On the dynamics of equations with infinite delay

• Autorzy: Dalibor Pražák
• Języki publikacji: EN

Abstrakty

EN

We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated.

Słowa kluczowe

EN

Equations with infinite delay; exponential attractor; fractal dimension; 37L25; 37L30; 34K17

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 4
• Strony: 635-647

Daty

wydano

2006-12-01

online

2006-12-01

Twórcy

autor

Dalibor Pražák

• Charles University Prague

Bibliografia

• [1] V.V. Chepyzhov, S. Gatti, M. Grasselli, A. Miranville and V. Pata: “Trajectory and global attractors for evolution equations with memory”, Appl. Math. Lett., Vol. 19(1), (2006), pp. 87–96. http://dx.doi.org/10.1016/j.aml.2005.03.007
• [2] V.V. Chepyzhov and A. Miranville: “On trajectory and global attractors for semilinear heat equations with fading memory”, Indiana Univ. Math. J., Vol. 55(1), (2006), pp. 119–167. http://dx.doi.org/10.1512/iumj.2006.55.2597
• [3] I. Chueshov and I. Lasiecka: “Attractors for second-order evolution equations with a nonlinear damping”, J. Dynam. Differential Equations, Vol. 16(2), (2004), pp. 469–512. http://dx.doi.org/10.1007/s10884-004-4289-x
• [4] C.M. Dafermos: “Asymptotic stability in viscoelasticity”, Arch. Rational Mech. Anal., Vol. 37, (1970), pp. 297–308. http://dx.doi.org/10.1007/BF00251609
• [5] A. Debussche and R. Temam: “Some new generalizations of inertial manifolds”, Discrete Contin. Dynam. Systems, Vol. 2(4), (1996), pp. 543–558.
• [6] A. Eden, C. Foias, B. Nicolaenko and R. Temam: Exponential attractors for dissipative evolution equations, Vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris, 1994.
• [7] S. Gatti, M. Grasselli, A. Miranville and V. Pata: “Memory relaxation of first order evolution equations”, Nonlinearity, Vol. 18(4), (2005), pp. 1859–1883. http://dx.doi.org/10.1088/0951-7715/18/4/023
• [8] S. Gatti, M. Grasselli, A. Miranville and V. Pata: “A construction of a robust family of exponential attractors,” Proc. Amer. Math. Soc., Vol. 134(1), (2006), pp. 117–127 (electronic). http://dx.doi.org/10.1090/S0002-9939-05-08340-1
• [9] J.K. Hale and G. Raugel: “Regularity, determining modes and Galerkin methods”, J. Math. Pures Appl. (9), Vol. 82(9), (2003), pp. 1075–1136.
• [10] J. Málek and J. Nečas: “A finite-dimensional attractor for three-dimensional flow of incompressible fluids”, J. Differ. Equations, Vol. 127(2), (1996), pp. 498–518. http://dx.doi.org/10.1006/jdeq.1996.0080
• [11] D. Pražák: “A necessary and sufficient condition for the existence of an exponential attractor” Cent. Eur. J. Math., Vol. 1(3), (2003), pp. 411–417. http://dx.doi.org/10.2478/BF02475219
• [12] D. Pražák: “On the dimension of the attractor for the wave equation with nonlinear damping”, Commun. Pure Appl. Anal., Vol. 4(1), (2005), pp. 165–174. http://dx.doi.org/10.3934/cpaa.2005.4.165
• [13] D. Pražák: “On reducing the 2d Navier-Stokes equations to a system of delayed ODEs”, In: Nonlinear elliptic and parabolic problems, Vol. 64 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 2005, pp. 403–411.
• [14] R. Temam: Infinite-dimensional dynamical systems in mechanics and physics, Vol. 68 of Applied Mathematical Sciences, 2nd ed., Springer-Verlag, New York, 1997.

Identyfikatory

DOI

10.2478/s11533-006-0024-7