﻿ 波动方程：拟稳定性和广义指数吸引子

# 波动方程：拟稳定性和广义指数吸引子Wave Equation: Quasi-Stability and Generalized Exponential Attractor

Abstract: This paper studies the long-term dynamical behavior of the solution of the wave equation. By using the methods developed by Chueshov and Lasiecha, we get the quasi-stability property of the system and obtain the existence of a global attractor which has finite fractal dimension. Result on exponential attractors of the system is also proved.

1. 引言

$\left\{\begin{array}{l}{u}_{tt}+\alpha {u}_{t}-\Delta u+f\left(u\right)=q\left(x\right),\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{in}\text{\hspace{0.17em}}\text{ }\Omega ,t>0,\\ u|{}_{\Gamma }={u}_{t}|{}_{\Gamma }=0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{on}\text{ }\text{\hspace{0.17em}}\Omega ,t>0,\\ u\left(0,x\right)={u}_{0}\left(x\right)\in {H}_{0}^{1}\left(\Omega \right),\\ {u}_{t}\left(0,x\right)={u}_{1}\left(x\right)\in {L}^{2}\left(\Omega \right),\end{array}$ (1)

$\underset{|s|\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{F\left(s\right)}{{s}^{2}}\ge 0，$ (2)

$\underset{|s|\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{sf\left(s\right)-{c}_{1}F\left(s\right)}{{s}^{2}}\ge 0，$ (3)

$|{f}^{\prime }\left(s\right)|\le {c}_{2}\left(1+{|s|}^{2}\right)，$ (4)

2. 半群

${\left(u,v\right)}_{0}={\int }_{\Omega }uv\text{d}x$${‖u‖}_{0}^{2}={\left(u,u\right)}_{0}$

${\left(u,v\right)}_{1}={\left(u,v\right)}_{0}+{\left(\nabla u,\nabla v\right)}_{0}$${‖u‖}_{1}^{2}={\left(u,u\right)}_{1}$

${\left(w,{w}^{\prime }\right)}_{H}={\left({w}_{1},{{w}^{\prime }}_{1}\right)}_{1}+{\left({w}_{2},{{w}^{\prime }}_{2}\right)}_{0}$${‖w‖}_{H}^{2}={‖{w}_{1}‖}_{1}^{2}+{‖{w}_{2}‖}_{0}^{2}$

$D\left(A\right)=\left\{u,-\Delta u\in {L}^{2}\left(\Omega \right),u|{}_{\Gamma }={u}_{t}|{}_{\Gamma }=0\right\}$

A的特征值为 ${\left\{{\lambda }_{i}\right\}}_{i\in {N}^{+}}$ 满足 $0\le {\lambda }_{1}\le {\lambda }_{2}\le \cdots \le {\lambda }_{m}\le \cdots$ ，且当 $m\to \infty$${\lambda }_{m}\to \infty$

$\left\{\begin{array}{l}\frac{\text{d}u}{\text{d}t}=v,\\ \frac{\text{d}v}{\text{d}t}+\alpha v-\Delta u+f\left(u\right)=q\left(x\right),\\ u|{}_{\Gamma }=v|{}_{\Gamma }=0,\\ u\left(0,x\right)={u}_{0}\left(x\right)\in {H}_{0}^{1}\left(\Omega \right),\\ v\left(0,x\right)={u}_{1}\left(x\right)\in {L}^{2}\left(\Omega \right),\end{array}$ (5)

$W=\left[\begin{array}{c}u\\ v\end{array}\right]$$L=\left[\begin{array}{c}-v\\ \alpha v-\Delta u\end{array}\right]$$G\left(W\right)=\left[\begin{array}{c}0\\ q\left(x\right)-f\left(u\right)\end{array}\right]$

$\frac{\text{d}W}{\text{d}t}+LW=G\left(W\right)$$W\left(0\right)={W}_{0}$(6)

$W\left(t\right)={\text{e}}^{-Lt}W\left(0\right)+{\int }_{0}^{\infty }{\text{e}}^{-L\left(t-s\right)}G\left(W\left(s\right)\right)\text{d}s,\text{\hspace{0.17em}}t\ge 0$

$T\left(t\right):W\left(0\right)={\left({u}_{0},{u}_{1}\right)}^{T}\to W\left(t\right)={\left(u\left(t\right),{u}_{t}\left(t\right)\right)}^{T},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }H\to H.$

3. 吸收集

${\phi }_{1}=u,{\phi }_{2}={u}_{t}+\epsilon u$ ，其中 $\epsilon =\frac{\alpha \left({\alpha }^{2}-1\right)}{3\left({\alpha }^{2}-1\right)+\sqrt{{\alpha }^{4}-1}}$ ，因此方程(1)可写作如下形式

$\left\{\begin{array}{l}\frac{\text{d}{\phi }_{1}}{\text{d}t}+\epsilon {\phi }_{1}-{\phi }_{2}=0,\\ \frac{\text{d}{\phi }_{2}}{\text{d}t}-\epsilon {\phi }_{2}+{\epsilon }^{2}{\phi }_{1}+\alpha \left({\phi }_{2}-\epsilon {\phi }_{1}\right)-\Delta {\phi }_{1}-f\left({\phi }_{1}\right)=q\left(x\right),\\ u|{}_{\Gamma }=v|{}_{\Gamma }=0,\\ u\left(0,x\right)={u}_{0}\left(x\right)\in {H}_{0}^{1}\left(\Omega \right),\\ v\left(0,x\right)={u}_{1}\left(x\right)\in {L}^{2}\left(\Omega \right),\end{array}$ (7)

$\theta =\left[\begin{array}{c}{\phi }_{1}\\ {\phi }_{2}\end{array}\right]$${L}_{\epsilon }\left(\theta \right)=\left[\begin{array}{c}\epsilon {\phi }_{1}-{\phi }_{2}\\ -\epsilon {\phi }_{2}+{\epsilon }^{2}{\phi }_{1}+\alpha \left({\phi }_{2}-\epsilon {\phi }_{1}\right)-\Delta {\phi }_{1}\end{array}\right]$$G\left(\theta \right)=\left[\begin{array}{c}0\\ q\left(x\right)-f\left({\phi }_{1}\right)\end{array}\right]$

$\frac{\text{d}\theta }{\text{d}t}+{L}_{\epsilon }\left(\theta \right)=G\left(\theta \right)$$\theta \left(0\right)={\left({\phi }_{1}\left(0\right),{\phi }_{2}\left(0\right)\right)}^{\text{T}}$(8)

${\left({L}_{\epsilon }\left(\theta \right),\theta \right)}_{H}\ge \frac{\epsilon }{2}{‖\theta ‖}_{H}^{2}+\frac{\alpha }{2}{‖{\phi }_{2}‖}_{0}^{2}$

${\left({L}_{\epsilon }\left(\theta \right),\theta \right)}_{H}={\left(\epsilon {\phi }_{1}-{\phi }_{2},{\phi }_{1}\right)}_{1}+{\left(-\epsilon {\phi }_{2}+{\epsilon }^{2}{\phi }_{1}+\alpha \left({\phi }_{2}-\epsilon {\phi }_{1}\right)-\Delta {\phi }_{1},{\phi }_{2}\right)}_{0}$

$\begin{array}{l}{\left({L}_{\epsilon }\left(\theta \right),\theta \right)}_{H}-\frac{\epsilon }{2}{‖\theta ‖}_{H}^{2}-\frac{\alpha }{2}{‖{\phi }_{2}‖}_{0}^{2}\\ =\epsilon {‖{\phi }_{1}‖}_{1}^{2}-{\left({\phi }_{1},{\phi }_{2}\right)}_{1}+\left({\epsilon }^{2}-\alpha \epsilon \right){\left({\phi }_{1},{\phi }_{2}\right)}_{0}-\epsilon {‖{\phi }_{2}‖}_{0}^{2}+\alpha {‖{\phi }_{2}‖}_{0}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{\left(\Delta {\phi }_{1},{\phi }_{2}\right)}_{0}-\frac{\epsilon }{2}{‖{\phi }_{1}‖}_{1}^{2}-\frac{\epsilon }{2}{‖{\phi }_{2}‖}_{0}^{2}-\frac{\alpha }{2}{‖{\phi }_{2}‖}_{0}^{2}\\ \ge \frac{\epsilon }{2}{‖{\phi }_{1}‖}_{0}^{2}-\left(1-{\epsilon }^{2}+\alpha \epsilon \right){‖{\phi }_{1}‖}_{0}{‖{\phi }_{2}‖}_{0}+\left(\frac{\alpha }{2}-\frac{3\epsilon }{2}\right){‖{\phi }_{2}‖}_{0}^{2}\end{array}$

$\alpha -3\epsilon \ge 0$$2\sqrt{\frac{\epsilon }{2}\cdot \frac{\alpha -3\epsilon }{2}}\ge 1-{\epsilon }^{2}+\alpha \epsilon$

${‖{T}_{\epsilon }\left(t\right)\phi \left(0\right)‖}_{H}^{2}\le {C}_{1}$

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\theta ‖}_{H}^{2}+\frac{\epsilon }{2}{‖\theta ‖}_{H}^{2}+\frac{\alpha }{2}{‖{\phi }_{2}‖}_{0}^{2}+{\left(F\left({\phi }_{1}\right),1\right)}_{0}+{\left(f\left({\phi }_{1}\right),{\phi }_{1}\right)}_{0}\le {\left(q\left(x\right),{\phi }_{2}\right)}_{0}$

${\left({\phi }_{1},f\left({\phi }_{1}\right)\right)}_{0}\ge {c}_{1}{\left(F\left({\phi }_{1}\right),1\right)}_{0}-\frac{1}{16}{‖\phi ‖}_{1}^{2}-k$(9)

${\left(F\left({\phi }_{1}\right),1\right)}_{0}\ge -\frac{1}{16\left(1+{c}_{1}\right)}{‖\phi ‖}_{1}^{2}-k$(10)

$\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({‖\theta ‖}_{H}^{2}+2{\left(F\left({\phi }_{1}\right),1\right)}_{0}\right)+\frac{7\epsilon }{16}{‖{\phi }_{1}‖}_{1}^{2}+\left(\frac{\epsilon }{2}+\frac{\alpha }{2}\right){‖{\phi }_{2}‖}_{0}^{2}+{c}_{1}\epsilon {\left(F\left({\phi }_{1}\right),1\right)}_{0}\le {\left(q,{\phi }_{2}\right)}_{0}+\epsilon k$

$\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({‖\phi ‖}_{H}^{2}+2{\left(F\left({\phi }_{1}\right),1\right)}_{0}\right)+\frac{7\epsilon }{16}{‖{\phi }_{1}‖}_{1}^{2}+\frac{7\epsilon }{16}{‖{\phi }_{2}‖}_{0}^{2}+{c}_{1}\epsilon {\left(F\left({\phi }_{1}\right),1\right)}_{0}\le \frac{1}{2\alpha }{‖q‖}_{0}^{2}+\epsilon k$

$\frac{\text{d}}{\text{d}t}\left({‖\theta ‖}_{H}^{2}+2{\left(F\left({\phi }_{1}\right),1\right)}_{0}\right)+\frac{7\epsilon }{8}\left({‖\theta ‖}_{H}^{2}+2{c}_{1}{\left(F\left({\phi }_{1}\right),1\right)}_{0}\right)\le \frac{1}{\alpha }{‖q‖}_{0}^{2}+2\epsilon k$

${‖\theta ‖}_{H}^{2}+2{\left(F\left({\phi }_{1}\right),1\right)}_{0}\ge \frac{7}{8}{‖\theta ‖}_{H}^{2}$

${‖\theta ‖}_{H}^{2}+2{\left(F\left({\phi }_{1}\right),1\right)}_{0}\le {‖\theta ‖}_{H}^{2}+2{\left(F\left({\phi }_{1}\right),1\right)}_{0}{\text{e}}^{-\frac{7{k}_{1}\epsilon }{8}t}+\left(\frac{8}{7\alpha {k}_{1}\epsilon }{‖q‖}_{0}^{2}+\frac{16k}{7{k}_{1}}\right)\left(1-{\text{e}}^{-\frac{7{k}_{1}\epsilon }{8}t}\right)$

${k}_{1}=\mathrm{min}\left\{1,4{c}_{1}\right\}$

${‖{\theta }_{0}‖}_{H}^{2}\le r$ ，由(4)和Sobolev’s嵌入定理，则存在 $K\left(r\right)>0$ ，使得

${‖\theta ‖}_{H}^{2}+2{\left(F\left({\phi }_{1}\right),1\right)}_{0}\le K\left(r\right)$

${‖{\theta }_{0}‖}_{H}^{2}\le \frac{8}{7}K\left(r\right){\text{e}}^{-\frac{{k}_{1}\epsilon }{4}t}+\left(\frac{64}{49\alpha {k}_{1}\epsilon }{‖q‖}_{0}^{2}+\frac{128k}{49{k}_{1}}\right)\left(1-{\text{e}}^{-\frac{{k}_{1}\epsilon }{4}t}\right)$

${C}_{1}=\frac{64}{49\alpha {k}_{1}\epsilon }{‖q‖}_{0}^{2}+\frac{128k}{49{k}_{1}}$

${T}_{\epsilon }\left(t\right){B}_{0}\subseteq {B}_{0}$

4. 动力系统的拟稳定性和渐进光滑性

$S\left(t\right){U}_{0}=\left(u,{u}_{t}\right)$${U}_{0}=\left(u\left(0\right),{u}_{t}\left(0\right)\right)\in H$(11)

$u\in C\left({R}^{+},X\right)\cap {C}^{1}\left({R}^{+},Y\right)$(12)

(i) $a\left(t\right)$$c\left(t\right)$$\left[0,+\infty \right)$ 上局部有界，

(ii) $b\left(t\right)\in {L}^{1}\left({R}^{+}\right)$${\mathrm{lim}}_{t\to \infty }b\left(t\right)=0$

(iii) 对任意的 ${u}^{1},{u}^{2}\in B$$t>0$ 下面的关系成立：

${‖S\left(t\right){U}^{1}-S\left(t\right){U}^{2}‖}_{H}^{2}\le a\left(t\right){‖{U}^{1}-{U}^{2}‖}_{H}^{2}$(13)

${‖S\left(t\right){U}^{1}-S\left(t\right){U}^{2}‖}_{H}^{2}\le b\left(t\right){‖{U}^{1}-{U}^{2}‖}_{H}^{2}+c\left(t\right)\underset{0(14)

$‖{T}_{\epsilon }{U}^{1}-{T}_{\epsilon }{U}^{2}‖\le {b}_{0}{\text{e}}^{-rt}{‖{U}^{1}-{U}^{2}‖}_{H}^{2}+{C}_{B}\left(1-{\text{e}}^{-rt}\right){‖{u}^{1}-{u}^{2}‖}_{0}^{2}$(15)

$\frac{\text{d}\Phi }{\text{d}t}+{L}_{\epsilon }\left(\Phi \right)=G\left(\Phi \right)$(16)

${U}^{1}\left(0\right)-{U}^{2}\left(0\right)=\left({\Phi }_{0},{\Phi }_{1}\right)$

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\Phi ‖}_{H}^{2}+\frac{\epsilon }{2}{‖\Phi ‖}_{H}^{2}+\frac{\alpha }{2}{‖\Phi ‖}_{0}^{2}\le -{\left(f\left({u}^{1}\right)-f\left({u}^{2}\right),{\Phi }_{2}\right)}_{0}\\ \le {‖f\left({u}^{1}\right)-f\left({u}^{2}\right)‖}_{0}{‖{\Phi }_{2}‖}_{0}\\ \le \frac{1}{2\alpha }{‖f\left({u}^{1}\right)-f\left({u}^{2}\right)‖}_{0}^{2}+\frac{\alpha }{2}{‖{\Phi }_{2}‖}_{0}^{2}\end{array}$

$\begin{array}{c}{‖f\left({u}^{1}\right)-f\left({u}^{2}\right)‖}_{0}^{2}={\int }_{\Omega }{\left(f\left({u}^{1}\right)-f\left({u}^{2}\right)\right)}^{2}\text{d}x\le {C}_{2}{\int }_{\Omega }{\left({|{u}^{1}|}^{3}-{|{u}^{2}|}^{3}\right)}^{2}\text{d}x\\ \le {\left({C}_{2}{\int }_{\Omega }{\left({|{u}^{1}|}^{2}+{u}^{1}{u}^{2}+{|{u}^{2}|}^{2}\right)}^{3}\text{d}x\right)}^{\frac{2}{3}}{\left({\int }_{\Omega }{\left({u}^{1}-{u}^{2}\right)}^{6}\text{d}x\right)}^{\frac{1}{3}}\\ \le {C}_{2}\left(\frac{1}{4k}{‖{|{u}^{1}|}^{2}+{u}^{1}{u}^{2}+{|{u}^{2}|}^{2}‖}_{{L}^{3}}^{2}+k{‖{u}^{1}-{u}^{2}‖}_{{L}^{6}}^{2}\right)\\ \le K{‖{u}^{1}-{u}^{2}‖}_{{L}^{6}}^{2}\le K{‖{u}^{1}-{u}^{2}‖}_{{H}_{0}^{1}}^{2}\end{array}$

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\Phi ‖}_{H}^{2}+\frac{\epsilon }{2}{‖\Phi ‖}_{H}^{2}\le K{‖{u}^{1}-{u}^{2}‖}_{{H}_{0}^{1}}^{2}$

${‖\Phi \left(t\right)‖}_{H}^{2}\le {‖\Phi \left(0\right)‖}_{H}^{2}{\text{e}}^{-\epsilon t}+\frac{2k}{\epsilon }{‖{u}^{1}-{u}^{2}‖}_{{H}_{0}^{1}}^{2}\left(1-{\text{e}}^{-\epsilon t}\right)$

${‖{T}_{\epsilon }{U}^{1}-{T}_{\epsilon }{U}^{2}‖}_{H}^{2}\le {\text{e}}^{-\epsilon t}{‖{U}^{1}-{U}^{2}‖}_{H}^{2}+\frac{2k}{\epsilon }\left(1-{\text{e}}^{-\epsilon t}\right){‖{u}^{1}-{u}^{2}‖}_{{H}_{0}^{1}}^{2}$

$B\subset H$ 关于 $T\left(t\right)$ 的有界正向不变集。设 $T\left(t\right){U}^{i}=\left({u}^{i},{u}^{i}+\epsilon {u}^{i}\right)$ 是方程(8)关于初值条件 ${U}^{i},i=1,2$ 在B中的解，我们定义半范数

${n}_{X}\left(\Phi \right)={‖\Phi ‖}_{{L}^{2}}^{2}$

${‖{T}_{\epsilon }{U}^{1}-{T}_{\epsilon }{U}^{2}‖}_{H}^{2}\le b\left(t\right){‖{U}^{1}-{U}^{2}‖}_{H}^{2}+c\left(t\right)\underset{0

$b\left(t\right)\in {L}^{1}\left({R}^{+}\right)$ 并且 $\underset{t\to \infty }{\mathrm{lim}}b\left(t\right)=0$

5. 全局吸引子和广义指数吸引子

${‖{U}_{t}\left(t\right)‖}_{\stackrel{˜}{H}}\le {C}_{B}$

${‖{T}_{\epsilon }\left({t}_{1}\right)U-{T}_{\epsilon }\left({t}_{2}\right)U‖}_{\stackrel{˜}{H}}\le {\int }_{{t}_{1}}^{{t}_{2}}{‖{U}_{t}\left(\tau \right)‖}_{\stackrel{˜}{H}}\text{d}\tau \le {C}_{B}{|{t}_{1}-{t}_{2}|}^{r},{t}_{1},{t}_{2}\in \left[0,T\right],U\in B$ (17)

[1] Temam, R. (1998) Infinite-Dimension Dynamical Systems in Mechanics and Physics. 2nd Edition, New York.

[2] Aviles, P. and Sandefur, J. (1985) Nonlinear Second Order Equations with Applications to Partial Differential Equations. Journal of Differential Equations, 58, 404-427.
https://doi.org/10.1016/0022-0396(85)90008-7

[3] Fitzgibbon, W.E. (1981) Strongly Damped Quasi-linear Evolution Equations. Journal of Mathematical Analysis & Applications, 79, 536-550.

[4] Zelik (1991) Asymptotic Regularity of Solutions of Singularly Perturbed Damped Wave Equations with Supercritical Nonlinearities.

[5] Frigeri, S. (2010) Attractors for Semilinear Damped Wave Equations with an Acoustic Boundary Condition. Journal of Evolution Equations, 10, 29-58.
https://doi.org/10.1007/s00028-009-0039-1

[6] Chueshov, I. and Lasiecka, I. (2012) Von Karman Evolution Equation: Well-Posedness and Longtime Dynamics. Springer, New York.

[7] Feng, B. (2017) On a Semilinear Timoshenko-Colean-Gurtin System: Quasi-Stability and Attractors. Discrete Continuous Dynamical Systems, 3, 4427-4451.

[8] Beale, J.T. (1967) Spectral Properties of an Acoustic Boundary Conditions. Indiana University Mathematics Journal, 25, 895-917.
https://doi.org/10.1512/iumj.1976.25.25071

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