Upper Semicontinuity of Strong Pullback Attractors for Delay Non-autonomous Dynamical Systems and Application to Double Time-delayed 2D MHD Equations

Abstract

The long term behavior of delay-dependent evolution processes generated by delay partial differential equations with non-autonomous forcing terms is analysed. First, we introduce some concepts on strong pullback attractors. We then establish a general theory concerning the existence of such attractors for abstract delay models. Moreover, we investigate the upper semicontinuity of strong pullback attractors in the regular space as the delay time tends to zero. Eventually, we apply the general results to double time-delayed 2D magnetohydrodynamics (MHD) equations. Since their solutions have no high regularity, we prove the strong pullback asymptotic compactness of solution operators via the flattening of solutions.

Keywords

Strong pullback attractor / upper semicontinuity / delay / MHD equations

References

[1] Anh CT, Son DT. Pullback attractors for non-autonomous 2D MHD equations on some unbounded domains. Ann. Polon. Math., 2015, 113(2): 129-154.

[2] Biskamp D. Nonlinear Magnetohydrodynamics, 1993, Cambridge: Cambridge University Press.

[3] Cao D, Song X, Sun C. Pullback attractors for 2D MHD equations on time-varying domains. Discrete Contin. Dyn. Syst., 2022, 42(2): 643-677.

[4] Caraballo T, Diop MA. Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion. Front. Math. China, 2013, 8(4): 745-760.

[5] Caraballo T, Łukaszewicz G, Real J. Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal., 2006, 64(3): 484-498.

[6] Caraballo T, Marín-Rubio P, Valero J. Autonomous and non-autonomous attractors for differential equations with delays. J. Differential Equations, 2005, 208(1): 9-41.

[7] Caraballo T, Real J. Asymptotic behaviour of two-dimensional Navier–Stokes equations with delays. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 2003, 459(2040): 3181-3194.

[8] Caraballo T, Real J. Attractors for 2D-Navier–Stokes models with delays. J. Differential Equations, 2004, 205(2): 271-297.

[9] Carvalho AN, Langa JA, Robinson JC. Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, 2013, New York: Springer.

[10] Davidson PA. An Introduction to Magnetohydrodynamics, 2001, Cambridge: Cambridge University Press.

[11] García-Luengo J, Marín-Rubio P. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Commun. Pure Appl. Anal., 2020, 19(4): 2127-2146.

[12] Garcíia-Luengo J, Maríin-Rubio P, Planas G. Some regularity results for a double time-delayed 2D-Navier–Stokes model. Discrete Contin. Dyn. Syst. Ser. B, 2019, 24(8): 3929-3946.

[13] García-Luengo J, Marín-Rubio P, Real J. Pullback attractors in V for non-autonomous 2D-Navier–Stokes equations and their tempered behaviour. J. Differential Equations, 2012, 252(8): 4333-4356.

[14] García-Luengo J, Marín-Rubio P, Real J. Pullback attractors for 2D Navier–Stokes equations with delays and their regularity. Adv. Nonlinear Stud., 2013, 13(2): 331-357.

[15] García-Luengo J, Marín-Rubio P, Real J. Some new regularity results of pullback attractors for 2D Navier–Stokes equations with delays. Commun. Pure Appl. Anal., 2015, 14(5): 1603-1621.

[16] Garrido-Atienza M J, Marín-Rubio P. Navier–Stokes equations with delays on unbounded domains. Nonlinear Anal., 2006, 64(5): 1100-1118.

[17] Kloeden PE. Upper semi continuity of attractors of delay differential equations in the delay. Bull. Austral. Math. Soc., 2006, 73(2): 299-306.

[18] Kloeden PE, Rasmussen M. Nonautonomous Dynamical Systems, 2011, Providence, RI: American Mathematical Society.

[19] Li Y, Zhang Q. Backward stability and divided invariance of an attractor for the delayed Navier–Stokes equation. Taiwanese J. Math., 2020, 24(3): 575-601.

[20] Ma Q, Wang S, Zhong C. Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ. Math. J., 2002, 51(6): 1541-1559.

[21] Marín-Rubio P, Real J. Attractors for 2D-Navier–Stokes equations with delays on some unbounded domains. Nonlinear Anal., 2007, 67(10): 2784-2799.

[22] Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math., 1983, 36(5): 635-664.

[23] Song X., Hou Y., Pullback attractors for 2D MHD equations with delays. J. Math. Phys., 2021, 62 (7): Paper No. 072704, 29 pp.

[24] Wang X, Lu K, Wang B. Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing. SIAM J. Appl. Dyn. Syst., 2015, 14(2): 1018-1047.

[25] Zhang Q, Li Y. Backward controller of a pullback attractor for delay Benjamin–Bona–Mahony equations. J. Dyn. Control Syst., 2020, 26(3): 423-441.

[26] Zhao C, Zhou S. Attractors of retarded first order lattice systems. Nonlinearity, 2007, 20(8): 1987-2006.