DU Ping,YANG Yutong,LIU Shuang,et al.Attractor for stochastic semilinear strongly damped waveequations in locally uniform spaces[J].Journal of Yanbian University,2018,44(01):7-13,18.
随机半线性强衰减波动方程在局部一致空间上的吸引子
- Title:
- Attractor for stochastic semilinear strongly damped wave equations in locally uniform spaces
- Keywords:
- strongly damped stochastic wave equation; unbounded domain; locally uniform space; stochastic attractor
- 分类号:
- O211.63; O175.29
- 文献标志码:
- A
- 摘要:
- 在无界区域Rn中考虑了具有可加噪声的随机强衰减半线性波动方程的Cauchy问题,在相空间X=W</sup>2,pl u(Rn)×L</sup>pl u(Rn)中证明了该方程的整体可解性和随机吸引子的存在性.为解决该方程相关联的半群S(t,ω)的弱渐近紧性问题,首先证明了集合B1:=S(1,ω)γ+(B0)在空间D(L)=W</sup>2,pl u(Rn)×W</sup>2,pl u(Rn)中的有界性,其中B0是半群S(t,ω)在相空间X中的吸收集; 然后利用紧嵌入定理 W</sup>2,pl u(Rn)×W</sup>2,pl u(Rn)W</sup>1,pρ(Rn)×W</sup>1,pρ(Rn)得到了集合B1在相空间X中的弱渐近紧性.
- Abstract:
- In this paper, we consider the Cauchy problem for the stochastic strongly damped semilinear wave equations with additive noisein the unbounded domain- Rn. The global solvability and the existence of the stochastic attractor to this problem are proved in the phase space X=W</sup>2,pl u(Rn)×L</sup>pl u(Rn). To study to the asymptotic compactness of the corresponding semigroup S(t,ω), we first prove the set B1:=S(1,ω)γ+(B0)is bounded in D(L)=W</sup>2,pl u(Rn)×W</sup>2,pl u(Rn), where B0 is the absorbing set of in S(t,ω)in X, then we use the compact embedding theorems W</sup>2,pl u(Rn)×W</sup>2,pl u(Rn)W</sup>1,pρ(Rn)×W</sup>1,pρ(Rn)obtain the compactness of the set B1 in the phase space X.
参考文献/References:
[1] Babin A V, Vishik M I. Attractors of partial differential evolution equations in an unbounded domain[J]. Proceedings of the Royal Society of Edinburgh, 1990,116(3/4):221-243.
[2] Arrieta J M, Rodríguez-Bemal A, Cholewa J W, et al. Linear parabolic equations in locally uniform spaces[J]. Math Models Method Appl Sci, 2004,14(2):253-293.
[3] Cholewa J W, Dlotko T. Strongly damped wave equation in uniform spaces[J]. Nonlinear Analysis Theory Methods and Applications, 2006,64(1):174-187.
[4] Shen Z, Zhou S, Shen W. One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation[J]. Journal of Differential Equations, 2014,248(6):1432-1457.
[5] Feireisl E. Bounded, locally compact global attractors for semilinear damped wave equations on RN[J]. Diff Inter Eqns, 1996,9(5):1147-1156.
[6] Wang Zhaojuan, Zhou Shengfan, Gu Anhui. Random attractor of the stochastic strongly damped wave equation[J]. Commun Nonlinear Sci Numer Simulat, 2002,17(4):1649-1658.
[7] Cholewa J W, Dlotko T. Global Attractors in Abstract Parabolic Problems[M]. Cambridge: Cambridge University Press, 2000,10(1):54-69.
备注/Memo
收稿日期: 2018-01-12
*通信作者: 韩英豪(1963—),男,理学博士,副教授,研究方向为无穷维动力系统.