Asymptotic behaviour of solutions to nonlinear shells in a supersonic flow Footnote¹

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References

• Adams , R. A. 1975 . Sobolev Spaces , Academic Press .
   Google Scholar
• Arrietta , J. , Carvalho , A. N. and Hale , J. 1992 . A damped hyperbolic equations with critical exponents . Comm. PDE , 17 : 841 – 866 .
   Web of Science ®Google Scholar
• Babin , A. and Vishik , I. 1986 . Unstable Invarient Sets of Semigroups of Nonlinear Operators and their Perturbations . Russian Math. Surveys , 41 ( 4 ) : 1 – 41 .
   Web of Science ®Google Scholar
• Bessaih , H. and Flandoli , F. Weak attractor for a dissipative Euler equation Preprint
   Google Scholar
• Bolotin , V. V. 1963 . Nonconservative Problems of the Theory of Elastic Stability , 274 – 312 . MacMillan .
   Google Scholar
• Chueshov , I. D. 1991 . Strong solutions and the attractors for von Karman Equations . Math. USSR Sbornik , 69 : 25 – 36 .
   Google Scholar
• Dowell , E. H. 1966 . Nonlinear Oscillations of a Fluttering Plate I . AIAA J. , 4 ( 7 ) : 1267 – 1275 .
   Google Scholar
• Dowell , E. H. 1967 . Nonlinear Oscillations of a Fluttering Plate II . AIAA J. , 5 ( 10 ) : 1856 – 1862 .
   Google Scholar
• Dowell , E. H. 1975 . Aeroelasticity of Plates and Shells , Noordhoff .
   Google Scholar
• Favini , A. , Lasiecka , I. , Horn , M. A. and Tataru , D. 1997 . Addendum to the paper: Global Existence, Uniqueness and Regularity of Solutions to a von Kármán system with nonlinear boundary dissipation . Differential and Integral Equations , 10 ( 1 ) : 197 – 200 .
   Google Scholar
• Favini , A. , Lasiecka , I. , Horn , M. A. and Tataru , D. 1996 . Global Existence, Uniqueness and Regularity of Solutions to a von Kármán system with nonlinear boundary dissipation . Differential and Integral Equations , 9 ( 2 ) : 267 – 294 .
   Google Scholar
• Feireisl , E. and Zuazua , E. 1993 . Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent . Communications on PDE's , 18 ( 2 ) : 1539 – 1555 .
   Web of Science ®Google Scholar
• Foias , C. , Sell , G. R. and Titi , E. S. 1989 . Exponential Tracking and Approximation of Inertial manifolds for dissipative Equations . Journal of Dynamics and Differential Equations , 1 ( 2 ) : 199 – 224 .
   Google Scholar
• Gazzola , F. and Sardella , M. 1998 . Attractors for families of processes in weak topologies of Banach spaces . Discrete and Continuous Dynamics , 4 ( 2 ) : 455 – 466 .
   Google Scholar
• Ghidaglia , J. M. 1988 . Weakly damped forces Kortweg-de Vries behave as a finite dimensional dynamical systems in the long time . J. Diff. Eq. , 74 ( 2 ) : 369 – 390 .
   Web of Science ®Google Scholar
• Ghidaglia , J. M. and Temam , R. 1987 . Regularity of the solutions of second order evolution equations and their attractors . Annali della Scuola Normale Superiore di Pisa , 14 ( 2 ) : 485 – 511 .
   Google Scholar
• Hale , J. K. 1988 . Asymptotic Behavior of Dissipative Systems , AMS .
   Google Scholar
• Hale , J. K. and Raugel , G. . Attractors for Dissipative Evolutionary Equations . International Conference on Differential Equations . Vol. 1, 2 , pp. 3 – 22 . River Edge, NJ : World Science Publishing .
   Google Scholar
• Hale , J. and Stavrakakis , N. 1988 . Compact attractors for weak dynamical systems . Appl. Analysis , 26 : 271 – 287 .
   Google Scholar
• Haraux , A. 1988 . Semilinear Hyperbolic Problems in Bounded Domains . Math., Harwood Acad. , 26 Reports 3, part 1
   Google Scholar
• Horn , M. A. , Lasiecka , I. and Tataru , D. 1994 . “ Wellposedness and Uniform Decay Rates of Weak Solutions to a von Karman system with Nonlinear Dissipative Boundary Conditions ” . In Lecture Notes in Pure and Applied Mathematics , Vol. 160 , 133 – 159 . Marcel Dekker .
   Google Scholar
• Ladyzhenskaya , O. A. 1985 . Finite-Dimensionality of Bounded Invariant Sets for Navier-Stokes Systems and Other Dissipative Systems . J. Soviet Math. , 28 : 714 – 26 .
   Google Scholar
• Lagnese , J. 1989 . Boundary Stabilization of Thin Plates , Philadelphia : SIAM .
   Google Scholar
• Lasiecka , I. 1989 . Finite dimensionality and compactness of attractors for von Kármán equations with nonlinear dissipation . Nonlinear Differential Equations and Applications , to appear
   Google Scholar
• Lasiecka , I. 1995 . Finite Dimensionality of Attractors Associated with von Kármán Plate Equations and Boundary Damping . Journal of Differential Equations , 117 ( 2 ) : 357 – 389 .
   Web of Science ®Google Scholar
• Lasiecka , I. and Heyman , W. S. 1995 . Asymptotic Behavior of Solutions in Nonlinear Dynamic Elasticity . Discrete and Continuous Dynamical Systems , 1 ( 2 ) : 237 – 252 .
   Google Scholar
• Lions , J. L. and Magenes , E. 1972 . Non-homogenous Boundary Value Problems and Applications , New York : Springer Verlag .
   Google Scholar
• Lions , J. 1969 . Quelques Methods do Resolution des Problemes aux Limits Nonlinearies , Paris : Dunod .
   Google Scholar
• Mallet-Paret , J. 1976 . Negatively invariant sets of compact maps and an extension of a theorem of Cartwright . J. Diff. Eq. , 22
   Web of Science ®Google Scholar
• Mane , R. 1981 . “ On the dimension of the compact invariant sets of certain nonlinear maps ” . In Lecture Notes in Mathematics , Springer Verlag .
   Google Scholar
• Mazya , V. and Saposnikova , M. 1985 . Multipliers in Spaces of Differentiable Function , Pittman .
   Google Scholar
• Raugel , G. 1992 . Une equation des ondes avec amortissment non lineaire dans le cas critique en dimensions trios . C. R. Acad. Sci. Paris , 314 : 177 – 182 .
   Google Scholar
• Tartar , L. 1972 . Interpolation nonlineaire et regularite . J. Fund. Anal. , 9 : 469 – 89 .
   Google Scholar
• Temam , R. 1988 . Infinite Dimensional Dynamical Systems in Mechanics and Physics , Springer Verlag .
   Google Scholar
• Triebel , H. 1983 . Theory of Function Spaces , Birkhauser .
   Google Scholar
• Triebel , H. 1978 . “ Interpolation Theory ” . In Function Spaces and Differential Operators , North Holland .
   Google Scholar
• Vishik , M. 1992 . Asymptotic Behaviour of Solutions of Evolutionary Equations , Cambridge Universty Press .
   Google Scholar
• Volmir , S. A. 1968 . A Translation of Flexible Plates and Shells . : 66 – 126 . AFFDL TR
   Google Scholar
• Washizu , K. 1968 . Variational Methods in Elasticity and Plasticity , Pergamon Press .
   Google Scholar