(N\geq3) $是有界光滑区域, $ i = \sqrt{-1} $, $ \lambda > 0 $, $ k > 0 $, $ \alpha, \beta\in{{\Bbb R}} $, $ \gamma > 0 $, $ g(\cdot, \cdot) $是作用在带有某种遗传特征的解上的算子(对于算子$ g(\cdot, \cdot) $的假设将在后面给出),依赖于时间的外力$ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $,初值$ \phi\in C([-h, 0]; L^{2}(\Omega)) $, $ h(> 0) $是时滞影响的长度.对于任意的$ t\geq\tau $, $ \tau\in{{\Bbb R}} $,我们用$ u_{t} $表示定义在$ [-h, 0] $上的函数$ u_{t}(\theta) = u(t+\theta) $, $ \theta\in[-h, 0] $.

对于算子$ g(\cdot, \cdot) $,参见文献[13-14],我们假设$ g(\cdot, \cdot): {{\Bbb R}} \times C_{L^{2}(\Omega)}\rightarrow L^{2}(\Omega) $,且

(Ⅰ) $ \forall \xi\in C_{L^{2}(\Omega)} $,函数$ t\in{{\Bbb R}} \mapsto g(t, \xi)\in L^{2}(\Omega) $可测;

(Ⅱ) $ \forall t\in{{\Bbb R}} $, $ g(t, 0) = 0 $;

(Ⅲ) $ \forall t\in{{\Bbb R}} $$ \forall\xi, \eta\in C_{L^{2}(\Omega)} $, $ \exists L_{g} > 0 $使得$ \|g(t, \xi)-g(t, \eta)\|_{2}\leq L_{g}\|\xi-\eta\|_{C_{L^{2}(\Omega)}}. $

分别用$ (\cdot, \cdot) $$ \|\cdot\|_{2} $表示$ L^{2}(\Omega) $中的内积和模, $ \|\cdot\|_{p} $表示$ L^{p}(\Omega) $中的模, $ ((\cdot, \cdot)) = (\nabla\cdot, \nabla\cdot) $$ \|\nabla\cdot\|_{2} $分别表示$ H_{0}^{1}(\Omega) $中的内积和模.用$ C_{X} $表示Banach空间$ C([-h, 0]; X) $,并赋予上确界模.对于任意的$ u\in C_{X} $,它的模表示为$ { }\|u\|_{C_{X}} = \max\limits_{t\in[-h, 0]}\|u(t)\|_{X} $. $ L^{p}(\Omega) $和它的共轭$ L^{p^{*}}(\Omega) $之间的内积也记为$ (\cdot, \cdot) $,其中$ p^{*} = p/(p-1) $ ($ 1 < p < \infty $).

复Ginzburg-Landau方程常作为一种重要的模型来描述非平衡流体动力系统中空间格局的形成或不稳定性的振幅演化(从平面Poiseuille叶流中Tollmien-Schlichting波的发展到反向旋转圆柱之间流动中Taylor涡的出现,以及对化学系统的研究)以及相变理论和超导现象,参见文献[12, 28-29].作为其特殊情形,非线性schrödinger方程是近年来被研究的具有广义非线性项的方程.因此,在理论物理和数学领域中的复Ginzburg-Landau方程日益受到数学家们的重视.

对于不含时滞项的复Ginzburg-Landau方程,其解的长时间行为已被许多学者进行研究,参见文献[1, 23, 25, 35-36]. Chepyzhov等[1]主要考虑周期边值条件和非自治情形; Kapustyan等[23]主要考虑解不唯一的复Ginzburg-Landau方程; Li等[25]证明了复Ginzburg-Landau方程在$ L^{2}(\Omega) $, $ L^{p}(\Omega) $ ($ p\geq2 $)$ H_{0}^{1}(\Omega) $中全局吸引子的存在性; Zhou等[35-36]考虑变区域上的复Ginzburg-Landau方程,证明了$ L^{2}(\Omega) $中拉回吸引子的存在性.

本文考虑带有时滞项的复Ginzburg-Landau方程解的适定性和拉回吸引子的存在性,并假设

$ \begin{eqnarray} \left(\frac{\alpha}{\lambda}, \frac{\beta}{k}\right)\in S\left(\frac{1}{c_{p}}, \frac{1}{c_{p}}\right), \end{eqnarray} $

其中$ S(x_{0}, y_{0}) = \{(x, y)\in{{\Bbb R}} ^{2}:\; |x|\leq x_{0}, |y|\leq y_{0}\} $$ c_{p} = \frac{p-2}{2\sqrt{p-1}} $, $ p > 2 $.

对于本文所考虑的问题,当验证解过程$ \{U(t, \tau)\}_{t\geq\tau} $的紧性时有两个主要困难.其一是方程(1.1)定义在复数域而不是实数域上,在复数域上一些(实数域上)已有的结果不再成立.例如,当我们对方程(1.1)做$ L^{2} $ -内积时,所选取的检验函数是$ u $的共轭$ \overline{u} $而不是$ u $本身.其二是本文中的问题带有时滞项$ g(t, u_{t}) $,这使得我们所研究问题的相空间是Banach空间$ C_{X} $而不是Banach空间$ X $.在Banach空间$ C_{X} $,已有的验证解过程$ \{U(t, \tau)\}_{t\geq\tau} $紧性的方法和技巧不再成立.为了克服上述困难,我们运用收缩函数的思想(参见文献[5, 20, 31, 34, 37])验证过程$ \{U(t, \tau)\}_{t\geq\tau} $的紧性,从而得到$ C_{L^{2}(\Omega)} $中拉回吸引子的存在性.

2 预备知识

本节首先给出拉回吸引子的一些基本概念和结果,参见文献[2-4].

$ \{U(t, \tau)\}_{t\geq \tau} $是度量空间$ X $上的过程(或双参数半群),即,一族映射$ U(t, \tau):X\rightarrow X $,满足$ U(\tau, \tau)x = x $, $ \forall x\in X $$ U(t, \tau) = U(t, s)U(s, \tau), \ \forall\tau\leq s\leq t. $

$ {\cal D} $为由非空参数集$ \widehat{D} = \{D(t); \; t\in {{\Bbb R}} \}\subset {\cal P}(X) $所组成的集类,其中$ {\cal P}(X) $表示$ X $的所有非空子集所组成的集族.

定义2.1 若对任意的$ t\in{{\Bbb R}} $, $ \widehat{D}\in{\cal D} $, $ \tau_{n}\rightarrow-\infty $和序列$ \{x_{n}\}_{n = 1}^{\infty}\subset D(\tau_{n}) $,序列$ \{U(t, \tau_{n})x_{n}\}_{n = 1}^{\infty} $$ X $中存在收敛子列,则称过程$ \{U(t, \tau)\}_{t\geq\tau} $拉回渐近紧.

定义2.2 若对任意的$ t\in{{\Bbb R}} $$ \widehat{D}\in{\cal D} $,存在$ \tau_{0} = \tau_{0}(t, \widehat{D})\leq t $使得

则称$ \widehat{B}\in {\cal D} $是过程$ \{U(t, \tau)\}_{t\geq\tau} $的拉回$ {\cal D} $ -吸收集.

定义2.3 若集族$ \widehat{{\cal A}} = \{{\cal A}(t); \; t\in{{\Bbb R}} \}\subset{\cal P}(X) $满足

$ (1) $对任意的$ t\in{{\Bbb R}} $, $ {\cal A}(t) $$ X $中紧,

$ (2) $对任意的$ t\in{{\Bbb R}} $$ \widehat{D}\in{\cal D} $, $ \widehat{{\cal A}} $$ X $中拉回$ {\cal D} $ -吸引,即

$ (3) $对任意的$ -\infty < \tau\leq t < +\infty $, $ \widehat{{\cal A}} $是不变的,即, $ U(t, \tau){\cal A}(\tau) = {\cal A}(t) $,则称$ \widehat{{\cal A}} = \{{\cal A}(t); \; t\in{{\Bbb R}} \} $是过程$ \{U(t, \tau)\}_{t\geq \tau} $$ X $中的拉回$ {\cal D} $ -吸引子.

而且, $ \widehat{{\cal A}} = \{{\cal A}(t); \; t\in{{\Bbb R}} \} $具有最小性;即,如果$ \widehat{C} = \{C(t); \; t\in{{\Bbb R}} \} $是由$ X $的闭子集$ C(t) $所组成的非空集族且满足

$ {\cal A}(t)\subset C(t) $.

下述定义和定理将用来证明拉回吸引子的存在性,参见文献[5, 20, 31, 34, 37].

定义2.4 设$ (X, \|\cdot\|_{X}) $是Banach空间, $ \widehat{B} = \{B(t); \; t\in{{\Bbb R}} \} $$ X $的子集族, $ \psi(\cdot, \cdot) $是定义在$ X\times X $上的函数.若对任意的序列$ \{x_{n}\}_{n = 1}^{\infty}\subset B(t) $,存在子列$ \{x_{n_{k}}\}_{k = 1}^{\infty}\subset\{x_{n}\}_{n = 1}^{\infty} $使得

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

$ B(t)\times B(t) $上收缩函数组成的集合为Contr$ (B(t)) $.

定理2.1 设Banach空间$ X $中的过程$ \{U(t, \tau)\}_{t\geq \tau} $存在拉回$ {\cal D} $ -吸收集$ \widehat{B} = \{B(t); \; t\in{{\Bbb R}} \} $.若对任给的$ \varepsilon > 0 $, $ t\in{{\Bbb R}} $,存在$ T = T(t, \widehat{B}, \varepsilon) = t-\tau $和依赖于$ t $$ T $的函数$ \psi_{t, T}(\cdot, \cdot)\in {\rm Contr}(B(t)) $使得

$ \{U(t, \tau)\}_{t\geq \tau} $$ X $中拉回$ {\cal D} $ -渐近紧.

定理2.2 设Banach空间$ X $中的连续过程$ \{U(t, \tau)\}_{t\geq\tau} $满足

(ⅰ) $ \{U(t, \tau)\}_{t\geq\tau} $$ X $中存在拉回$ {\cal D} $ -吸收集$ \widehat{B}_{0} $,

(ⅱ) $ \{U(t, \tau)\}_{t\geq \tau} $$ \widehat{B}_{0} $中拉回$ {\cal D} $ -渐近紧,则过程$ \{U(t, \tau)\}_{t\geq \tau} $$ X $中存在拉回$ {\cal D} $ -吸引子.

引理2.1[30] 设$ q\in(1, \infty) $,则对任意非零且互不相等的$ z, w\in C $,有

引理2.2[30]  (ⅰ)设$ q\in[2, +\infty) $,则对任意的$ z, w\in C $,有

(ⅱ)设$ q\in(1, 2) $,则对任意非零的$ z, w\in C $,有

3 解的适定性

本节我们利用Faedo-Galerkin方法证明方程(1.1)解的适定性,下面首先定义弱解.

定义3.1 若函数$ u\in C([\tau-h, T]; L^{2}(\Omega))\cap L^{2}(\tau, T; H_{0}^{1}(\Omega)) \cap L^{p}(\tau, T; L^{p}(\Omega)) $满足当$ t\in[\tau-h, \tau] $时, $ u(t) = \phi(t-\tau) $,且对任意的$ \varphi\in H_{0}^{1}(\Omega)\cap L^{p}(\Omega) $,有

$ {\cal D}'(\tau, \infty) $中成立,则称$ u $为方程(1.1)的弱解.

定理3.1 设$ g(\cdot, \cdot) $满足假设(Ⅰ)–(Ⅲ), $ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C_{L^{2}(\Omega)} $,则对任意的$ \tau\in{{\Bbb R}} $$ T > \tau $,方程(1.1)存在弱解$ u(\cdot) = u(\cdot; \tau, \phi) $.

 设$ \{w_{j}\}_{j\geq 1}\subset H_{0}^{1}(\Omega)\cap L^{p}(\Omega) $$ L^{2}(\Omega) $中的Hilbert基且span$ \{w_{j}\}_{j\geq 1} $$ H_{0}^{1}(\Omega)\cap L^{p}(\Omega) $中稠密.设$ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C([-h, 0];\; \mbox{span}\{w_{j}\}_{j = 1}^{n}) $,给定$ n\in{\Bbb N} $ (特别地, $ \phi(0)\in H_{0}^{1}(\Omega)\cap L^{p}(\Omega) $).

为了证明解的存在性,设$ \widetilde{u}^{m}(t, x) = \sum\limits_{j = 1}^{m}\gamma_{mj}(t)\omega_{j}(x) $且满足下述逼近系统

$ \begin{eqnarray} \left\{\begin{array}{ll} { } \frac{\rm d}{{\rm d}t}(\widetilde{u}^{m}(t), \overline{\omega_{j}}) -(\lambda+{\rm i}\alpha)(\Delta\widetilde{u}^{m}(t), \overline{\omega_{j}}) +(k+{\rm i}\beta)(|\widetilde{u}^{m}(t)|^{p-2}\widetilde{u}^{m}(t), \overline{\omega_{j}}) \\ {\qquad}{\qquad}{\qquad}\, -\gamma(\widetilde{u}^{m}(t), \overline{\omega_{j}}) = (g(t, \widetilde{u}^{m}_{t}), \overline{\omega_{j}})+(f(t), \overline{\omega_{j}}), \\ \widetilde{u}_{\tau}^{m} = \phi, \end{array}\right. \end{eqnarray} $

其中$ m\geq n $, $ 1\leq j\leq m $, $ \overline{\omega_{j}} $$ \omega_{j} $的共轭.

众所周知上述有限维时滞系统是适定的,参见文献[19].我们将给出先验估计来说明对任意的$ T > \tau $,其解在每个区间$ [\tau-h, T] $上的存在性.

步骤1 第一先验估计.

在(3.1)式两边乘以$ \overline{\gamma_{mj}(t)} $并对$ j = 1 $$ m $求和,可得

其中$ \overline{\gamma_{mj}(t)} $$ \overline{\widetilde{u}^{m}(t)} $分别是$ \gamma_{mj}(t) $$ \widetilde{u}^{m}(t) $的共轭.

由假设(Ⅱ)–(Ⅲ), Hölder不等式和Young不等式得

进而

$ \begin{eqnarray} &&\frac{\rm d}{{\rm d}t}\|\widetilde{u}^{m}(t)\|_{2}^{2} +2\lambda\|\nabla\widetilde{u}^{m}(t)\|_{2}^{2}+2k\|\widetilde{u}^{m}(t)\|_{p}^{p} \\ &\leq& (1+2L_{g}+2\gamma)\|\widetilde{u}^{m}_{t}\|_{C_{L^{2}(\Omega)}}^{2} +\|f(t)\|_{2}^{2}, \quad\; {\rm a.e.}\; t>\tau. \end{eqnarray} $

对(3.2)式关于时间$ t $$ [\tau, t] $上积分,可得

$ \begin{eqnarray} &&\|\widetilde{u}^{m}(t)\|_{2}^{2} +2\lambda\int_{\tau}^{t}\|\nabla\widetilde{u}^{m}(s)\|_{2}^{2}{\rm d}s +2k\int_{\tau}^{t}\|\widetilde{u}^{m}(s)\|_{p}^{p}{\rm d}s \\ &\leq&\|\widetilde{u}^{m}(\tau)\|_{2}^{2} +(1+2L_{g}+2\gamma)\int_{\tau}^{t}\|\widetilde{u}^{m}_{s}\|_{C_{L^{2}(\Omega)}}^{2}{\rm d}s +\int_{\tau}^{t}\|f(s)\|_{2}^{2}{\rm d}s, \quad\forall t>\tau. \end{eqnarray} $

特别地,用$ t+\theta $代替$ t $,其中$ \theta\in [-h, 0] $,可得

利用Gronwall引理可得

$ \begin{eqnarray} \|\widetilde{u}^{m}_{t}\|_{C_{L^{2}(\Omega)}}^{2} \leq \left(\|\phi\|_{C_{L^{2}(\Omega)}}^{2}+\int_{\tau}^{t}\|f(s)\|_{2}^{2}{\rm d}s\right) e^{(1+2L_{g}+2\gamma)(t-\tau)}, \quad\forall t\geq\tau. \end{eqnarray} $

综合(3.3)式和(3.4)式可知,对任意的$ T > \tau $,有

$ \begin{eqnarray} \{\widetilde{u}^{m}\}_{m\geq n}\ \mbox{在}\ L^{\infty}(\tau, T; L^{2}(\Omega))\cap L^{2}(\tau, T;H_{0}^{1}(\Omega))\cap L^{p}(\tau, T;L^{p}(\Omega))\ \mbox{中有界.} \end{eqnarray} $

从而,存在$ \widetilde{u}\in L^{\infty}(\tau, T; L^{2}(\Omega)) \cap L^{2}(\tau, T; H_{0}^{1}(\Omega))\cap L^{p}(\tau, T; L^{p}(\Omega)) $$ \{\widetilde{u}^{m}\} $的子列(仍记作$ \{\widetilde{u}^{m}\} $)使得

$ \begin{eqnarray} \left\{\begin{array}{ll} \widetilde{u}^{m}\; \mbox{在}\; L^{\infty}(\tau, T; L^{2}(\Omega)) \; \mbox{中弱$\star$ -收敛到}\; \widetilde{u}, \\ \widetilde{u}^{m}\; \mbox{在}\; L^{2}(\tau, T; H_{0}^{1}(\Omega))\; \mbox{中弱收敛到}\; \widetilde{u}, \\ \widetilde{u}^{m}\; \mbox{在} \; L^{p}(\tau, T; L^{p}(\Omega))\; \mbox{中弱收敛到}\; \widetilde{u}, \\ |\widetilde{u}^{m}|^{p-2}\cdot\widetilde{u}^{m} \; \mbox{在}\; L^{p^{\ast}}(\tau, T; L^{p^{\ast}}(\Omega))\; \mbox{中弱收敛到}\; |\widetilde{u}|^{p-2}\cdot\widetilde{u}. \end{array}\right. \end{eqnarray} $

步骤2  关于时间导数的一致估计.

在(3.1)式两边乘以$ \overline{\gamma'_{mj}(t)} $并对$ j = 1 $$ m $求和,可得

其中$ \overline{(\widetilde{u}^{m})'(t)} $$ (\widetilde{u}^{m})'(t) $的共轭.

进而,利用假设(Ⅱ)–(Ⅲ), Hölder和Young不等式可得

$ \begin{eqnarray} &&\|(\widetilde{u}^{m})'(t)\|_{2}^{2} +\lambda\frac{\rm d}{{\rm d}t}\|\nabla\widetilde{u}^{m}(t)\|_{2}^{2} +\frac{2k}{p}\frac{\rm d}{{\rm d}t}\|\widetilde{u}^{m}(t)\|_{p}^{p} -\gamma\frac{\rm d}{{\rm d}t}\|\widetilde{u}^{m}(t)\|_{2}^{2} \\ &\leq& 2L^{2}_{g}\|\widetilde{u}_{t}^{m}\|_{C_{L^{2}(\Omega)}}^{2}+2\|f(t)\|_{2}^{2}, \quad {\rm a.e.} \; t>\tau. \end{eqnarray} $

对(3.7)式关于时间$ t $$ [\tau, t] $上积分,可得

$ \begin{eqnarray} &&\lambda\|\nabla\widetilde{u}^{m}(t)\|_{2}^{2} +\frac{2k}{p}\|\widetilde{u}^{m}(t)\|_{p}^{p}-\gamma\|\widetilde{u}^{m}(t)\|_{2}^{2} +\int_{\tau}^{t}\|(\widetilde{u}^{m})'(s)\|_{2}^{2}{\rm d}s \\ &\leq&\lambda\|\nabla\widetilde{u}^{m}(\tau)\|_{2}^{2} +\frac{2k}{p}\|\widetilde{u}^{m}(\tau)\|_{p}^{p}-\gamma\|\widetilde{u}^{m}(\tau)\|_{2}^{2} +2L^{2}_{g}\int_{\tau}^{t}\|\widetilde{u}_{s}^{m}\|_{C_{L^{2}(\Omega)}}^{2}{\rm d}s \\ &&+2\int_{\tau}^{t}\|f(s)\|_{2}^{2}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray} $

由于对任意的$ m\geq n $, $ \widetilde{u}_{\tau}^{m} = \phi $$ \widetilde{u}^{m}(\tau) = \phi(0)\in H_{0}^{1}(\Omega)\cap L^{p}(\Omega) $,综合(3.4)式和(3.8)式可知对任意的$ T > \tau $,有

$ \begin{eqnarray} &&\{\widetilde{u}^{m}(t)\}_{m\geq n}\ \mbox{在}\ L^{\infty}(\tau, T; H_{0}^{1}(\Omega)\cap L^{p}(\Omega))\ \mbox{中有界, } \end{eqnarray} $

$ \begin{eqnarray} &&\{(\widetilde{u}^{m})'(t)\}_{m\geq n}\ \mbox{在}\ L^{2}(\tau, T; L^{2}(\Omega))\ \mbox{中有界.} \end{eqnarray} $

从而,存在$ \widetilde{u} $ (步骤1中的$ \widetilde{u} $)使得$ \widetilde{u}\in L^{\infty}(\tau, T; H_{0}^{1}(\Omega) \cap L^{p}(\Omega)) $, $ \widetilde{u}'\in L^{2}(\tau, T; L^{2}(\Omega) $.

对任意给定的$ T > \tau $,由于

$ \begin{eqnarray} \|\widetilde{u}^{m}(t_{2})-\widetilde{u}^{m}(t_{1})\|_{2}^{2}& = & \bigg\|\int_{t_{1}}^{t_{2}}(\widetilde{u}^{m})'(s){\rm d}s\bigg\|_{2}^{2} \\ &\leq&\|(\widetilde{u}^{m})'\|_{L^{2}(\tau, T;L^{2}(\Omega))}^{2}|t_{2}-t_{1}|, \quad\forall t_{1}, t_{2}\in [\tau, T], \end{eqnarray} $

由(3.9)式, Sobolev紧嵌入$ H_{0}^{1}(\Omega)\cap L^{p}(\Omega)\hookrightarrow L^{2}(\Omega) $$ (p > 2) $, (3.10)式和(3.11)式,以及Ascoli-Arzelà定理,可知存在$ \{\widetilde{u}^{m}\} $的子列(仍记作$ \{\widetilde{u}^{m}\} $)使得

$ \begin{eqnarray} \mbox{在}\ C([\tau-h, T];L^{2}(\Omega))\ \mbox{中}\ \widetilde{u}^{m}\rightarrow\widetilde{u} \end{eqnarray} $

$ \Omega\times(\tau, \infty) $上几乎处处成立.

结合(3.6)式和(3.12)式,对$ \{\widetilde{u}^{m}\} $满足的方程两边取极限,再由span$ \{\omega_{j}\}_{j\geq 1} $$ H_{0}^{1}(\Omega)\cap L^{p}(\Omega) $中稠密,可知$ \widetilde{u} $是方程(1.1)的弱解.

步骤3  利用稠密性来证明更一般的结果.

对每个$ n\in{\Bbb N} $,定义$ \phi^{n}(\theta) = \sum\limits_{j = 1}^{n}(\phi(\theta), \omega_{j})\omega_{j} $, $ \theta\in[-h, 0] $. (由于$ \{\omega_{j}\}_{j\geq1} $$ L^{2}(\Omega) $中的Hilbert基,用反证法容易证明在$ C_{L^{2}(\Omega)} $$ \phi_{n}\rightarrow\phi $).

$ u^{n} $是方程(1.1)的对应于初值$ u_{\tau}^{n} = \phi^{n} $的解,则$ u^{n} $的能量等式为

其中$ \overline{u^{n}(s)} $$ u^{n}(s) $的共轭.

类似于步骤1的推导过程,可知对任意的$ T > \tau $,有

其中$ q = p/(p-1) $$ p $的共轭.

所以,对任意的$ T > \tau $,存在$ u\in L^{\infty}(\tau-h, T; L^{2}(\Omega))\cap L^{2}(\tau, T; H_{0}^{1}(\Omega))\cap L^{p}(\tau, T; L^{p}(\Omega)) $, $ \chi\in L^{q}(\tau, T; L^{q}(\Omega)) $以及$ \{u^{n}\} $的子列(仍记作$ \{u^{n}\} $)使得

$ \begin{eqnarray} \left\{\begin{array}{ll} u^{n}\; \mbox{在}\; L^{\infty}(\tau-h, T; L^{2}(\Omega))\; \mbox{中弱$\star$ -收敛到}\; u, \\ u^{n}\; \mbox{在}\; L^{2}(\tau, T; H_{0}^{1}(\Omega))\; \mbox{中弱收敛到}\; u, \\ u^{n}\; \mbox{在}\; L^{p}(\tau, T; L^{p}(\Omega))\; \mbox{中弱收敛到}\; u, \\ |u^{n}|^{p-2}u^{n}\; \mbox{在}\; L^{q}(\tau, T; L^{q}(\Omega))\; \mbox{中弱收敛到}\; \chi. \end{array}\right. \end{eqnarray} $

$ u^{n}-u^{m} $的能量等式为

$ \begin{eqnarray} &&\|u^{n}(t)-u^{m}(t)\|_{2}^{2}+2\lambda\int_{\tau}^{t}\|\nabla u^{n}(s)-\nabla u^{m}(s)\|_{2}^{2}{\rm d}s \\ &&+2{\rm Re}\left[(k+{\rm i}\beta)\int_{\tau}^{t}(|u^{n}(s)|^{p-2}u^{n}(s)-|u^{m}(s)|^{p-2}u^{m}(s), \overline{u^{n}(s)-u^{m}(s)}){\rm d}s\right] \\ &\leq&\|u^{n}(\tau)-u^{m}(\tau)\|_{2}^{2}+2\gamma\int_{\tau}^{t}\|u^{n}(s)-u^{m}(s)\|_{2}^{2}{\rm d}s +2L_{g}\int_{\tau}^{t}\|u^{n}_{s}-u^{m}_{s}\|_{C_{L^{2}(\Omega)}}{\rm d}s, \; \; \forall t\geq\tau. {}\\ \end{eqnarray} $

在(3.14)式中用$ t+\theta $代替$ t $, $ \theta\in[-h, 0] $可得

$ \begin{eqnarray} &&\|u^{n}_{t}-u^{m}_{t}\|_{C_{L^{2}(\Omega)}}^{2} +2\lambda\int_{\tau}^{t}\|\nabla u^{n}(s)-\nabla u^{m}(s)\|_{2}^{2}{\rm d}s \\ &\leq&\|\phi^{n}-\phi^{m}\|_{C_{L^{2}(\Omega)}}^{2} +2(\gamma+L_{g})\int_{\tau}^{t}\|u^{n}_{s}-u^{m}_{s}\|_{C_{L^{2}(\Omega)}}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray} $

对(3.15)式应用Gronwall引理并代入(3.14)式,可知对任意的$ T > \tau $,有

所以,在$ \Omega\times(\tau, \infty) $上, $ u^{n}\rightarrow u $几乎处处成立.

因此,类似于前面,由(3.13)式和文献[24,引理1.3, p12]可得$ \chi = |u|^{p-2}u $,并由(3.13)式再对$ u^{n} $所满足的方程取极限得出$ u $是方程(1.1)的弱解.

定理3.2 设$ g(\cdot, \cdot) $满足假设(Ⅰ)–(Ⅲ), $ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C_{L^{2}(\Omega)} $且(1.2)式成立,则方程(1.1)的弱解唯一.而且,对于两个初值不同的解$ u(t) $$ v(t) $,有下述$ \rm Lipschitz $连续

$ \begin{eqnarray} \| w_{t}\|_{C_{L^{2}(\Omega)}}^{2} \leq\|w_{\tau}\|^{2}_{C_{L^{2}(\Omega)}}e^{2(L_{g}+\gamma)(t-\tau)}, \quad\; \forall t\geq\tau, \end{eqnarray} $

其中$ w(t) = u(t)-v(t) $.

 记$ w(t) = u(t)-v(t) $,则$ w(t) $满足

$ \begin{eqnarray} \partial_{t}w-(\lambda+{\rm i}\alpha)\Delta w+(k+{\rm i}\beta)(|u|^{p-2}u-|v|^{p-2}v)-\gamma w = g(t, u_{t})-g(t, v_{t}), \end{eqnarray} $

其初值为$ w(x, \tau+\theta) = \phi_{1}(x, \theta)-\phi_{2}(x, \theta) $, $ \theta\in[-h, 0] $.

$ \overline{w} $与(3.17)式在$ \Omega $上作$ L^{2} $ -内积,可得

其中$ \overline{w} $$ w $的共轭.

由假设(Ⅲ)和Hölder不等式得

综合(1.2)式,引理2.1和引理2.2可得

从而

对上式关于时间$ t $$ [\tau, t] $上积分,可得

特别地,用$ t+\theta $代替$ t $, $ \theta\in[-h, 0] $可得

对上式应用Gronwall引理得出(3.16)式,唯一性成立.

因此,由定理3.2可以定义$ C_{L^{2}(\Omega)} $中的解过程$ \{U(t, \tau)\}_{t\geq\tau} $

$ \begin{eqnarray} U(t, \tau): C_{L^{2}(\Omega)}\rightarrow C_{L^{2}(\Omega)}, \quad U(t, \tau)\phi: = u(t;\tau, \phi) = u(t), \quad\forall t\geq\tau. \end{eqnarray} $

而且,由(3.18)式所定义的过程$ \{U(t, \tau)\}_{t\geq\tau} $$ C_{L^{2}(\Omega)} $中满足Lipschitz连续

4 拉回吸引子

本节我们将证明过程$ \{U(t, \tau)\}_{t\geq\tau} $$ C_{L^{2}(\Omega)} $中存在拉回$ {\cal D} $ -吸引子.

4.1 拉回吸收集

下述引理将用来证明过程$ \{U(t, \tau)\}_{t\geq\tau} $$ C_{L^{2}(\Omega)} $中存在拉回$ {\cal D} $ -吸收集.

引理4.1 设$ g(\cdot, \cdot) $满足假设(Ⅰ)–(Ⅲ), $ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C_{L^{2}(\Omega)} $,则方程(1.1)的弱解$ u(t) $满足下述估计

$ \begin{eqnarray} \|u_{t}\|_{C_{L^{2}(\Omega)}}^{2}\leq e^{2(\lambda\lambda_{1}+\widetilde{k})h-\sigma(t-\tau)} \|\phi\|_{C_{L^{2}(\Omega)}}^{2}+e^{2(\lambda\lambda_{1}+\widetilde{k})h-\sigma t}\int_{\tau}^{t} e^{\sigma s}\|f(s)\|_{2}^{2}{\rm d}s, \quad\forall t\geq\tau, \end{eqnarray} $

其中$ \sigma = 2(\lambda\lambda_{1}+\widetilde{k})-(2L_{g}+2\gamma+1)e^{2(\lambda\lambda_{1}+\widetilde{k})h} $.

 让$ \overline{u} $和方程(1.1)在$ \Omega $上做$ L^{2} $ -内积,可得

其中$ \overline{u} $$ u $的共轭.

利用Hölder不等式和假设(Ⅲ)可得

由Poincaré不等式和$ 2k\|u\|^{p}_{p}\geq2\widetilde{k}\|u\|^{p}_{2}\geq2\widetilde{k}(\|u\|^{2}_{2}-c_{0}) $可得

$ \begin{eqnarray} \frac{\rm d}{{\rm d}t}\|u\|_{2}^{2}+2(\lambda\lambda_{1}+\widetilde{k})\|u\|_{2}^{2} \leq(2L_{g}+2\gamma+1)\|u_{t}\|^{2}_{C_{L^{2}(\Omega)}}+\|f(t)\|_{2}^{2}+2\widetilde{k}c_{0}, \end{eqnarray} $

其中$ \lambda_{1} $$ -\Delta $$ H_{0}^{1}(\Omega) $中的第一特征值.

在(4.2)式两边乘以$ e^{2(\lambda\lambda_{1}+\widetilde{k})t} $并关于时间$ t $$ [\tau, t] $上积分,得

特别地,用$ t+\theta $代替$ t $, $ \theta\in[-h, 0] $可得

由Gronwall引理得

因此

其中$ \sigma = 2(\lambda\lambda_{1}+\widetilde{k})-(2L_{g}+2\gamma+1)e^{2(\lambda\lambda_{1}+\widetilde{k})h} $.

为了得到$ C_{L^{2}(\Omega)} $中的拉回$ {\cal D}_{\sigma} $ -吸收集,我们进一步假设

$ \begin{eqnarray} 0<2(\lambda\lambda_{1}+\widetilde{k})-(2L_{g}+2\gamma+1)e^{2(\lambda\lambda_{1}+\widetilde{k})h} = \sigma, \end{eqnarray} $

$ \begin{eqnarray} \int_{-\infty}^{t}e^{\sigma s}\|f(s)\|_{2}^{2}{\rm d}s<\infty, \quad\forall t\in{{\Bbb R}} . \end{eqnarray} $

接下来,我们给出$ {\cal D}_{\sigma} $的定义.

定义4.1 对任意的$ \sigma > 0 $,用$ {\cal D}_{\sigma} $表示满足

的所有非空子集族$ \widehat{D} = \{D(t):t\in{{\Bbb R}} \}\subset{\cal P}(C_{L^{2}(\Omega)}) $所组成的类.

推论4.1 设$ g(\cdot, \cdot) $满足假设(Ⅰ)–(Ⅲ), $ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C_{L^{2}(\Omega)} $,且(4.3)式和(4.4)式成立,则中心在原点,半径为$ \rho(t) $的闭球所组成的集族$ \widehat{D}_{0} = \{D_{0}(t): t\in{{\Bbb R}} \} $构成过程$ \{U(t, \tau)\}_{t\geq\tau} $$ C_{L^{2}(\Omega)} $中的拉回$ {\cal D}_{\sigma} $ -吸收集,其中$ D_{0}(t) = \overline{B}_{C_{L^{2}}}(0, \rho(t)) $,

$ \begin{eqnarray} \rho^{2}(t) = 1+e^{2(\lambda\lambda_{1}+\widetilde{k})h-\sigma t}\int_{-\infty}^{t}e^{\sigma s}\|f(s)\|_{2}^{2}{\rm d}s. \end{eqnarray} $

而且, $ \widehat{D}_{0}\in {\cal D}_{\sigma} $.

 由引理4.1中(4.1)式易得过程$ \{U(t, \tau)\}_{t\geq \tau} $拉回$ {\cal D}_{\sigma} $ -吸收集的存在性.

再由(4.5)式知,当$ t\rightarrow-\infty $时, $ e^{\sigma t}\rho^{2}(t)\rightarrow 0 $.从而$ \widehat{D}_{0}\in{\cal D}_{\sigma} $.

4.2 拉回吸引子

我们将运用收缩函数方法证明过程$ \{U(t, \tau)\}_{t\geq\tau} $$ C_{L^{2}(\Omega)} $中拉回渐近紧.首先,我们给出下述引理,参见文献[24].

引理4.2 设$ g(\cdot, \cdot) $满足假设(Ⅰ)–(Ⅲ), $ f(\cdot)\in L_{loc}^{2}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C_{L^{2}(\Omega)} $, $ \{u^{n}(t)\}_{n = 1}^{\infty} $是方程(1.1)对应于初值$ \phi^{n}\in C_{L^{2}(\Omega)} $$ (n = 1, 2, \cdot\cdot\cdot) $的解序列,则$ \{u^{n}(t)\}_{n = 1}^{\infty} $$ L^{2}(\tau, T; L^{2}(\Omega)) $中存在收敛子列.

定理4.1 设$ g(\cdot, \cdot) $满足假设(Ⅰ)–(Ⅲ), $ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C_{L^{2}(\Omega)} $且(1.2)式成立,则过程$ \{U(t, \tau)\}_{t\geq \tau} $$ C_{L^{2}(\Omega)} $中拉回$ {\cal D}_{\sigma} $ -渐近紧.

 设$ u^{i}(t) $$ (i = 1, 2) $是方程(1.1)对应于初值$ \phi^{i}(x, \theta)\in D_{0}(\tau) $, $ \theta\in[-h, 0] $的解,即, $ u^{i}(t) $满足

初值为

$ \omega(t) = u^{1}(t)-u^{2}(t) $,则$ \omega(t) $满足

$ \begin{eqnarray} &&\partial_{t}\omega-(\lambda+{\rm i}\alpha)\Delta\omega +(k+{\rm i}\beta)(|u^{1}|^{p-2}u^{1}-|u^{2}|^{p-2}u^{2})-\gamma\omega \\ & = &g(t, u^{1}_{t})-g(t, u^{2}_{t}), \quad(x, t)\in\Omega\times(\tau, \infty), \end{eqnarray} $

初值为

$ \overline{\omega(t)} $与(4.6)式在$ \Omega $上做$ L^{2} $ -内积,可得

$ \begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\|\omega\|_{2}^{2}+(\lambda+{\rm i}\alpha)\|\nabla\omega\|_{2}^{2} +(k+{\rm i}\beta)\int_{\Omega}(|u^{1}|^{p-2}u^{1}-|u^{2}|^{p-2}u^{2}) \overline{\omega}{\rm d}x-\gamma\|\omega\|_{2}^{2} \\ & = &(g(t, u^{1}_{t})-g(t, u^{2}_{t}), \overline{\omega}), \end{eqnarray} $

其中$ \overline{\omega} $$ \omega $的共轭.

对(4.7)式利用假设(Ⅲ)和Hölder不等式可得

$ \begin{eqnarray} &&\frac{1}{2}\frac{\rm d}{{\rm d}t}\|\omega\|_{2}^{2}+\lambda\|\nabla\omega\|_{2}^{2} +{\rm Re}\left[(k+{\rm i}\beta)\int_{\Omega}(|u^{1}|^{p-2}u^{1}-|u^{2}|^{p-2}u^{2}) \overline{\omega}{\rm d}x\right]-\gamma\|\omega\|_{2}^{2} \\ &\leq &L_{g}\|\omega_{t}\|_{C_{L^{2}(\Omega)}}\|\omega\|_{2}, \end{eqnarray} $

结合(1.2)式,引理2.1和引理2.2可得

$ \begin{eqnarray} &&{\rm Re}\left[(k+{\rm i}\beta)\int_{\Omega}(|u^{1}|^{p-2}u^{1}-|u^{2}|^{p-2}u^{2}) \overline{\omega}{\rm d}x\right] \\ & = &k{\rm Re}\left[\int_{\Omega}(|u^{1}|^{p-2}u^{1}-|u^{2}|^{p-2}u^{2}) \overline{\omega}{\rm d}x\right]-\beta {\rm Im}\left[\int_{\Omega}(|u^{1}|^{p-2}u^{1}-|u^{2}|^{p-2}u^{2}) \overline{\omega}{\rm d}x\right] \\ &\geq& k{\rm Re}\left[\int_{\Omega}(|u^{1}|^{p-2}u^{1}-|u^{2}|^{p-2}u^{2}) \overline{\omega}{\rm d}x\right]\left(1-\frac{|\beta|}{k}c_{p}\right) \geq0. \end{eqnarray} $

结合(4.7)式–(4.9)式,再利用Poincaré不等式可得

$ \begin{eqnarray} \frac{\rm d}{{\rm d}t}\|\omega\|_{2}^{2}+2\lambda\lambda_{1}\|\omega\|_{2}^{2} \leq 2L_{g}\|\omega_{t}\|_{C_{L^{2}(\Omega)}}\|\omega\|_{2}+2\gamma\|\omega\|_{2}^{2}. \end{eqnarray} $

在(4.10)式两边同乘以$ e^{2\lambda\lambda_{1}t} $

$ \begin{eqnarray} \frac{\rm d}{{\rm d}t}(e^{2\lambda\lambda_{1}t}\|\omega\|_{2}^{2}) \leq 2L_{g}e^{2\lambda\lambda_{1}t}\|\omega_{t}\|_{C_{L^{2}(\Omega)}}\|\omega\|_{2} +2\gamma e^{2\lambda\lambda_{1}t}\|\omega\|_{2}^{2}. \end{eqnarray} $

对(4.11)式关于时间$ t $$ [\tau, t] $上积分,可得

进而

$ \begin{eqnarray} \|\omega(t)\|_{2}^{2}&\leq &e^{-2\lambda\lambda_{1}(t-\tau)}\|\omega(\tau)\|_{2}^{2} +2L_{g}e^{-2\lambda\lambda_{1}t} \int_{\tau}^{t}e^{2\lambda\lambda_{1}s}\|\omega_{s}\|_{C_{L^{2}(\Omega)}}\|\omega(s)\|_{2}{\rm d}s \\ &&+2\gamma\int_{\tau}^{t}\|\omega(s)\|_{2}^{2}{\rm d}s, \quad\forall t\geq\tau. \end{eqnarray} $

对(4.12)式运用Hölder不等式,并用$ t+\theta $代替$ t $, $ \theta\in [-h, 0] $,可得

$ T = t-\tau $,且

综合定义2.4,引理4.2以及引理4.1中的(4.1)式即可知$ \psi_{t, T}(\cdot, \cdot) $是收缩函数.从而对任给的$ \varepsilon > 0 $$ t\in{{\Bbb R}} $,只要取$ \tau_{0} = t-h-\frac{1}{2\lambda\lambda_{1}}\ln\frac{\|w_{\tau}\|_{C_{L^{2}(\Omega)}}^{2}}{\varepsilon} $,当$ \tau\leq\tau_{0} $时,由定理2.1即可证得过程$ \{U(t, \tau)\}_{t\geq\tau} $$ C_{L^{2}(\Omega)} $中拉回$ {\cal D}_{\sigma} $ -渐近紧.

结合引理4.1,推论4.1,引理4.2,定理4.1和定理2.2,我们得到本文的主要结果.

定理4.2 设$ g(\cdot, \cdot) $满足假设(Ⅰ)–(Ⅲ), $ f(\cdot)\in L^{2}_{loc}({{\Bbb R}}; L^{2}(\Omega)) $, $ \phi\in C_{L^{2}(\Omega)} $且(1.2)式成立,则过程$ \{U(t, \tau)\}_{t\geq\tau} $$ C_{L^{2}(\Omega)} $中存在拉回$ {\cal D}_{\sigma} $ -吸引子$ \widehat{{\cal A}} = \{{\cal A}(t); \; t\in {{\Bbb R}} \} $;即, $ {\cal A}(t) $紧, $ \widehat{{\cal A}} $非空、不变且在$ C_{L^{2}} $ -拓扑下拉回吸引每个集合$ \widehat{D}\in{\cal D}_{\sigma} $.

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