Time-dependent global attractors for the strongly damped wave equations with lower regular forcing term
DOI:
https://doi.org/10.12775/TMNA.2022.022Keywords
Strongly damped wave equations, critical exponential growth, time-dependent global attractorsAbstract
In this paper, based on a new theoretical framework of time-dependent global attractors (Conti, Pata and Temam \cite{CPT13}), we consider the strongly damped wave equations $\varepsilon(t)u_{tt}-\Delta u_{t}-\Delta u+f(u)=g(x)$ and establish the existence of attractors in $\mathcal{H}_{t}=H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ and $\mathcal{V}_{t}=H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)$, respectively.References
J. Arrieta, A.N. Carvalho and J.K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations 17 (1992), 841–866.
A.N. Carvalho and J.W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc. 66 (2002), 443–463.
A.N. Carvalho and J.W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math. 207 (2002), 287–310.
A.N. Carvalho, J.A. Langa and J.C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer, New York, 2013.
J.W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal. 64 (2006), 174–187.
M. Conti and V. Pata, Asymptotic structure of the attractor for processes on timedependent spaces, Nonlinear Anal. Real World Appl. 19 (2014), 1–10.
M. Conti, V. Pata and M. Squassina, Strongly damped wave equations on R3 with critical nonlinearities, Commun. Appl. Anal. 9 (2005), 161–176.
M. Conti, V. Pata and R. Temam, Attractors for process on time-dependent spaces: Applications to wave equations, J. Differential Equations 255 (2013), 1254–1277.
F. Di Plinio, G.S. Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst. 29 (2011), 141–167.
Y.L. Du, X. Li and C.Y. Sun, On the asymptotic behavior of strongly damped wave equations, Topol. Methods Nonlinear Anal. 44 (2014), 161–175.
J.M. Ghidaglia and A. Marzocchi, Longtime behavior of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal. 22 (1991), 879–895.
F.J. Meng, M.H. Yang and C.K. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), 205–225.
V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys. 253 (2005), 511–533.
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), 1495–1506.
C.Y. Sun, D.M. Cao and J.Q. Duan, Non-autonomous wave dynamics with memoryasymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), 743–761.
C.Y. Sun and M.H. Yang, Dynamics of the nonclassical diffusion equations, Asymptot. Anal. 59 (2008), 51–81.
Y. Sun and Z.J. Yang, Longtime dynamics for a nonlinear viscoelastic equation with time-dependent memory kernel, Nonlinear Anal. Real World Appl. 64 (2022), 103432.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York, 1997.
Y.L. Xiao, Attractors for a nonclassical diffusion equation, Acta Math. Appl. Sin. (Engl. Ed.) 18 (2002), 273–276.
Y.Q. Xie, Q.S. Li and K.X. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal. Real World Appl. 31 (2016), 23–37.
M.H. Yang and C.Y. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc. 361 (2009), 1069–1101.
M.H. Yang and C.Y. Sun, Attractors for strongly damped wave equations, Nonlinear Anal. Real World Appl. 10 (2009), 1097–1100.
M.H. Yang and C.Y. Sun, Exponential attractors for the strongly damped wave equations, Nonlinar Anal. Real World Appl. 11 (2010), 913–919.
S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal. 3 (2004), 921–934.
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Xinyu Mei, Tao Sun, Yongqin Xie, Kaixuan Zhu
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
