A remark on the damped wave equation

Communications on Pure and Applied Analysis

Volumn 5, Issue 3, 2006, Pages 611-616

  Pata, Vittorino ^a   Zelik, Sergey ^b  

^a POLITECNICO DI MILANO   (Italy)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Critical nonlinearity; Damped wave equation; Global attractor

Indexed keywords

EID: 33749382562     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2006.5.611     Document Type: Article

References (21)

1. J. Arrieta, A.N. Carvalho and J.K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866.
2. A.V. Babin and M.I. Vishik, Regular attractors of semigroups and evolution equation, J. Math. Pures Appl., 62 (1983), 441-491.
3. A.V. Babin and M.I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam, 1992.
4. A. Eden, A.J. Milani and B. Nicolaenko, Finite dimensional exponential attractors for semilinear wave equations with damping, J. Math. Anal. Appl., 169 (1992), 408-419.
5. P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dynam. Systems, 10 (2004), 211-238.
6. S. Gatti, M. Grasselli, A. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation, J. Math. Anal. Appl., 312 (2006), 230-247.
7. J.M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319.
8. M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., to appear.
9. M. Grasselli and V. Pata, On the damped semilinear wave equation with critical exponent, Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Contin. Dynam. Systems, (suppl.) (2003), 351-358.
10. M. Grasselli and V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal., 3 (2004), 849-881.
11. J.K. Hale, Asymptotic behavior and dynamics in infinite dimensions, in "Nonlinear Differential Equations" (J.K. Hale and P. Martìnez-Amores Eds), Pitman, Boston, 1985.
12. J.K. Hale, Asymptotic behavior of dissipative systems, Amer. Math. Soc., Providence, 1988.
13. J.K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214.
14. A. Haraux, Two remarks on dissipative hyperbolic problems, in "Séminaire du Collége de France" (J.-L. Lions Ed.) Pitman, Boston, 1985.
15. A. Haraux and M. Jendoubi Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9 (1999), 95-124.
16. M. Jendoubi Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144 (1998), 302-312.
17. O.A. Ladyzhenskaya, Attractors of nonlinear evolution problems with dissipation, J. Soviet Math., 40 (1988), 632-640.
18. V. Pata and S. Zelik Attractors and their regularity for 2-D wave equations with nonlinear damping, submitted.
19. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer, New York, 1988.
20. S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.
21. S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete Contin. Dynam. Systems, 11 (2004), 351-392.