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A note on the dimension of the global attractor for an abstract semilinear hyperbolic problem. (English) Zbl 1179.37104
Summary: We study a semilinear hyperbolic problem, written as a second-order evolution equation in an infinite-dimensional Hilbert space. Assuming existence of the global attractor, we estimate its fractal dimension explicitly in terms of the data. Despite its elementary character, our technique gives reasonable results. Notably, we require no additional regularity, although nonlinear damping is allowed.

37L30Attractors and their dimensions, Lyapunov exponents
Full Text: DOI
[1] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., (1997) · Zbl 0871.35001
[2] Pata, V.; Zelik, S.: Attractors and their regularity for 2-D wave equations with nonlinear damping, Adv. math. Sci. appl. 17, No. 1, 225-237 (2007) · Zbl 1145.35045
[3] Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R.: Exponential attractors for dissipative evolution equations, RAM: research in applied mathematics 37 (1994) · Zbl 0842.58056
[4] Efendiev, M.; Miranville, A.; Zelik, S.: Exponential attractors for a nonlinear reaction--diffusion system in R3, C. R. Acad. sci. Paris sér. I math. 330, No. 8, 713-718 (2000) · Zbl 1151.35315 · doi:10.1016/S0764-4442(00)00259-7
[5] Málek, J.; Nečas, J.: A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. differential equations 127, No. 2, 498-518 (1996) · Zbl 0851.35107 · doi:10.1006/jdeq.1996.0080
[6] Málek, J.; Pražák, D.: Large time behavior via the method of l-trajectories, J. differential equations 181, No. 2, 243-279 (2002) · Zbl 1187.37113 · doi:10.1006/jdeq.2001.4087
[7] Chueshov, I.; Lasiecka, I.: Attractors for second-order evolution equations with a nonlinear damping, J. dynam. Differential equations 16, No. 2, 469-512 (2004) · Zbl 1072.37054 · doi:10.1007/s10884-004-4289-x
[8] Bucci, F.; Chueshov, I.; Lasiecka, I.: Global attractor for a composite system of nonlinear wave and plate equations, Commun. pure appl. Anal. 6, No. 1, 113-140 (2007) · Zbl 1220.35172 · doi:10.3934/cpaa.2007.6.113
[9] Pata, V.; Zelik, S.: Global and exponential attractors for 3-D wave equations with displacement dependent damping, Math. methods appl. Sci. 29, No. 11, 1291-1306 (2006) · Zbl 1101.35020 · doi:10.1002/mma.726
[10] Feireisl, E.: Global attractors for semilinear damped wave equations with supercritical exponent, J. differential equations 116, No. 2, 431-447 (1995) · Zbl 0819.35097
[11] Pražák, D.: On the dimension of the attractor for the wave equation with nonlinear damping, Commun. pure appl. Anal. 4, No. 1, 165-174 (2005) · Zbl 1070.37057
[12] Davies, E. B.: Spectral theory and differential operators, Cambridge studies in advanced mathematics 42 (1995)