Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation

Communications in Partial Differential Equations

Volumn 29, Issue 11-12, 2004, Pages 1847-1876

  Chueshov, Igor ^a   Eller, Matthias ^b   Lasiecka, Irena ^c  

^a University of Kharkov   (Ukraine)

^b GEORGETOWN UNIVERSITY   (United States)

^c University of Virginia   (United States)

Author keywords

Boundary dissipation; Finite dimensionality; Global attractors; Nonlinear wave equation

Indexed keywords

EID: 10844270396     PISSN: 03605302     EISSN: None     Source Type: Journal    
DOI: 10.1081/PDE-200040203     Document Type: Article

References (40)

1. Babin, A. V., Vishik, M. I. (1992). Attractors of Evolution Equations. Amsterdam: North-Holland.
2. Barbu, V. (1976). Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff.
3. Brezis, H. (1973). Operateurs Maximaux Monotones. Amsterdam: North-Holland.
4. Chueshov, I. (1998). Theory of functionals that uniquely determine asymptotic dynamics of infinite-dimensional dissipative systems. Russian Math. Surveys 53:731-776.
5. Chueshov, I. D. (1999). Introduction to the Theory of Infinite-Dimensional Dissipative Systems. Kharkov: Acta. In: Russian. English translation: Kharkov: Acta, 2002. See also http://www.emis.de/monographs/ Chueshov/
6. Chueshov, I. D. (1999). Introduction to the Theory of Infinite-Dimensional Dissipative Systems. Kharkov: Acta. In: Russian. English translation: Kharkov: Acta, 2002. See also http://www.emis.de/monographs/ Chueshov/
7. Chueshov, I., Kalantarov, V. (2001). Determining functionals for nonlinear damped wave equations. Matem. Fizika. Analyz. Geometriya 8:215-227.
8. Chueshov, I., Lasiecka, I. (2003). Determining functionals for a class of second order in time evolution equations with applications to von Karman equations. In: V. Barbu et al. eds. Analysis and Optimization of Differential Systems. Boston-Dordrecht-London: Kluwer, pp. 109-122.
9. Chueshov, I., Lasiecka, I. (2004). Global attractors for von Karman evolutions with a nonlinear boundary dissipation. J. Diff. Equations 198:196-231.
10. Chueshov, I., Lasiecka, I. Kolmogorov's ε-entropy for a class of invariant sets and dimension of global attractors for second order in time evolution equations with nonlinear damping. In: Proceedings Lecture Notes in Pure and Applied Mathematics. Marcel Dekker. To appear in 2004.
11. Chueshov, I., Eller, M., Lasiecka, I. (2002). On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Comm. PDE 27:1901-1951.
12. Eden, A., Milani, A., Nicolaenko, B. (1992). Finite dimensional exponential attractors for semilinear wave equations with damping. J. Math. Anal. Appl. 169:408-419.
13. Eden, A., Foias, C., Nicolaenko, B., Temam, R. (1994). Exponential Attractors for Dissipative Evolution Equations. Masson, Paris: Wiley.
14. Fereisel, E. (1992). Attractors for wave equation with nonlinear dissipation and critical exponent. C.R. Acad.Sc. Paris, Ser I 315:551-555.
15. Fereisel, E. (1995). Global attractors for semilinear damped wave equations with supercritical exponent. J. Diff. Equations 116:431-447.
16. Foias, C., Prodi, G. (1967). Sur le comportement global des solutions nonstationnaires des équations de Navier-Stokes en dimension deux. Rend. Sem. Mat. Univ. Padova 39:1-34.
17. Ghidaglia, J. M., Temam, R. (1987). Attractors of damped nonlinear hyperbolic equations. J. Math. Pure et Appl. 66:273-319.
18. Hale, J. (1988). Asymptotic Behavior of Dissipative Systems. Providence: AMS.
19. Hale, J. K., Raugel, G. (1993). Attractors for dissipative evolutionary equations. In: International Conference on Differential Equations. Vol. 1, 2. River Edge, NJ: World Science Publishing, pp. 3-22.
20. Haraux, A. (1987). Semilinear Hyperbolic Problems in Bounded Domains. Mathematical Reports, Vol. 3. New York: Harwood Gordon Breach.
21. Kostin, I. N. (1998). Rate of attraction to a non-hyperbolic attractor. Asympt. Anal. 16:203-222.
22. Ladyzhenskaya, O. (1975). A dynamical system generated by the Navier-Stokes equations. J. Soviet Math. 3:458-479.
23. Ladyzhenskaya, O. (1985). Finite dimensionality of bounded invariant sets for Navier Stokes systems and other dissipative systems. J. Soviet Math. 28:714-726.
24. Lagnese, J. (1983). Decay of the solution of the wave equation in a bounded region with boundary dissipation. J. Diff. Equations 50:163-182.
25. Lagnese, J. (1989). Boundary Stabilization of Thin Plates. Philadelphia: SIAM.
26. Lagnese, J., Lions, J. L. (1988). Modeling, Analysis and Control of Thin Plates. Paris: Masson.
27. Lasiecka, I., Triggiani, R. (1992). Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25:189-224.
28. Lasiecka, I., Tataru, D. (1993). Uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation. Diff. Integral Equations 6:507-533.
29. Lasiecka, I., Ruzmaikina, A. (2002). Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation. J. Math. Anal. Appl. 270:16-50.
30. Mallet-Paret, J. (1976). Negatively invariant sets of compact maps and an extension of a theorem of Carwright. J. Diff. Equations 22:331-348.
31. Málek, J., Pražak, D. (2002). Large time behavior via the method of l-trajectories. J. Diff. Equations 181:243-279.
32. Pražak, D. (2002). On finite fractal dimension of the global attractor for the wave equation with nonlinear damping. J. Dyn. Diff. Equations 14:764-776.
33. Raugel, G. (1992). Une equation des ondes avec amortissment non lineaire dans le cas critique en dimensions trois. C. R. Acad. Sci. Paris 314:177-182.
34. Ruiz, A. (1992). Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures et Appl. 71:455-467.
35. Sell, G. R., You, Y. (2002). Dynamics of Evolutionary Equations. New York: Springer.
36. Showalter, R. (1997). Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. Providence: AMS.
37. Shengfan, Z. (1999). Dimension of the global attractor for damped nonlinear wave equation. Proceedings AMS 127:3623-3631.
38. p(0, T; B). Annali di Matematica Pura ed Applicata. Serie 4 148:65-96.
39. Temam, R. (1988). Infinite Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer Verlag.
40. You, Y. (2004). Global dynamics of nonlinear wave equations with cubic nonmonotone damping. Dyn. Partial Diff. Equations 1(1):65-87.