Attractors and long time behavior of von Karman thermoelastic plates

Applied Mathematics and Optimization

Volumn 58, Issue 2, 2008, Pages 195-241

  Chueshov, Igor ^a   Lasiecka, Irena ^b  

^a University of Kharkov   (Ukraine)

^b University of Virginia   (United States)

Author keywords

Attractors; Dimension; Rate of attraction; Thermoelastic plates; Von Karman evolutions

Indexed keywords

AIRY'S STRESS FUNCTION; ASYMPTOTIC BEHAVIOR OF SOLUTIONS; ATTRACTORS; DIMENSION; FINITE DIMENSIONAL; GLOBAL ATTRACTORS; HEAT EQUATIONS; LONG TIME; LONG-TIME BEHAVIOR; MECHANICAL DISSIPATION; RATE OF ATTRACTION; ROTATIONAL INERTIA; THERMOELASTIC PLATES; UPPER SEMI-CONTINUITY; VON KARMAN; VON KARMAN EVOLUTIONS;

ASYMPTOTIC ANALYSIS; DYNAMICS; PLATES (STRUCTURAL COMPONENTS); STRESSES; THERMOELASTICITY;

PARAMETER ESTIMATION;

EID: 51849118403     PISSN: 00954616     EISSN: 14320606     Source Type: Journal    
DOI: 10.1007/s00245-007-9031-8     Document Type: Article

References (55)

1. Avalos, G.: Exact controllability of thermoelastic system with control in thermal component only. Differ. Integral Equ. 22, 1-15 (2000)
2. Avalos, G., Lasiecka, I.: Exponential stability of thermoelastic system without mechanical dissipation-invited paper for the special volume dedicated to memory of Pierre Grisvard. Rend. Inst. Mat. Univ. Trieste 28, 1-28 (1997)
3. Avalos, G., Lasiecka, I.: Exponential stability of a thermoelastic plates with free boundary conditions and without mechanical dissipation. SIAM J. Math. Anal. 29, 155-182 (1998)
4. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)
5. Ball, J.M.: Global attractors for semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31-52 (2004)
6. Benabdallah, A., Naso, G.: Null controllability of thermoelastic plates. Abstr. Appl. Anal. 7, 585-599 (2002)
7. Benabdallah, A., Teniou, D.: Exponential decay rates for a full von Karman model with thermal effects. Electron. J. Differ. Equ. 7, 1-13 (1998)
8. Bucci, F., Chueshov, I.: Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Preprint (2007)
9. Bucci, F., Chueshov, I., Lasiecka, I.: Global attractor for a composite system of nonlinear wave and plate equations. Commun. Pure Appl. Anal. 6, 113-140 (2007)
10. Ceron, S., Lopes, O.: α-contractions and attractors for dissipative semilinear hyperbolic equations and systems. Ann. Math. Pura Appl. IV 160, 193-206 (1991)
11. Chepyzhov, V., Vishik, M.: Attractors for Equations of Mathematical Physics. AMS, Providence (2002)
12. Chueshov, I.: Finite-dimensionality of the attractor in some problems of the nonlinear theory shells. Math. USSR Sb. 61, 411-420 (1988)
13. Chueshov, I.: Strong solutions and the attractors for von Karman equations. Math. USSR Sb. 69, 25-36 (1991)
14. Chueshov, I.D.: On some continuity property of attractor in a problem on oscillations of shallow shell. In: Marchenko, V.A. (ed.) Dynamical Systems and Complex Analysis, pp. 85-91. Naukova Dumka, Kiev (1992) (in Russian)
15. Chueshov, I.: Introduction to the Theory of Infinite-Dimensional Dissipative Systems. Acta, Kharkov (1999) (in Russian); English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/
16. Chueshov, I., Lasiecka, I.: Inertial manifolds for von Karman plate equations. Appl. Math. Optim. 46, 179-206 (2002)
17. Chueshov, I., Lasiecka, I.: Global attractors for von Karman evolutions with a nonlinear boundary dissipation. J. Differ. Equ. 198, 196-221 (2004)
18. Chueshov, I., Lasiecka, I.: Attractors for second-order evolution equations with a nonlinear damping. J. Dyn. Differ. Equ. 16, 469-512 (2004)
19. 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: Imanuvilov, O. et al. (eds.) Control Theory of Partial Differential Equations. A Series of Lectures in Pure and Applied Mathematics, vol. 242. pp. 51-69. Chapman & Hall/CRC, Boca Raton (2005)
20. Chueshov, I., Lasiecka, I.: Long time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents. AMS Contemp. Math. 426, 153-193 (2007)
21. Chueshov, I., Lasiecka, I.: Long-time behavior of second order evolution equations with nonlinear damping. Memoirs of AMS (in press)
22. Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations (book in preparation under the contract with Springer)
23. Chueshov, I., Eller, M., Lasiecka, I.: On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation. Commun. Partial Differ. Equ. 27, 1901-1951 (2002)
24. Dafermos, C.: On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rat. Mech. Anal. 29, 241-271 (1968)
25. Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations. Research in Appl. Math., vol. 377. Masson, Paris (1994)
26. n. C. R. Acad. Sci. Paris Ser. I 330, 713-718 (2000)
27. Favini, A., Horn, M., Lasiecka, I., Tataru, D.: Global existence, uniqueness and regularity of solutions to a von Karman equation with nonlinear boundary dissipation. Differ. Int. Equ. 9, 267-294 (1996); and Addendum to this paper Differ. Int. Equ. 10, 197-200 (1997)
28. Grisvard, P.: Elliptic Problems in Non-smooth Domains. Pittman, Boston (1985)
29. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Am. Math. Soc., Providence (1988)
30. Hale, J.K., Lin, X.B., Raugel, G.: Upper semicontinuity of attractors for approximations of semigroups and partial differential equations. Math. Comput. 50, 89-123 (1988)
31. Hansen, S., Zhang, B.: Boundary control of thermoelastic beam. J. Math. Anal. Appl. 210, 182-205 (1997)
32. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1981)
33. Jiang, S., Racke, R.: Evolution Equations in Thermoelasticity. Chapman & Hall, London (2000)
34. Kato, T.: Perturbation Theory of Linear Operators. Springer, New York (1966)
35. Koch, H., Lasiecka, I.: Backward uniqueness in linear thermo-elasticity with variable coefficients. Functional Analysis and Evolution Equations, in memory of Gunter Lumer. Birkhauser (2007, to appear). Festschrift for Gunter Lumer, Birkhauser (2007, to appear)
36. Kim, J.U.: On the energy decay of linear thermoelastic plate. J. Math. Anal. Appl. 23, 889-899 (1992)
37. Ladyzhenskaya, O.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)
38. Lagnese, J.: Boundary Stabilization of Thin Plates. SIAM, Philadelphia (1989)
39. Lagnese, J.: The reachability problem for thermoelastic plate. Arch. Ration. Mech. Anal. 112, 223-267 (1990)
40. Lagnese, J., Lions, J.L.: Modeling, Analysis and Control of Thin Plates. Collection RMA. Masson, Paris (1988)
41. Lasiecka, I.: Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation. Commun. Partial Differ. Equ. 24, 1801-1847 (1999)
42. Lasiecka, I.: Finite dimensionality and compactness of attractors for von Karman equations with nonlinear dissipation. Nonlinear Differ. Equ. 6, 437-472 (1999)
43. Lasiecka, I., Triggiani, R.: Structural decomposition of thermoelastic semigroups with rotational forces. Semigroup Forum 60, 16-60 (2000)
44. Lasiecka, I., Triggiani, R.: Control Theory for PDEs, vol. 1. Cambridge Univ. Press, Cambridge (2000)
45. Lasiecka, I., Renardy, M., Triggiani, R.: Backward uniqueness of thermoelastic plates with rotational forces. Semigroup Forum 62, 217-242 (2001)
46. Lions, J.L.: Quelques Methods de Resolution des Problemes aux Limites Nonlineaires. Dunod, Paris (1969)
47. Liu, Z., Renardy, M.: A note on the equation of thermoelastic plate. Appl. Math. Lett. 8, 1-6 (1995)
48. Málek, J., Nečas, J.: A finite dimensional attractor for three-dimensional flow of incompressible fluids. J. Differ. Equ. 127, 498-518 (1996)
49. Málek, J., Pražak, D.: Large time behavior via the method of l-trajectories. J. Differ. Equ. 181, 243-279 (2002)
50. Milani, A., Koksch, N.: An Introduction to Semiflows. Chapman & Hall, London (2005)
51. Pazy, A.: Semigroups of Linear Operators and Applications to PDE's. Springer, New York (1986)
52. Raugel, G.: Global attractors in partial differential equations. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 885-982. Elsevier, Amsterdam (2002)
53. Sell, G., You, Y.: Dynamics of Evolution Equations. Springer, New York (2002)
54. p (0,T;B). Annal. Mat. Pura Appl. Ser. 4 148, 65-96 (1987)
55. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)