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Finite-dimensional attractors for the Kirchhoff models. (English) Zbl 1309.74015

Summary: The paper studies the existence of the finite-dimensional global attractor and exponential attractor for the dynamical system associated with the Kirchhoff models arising in elasto-plastic flow \(u_{tt}- \mathrm{div} \{|\nabla u|^{m-1}\nabla u\}-\Delta u_t + \Delta^2u+h(u_t)+g(u)=f(x)\). By using the method of \(\ell\)-trajectories and the operator technique, it proves that under subcritical case, \(1\leq m< \frac{N+2}{(N-2)^+}\), the above-mentioned dynamical system possesses in different phase spaces a finite-dimensional (weak) global attractor and a weak exponential attractor, respectively. For application, the fact shows that for the concerned elasto-plastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the fractal dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.{
©2010 American Institute of Physics}

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74N15 Analysis of microstructure in solids
70K20 Stability for nonlinear problems in mechanics
35B41 Attractors
35B44 Blow-up in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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[1] Babin, A. V.; Vishik, M. I., Attractors of Evolutionary Equations (1992) · Zbl 0778.58002
[2] Chen, F. -X.; Guo, B. -L.; Wang, P., Long time behavior of strongly damped nonlinear wave equations, J. Differ. Equations, 147, 231 (1998) · Zbl 0912.35111 · doi:10.1006/jdeq.1998.3447
[3] Chen, G. -W.; Yang, Z. -J., Existence and nonexistence of global solutions for a class of nonlinear wave equation, Math. Methods Appl. Sci., 23, 615 (2000) · Zbl 1007.35046 · doi:10.1002/(SICI)1099-1476(20000510)23:7<615::AID-MMA133>3.0.CO;2-E
[4] Chueshov, I.; Lasiecka, I., Existence, uniqueness of weak solutions and global attractors for a class of 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15, 777 (2006) · Zbl 1220.35014 · doi:10.3934/dcds.2006.15.777
[5] Chueshov, I.; Lasiecka, I., Attractors for second order evolution equations with a nonlinear damping, J. Dyn. Differ. Equ., 16, 469 (2004) · Zbl 1072.37054 · doi:10.1007/s10884-004-4289-x
[6] Chueshov, I.; Lasiecka, I., Long-time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents, Control Methods in PDE-Dynamical Systems, 426, 153-192 (2007) · Zbl 1130.35014
[7] Chueshov, I.; Lasiecka, I., Long-time behavior of second order evolution equations with nonlinear damping, 195, 912 (2008) · Zbl 1151.37059
[8] Dai, Z. -D.; Guo, B. -L., Exponential attractors of the strongly damped nonlinear wave equations, Recent Advances in Differential Equations, Kunming, 1997, 386, 149-159 (1998) · Zbl 0929.35014
[9] Feireisl, E., Global attractors for semi-linear damped wave equations with supercritical exponent, J. Differ. Equations, 116, 431 (1995) · Zbl 0819.35097 · doi:10.1006/jdeq.1995.1042
[10] Ghidaglia, J. M.; Marzocchi, A., Longtime behavior of strongly damped wave equation, global attractors and their dimension, SIAM J. Math. Anal., 22, 879 (1991) · Zbl 0735.35015 · doi:10.1137/0522057
[11] Greenberg, J. M.; MacCamy, R. C.; Mizel, V. J., On the existence, uniqueness and stability of solutions of the equation \(<mml:math display=''inline`` overflow=''scroll``>\), J. Math. Mech., 17, 707 (1968)
[12] Guesmia, A., Existence globale et stabilisation interne non linéaire d’un système de Petrovsky, Bull. Belg. Math. Soc., 5, 583 (1998) · Zbl 0916.93034
[13] Hale, J., Asymptotic Behavior of Dissipative Systems (1988) · Zbl 0642.58013
[14] Kalantarov, V.; Zelik, S., Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differ. Equations, 247, 1120 (2009) · Zbl 1183.35053 · doi:10.1016/j.jde.2009.04.010
[15] Lagnese, J., Boundary Stabilization of Thin Plates (1989) · Zbl 0696.73034
[16] Lagnese, J.; Lions, J. L., Modeling, analysis and control of thin plates, Collection RMA (1988) · Zbl 0662.73039
[17] Lasiecka, I.; Ruzmaikina, A. A., Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation, J. Math. Anal. Appl., 270, 16 (2002) · Zbl 1002.35089 · doi:10.1016/S0022-247X(02)00006-9
[18] Levandosky, S. P., Decay estimates for the fourth order wave equations, J. Differ. Equations, 143, 360 (1998) · Zbl 0901.35058 · doi:10.1006/jdeq.1997.3369
[19] Lianjun, A.; Pierce, A., A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55, 136 (1995) · Zbl 0815.73022 · doi:10.1137/S0036139993255327
[20] Málek, J.; Pražák, D., Large time behavior via the method of \(<mml:math display=''inline`` overflow=''scroll``>\)-trajectories, J. Differ. Equations, 181, 243 (2002) · Zbl 1187.37113 · doi:10.1006/jdeq.2001.4087
[21] Massatt, P., Limiting behavior for strongly damped nonlinear wave equations, J. Differ. Equations, 48, 334 (1983) · Zbl 0561.35049 · doi:10.1016/0022-0396(83)90098-0
[22] Messaoudi, S. A., Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265, 296 (2002) · Zbl 1006.35070 · doi:10.1006/jmaa.2001.7697
[23] Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics (1997) · Zbl 0871.35001
[24] Nakao, M.; Yang, Z. -J., Global attractors for some quasi-linear wave equations with a strong dissipation, Adv. Math. Sci. Appl., 17, 89 (2007) · Zbl 1145.35044
[25] Pražák, D., On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dyn. Differ. Equ., 14, 763 (2002) · Zbl 1030.35017 · doi:10.1023/A:1020756426088
[26] Pražák, D., A necessary and sufficient condition for the existence of an exponential attractor, Cent. Eur. J. Math., 1, 411 (2003) · Zbl 1030.37053 · doi:10.2478/BF02475219
[27] Varlamov, V. V., On the initial boundary value problem for the damped Boussinesq equation, Discrete Contin. Dyn. Syst., 4, 431 (1998) · Zbl 0952.35103 · doi:10.3934/dcds.1998.4.431
[28] Xuing, J. -C., Point Set Topology Lecture Notes (1981)
[29] Yang, Z. -J.; Chen, G. -W., Initial value problem for a nonlinear wave equation with damping term, Acta Math. Appl. Sin., 23, 45 (2000) · Zbl 0952.35079
[30] Yang, Z. -J., Cauchy problem for quasi-linear wave equations with viscous damping, J. Math. Anal. Appl., 320, 859 (2006) · Zbl 1112.35128 · doi:10.1016/j.jmaa.2005.07.051
[31] Yang, Z. -J., Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equation with dissipative term, J. Differ. Equations, 187, 520 (2003) · Zbl 1030.35125 · doi:10.1016/S0022-0396(02)00042-6
[32] Yang, Z. -J., Cauchy problem for a class of nonlinear dispersive wave equations arising in elasto-plastic flow, J. Math. Anal. Appl., 313, 197 (2006) · Zbl 1091.35103 · doi:10.1016/j.jmaa.2005.04.086
[33] Yang, Z. -J.; Guo, B. -L., Cauchy problem for the multi-dimensional Boussinesq type equation, J. Math. Anal. Appl., 340, 64 (2008) · Zbl 1132.35315 · doi:10.1016/j.jmaa.2007.08.017
[34] Yang, Z. -J.; Jin, B. -X., Global attractor for a class of Kirchhoff models, J. Math. Phys., 50, 032701 (2009) · Zbl 1202.37114 · doi:10.1063/1.3085951
[35] Yang, Z. -J., Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Methods Appl. Sci., 32, 1082 (2009) · Zbl 1167.35558 · doi:10.1002/mma.1106
[36] Yang, Z. -J., Global attractors and their Hausdorff dimensions for a class of Kirchhoff models, J. Math. Phys., 51, 032701 (2010) · Zbl 1309.74016 · doi:10.1063/1.3303633
[37] Zhenya, Y.; Hongqing, Z., Similarity reductions for a nonlinear wave equation with damping term, Acta Phys. Sin., 49, 2113 (2000) · Zbl 0965.35153
[38] Zhenya, Y., Similarity reduction and integrability for the nonlinear wave equations from EPM model, Commun. Theory Phys., 35, 647 (2001)
[39] Zhou, S. -F., Dimensions of the global attractor for damped nonlinear wave equation, Proc. AMS, 127, 3623 (1999) · Zbl 0940.35038
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