References
- A.N. Carvalho, J.A. Langa, and J.C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences Vol. 182, Springer, New York, 2013.
- M. Chekroun and N.E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Comm. Math. Phys. 316(3) (2012), pp. 723–761.
- C. Foias, O. Manley, R. Rosa, and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.
- J. García-Luengo, P. Marín-Rubio, and J. Real, Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differ. Equ. 252 (2012), pp. 4333–4356.
- Y.F. Guo, Dynamics and invariant manifolds for a nonlocal nonautonomous Swift-Hohenberg equation, J. Inequal. Appl. 2015 (2015), p. 366.
- Y.F. Guo, J.Q. Duan, and D.L. Li, Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation, Discrete Contin. Dyn. Syst. 9(6) (2016), pp. 1701–1715.
- J.K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.
- Y. He, C.Q. Li, and J.T. Wang, Invariant measures and statistical solutions for the nonautonomous discrete modified Swift-Hohenberg equation, Bull. Malays. Math. Sci. Soc. 44 (2021), pp. 1–19.
- D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer Verlag, Berlin, New York, 1981.
- P.C. Hohenberg and J.B. Swift, Effects of additive noise at the onset of Rayleigh-Bénard's convection, Phys. Rev. Lett. 46(8) (1992), pp. 4773–4785.
- P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. A. 463 (2007), pp. 163–181.
- R.E. LaQuey, S.M. Mahajan, P.H. Rutherford, and W.M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett. 34(7) (1975), pp. 391–394.
- J. Lega, J.V. Moloney, and A.C. Newell, Swift-Hohenberg equation for lasers, Phys. Rev. Lett. 73(22) (1994), pp. 2978–2981.
- C.Q. Li, D.S. Li, and J.T. Wang, A remark on attractor bifurcation, Dyn. Partial Differ. Equ. 18 (2021), pp. 157–172.
- C.Q. Li and J.T. Wang, Bifurcation from infinity of the Schrödinger equation via invariant manifolds in nonlinear analysis, Nonlinear Anal. 213 (2021), p. 112490.
- D.S. Li and C.K. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differ. Equ. 149(2) (1998), pp. 191–210.
- X. Li, W.X. Shen, and C.Y. Sun, Invariant measures for complex-valued dissipative dynamical systems and applications, Discrete Contin. Dyn. Syst. Ser. B 22(6) (2017), pp. 2427–2446.
- Y.J. Li, H.Q. Wu, and T.G. Zhao, Random pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg equation with multiplicative noise, J. Math. Phys. 61 (2020), pp. 1–13.
- Y.J. Li and C.K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput. 190(2) (2007), pp. 1020–1029.
- G.G. Lin, H.J. Gao, J.Q. Duan, and V.J. Ervin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, J. Math. Phys. 41 (2000), pp. 2077–2089.
- G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B 9(3/4) (2008), pp. 643–659.
- G. Łukaszewicz, J. Real, and J.C. Robinson, Invariant measures for dissipative dynamical systems and generalised Banach limits, J. Dynam. Differ. Equ. 23(2) (2011), pp. 225–250.
- G. Łukaszewicz and J.C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst. 34(10) (2014), pp. 4211–4222.
- S.H. Park and J.Y. Park, Pullback attractor for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl. 67(3) (2014), pp. 542–548.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
- Y. Pomeau and P. Manneville, Wave length selection in cellular flows, Phys. Lett. A. 75(4) (1980), pp. 296–298.
- A.J. Roberts, Planform evolution in convection: An embedded center manifold, J. Aust. Math. Soc. 34(2) (1992), pp. 174–198.
- A.J. Roberts, The Swift-Hohenberg equation requires nonlocal modifications to model spatial pattern evolution of physical problems, Physics, 1994.
- K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, Heildelberg, New York, 1980.
- J.B. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A. 15(1) (1977), pp. 319–328.
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.
- J.T. Wang, C.Q. Li, L. Yang, and M. Jia, Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise, J. Math. Phys. 62 (2021), p. 111507.
- J.T. Wang and D.S. Li, On relative category and Morse decompositions for infinite-dimensional dynamical systems, Topol. Appl. 291 (2021), p. 107624.
- J.T. Wang, D.S. Li, and J.Q. Duan, On the shape Conley index theory of semiflows on complete metric spaces, Discrete Contin. Dyn. Syst. 36(3) (2015), pp. 1629–1647.
- J.T. Wang, D.S. Li, and J.Q. Duan, Compactly generated shape index theory and its application to a retarded nonautonomous parabolic equation, Topol. Meth. Nonlinear Anal. (2018). in press.
- J.T. Wang, L. Yang, and J.Q. Duan, Recurrent solutions of a nonautonomous modified Swift-Hohenberg equation, Appl. Math. Comput. 379 (2020), p. 125270.
- J.T. Wang, X.Q. Zhang, and C.D. Zhao, Statistical solutions for a nonautonomous modified Swift-Hohenberg equation, Math. Methods Appl. Sci. 44 (2021), pp. 14502–14516.
- J.T. Wang, C.D. Zhao, and T. Caraballo, Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays, Commun. Nonlinear Sci. Numer. Simul. 91 (2020), p. 105459.
- W. Wang, J.H. Sun, and J.Q. Duan, Ergodic dynamics of the stochastic Swift-Hohenberg system, Nonlinear Anal. Real World Appl. 6 (2005), pp. 273–295.
- X.M. Wang, Upper-semicontinuity of stationary statistical properties of dissipative systems, Discrete Contin. Dyn. Syst. 23(1/2) (2009), pp. 521–540.
- Z. Wang and X.Y. Du, Pullback attractors for modified Swift-Hohenberg equation on unbounded domains with non-autonomous deterministic and stochastic forcing terms, J. Appl. Anal. Comput. 7(1) (2017), pp. 207–223.
- Q.K. Xiao and H.J. Gao, Bifurcation analysis of a modified Swift-Hohenberg equation, Nonlinear Anal. Real World Appl. 11 (2010), pp. 4451–4464.
- L. Xu and Q.Z. Ma, Existence of the uniform attractors for a non-autonomous modified Swift-Hohenberg equation, Adv. Differ. Equ. 2015(1) (2015), pp. 1–11.
- C.D. Zhao, Y.J. Li, and G. Łukaszewicz, Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids, Z. Angew. Math. Phys. 71(4) (2020), pp. 1411–14124.
- C.D. Zhao, Y.J. Li, and T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differ. Equ. 269 (2020), pp. 467–494.
- C.D. Zhao and L. Yang, Pullback attractor and invariant measures for the globally modified Navier-Stokes equations, Commun. Math. Sci. 15(6) (2017), pp. 1565–1580.
- Z.Q. Zhu and C.D. Zhao, Pullback attractor and invariant measures for the three-dimensional regularized MHD equations, Discrete Contin. Dyn. Syst. 38(3) (2018), pp. 1461–1477.