Global attractor for a composite system of nonlinear wave and plate equations

Communications on Pure and Applied Analysis

Volumn 6, Issue 1, 2007, Pages 113-140

  Bucci, Francesca ^a   Chueshov, Igor ^b   Lasiecka, Irena ^c  

^a UNIVERSITY OF FLORENCE   (Italy)

^b Department of Mathematics and Mechanics   (Ukraine)

^c University of Virginia   (United States)

Author keywords

Critical exoonents; Finite fractal dimension; Global attractor; Nonlinear damping; Wave plate model

Indexed keywords

EID: 34248136495     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2007.6.113     Document Type: Article

References (41)

1. G. Avalos and I. Lasiecka, Uniform decay rates for solutions to a structural acoustic model with nonlinear dissipation, Appl. Math. Comput. Sci., 8 (1998), no. 2, 287-312.
2. G. Avalos and I. Lasiecka, Exact controllability of structural acoustic interactions, J. Math. Pures Appl., 82 (2003), 1047-1073.
3. G. Avalos and I. Lasiecka, Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls, Adv. Differential Equations, 10 (2005), 901-930.
4. A.V. Babin and M.I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications 25, North-Holland Publishing Co., Amsterdam, 1992.
5. V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, 1976.
6. J.T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.
7. A. Boutet de Monvel and I. Chueshov, The problem on interaction of von Karman plate with subsonic flow of gas, Math. Methods Appl. Sci., 22 (1999), 801-810.
8. L. Boutet de Monvel and I. Chueshov, Non-linear oscillations of a plate in a flow of gas, C.R. Acad. Sci. Paris, Ser.I, 322 (1996), 1001-1006.
9. L. Boutet de Monvel and I. Chueshov, Oscillations of von Karman plate in a potential flow of gas, Izvestiya: Mathematics, 63 (1999), 219-244.
10. H. Brezis, Problèmes unilatéraux, J. Math. Pures Appl., 51 (1972), 1-168.
11. H. Brezis, "Operateurs Maximaux Monotones et semi-groupes des contractions dans les espaces de Hilbert," North-Holland Mathematics Studies, Vol. 5, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1973.
12. F. Bucci, Uniform decay rates of solutions to a system of coupled PDEs with nonlinear internal dissipation, Differential Integral Equationsm, 16 (2003), no. 7, 865-896.
13. J. Cagnol, I. Lasiecka and C. Lebiedzik, Uniform stability in structural acoustic models with flexible curved walls, J. Differential Equations, 186 (2002), 88-121.
14. I. Chueshov, Construction of solutions in a problem of the oscillations of a shell in a potential subsonic flow, in "Operator Theory, Subharmonic Functions," (ed. V.A. Marchenko), Naukova dumka, Kiev (1991), 147-154 (in Russian).
15. I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," University Lectures in Contemporary Mathematics, Kharkov, 2002 (from the Russian edition (Acta, 1999)); see also http://www.emis.de/monographs/Chueshov/.
16. I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), no. 9-10, 1901-1951.
17. I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 29 (2004), no. 11-12, 1847-1876.
18. I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), no. 2, 469-512.
19. I. Chueshov and I. Lasiecka, Long-time behaviour of second order evolution equations with nonlinear damping, Scuola Normale Superiore of Pisa, Preprint no. 14, 2004; see http:math.sns.it/papers/chulas04.
20. I. Chueshov and I. Lasiecka, "Long-time behaviour of second order evolution equations with nonlinear damping," a version of [19] revised for Memoirs of AMS.
21. I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping, J. Differential Equations (to appear).
22. I. Chueshov and I. Lasiecka, Long time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents, AMS Contemporary Mathematics (to appear).
23. A. Eden, C. Foias, B. Nicolaenko and R. Temam, "Exponential Attractors for Dissipative Evolution Equations," Research in Appl. Math. 37, Masson, Paris, 1994.
24. P. Fabric, C. Galushinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singular perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238.
25. E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponents, C.R. Acad. Sci. Paris, Ser.I, 315 (1992), 551-555.
26. M. Grobbelaar-Van Dalsen, On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam, Journal of Mathematical Analysis and Applications, 320 (2006), no. 1, 121-144.
27. J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.
28. M.S. Howe, "Acoustics of fluid-structure interactions," Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1998.
29. A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.
30. I. Lasiecka, Boundary stabilization of a three-dimensional structural acoustic model, J. Math. Pures Appl., 78 (1999), 203-322.
31. I. Lasiecka, Existence and uniqueness of the solutions to second order abstract equations with nonlinear and nonmonotone boundary conditions, Nonlinear Anal., 23 (1994), 797-823.
32. I. Lasiecka, "Mathematical Control Theory of Coupled PDE's," CBMS-NSF Regional Conference Series in Applied Mathematics 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2002.
33. I. Lasiecka and A. Ruzmaikina, Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation, J. Math. Anal. Appl., 270 (2002), 16-50.
34. I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories," Vol. I: Abstract Parabolic Systems; Vol. II: Abstract Hyperbolic-like Systems over a Finite Time Horizon, Encyclopedia of Mathematics and its Applications, Voll. 74-75, Cambridge University Press 2000, 1067 pp.
35. W. Littman and S. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl., 152 (1988), no. 4, 281-330.
36. P.M. Morse and K.U. Ingard, "Theoretical Acoustics," McGraw-Hill, New York, 1968.
37. D. Pražák, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dynam. Differential Equations, 14 (2002), 763-776.
38. R. Showalter, "Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations", Mathematical Surveys and Monographs 49, American Mathematical Society, Providence, RI, 1997.
39. C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations, 227 (2006), 427-443.
40. R. Temam, "Infinite-dimensional dynamical systems in mechanics and physics," Applied Mathematical Sciences 68, Springer-Verlag, New York, 1997.
41. H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library 18, North-Holland Publishing Co., Amsterdam-New York, 1978.