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Upper semicontinuity of attractors for lattice systems under singular perturbations. (English) Zbl 1189.34021
Summary: Consider the following first order lattice system $$\dot u_m+(2u_m-u_{m-1}-u_{m+1})+\lambda_mu_m+f_m(u_m)=g_m,\quad m\in\Bbb Z,$$ which is perturbed by the $\varepsilon$-small two order term $$\varepsilon\ddot u_m+\dot u_m+(2u_m-u_{m-1}-u_{m+1}+\lambda_mu_m+f_m(u_m)=g_m,\quad m\in \Bbb Z.$$ Under certain conditions on $f_m$, $\lambda_m$ and $g_m$, the original systems and the $\varepsilon$-small perturbed systems have global attractors $\cal A$ in $\ell^2$ and ${\cal A}_\varepsilon$ in $\ell^2\times \ell^2$, respectively, and $\cal A$ can be naturally embedded into a compact set ${\cal A}_0$ in $\ell^2\times \ell^2$. We prove the upper semicontinuity of ${\cal A}_0$ with respect to the attractors ${\cal A}_\varepsilon$ at zero by showing that for any neighborhood ${\cal O}({\cal A}_0)$ of ${\cal A}_0$, ${\cal A}_\varepsilon$ enters ${\cal O}({\cal A}_0)$ if $\varepsilon$ is small enough.
34A33Lattice differential equations
34E15Asymptotic singular perturbations, general theory (ODE)
Full Text: DOI
[1] Chepyzhov, V. V.; Vishik, M. I.: Attractors for equations of mathematical physics, vol. 49, (2002) · Zbl 0986.35001
[2] Hale, J. K.: Asymptotic behavior of dissipative systems, (1988) · Zbl 0642.58013
[3] Ladyzhenskaya, O.: Attractors for semigroups and evolution equations, (1991) · Zbl 0755.47049
[4] Robinson, J. C.: Infinite-dimensional dynamical systems, (2001) · Zbl 1026.37500
[5] Temam, R.: Infinite dimensional dynamical systems in mechanics and physics, (1997) · Zbl 0871.35001
[6] Crauel, H.; Flandoli, F.: Attractors for random dynamical systems, Probab. related fields 100, 365-393 (1994) · Zbl 0819.58023 · doi:10.1007/BF01193705
[7] Arnold, L.: Random dynamical systems, (1998) · Zbl 0906.34001
[8] Schmalfuss, B.: Backward cocycles and attractors of stochastic differential equations, International seminar on applied mathematics-nonlinear dynamics: attractor approximation and global behavior, 185-192 (1992)
[9] Babin, A. V.; Vishik, M. I.: Attractors of evolution equations, (1992) · Zbl 0778.58002
[10] Caraballo, T.; Langa, J. A.: On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. cont. Disc. impulsive syst. 10, 491-513 (2003) · Zbl 1035.37013
[11] Elliott, C. M.; Kostin, I. N.: Lower semicontinuity of a nonhyperbolic attractor for the viscous Cahn--Hilliard equation, Nonlinearity 9, 687-702 (1996) · Zbl 0888.35047 · doi:10.1088/0951-7715/9/3/005
[12] Hale, J. K.; Xin, L.; Raugel, G.: Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. comput. 5, 89-123 (1988) · Zbl 0666.35013 · doi:10.2307/2007916
[13] Hale, J. K.; Raugel, G.: Upper semicontinuity of the attractors for a singularly perturbed hyperbolic equation, J. differential equations 73, 197-214 (1988) · Zbl 0666.35012 · doi:10.1016/0022-0396(88)90104-0
[14] Hale, J. K.; Raugel, G.: Lower-semicontinuity of attractors of gradient systems and applications, Ann. of math. Pure appl. 154, 278-326 (1989) · Zbl 0712.47053 · doi:10.1007/BF01790353
[15] Kostin, I. N.: Lower semicontinuity of a nonhyperbolic attractor, J. London math. Soc. 52, 568-582 (1995) · Zbl 0856.35066
[16] Lu, K. N.; Wang, B. X.: Upper semicontinuity of attractors for the Klein--Gordon--Schrödinger equation, Inter. J. Bifur. chaos 15, 157-168 (2005) · Zbl 1067.35109 · doi:10.1142/S0218127405012077
[17] Lv, Y.; Wang, W.: Limiting dynamics for stochastic wave equations, J. differential equations 244, 1-23 (2008) · Zbl 1127.37042 · doi:10.1016/j.jde.2007.10.009
[18] Lv, Y.; Wang, W.: Dynamics of the nonlinear stochastic heat equation with singular perturbation, J. math. Anal. appl. 333, 695-711 (2007) · Zbl 1123.60047 · doi:10.1016/j.jmaa.2006.11.046
[19] Vleck, E. V.; Wang, B.: Attractors for lattice Fitzhugh-Nagumo systems, Physica D 212, 317-336 (2005) · Zbl 1086.34047 · doi:10.1016/j.physd.2005.10.006
[20] Zhao, C.; Zhou, S.: Attractors of retarded first order lattice systems, Nonlinearity 20, 1987-2006 (2007) · Zbl 1130.34053 · doi:10.1088/0951-7715/20/8/010
[21] Zhao, C.; Zhou, S.: Limit behavior of global attractors for the complex Ginzburg-Landau equation on infinite lattices, Appl. math. Lett. 21, 628-635 (2008) · Zbl 1138.37048 · doi:10.1016/j.aml.2007.07.016
[22] Zhou, S.: Attractors for first order dissipative lattice dynamical systems, Physica D 178, 51-61 (2003) · Zbl 1011.37047 · doi:10.1016/S0167-2789(02)00807-2
[23] Zhou, S.: Attractors and approximations for lattice dynamical systems, J. differential equations 200, 342-368 (2004) · Zbl 1173.37331 · doi:10.1016/j.jde.2004.02.005
[24] Bates, P. W.; Lu, K.; Wang, B.: Attractors for lattice dynamical systems, Internat. J. Bifur. chaos 11, No. 1, 143-153 (2001) · Zbl 1091.37515 · doi:10.1142/S0218127401002031
[25] Chow, S. N.; Paret, J. M.: Pattern formation and spatial chaos in lattice dynamical systems, IEEE trans. Circuits syst. 42, 746-751 (1995)
[26] Zhou, S.: Attractors for second order lattice dynamical systems, J. differential equations 179, 605-624 (2002) · Zbl 1002.37040 · doi:10.1006/jdeq.2001.4032
[27] Karachalios, N. I.; Yannacopoulos, A. N.: Gloabl existence and compact attractors for the discrete nonlinear Schrödinger equation, J. differential equations 217, 88-123 (2005) · Zbl 1084.35092 · doi:10.1016/j.jde.2005.06.002
[28] Bates, P. W.; Lisei, H.; Lu, K.: Attractors for stochastic lattice dynamical systems, Stoch. dyn. 6, 1-21 (2006) · Zbl 1105.60041 · doi:10.1142/S0219493706001621
[29] Lv, Y.; Sun, J. H.: Dynamical behavior for stochastic lattice systems, Chaos solitons fractals 27, 1080-1090 (2006) · Zbl 1134.37350 · doi:10.1016/j.chaos.2005.04.089
[30] Caraballo, T.; Langa, J. A.; Robinson, J. C.: Upper semicontinuity of attractors for small random perturbations of dynamical systems, Commun. partial differential equation 23, 1557-1581 (1998) · Zbl 0917.35169 · doi:10.1080/03605309808821394
[31] Caraballo, T.; Langa, J. A.; Robinson, J. C.: Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete contin. Dyn. syst. 6, 875-892 (2000) · Zbl 1011.37031 · doi:10.3934/dcds.2000.6.875
[32] Robinson, J. C.: Stability of random attractors under perturbation and approximation, J. differential equations 186, 652-669 (2002) · Zbl 1020.37033 · doi:10.1016/S0022-0396(02)00038-4