Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

  • Eduard Feireisl


We prove that the solution semigroup
$$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$
generated by the evolutionary problem
$$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$
possesses a global attractorA in the energy spaceEo=V×L2(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE1=D(LV exponentially with growing time.

Key words

Semilinear hyperbolic equation attractor asymptotic behavior 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Babin, A. V., and Vishik, M. I. (1992).Attractors of Evolution Equations, North-Holland, Amsterdam.Google Scholar
  2. Ceron, S., and Lopes, O. (1984). Existence of forced periodic solutions of dissipative semilinear hyperbolic equations and systems. Preprint, UNICAMP, Sao Paulo.Google Scholar
  3. Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1990a). Ensembles inertiels pour des équations d'évolution dissipatives.C. R. Acad. Sci. Paris 310 (Sér. I), 559–562.Google Scholar
  4. Eden, A., Milani, A. J., and Nicolaenko, B. (1990b). Finite-dimensional exponential attractors for semilinear wave equations with damping.J. Math. Anal. Appl. 169(2), 408–419.Google Scholar
  5. Ekeland, I., and Temam, R. (1976).Convex Analysis and Variational Problems, North-Holland, Amsterdam.Google Scholar
  6. Feireisl, E. (1992). Attractors for wave equations with nonlinear dissipation and critical exponent.C. R. Acad. Sci. Paris 315 (Sér. I), 551–555.Google Scholar
  7. Feireisl, E., and Zuazua, E. (1993). Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent.Commun. Partial Differential Equations 18 (9–10), 1539–1555.Google Scholar
  8. Ghidaglia, J. M. (1990). Upper bounds on the Lyapunov exponents for dissipative perturbations of infinite dimensional Hamiltonian systems. In Balabane, M., Lochak, P., and Sulem, C. (eds.),Integrable Systems and Applications, Proceedings of a Workshop, Oleron, 1988, Lecture Notes in Physics 342, Springer.Google Scholar
  9. Ghidaglia, J. M., and Temam, R. (1987). Attractors for damped nonlinear hyperbolic equations.J. Math. Pures Appl. 66, 273–319.Google Scholar
  10. Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems, Math. Surv. Monogr. 25, Am. Math. Soc., Providence, RI.Google Scholar
  11. Haraux, A. (1987).Semi-Linear Hyperbolic Problems in Bounded Domains, Mathematical Reports 3, Harwood Gordon Breach.Google Scholar
  12. Haraux, A. (1991).Systèmes dynamiques dissipatifs et applications, RMA 17, Masson.Google Scholar
  13. Raugel, G. (1992). Une équation des ondes avec amortissement non linéaire dans le cas critique en dimension trois,C. R. Acad. Sci. Paris 314 (Sér. I), 177–182.Google Scholar
  14. Temam, R. (1988).Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Eduard Feireisl
    • 1
  1. 1.Institute of Mathematics čSAVPraha 1Czech Republic

Personalised recommendations