# 有界域中带阻尼波动方程的全局吸引子Global Attractor for a Damped Wave Equation in Bounded Domain

DOI: 10.12677/PM.2021.114070, PDF, HTML, XML, 下载: 21  浏览: 73

Abstract: This note considers the long-time behavior of a wave equation with damping. Based on the known results in the references, by analyzing the model, we obtain the existence of a global attractor for this kind of model.

1. 引言

2. 预备知识

1) $K\left(t\right)B$ 是相对紧的对于所有的 $t\ge {t}_{B}$

2) ${s}_{B}\left(t\right)=\underset{x\in B}{\mathrm{sup}}{‖S\left(t\right)x‖}_{X}<\infty$ 对所有的 $t\ge {t}_{B}$ 成立，且 ${s}_{B}\left(t\right)\to 0$$t\to \infty$

3. 波动方程模型以及全局吸引子的构造

$\left\{\begin{array}{l}{u}_{tt}+\eta {\left(-\Delta \right)}^{\theta }{u}_{t}+\left(-\Delta \right)u=f\left(u\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0,x\in \Omega ,\\ u\left(x,0\right)={u}_{0}\left(x\right),{u}_{t}\left(0,x\right)={v}_{0}\left(x\right),\text{\hspace{0.17em}}x\in \Omega ,\\ u\left(x,t\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge 0,x\in \partial \Omega ,\end{array}$ (3.1)

$A=-\Delta$ 是带有Dirichlet边界条件，易知A是正定的、自伴算子且有定义域 $D\left(A\right)={H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)$，因此 $-A$$X={L}^{2}$ 上生成了一个解析半群(具体可参见 [4] , 6.5节)。记 ${X}^{\alpha }$ 是与算子A相关的分数阶空间，也就是 ${X}^{\alpha }=D\left({A}^{\alpha }\right)$ 且赋予了图像范数，详细可见( [4] , 6.4节)，并且记 ${X}^{0}=X,{X}^{\frac{1}{2}}={H}_{0}^{1}\left(\Omega \right),{X}^{1}={H}^{2}\left(\Omega \right)\cap {H}_{0}^{1}\left(\Omega \right)$

$\left\{\begin{array}{l}\frac{\text{d}}{\text{d}t}\left[\begin{array}{l}u\\ v\end{array}\right]+{\mathcal{A}}_{\left(\theta \right)}\left[\begin{array}{l}u\\ v\end{array}\right]=\mathcal{F}\left(\left[\begin{array}{l}u\\ v\end{array}\right]\right),t>0,\\ \left[\begin{array}{l}u\left(0\right)\\ v\left(0\right)\end{array}\right]=\left[\begin{array}{l}{u}_{0}\\ {v}_{0}\end{array}\right]\in Y\end{array}$ (3.2)

$D\left({\mathcal{A}}_{\left(\theta \right)}\right)={Y}_{\left(\theta \right)}^{1}=\left\{\left[\begin{array}{l}\phi \\ \xi \end{array}\right]:\phi \in {X}^{\frac{3}{2}-\theta },\xi \in {X}^{\frac{1}{2}},{A}^{1-\theta }\phi +\eta \xi \in {X}^{\theta }\right\}$，对于 $\theta \in \left[0,1\right]$

${\mathcal{A}}_{\left(\theta \right)}\left[\begin{array}{c}\phi \\ \xi \end{array}\right]=\left[\begin{array}{cc}0& -I\\ A& \eta {A}^{\theta }\end{array}\right]\left[\begin{array}{c}\phi \\ \xi \end{array}\right]=\left[\begin{array}{c}-\xi \\ {A}^{\theta }\left({A}^{1-\theta }\phi +\eta \xi \right)\end{array}\right]$，对 $\left[\begin{array}{l}\phi \\ \xi \end{array}\right]\in D\left({\mathcal{A}}_{\left(\theta \right)}\right)$$\mathcal{F}\left(\left[\begin{array}{c}u\\ v\end{array}\right]\right)=\left[\begin{array}{c}0\\ F\left(u\right)\end{array}\right]$

$\left\{\begin{array}{l}{u}_{tt}+\eta {A}^{\theta }{u}_{t}+Au=0,t>0,\\ u\left(0\right)={u}_{0},{u}_{t}\left(0\right)={v}_{0},\end{array}$

$\theta =0$ 时对应于弱阻尼波动方程，当 $\theta =\left(0,1\right]$ 时对应于强阻尼波动方程。对每个 $\theta =\left[0,1\right]$，算子 $-{\mathcal{A}}_{\left(\theta \right)}$ 生成了一个强连续的半群 $\left\{{\text{e}}^{-{\mathcal{A}}_{\left(\theta \right)}t}:t\ge 0\right\}\subseteq \mathcal{L}\left(Y\right)$。若 $\theta =0$$\left\{{\text{e}}^{-{\mathcal{A}}_{\left(0\right)}t}:t\ge 0\right\}$ 可以成为一个群，对于 $\theta >0$$\left\{{\text{e}}^{-{\mathcal{A}}_{\left(\theta \right)}t}:t\ge 0\right\}$ 不能成为一个群，但是可以用如下方式描述：

1) 若 $\theta \in \left(0,\frac{1}{2}\right)$$-{\mathcal{A}}_{\left(\theta \right)}$ 生成了一个Gevrey类 $\delta >\frac{1}{2\theta }$ 的半群，因此它是可微的(参见 [5] )，

2) 若 $\theta \in \left[\frac{1}{2},1\right]$$-{\mathcal{A}}_{\left(\theta \right)}$ 生成了一个解析半群(参见 [6] )。在 [7] 中作者给出了 ${\mathcal{A}}_{\left(\theta \right)}$ 的一个分数次定义域的刻画，也就是分数阶空间 ${Y}_{\left(\theta \right)}^{\alpha }$$\alpha \in \left[\frac{1}{2},1\right]$

$|f\left(u\right)-f\left({u}^{\prime }\right)|\le c|u-{u}^{\prime }|\left(1+{|u|}^{\rho -1}+{|{u}^{\prime }|}^{\rho -1}\right)$ (3.3)

$\underset{|u|\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{f\left(u\right)}{u}\le 0.$ (3.4)

${‖\left[\begin{array}{l}u\left(t,{u}_{0},{v}_{0}\right)\\ v\left(t,{u}_{0},{v}_{0}\right)\end{array}\right]‖}_{Y}\le c+{c}^{\prime }\mathcal{L}\left(\left[\begin{array}{l}{u}_{0}\\ {v}_{0}\end{array}\right]\right)\le C\left({‖\left[\begin{array}{l}{u}_{0}\\ {v}_{0}\end{array}\right]‖}_{Y}\right)$ (3.5)

$\mathcal{L}\left(\left[\begin{array}{l}{w}_{1}\\ {w}_{2}\end{array}\right]\right)=\frac{1}{2}{‖{w}_{2}‖}_{{L}^{2}\left(\Omega \right)}^{2}+\frac{1}{2}{‖\nabla {w}_{1}‖}_{{L}^{2}\left(\Omega \right)}^{2}-{\int }_{\Omega }{\int }_{0}^{{w}_{1}}f\left(s\right)\text{d}s,{w}_{1},{w}_{2}\in Y,$ (3.6)

$C:{ℝ}^{+}\to {ℝ}^{+}$ 是局部有界函数且与 $\eta$ 选取无关。我们记 $T\left(t\right):Y\to Y$ 是方程(3.1)的解算子。

 [1] Arrieta, J.M. Carvalho, A.N. and Hale, J.K. (1992) A Damped Hyperbolic Equation with Critical Exponent. Communications in Partial Differential Equations, 17, 841-866. https://doi.org/10.1080/03605309208820866 [2] Carvalho, A.N. and Cholewa, J.W. (2002) Attractors for Strongly Damped Wave Equation with Critical Nonlinearities. Pacific Journal of Mathematics, 207, 287-310. https://doi.org/10.2140/pjm.2002.207.287 [3] Carvalho, A.N. and Cholewa, J.W. (2002) Local Well Posedness for Strongly Damped Wave Equation with Critical Nonlinearities. Bulletin of the Australian Mathe-matical Society, 66, 443-463. https://doi.org/10.1017/S0004972700040296 [4] Carvalho, A.N., Langa, J.A. and Robinson, J.C. (2013) Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems. Applied Mathematical Sciences, 182. https://doi.org/10.1007/978-1-4614-4581-4 [5] Chen, S.P. and Triggiani, R. (1990) Gevrey Class Semigroups Arising from Elastic Systems with Gentle Dissipation: The Case 0 < α < 1/2. Proceedings of American Mathematical Society, 110, 401-415. https://doi.org/10.2307/2048084 [6] Chen, S.P. and Triggiani, R. (1988) Proof of Two Conjectures by G. Chen and D. L. Russell on Structural Damping for Elastic Systems. Approximation and Optimization (Havana, 1987), Lecture Notes in Math., vol. 1354, Springer, Berlin, 234-256. https://doi.org/10.1007/BFb0089601 [7] Chen, S.P. and Triggiani, R. (1989) Proof of Extensions of Two Conjectures on Structural Damping for Elastic Systems. Pacific Journal of Mathematics, 136, 15-55. https://doi.org/10.2140/pjm.1989.136.15 [8] Carvalho, A.N. and Cholewa, J.W. (2008) Regularity of Solutions on the Global Attractor for a Semilinear Damped Wave Equation. Journal of Mathematical Analysis and Applications, 337, 932-948. https://doi.org/10.1016/j.jmaa.2007.04.051 [9] Carvalho, A.N., Cholewa, J.W. and Dlotko, T. (2008) Strongly Damped Wave Problems: Bootstrapping and Regularity of Solutions. Journal of Differential Equations, 244, 2310-2333. https://doi.org/10.1016/j.jde.2008.02.011 [10] Ball, J.M. (1997) Continuity Properties and Global Attractors of Gener-alized Semiflows and the Navier-Stokes Equations. Journal of Nonlinear Science, 7, 475-502. https://doi.org/10.1007/s003329900037 [11] Crauel, H. and Flandoli, F. (1994) Attractors for Random Dynamical Sys-tems. Probability Theory and Related Fields, 100, 365-393. https://doi.org/10.1007/BF01193705 [12] Freitas, M.M., Kalita, P. and Langa, J.A. (2018) Continuity of Non-Autonomous Attractors for Hyperbolic Perturbation of Parabolic Equa-tions. Journal of Differential Equations, 264, 1886-1945. https://doi.org/10.1016/j.jde.2017.10.007