西南大学学报 (自然科学版)  2018, Vol. 40 Issue (9): 59-66.  DOI: 10.13718/j.cnki.xdzk.2018.09.010
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  • 无界域上非自治Reaction-Diffusion方程的后向紧动力学    [PDF全文]
    佘连兵1, 李信韬1, 李扬荣2     
    1. 六盘水师范学院 数学与信息工程学院, 贵州 六盘水 553004;
    2. 西南大学 数学与统计学院, 重庆 400715
    摘要:在非自治外力项是后向λ-缓增有限的和后向尾部渐近趋于零的假设条件下,运用cut-off函数、后向Granwall不等式、后向Granwall-type不等式获得了无界域上非自治Reaction-Diffusion方程拉回吸引子的后向紧性.
    关键词非自治动力系统    后向紧动力    cut-off函数    无界域    

    在物理学、化学、生物学、经济学及各种工程问题中提出的大量反应扩散问题,日益受到人们的重视.在数学上通常把半线性抛物型方程叫作Reaction-Diffusion方程,对Reaction-Diffusion方程的研究一直以来都受到广大学者的格外关注,而对非自治Reaction-Diffusion方程拉回吸引子的研究是其中一个重要课题[1-4].

    最近,文献[5-6]研究了有界域上非自治Reaction-Diffusion方程拉回吸引子的后向紧性,这种紧性反映了非自治动力系统的半全局性质,体现了与自治系统片段紧性的差异.文献[7]建立了无界管道上非自治Benjamin-Bona-Mahony方程后向紧拉回吸引子的存在性理论.文献[8-9]分别研究了非自治3D Navier-Stokes方程和非自治的波动方程的后向紧吸引子的存在性.本文研究在$\mathbb{R}$N上如下非自治Reaction-Diffusion方程拉回吸引子的后向紧动力:

    $ \left\{ \begin{array}{l} {u_t} + \lambda u - \Delta u = f\left( {x,u} \right) + g\left( {t,x} \right)\\ u\left( {s,s,x} \right) = {u_0}\left( x \right) \end{array} \right. $ (1)

    其中:x∈$\mathbb{R}$Nts∈$\mathbb{R}$,λ>0.众所周知,在无界域上Sobolev嵌入不再是紧的,这给后向一致先验估计带来困难,为了克服这个难点,采用cut-off函数的技巧,结合后向Granwall不等式和后向Granwall-type不等式,在假设非自治外力项是后向λ-缓增有限的和后向尾部渐近趋于零的情况下,对方程的解进行后向一致估计,证明了非自治Reaction-Diffusion方程的吸引子的后向紧性.

    1 Banach空间上的非自治过程的后向紧动力学

    定义1  若定义在Banach空间X上的一族映射S(ts):XX,∀ts,满足对于任意的trs

    $ \begin{array}{*{20}{c}} {S\left( {s,s} \right) = I}&{S\left( {t,s} \right) = S\left( {t,r} \right)S\left( {r,s} \right)} \end{array} $

    则称S(·,·)是X上的一个非自治过程.

    定义2  设$\mathscr{A}$={$\mathscr{A}$(t)}t∈$\mathbb{R}$ 是Banach空间X中的一个非自治集,对任意的t1t2∈$\mathbb{R}$,当t1t2时,有$\mathscr{A}$(t1)⊂$\mathscr{A}$(t2),则称$\mathscr{A}$是单调递增的;当t1t2时,有$\mathscr{A}$(t1)⊃$\mathscr{A}$(t2),则称$\mathscr{A}$是单调递减的.

    关于非自治动力系统的详细介绍可见专著[3],对任意的t∈$\mathbb{R}$,$\mathscr{A}$(t)在X中是紧的,但并不表示$\mathop \cup \limits_{s \leqslant t} \mathscr{A}\left( s \right)$X中也是紧的,因此下面关于后向紧的拉回吸引子的定义是有意义的.

    定义3  设S(·,·)是定义在Banach空间X上的一个非自治过程,若X中的一个非自治集$\mathscr{A}$={$\mathscr{A}$(t)}t∈$\mathbb{R}$ 满足:

    1) $\mathscr{A}$是后向预紧的,即对∀t∈$\mathbb{R}$,$\mathop \cup \limits_{s \leqslant t} \mathscr{A}\left( s \right)$X中是预紧的;

    2) $\mathscr{A}$是不变的,即对于所有的tτ,有S(tτ)$\mathscr{A}$(τ)=$\mathscr{A}$(t);

    3) $\mathscr{A}$是拉回吸引的,即对于X中所有的有界集B,有

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{\tau \to + \infty } {\text{dis}}{{\text{t}}_X}\left( {S\left( {t,t - \tau } \right)B,\mathscr{A}\left( t \right)} \right) = 0}&{\forall t \in \mathbb{R}} \end{array} $

    其中

    $ {\rm{dis}}{{\rm{t}}_X}\left( {A,B} \right) = \mathop {\sup }\limits_{a \in A} \mathop {\inf }\limits_{b \in B} {\left\| {a - b} \right\|_X} $

    是Hausdorff半距离,则称$\mathscr{A}$是一个关于非自治过程S(·,·)的后向紧拉回吸引子.

    下面引用文献[6-7]中的一个存在性定理:

    定理1  设S(·,·)是定义在Banach空间X上的一个非自治过程,若满足

    (i) S(·,·)在X上有一个单调递增的有界的吸收集$\mathscr{K}$={$\mathscr{K}$(t)}t∈$\mathbb{R}$

    (ii) S(·,·)是后向Omega-limit紧的,即

    $ \mathop {\lim }\limits_{{\tau _0} \to + \infty } {\kappa _X}\left( {\bigcup\limits_{\tau \ge {\tau _0}} {\bigcup\limits_{s \ge t} {S\left( {s,s - \tau } \right)D} } } \right) = 0 $

    其中κX(·)是文献[10]中介绍的Kuratowski测度.

    S(·,·)有一个后向紧的拉回吸引子$\mathscr{A}$={$\mathscr{A}$(t)}t∈$\mathbb{R}$ ,其中

    $ \begin{array}{*{20}{c}} {\mathscr{A}\left( t \right) = {\omega _X}\left( {\mathscr{K}\left( t \right),t} \right) = \bigcap\limits_{{\tau _0} > 0} {\bigcup\limits_{\tau \geqslant {\tau _0}} {S\left( {t,t - \tau } \right)\mathscr{K}{{\left( t \right)}^{\bar X}}} } }&{t \in \mathbb{R}} \end{array} $ (2)

    下面介绍后向Granwall不等式和后向Granwall-type不等式,其证明完全类似于文献[12].

    引理1(后向Gronwall不等式)  设t∈$\mathbb{R}$和τ>0,ygh是[s-τs](st)上的非负可积函数,且y′是[s-τs](st)上的可积函数,若

    $ \begin{array}{*{20}{c}} {y'\left( r \right) + g\left( r \right)y\left( r \right) \le h\left( r \right)r \in \left[ {s - \tau ,s} \right]}&{s \le t} \end{array} $

    $ \mathop {\sup }\limits_{s \le t} y\left( s \right) \le \mathop {\sup }\limits_{s \le t} y\left( {s - \tau } \right){{\rm{e}}^{ - \int_{s - \tau }^s {g\left( r \right){\rm{d}}r} }} + \mathop {\sup }\limits_{s \le t} \int_{s - \tau }^s {h\left( r \right){{\rm{e}}^{\int_s^r {g\left( r \right){\rm{d}}r} }}{\rm{d}}r} $ (3)

    g=a>0是一个常数,则

    $ \mathop {\sup }\limits_{s \le t} y\left( s \right)\mathop {\sup }\limits_{s \le t} y\left( {s - \tau } \right){{\rm{e}}^{ - a\tau }} + \mathop {\sup }\limits_{s \le t} \int_{s - \tau }^s {h\left( r \right){{\rm{e}}^{a\left( {r - s} \right)}}{\rm{d}}r} $ (4)

    引理2  (后向Gronwall-type不等式)  设yy′,y1y2是$\mathbb{R}$上的局部可积函数,且yy1y2是非负函数,对于每个t∈$\mathbb{R}$,有

    $ \begin{array}{*{20}{c}} {y'\left( s \right) + by\left( s \right) + {y_1}\left( s \right) \le {y_2}\left( s \right)}&{s \le t} \end{array} $

    (i) 若b∈$\mathbb{R}$是一个给定的常数,则对每一个t∈$\mathbb{R}$和μ>0,有

    $ \mathop {\sup }\limits_{s \le t} y\left( s \right) \le \frac{1}{\mu }\mathop {\sup }\limits_{s \le t} \int_{s - \mu }^s {{{\rm{e}}^{b\left( {r - s} \right)}}y\left( r \right){\rm{d}}r} + \mathop {\sup }\limits_{s \le t} \int_{s - \mu }^s {{{\rm{e}}^{b\left( {r - s} \right)}}{y_2}\left( r \right){\rm{d}}r} $ (5)

    (ii) 若b≥0是一个给定的常数,则对每一个t∈$\mathbb{R}$和μ>0,有

    $ \mathop {\sup }\limits_{s \le t} y\left( s \right) + \mathop {\sup }\limits_{s \le t} \int_{s - \mu }^s {{y_1}\left( r \right){\rm{d}}r} \le \frac{{{{\rm{e}}^{ - b\mu }}}}{\mu }\mathop {\sup }\limits_{s \le t} \int_{s - 3\mu }^s {y\left( r \right){\rm{d}}r} + \mathop {\sup }\limits_{s \le t} \int_{s - 3\mu }^s {{y_2}\left( r \right){\rm{d}}r} $ (6)
    2 无界域上非自治Reaction-Diffusion方程的后向紧动力学

    本节将应用上一节的存在性理论证明方程(1)在下面的假设条件下存在唯一的后向紧拉回吸引子.为了计算方便,设c是变化的正常数.

    下面给出方程(1)中关于fg的假设条件:

    (A) 设p>2,β1β2β3>0,f(·,·)∈C1($\mathbb{R}$×$\mathbb{R}$,$\mathbb{R}$)满足

    $ \begin{array}{*{20}{c}} {f\left( {x,s} \right)s \leqslant - {\beta _1}{{\left| s \right|}^p} + {\psi _1}}&{{\psi _1} \in {L^1}\left( {{\mathbb{R}^N}} \right) \cap {L^2}\left( {{\mathbb{R}^N}} \right)} \end{array} $
    $ \begin{array}{*{20}{c}} {\left| {f\left( {x,s} \right)} \right| \leqslant {\beta _2}{{\left| s \right|}^{p - 1}} + {\psi _2}}&{{\psi _2} \in {L^2}\left( {{\mathbb{R}^N}} \right)} \end{array} $
    $ \begin{array}{*{20}{c}} {\frac{{\partial f}}{{\partial s}}\left( {x,s} \right) \leqslant {\beta _3},\left| {\frac{{\partial f}}{{\partial x}}\left( {x,s} \right)} \right| \leqslant {\psi _3}}&{{\psi _3} \in {L^2}\left( {{\mathbb{R}^N}} \right)} \end{array} $

    (B0) gLloc2($\mathbb{R}$,L2($\mathbb{R}$N)).

    (B1) g是后向λ-缓增有限的:

    $ M\left( t \right) = \mathop {\sup }\limits_{s \leqslant t} \int_{ - \infty }^s {{{\text{e}}^{\lambda \left( {r - s} \right)}}\left\| {g\left( {r, \cdot } \right)} \right\|_{{L^2}}^2{\text{d}}r} < \infty $

    其中:t∈$\mathbb{R}$,λ给定于(6)式中.

    (B2) g是后向尾部趋于0的:

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{k \to + \infty } \mathop {\sup }\limits_{s \leqslant t} \int_{{\mathbb{R}^N}\left( {\left| x \right| \geqslant k} \right)} {{{\left| {g\left( {s,x} \right)} \right|}^2}{\text{d}}x = 0} }&{\forall t \in \mathbb{R}} \end{array} $

    由文献[4]知,对∀s∈$\mathbb{R}$,当条件(A)和(B0)满足时,方程(1)有唯一的连续的解

    $ u\left( { \cdot ,s,{u_0}} \right) \in C\left( {\left[ {s, + \infty } \right),{L^2}\left( {{\mathbb{R}^N}} \right)} \right) \cap L_{{\text{loc}}}^2\left( {\left[ {s, + \infty } \right),{H^1}\left( {{\mathbb{R}^N}} \right)} \right) \cap L_{{\text{loc}}}^p\left( {\left( {s, + \infty } \right),{L^p}\left( {{\mathbb{R}^N}} \right)} \right) $ (7)

    特别地,u(ssu0)=u0,故可定义如下的非自治过程S(·,·):L2($\mathbb{R}$N)→L2($\mathbb{R}$N):

    $ \begin{array}{*{20}{c}} {S\left( {t,t - \tau } \right){u_0} = u\left( {t,t - \tau ,{u_0}} \right),\forall {u_0} \in {L^2}\left( {{\mathbb{R}^N}} \right)}&{\forall \tau \geqslant 0} \end{array} $ (8)

    引理3  若条件(A),(B0),(B1)满足,则对每个t∈$\mathbb{R}$和每个L2($\mathbb{R}$N)中的有界集B,存在一个τ0>9,使得当ττ0时有,

    $ \begin{gathered} \mathop {\sup }\limits_{s \leqslant t} \left\| {u\left( {s,s - \tau ,{u_0}} \right)} \right\|_2^2 + \theta \mathop {\sup }\limits_{s \leqslant t} \int_{ s - \tau }^s {{{\text{e}}^{\lambda \left( {r - s} \right)}}\left\| {u\left( {r,s - \tau ,{u_0}} \right)} \right\|_{{H^1}}^2{\text{d}}r} + \hfill \\ 2{\beta _1}\mathop {\sup }\limits_{s \leqslant t} \int_{ s - \tau }^s {{{\text{e}}^{\lambda \left( {r - s} \right)}}\left\| {u\left( {r,s - \tau ,{u_0}} \right)} \right\|_p^p{\text{d}}r} \leqslant 1 + \frac{2}{\lambda }M\left( t \right) + \frac{{2{{\left\| {{\psi _1}} \right\|}_1}}}{\lambda } \hfill \\ \end{gathered} $ (9)

    其中$\theta = \min \left( {\frac{\lambda }{2},2} \right)$M(t)由条件(B1)给出.

      让方程(1)与u在$\mathbb{R}$N上做内积可得

    $ \frac{{\text{d}}}{{{\text{d}}t}}\left\| u \right\|_2^2 + 2\lambda \left\| u \right\|_2^2 + 2\left\| {\nabla u} \right\|_2^2 = \int_{{\mathbb{R}^N}} {2f\left( {x,u} \right)u{\text{d}}x} + \int_{{\mathbb{R}^N}} {2g\left( {t,x} \right)u{\text{d}}x} $ (10)

    由Young不等式和条件(A)知

    $ \begin{gathered} \int_{{\mathbb{R}^N}} {2f\left( {x,u} \right)u{\text{d}}x} + \int_{{\mathbb{R}^N}} {2g\left( {t,x} \right)u{\text{d}}x} \leqslant \hfill \\ - 2{\beta _1}\left\| u \right\|_p^p + 2{\left\| {{\psi _1}} \right\|_1} + \frac{\lambda }{2}\left\| u \right\|_2^2 + \frac{2}{\lambda }\left\| {g\left( {t, \cdot } \right)} \right\|_2^2 \hfill \\ \end{gathered} $ (11)

    $ \theta = \min \left( {\frac{\lambda }{2},2} \right) $

    由(10)式和(11)式知

    $ \frac{{\text{d}}}{{{\text{d}}t}}\left\| u \right\|_2^2 + \lambda \left\| u \right\|_2^2 + \theta \left\| u \right\|_{{H^1}}^2 + 2{\beta _1}\left\| u \right\|_p^p \leqslant \frac{2}{\lambda }\left\| {g\left( {t, \cdot } \right)} \right\|_2^2 + 2{\left\| {{\psi _1}} \right\|_1} $ (12)

    由后向Gronwall不等式知,存在一个τ0>9,当ττ0u0B时,有

    $ \begin{align} & \mathop {\sup }\limits_{s \leqslant t} \left\| {u\left( {s,s - \tau ,{u_0}} \right)} \right\|_2^2 + \theta \mathop {\sup }\limits_{s \leqslant t} \int_{ s - \tau }^s {{{\text{e}}^{\lambda \left( {r - s} \right)}}\left\| {u\left( {r,s - \tau ,{u_0}} \right)} \right\|_{{H^1}}^2{\text{d}}r} + \hfill \\ & 2{\beta _1}\mathop {\sup }\limits_{s \leqslant t} \int_{s - \tau }^s {{{\text{e}}^{\lambda \left( {r - s} \right)}}\left\| {u\left( {r,s - \tau ,{u_0}} \right)} \right\|_p^p{\text{d}}r} \leqslant \hfill \\ & {{\text{e}}^{ - \lambda \tau }}\left\| {{u_0}} \right\|_2^2 + \frac{2}{\lambda }\mathop {\sup }\limits_{s \leqslant t} \int_{s - \tau }^s {{{\text{e}}^{\lambda \left( {r - s} \right)}}\left\| {g\left( {r, \cdot } \right)} \right\|_2^2{\text{d}}r} + 2{\left\| {{\psi _1}} \right\|_1}\mathop {\sup }\limits_{s \leqslant t} \int_{s - \tau }^s {{{\text{e}}^{\lambda \left( {r - s} \right)}}{\text{d}}r} \leqslant \hfill \\ & 1 + \frac{2}{\lambda }M\left( t \right) + \frac{{2{{\left\| {{\psi _1}} \right\|}_1}}}{\lambda } \hfill \\ \end{align} $

    于是可得(9)式.

    引理4  若条件(A),(B0),(B1)满足,则对每个t∈$\mathbb{R}$和每个L2($\mathbb{R}$N)中的有界集B,存在一个τ0>9,使得当ττ0时有

    $ \mathop {\sup }\limits_{{u_0} \in B} \mathop {\sup }\limits_{s \leqslant t} \left\| {u\left( {s,s - \tau ,{u_0}} \right)} \right\|_p^p \leqslant c{{\text{e}}^{\lambda t}}\left( {1 + M\left( t \right)} \right) $ (13)

      让方程(1)与|u|p-2u在$\mathbb{R}$N上做内积,可得

    $ \begin{gathered} \frac{1}{p}\frac{{\text{d}}}{{{\text{d}}t}}\left\| u \right\|_p^p + \lambda \left\| u \right\|_p^p + \left( {p - 1} \right)\int_{{\mathbb{R}^N}} {{{\left| u \right|}^{p - 2}}\left| {\nabla u} \right|{\text{d}}x} = \hfill \\ \int_{{\mathbb{R}^N}} {f\left( {x,u} \right){{\left| u \right|}^{p - 2}}u{\text{d}}x} + \int_{{\mathbb{R}^N}} {g\left( {t,x} \right){{\left| u \right|}^{p - 2}}u{\text{d}}x} \hfill \\ \end{gathered} $ (14)

    对非线性项估计如下:

    $ \begin{align} & \int_{{\mathbb{R}^N}} {f\left( {x,u} \right){{\left| u \right|}^{p - 2}}u{\text{d}}x} \leqslant \hfill \\ & - {\beta _1}\left\| u \right\|_{2p - 2}^{2p - 2} + \int_{{\mathbb{R}^N}} {\left| {{\psi _1}} \right|{{\left| u \right|}^{p - 2}}{\text{d}}x} \leqslant \hfill \\ & - \frac{{{\beta _1}}}{2}\left\| u \right\|_{2p - 2}^{2p - 2} + c\left\| {{\psi _1}} \right\|_{2 - \frac{2}{p}}^{2 - \frac{2}{p}} \leqslant \hfill \\ & - \frac{{{\beta _1}}}{2}\left\| u \right\|_{2p - 2}^{2p - 2} + c{\left\| {{\psi _1}} \right\|_1} + c\left\| {{\psi _1}} \right\|_2^2 \leqslant \hfill \\ & - \frac{{{\beta _1}}}{2}\left\| u \right\|_{2p - 2}^{2p - 2} + c \hfill \\ \end{align} $ (15)

    由Young不等式知

    $ \int_{{\mathbb{R}^N}} {g\left( {t,x} \right){{\left| u \right|}^{p - 2}}u{\text{d}}x} \leqslant \frac{{{\beta _1}}}{4}\left\| u \right\|_{2p - 2}^{2p - 2} + c\left\| {g\left( {t, \cdot } \right)} \right\|_2^2 $ (16)

    由(14)式和(16)式可得

    $ \frac{{\text{d}}}{{{\text{d}}t}}\left\| u \right\|_p^p + \frac{{{\beta _1}p}}{4}\left\| u \right\|_{2p - 2}^{2p - 2} \leqslant c\left\| {g\left( {t, \cdot } \right)} \right\|_2^2 + c $ (17)

    对(17)式运用后向Granwall-type不等式(取μ=3和b=0),结合(9)式和条件(B1)可知,当ττ0>9,u0B时,有

    $ \begin{align} & \underset{s\le t}{\mathop{\sup }}\,\left\| u\left( s,s-\tau ,{{u}_{0}} \right) \right\|_{p}^{p}+\frac{{{\beta }_{1}}p}{4}\underset{s\le t}{\mathop{\sup }}\,\left\| u\left( s,s-\tau ,{{u}_{0}} \right) \right\|_{2p-2}^{2p-2}\le \\ & \underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{\left\| u\left( r \right) \right\|_{p}^{p}\text{d}r}+c\underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{\left\| g\left( r,\cdot \right) \right\|_{2}^{2}\text{d}r}+c\underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{1\text{d}r}\le \\ & {{\text{e}}^{\lambda t}}\underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{{{\text{e}}^{\lambda \left( r-t \right)}}\left\| u\left( r \right) \right\|_{p}^{p}\text{d}r}+c{{\text{e}}^{\lambda t}}\underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{{{\text{e}}^{\lambda \left( r-t \right)}}\left\| g\left( r,\cdot \right) \right\|_{2}^{2}\text{d}r}+c\le \\ & c{{\text{e}}^{\lambda t}}\left( 1+M\left( t \right) \right) \\ \end{align} $

    故(13)式成立.

    利用上面的结果可得在H1($\mathbb{R}$N)上的一个后向一致估计.

    引理5  若条件(A),(B0),(B1)满足,则对每个t∈$\mathbb{R}$和每个L2($\mathbb{R}$N)中的有界集B,存在一个τ0>9,使得当ττ0时,有

    $ \mathop {\sup }\limits_{{u_0} \in B} \mathop {\sup }\limits_{s \leqslant t} \left\| {u\left( {s,s - \tau ,{u_0}} \right)} \right\|_{{H^1}}^2 + \mathop {\sup }\limits_{{u_0} \in B} \mathop {\sup }\limits_{s \leqslant t} \int_{s - 1}^s {\left\| {{u_r}\left( {r,s - \tau ,{u_0}} \right)} \right\|_2^2} \leqslant c{{\text{e}}^{\lambda t}}\left( {1 + M\left( t \right)} \right) $ (18)

      让方程(1)与ut在$\mathbb{R}$N上做内积可得

    $ {\left\| {{u_t}} \right\|^2} + \frac{\lambda }{2}\frac{{\text{d}}}{{{\text{d}}t}}{\left\| u \right\|^2} + \frac{1}{2}\frac{{\text{d}}}{{{\text{d}}t}}{\left\| {\nabla u} \right\|^2} = \int_{{\mathbb{R}^N}} {f\left( {x,u} \right){u_t}{\text{d}}x} + \int_{{\mathbb{R}^N}} {g\left( {t,x} \right){u_t}{\text{d}}x} $ (19)

    由条件(A)和Young不等式可知

    $ \int_{{\mathbb{R}^N}} {f\left( {x,u} \right){u_t}{\text{d}}x} \leqslant {\beta _2}\int_{{\mathbb{R}^N}} {{u_t}{{\left| u \right|}^{p - 1}}{\text{d}}x} + \int_{{\mathbb{R}^N}} {{u_t}{\psi _2}{\text{d}}x} \leqslant \frac{1}{4}\left\| {{u_t}} \right\|_2^2 + c\left\| u \right\|_{2p - 2}^{2p - 2} + c $ (20)
    $ \int_{{\mathbb{R}^N}} {g\left( {t,x} \right){u_t}{\text{d}}x} \leqslant \frac{1}{4}\left\| {{u_t}} \right\|_2^2 + c\left\| {g\left( {t, \cdot } \right)} \right\|_2^2 $ (21)

    由(19)式和(20)式可知

    $ \frac{{\text{d}}}{{{\text{d}}t}}\left\| u \right\|_{{H^1}}^2 + \left\| {{u_t}} \right\|_2^2 \leqslant c\left\| u \right\|_{2p - 2}^{2p - 2} + c\left\| {g\left( {t, \cdot } \right)} \right\|_2^2 + c $ (22)

    对(22)式运用后向Granwall-type不等式(取μ=1和b=0),结合(9)式、(13)式和条件(B1),可知当ττ0>9,u0B时,有

    $ \begin{align} & \underset{s\le t}{\mathop{\sup }}\,\left\| u\left( s,s-\tau ,{{u}_{0}} \right) \right\|_{{{H}^{1}}}^{2}+\underset{s\le t}{\mathop{\sup }}\,\int_{s-1}^{s}{\left\| {{u}_{r}}\left( r,s-\tau ,{{u}_{0}} \right) \right\|_{2}^{2}}\le \\ & c\underset{s\le t}{\mathop{\sup }}\,\int_{s-3}^{s}{\left\| u\left( r,s-\tau ,{{u}_{0}} \right) \right\|_{{{H}^{1}}}^{2}\text{d}r}+c\underset{s\le t}{\mathop{\sup }}\,\int_{s-3}^{s}{\left\| u\left( r,s-\tau ,{{u}_{0}} \right) \right\|_{2p-2}^{2p-2}\text{d}r}+ \\ & c\underset{s\le t}{\mathop{\sup }}\,\int_{s-3}^{s}{\left\| g\left( r,\cdot \right) \right\|_{2}^{2}\text{d}r}+c\le \\ & c{{\text{e}}^{\lambda t}}\underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{{{\text{e}}^{\lambda \left( r-s \right)}}\left\| u\left( r,s-\tau ,{{u}_{0}} \right) \right\|_{{{H}^{1}}}^{2}\text{d}r}+ \\ & c{{\text{e}}^{\lambda t}}\underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{{{\text{e}}^{\lambda \left( r-s \right)}}\left\| u\left( r,s-\tau ,{{u}_{0}} \right) \right\|_{2p-2}^{2p-2}\text{d}r}+ \\ & c{{\text{e}}^{\lambda t}}\underset{s\le t}{\mathop{\sup }}\,\int_{s-9}^{s}{{{\text{e}}^{\lambda \left( r-s \right)}}\left\| g\left( r,\cdot \right) \right\|_{2}^{2}\text{d}r}+c\le \\ & c{{\text{e}}^{\lambda t}}\left( 1+M\left( t \right) \right) \\ \end{align} $

    由条件(B2),运用cut-off函数可以证明当时间和空间都趋于无穷时,方程(1)的解在L2(Qkc)上是后向渐近趋于0的,这里Qkc=$\mathbb{R}$N\QkQk={xx∈$\mathbb{R}$N,|x|<k}.

    引理6  若条件(A),(B0),(B1)满足,则对每个t∈$\mathbb{R}$和每个L2($\mathbb{R}$N)中的有界集B,存在一个τ0>9,使得当ττ0时有

    $ \mathop {\lim }\limits_{\tau \to + \infty } \mathop {\lim }\limits_{k \to + \infty } \mathop {\sup }\limits_{{u_0} \in B} \mathop {\sup }\limits_{s \leqslant t} \int_{{Q_k}}^c {{{\left| {u\left( {s,s - \tau ,{u_0}} \right)} \right|}^2}{\text{d}}x} = 0 $ (23)

      对于x∈$\mathbb{R}$Nk≥1,定义${\varphi _k}\left( x \right) = {\varphi _k}\left( {\frac{{{{\left| x \right|}^2}}}{{{k^2}}}} \right)$,这里φ(·):$\mathbb{R}$+→[0, 1]是一个光滑函数:

    $ \varphi \left( s \right) = \left\{ {\begin{array}{*{20}{c}} \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} &\begin{gathered} 0 \leqslant s \leqslant 1 \hfill \\ s \geqslant 2 \hfill \\ \end{gathered} \end{array}} \right. $

    易证明

    $ {\left\| {\nabla {\varphi _k}} \right\|_\infty } \leqslant \frac{c}{k} $

    让(1)式与φku在$\mathbb{R}$N做内积可得

    $ \begin{gathered} \frac{1}{2}\frac{{\text{d}}}{{{\text{d}}t}}\int_{{\mathbb{R}^N}} {{\varphi _k}{{\left| u \right|}^2}{\text{d}}x} + \lambda \int_{{\mathbb{R}^N}} {{\varphi _k}{{\left| u \right|}^2}{\text{d}}x} = \hfill \\ \int_{{\mathbb{R}^N}} {{\varphi _k}u\Delta u{\text{d}}x} \int_{{\mathbb{R}^N}} {f\left( {x,u} \right){\varphi _k}u{\text{d}}x} + \int_{{\mathbb{R}^N}} {g\left( {t,x} \right){\varphi _k}u{\text{d}}x} \hfill \\ \end{gathered} $ (24)

    由条件(A)和Young不等式可知

    $ \int_{{\mathbb{R}^N}} {{\varphi _k}u\Delta u{\text{d}}x} = \int_{{\mathbb{R}^N}} {{\varphi _k}{{\left| {\nabla u} \right|}^2}{\text{d}}x} - \int_{{\mathbb{R}^N}} {\nabla {\varphi _k}u\nabla u{\text{d}}x} \leqslant \frac{c}{k}\left\| u \right\|_{{H^1}}^2 $ (25)
    $ \int_{{\mathbb{R}^N}} {f\left( {x,u} \right){\varphi _k}u{\text{d}}x} \leqslant - {\beta _1}\int_{{\mathbb{R}^N}} {{\varphi _k}{{\left| u \right|}^p}{\text{d}}x} + \int_{{Q_k}}^c {\left| {{\psi _1}} \right|{\text{d}}x} \leqslant \int_{{Q_k}}^c {\left| {{\psi _1}} \right|{\text{d}}x} $ (26)
    $ \int_{{\mathbb{R}^N}} {g\left( {t,x} \right){\varphi _k}u{\text{d}}x} \leqslant \frac{\lambda }{2}\int_{{\mathbb{R}^N}} {{\varphi _k}{{\left| u \right|}^2}{\text{d}}x} + c\int_{{Q_k}}^c {{{\left| {g\left( {t,x} \right)} \right|}^2}{\text{d}}x} $ (27)

    由(24)式和(27)式可知

    $ \begin{gathered} \frac{{\text{d}}}{{{\text{d}}t}}\int_{{\mathbb{R}^N}} {{\varphi _k}{{\left| u \right|}^2}{\text{d}}x} + \lambda \int_{{\mathbb{R}^N}} {{\varphi _k}{{\left| u \right|}^2}{\text{d}}x} \leqslant \hfill \\ \frac{c}{k}\left\| u \right\|_{{H^1}}^2 + 2\int_{{Q_k}}^c {\left| {{\psi _k}} \right|{\text{d}}x} + c\int_{{Q_k}}^c {{{\left| {g\left( {t,x} \right)} \right|}^2}{\text{d}}x} \hfill \\ \end{gathered} $ (28)

    由后向Gronwall不等式可知,当ττ0u0B

    $ \begin{align} & \underset{s\le t}{\mathop{\sup }}\,\int_{{{\mathbb{R}}^{N}}}{{{\varphi }_{k}}{{\left| u\left( s,s-\tau ,{{u}_{0}} \right) \right|}^{2}}\text{d}x}\le \\ & {{\text{e}}^{-\lambda \tau }}\int_{{{\mathbb{R}}^{N}}}{{{\varphi }_{k}}{{\left| {{u}_{0}} \right|}^{p}}\text{d}x}+\frac{c}{k}\underset{s\le t}{\mathop{\sup }}\,\int_{s-\tau }^{s}{{{\text{e}}^{\lambda \left( r-s \right)}}\left\| u\left( r,s-\tau ,{{u}_{0}} \right) \right\|_{{{H}^{1}}}^{2}\text{d}r}+ \\ & 2\int_{{{Q}_{k}}}^{c}{\left| {{\varphi }_{k}} \right|\text{d}x}\underset{s\le t}{\mathop{\sup }}\,\int_{s-\tau }^{s}{{{\text{e}}^{\lambda \left( r-s \right)}}\text{d}r}+ \\ & c\underset{s\le t}{\mathop{\sup }}\,\int_{s-\tau }^{s}{{{\text{e}}^{\lambda \left( r-s \right)}}\int_{{{Q}_{k}}}^{c}{{{\left| g\left( r,x \right) \right|}^{2}}\text{d}x}}\text{d}r\le \\ & {{\text{e}}^{-\lambda \tau }}{{\left\| {{u}_{0}} \right\|}^{2}}+\frac{c}{k}\left( 1+M\left( t \right) \right)+\frac{2}{\lambda }\int_{{{Q}_{k}}}^{c}{\left| {{\varphi }_{k}} \right|\text{d}x}+ \\ & \frac{c}{\lambda }\underset{s\le t}{\mathop{\sup }}\,\int_{{{Q}_{k}}}^{c}{{{\left| g\left( s,x \right) \right|}^{2}}\text{d}x} \\ \end{align} $

    B的有界性、条件(B1),(B2)和Lesbegue定理可知(23)式成立.

    定理1  在L2($\mathbb{R}$N)上,非自治的Reaction-Diffusion方程(1)在条件(A),(B0),(B1),(B2)下有一个增的有界的吸收集$\mathscr{K}$={$\mathscr{K}$(t)}t∈$\mathbb{R}$ ,且有唯一的后向紧拉回吸引子$\mathscr{A}$={$\mathscr{A}$(t)}t∈$\mathbb{R}$ ,其中

    $ \begin{array}{*{20}{c}} {\mathscr{K}\left( t \right) = \left\{ {w \in {H^1}:\left\| w \right\|_{{H^1}}^2 \leqslant {M_0}\left( t \right) = 1 + \frac{2}{\lambda }M\left( t \right) + \frac{{2{{\left\| {{\psi _1}} \right\|}_1}}}{\lambda}} \right\}}&{t \in \mathbb{R}} \end{array} $ (29)
    $ \begin{array}{*{20}{c}} {\mathscr{A}\left( t \right) = {\omega _X}\left( {\mathscr{K}\left( t \right),t} \right) = \bigcap\limits_{{\tau _0} > 0} {\overline {\bigcup\limits_{\tau \geqslant {\tau _0}} {S\left( {t,t - \tau } \right)\mathscr{K}{{\left( t \right)}^X}} } } }&{t \in \mathbb{R}} \end{array} $ (30)

      由定理1可知,只需证明(8)式定义的过程S(·,·)在L2($\mathbb{R}$N)的拓扑下有一个增的有界的吸收集且是后向Omega-limit紧的.事实上,由条件(A)和(B1)可知(29)式中的M0(t)是一个关于时间t的有限的增函数,于是由引理3知(29)式中的$\mathscr{K}$={$\mathscr{K}$(t)}t∈$\mathbb{R}$S(·,·)在L2($\mathbb{R}$N)上的一个增的有界的吸收集.下面证明S(·,·)在L2($\mathbb{R}$N)的拓扑下是后向Omega-limit紧的.对每一个t∈$\mathbb{R}$和L2($\mathbb{R}$N)中有界集B,定义

    $ E\left( \tau \right) = \bigcup\limits_{r \geqslant \tau } {\bigcup\limits_{s \leqslant t} {S\left( {s,s - r} \right)B,\tau } } \geqslant 0 $

    由引理6可知,∀ε>0,存在一个τ1τ0K≥1使得

    $ \mathop {\sup }\limits_{{u_0} \in B} \mathop {\sup }\limits_{s \leqslant t} {\left\| {u\left( {s,s - \tau ,{u_0}} \right)} \right\|_{{L_2}\left( {Q_K^c} \right)}} < \varepsilon ,u \in E\left( {{\tau _1}} \right) $

    于是由文献[10]中Kuratowski测度的性质可知

    $ {\kappa _{{L^2}\left( {Q_K^c} \right)}}E\left( {{\tau _1}} \right) < 2\varepsilon $ (31)

    运用Sobolev紧嵌入:H1(QK)$\circlearrowleft$ L2(QK)到(18)式可知,E(τ1)|QK(E(τ1)在QK上的限制)在L2(QK)上是预紧的,于是由文献[10]中Kuratowski测度的性质可知

    $ {\kappa _{{L^2}\left( {{Q_K}} \right)}}E\left( {{\tau _1}} \right)\left| {_{{Q_K}}} \right. < \varepsilon $ (32)

    于是由(31)式和(32)式可知

    $ {\kappa _{{L^2}\left( {{\mathbb{R}^N}} \right)}}E\left( {{\tau _1}} \right) = {\kappa _{{L^2}\left( {Q_K^c} \right)}}E\left( {{\tau _1}} \right) + {\kappa _{{L^2}\left( {{Q_K}} \right)}}E\left( {{\tau _1}} \right)\left| {_{{Q_K}}} \right. < 3\varepsilon $

    E(τ)关于τ是单调递减的,故当ττ1时,

    $ {\kappa _{{L^2}\left( {{\mathbb{R}^N}} \right)}}E\left( \tau \right) \leqslant {\kappa _{{L^2}\left( {{\mathbb{R}^N}} \right)}}E\left( {{\tau _1}} \right) < 3\varepsilon $

    也即是

    $ \mathop {\lim }\limits_{\tau \to + \infty } {\kappa _{{L^2}\left( {{\mathbb{R}^N}} \right)}}E\left( \tau \right) = 0 $

    再由文献[10]中Kuratowski测度的性质可知(8)式中定义的过程S(·,·)在L2($\mathbb{R}$N)的拓扑下是后向Omega-limit紧的,于是由定理1可知定理2的结论成立.

    参考文献
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    The Backward Compact Dynamics for Non-Autonomous Reaction-Diffusion Equations on Unbounded Domains
    SHE Lian-bing1, LI Xin-tao1, LI Yang-rong2     
    1. School of Mathematics and Information Engineering, Liupanshui Normal College, LiupanshuiGuizhou 553004, China;
    2. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: The backward compactness of attractors for non-autonomous reaction-diffusion equations on unbounded domains is obtained under the conditions of both backward λ-tempered finiteness and backward tail-smallness for the non-autonomous force by using a cut-off function, a backward Granwall inequality and a backward Granwall-type inequality.
    Key words: non-autonomous dynamical system    backward compact dynamics    cut-off function    unbounded domain    
    X