Perturbation de la dynamique des équations des ondes amorties
Journées équations aux dérivées partielles (2006), article no. 6, 16 p.
DOI: 10.5802/jedp.33
Joly, Romain 1

1 Université Paris Sud, Analyse Numérique et EDP, UMR CNRS 8628, Bâtiment 425, F-91405 Orsay Cedex, France
@article{JEDP_2006____A6_0,
     author = {Joly, Romain},
     title = {Perturbation de la dynamique des \'equations des ondes amorties},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {6},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2006},
     doi = {10.5802/jedp.33},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/jedp.33/}
}
TY  - JOUR
AU  - Joly, Romain
TI  - Perturbation de la dynamique des équations des ondes amorties
JO  - Journées équations aux dérivées partielles
PY  - 2006
DA  - 2006///
PB  - Groupement de recherche 2434 du CNRS
UR  - http://archive.numdam.org/articles/10.5802/jedp.33/
UR  - https://doi.org/10.5802/jedp.33
DO  - 10.5802/jedp.33
LA  - fr
ID  - JEDP_2006____A6_0
ER  - 
%0 Journal Article
%A Joly, Romain
%T Perturbation de la dynamique des équations des ondes amorties
%J Journées équations aux dérivées partielles
%D 2006
%I Groupement de recherche 2434 du CNRS
%U https://doi.org/10.5802/jedp.33
%R 10.5802/jedp.33
%G fr
%F JEDP_2006____A6_0
Joly, Romain. Perturbation de la dynamique des équations des ondes amorties. Journées équations aux dérivées partielles (2006), article  no. 6, 16 p. doi : 10.5802/jedp.33. http://archive.numdam.org/articles/10.5802/jedp.33/

[1] S.B. Angenent, The Morse-Smale property for a semilinear parabolic equation, Journal of Differential Equations no 62 (1986), pp 427-442. | MR | Zbl

[2] K. Ammari et M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control, Optimisation and Calculus of Variations no 6 (2001), pp 361-386. | Numdam | MR | Zbl

[3] J.M. Arrieta, J.K. Hale et Q. Han, Eigenvalue problems for nonsmoothly perturbed domains, Journal of Differential Equations no 91 (1991), pp. 24-52. | MR | Zbl

[4] A.V. Babin et M.I. Vishik, Uniform asymptotic solutions of a singularly perturbed evolutionary equation, Journal de Mathématiques Pures et Appliquées no 68 (1989), pp 399-455. | MR | Zbl

[5] A.V. Babin et M.I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications no 25 (1992), North-Holland. | MR | Zbl

[6] H.T. Banks, K. Ito et C. Wang, Exponential stable approximations of weakly damped wave equations, Estimation and control of distributed parameter systems (Vorau 1990), vol. 100 of Internat. Ser. Numer. Math. (1991), pp. 1-33, Birkäuser, Basel. | MR | Zbl

[7] C. Bardos, G. Lebeau et J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM Journal on Control and Optimization no 30 (1992), pp. 1024-1065. | MR | Zbl

[8] P. Brunovský et P. Polàčik, The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension, Journal of Differential Equations no 135 (1997), pp. 129-181. | MR | Zbl

[9] P. Brunovský et G. Raugel, Genericity of the Morse-Smale property for damped wave equations, Journal of Dynamics and Differential Equations no 15 (2003), pp. 571-658. | MR | Zbl

[10] I. Chueshov, M. Eller et I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Communications in Partial Differential Equations no 27 (2002), pp 1901-1951. | MR | Zbl

[11] S. Cox et E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana University Mathematics Journal no 44 (1995), pp. 545-573. | MR | Zbl

[12] I.S. Ciuperca, Spectral properties of Schrödinger operators on domains with varying order of thinness, Journal of Dynamics and Differential Equations no 10 (1998), pp. 73-108. | MR | Zbl

[13] C.M. Elliot et A.M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM Journal of Numerical Analysis no 30 (1993), pp. 1622-1663. | MR | Zbl

[14] C. Fabre et J.P. Puel, Pointwise controllability as limit of internal controllability for the wave equation in one space dimension, Portugaliae Mathematica no 51 (1994), pp 335-350. | MR | Zbl

[15] R. Glowinski, C.H. Li et J-L. Lions, A numerical approach to the exact boundary controllability of the wave equation, Japan Journal of Applied Mathematics no 7 (1990), pp. 1-76. | MR | Zbl

[16] J.K. Hale, Asymptotic behavior of dissipative systems, Mathematical Survey no 25 (1988), American Mathematical Society. | MR | Zbl

[17] J.K. Hale, Numerical dynamics, chaotic numerics (Geelong, 1993), Contemporary Mathematics no 172, American Mathematical Society (1994). | MR | Zbl

[18] J.K. Hale, X.B. Lin et G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Mathematics of Computation no 50 (1988), pp 89-123. | MR | Zbl

[19] J.K. Hale, L. Magalhães et W. Oliva, An introduction to infinite dimensional dynamical systems, Applied Mathematical Sciences no 47 (1984), Springer-Verlag. Seconde édition (2002), Dynamics in infinite dimensions. | MR

[20] J.K. Hale et G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Annali di Matematica Pura ed Applicata (IV) no CLIV (1989), pp 281-326. | MR | Zbl

[21] J.K. Hale et G. Raugel, Attractors for dissipative evolutionary equations, International Conference on Differential Equations vol 1, 2 (1991) World Sci. Publishing, pp. 3-22. | MR | Zbl

[22] J.K. Hale et G. Raugel, Reaction-diffusion equations on thin domains, Journal de Mathématiques Pures et Appliquées no 71 (1992), pp. 33-95. | MR | Zbl

[23] J.K. Hale et G. Raugel, A reaction-diffusion equation on a thin L-shaped domain, Proceedings of the Royal Society of Edimburgh no 125A (1995), pp. 283-327. | MR | Zbl

[24] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugaliae Mathematica no 46 (1989), pp 246-257. | MR | Zbl

[25] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics no 840 (1981), Springer-Verlag. | MR | Zbl

[26] D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, Journal of Differential Equations no 59 (1985), pp. 165-205. | MR | Zbl

[27] J.A. Infante et E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation, M2AN Mathematical Modelling and Numerical Analysis no 33 (1999), pp. 407-438. | Numdam | MR | Zbl

[28] R. Joly, Dynamique des équations des ondes avec amortissement variable, thèse (Orsay, 2005).

[29] R. Joly, Generic transversality property for a class of wave equations with variable damping, Journal de Mathématiques Pures et Appliquées no 84 (2005), pp. 1015-1066. | MR | Zbl

[30] R. Joly, Convergence of the wave equation damped on the interior to the one damped on the boundary, à paraître dans le Journal of Differential Equations. | Zbl

[31] M. Kazeni et M.V. Klibanov, Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities, Applicable Analysis no 50 (1993), pp. 93-102. | MR | Zbl

[32] V. Komornik et E. Zuazua, A direct method for the boundary stabilization of the wave equation, Journal de Mathématiques Pures et Appliquées no 69 (1990), pp 33-55. | MR | Zbl

[33] I. Lasiecka et D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations vol 6 (1993), pp 507-533. | MR | Zbl

[34] I. Lasiecka, R. Triggiani et X. Zhang, Nonconservative wave equations with unobserved Neumann B.C. : global uniqueness and observability in one shot, Contemporary Mathematics no 268 (2000), pp. 227-325. | MR | Zbl

[35] Z. Liu et S. Zheng, Semigroups associated with dissipative systems, Chapman and Hall/CRC Resarch Notes in Mathematics no 398 (1999). | MR | Zbl

[36] J. Palis, On Morse-Smale dynamical systems, Topology no 8 (1968), pp. 385-404. | MR | Zbl

[37] J. Palis et S. Smale, Structural stability theorems, Global Analysis (Berkeley, 1968), pp. 223–231, Proc. Sympos. Pure Math. no 14, American Mathematical Society (1970). | MR | Zbl

[38] M. Prizzi et K.P. Rybakowski, Inertial manifolds on squeezed domains, Journal of Dynamics and Differential Equations no 15 (2003), pp. 1-48. | MR | Zbl

[39] K. Ramdani, T. Takahashi et M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations, prépublication.

[40] J. Rauch, Qualitative behavior of dissipative wave equations on bounded domains, Archive for Rational Mechanics and Analysis vol 62 (1976), pp. 77-85. | MR | Zbl

[41] G. Raugel, chapitre 17 de Handbook of dynamical systems vol.2 (2002), edité par B. Fiedler, Elsevier Science.

[42] G. Raugel, Dynamics of Partial Differential Equations on Thin Domains, CIME Course, Montecatini Terme, Lecture Notes in Mathematics no 1609 (1995), pp. 208-315, Springer Verlag. | MR | Zbl

[43] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, Journal de Mathématiques Pures et Appliquées no 71 (1992), pp 455-467. | MR | Zbl

[44] A.M. Stuart, Convergence and stability in the numerical approximation of dynamical systems, The state of the art in numerical analysis (York, 1996), Inst. Math. Appl. Conf. Ser. New Ser. no 63, pp. 145-169, Oxford Univ. Press, New York, 1997. | MR | Zbl

[45] A.M. Stuart et A.R. Humphries, Dynamical systems and numerical analysis, Cambridge Monographs on Applied and Computational Mathematics no 2 (1996), Cambridge University Press, Cambridge. | MR | Zbl

[46] D. Tataru, Uniform decay rates and attractors for evolution PDE’s with boundary dissipation, Journal of Differential Equations no 121 (1995), pp. 1-27. | Zbl

[47] L. Tcheougoué Tebou et E. Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numerische Mathematik no 95(2003), pp. 563-598. | MR | Zbl

[48] E. Yanagida, Existence of stable stationary solutions of scalar reaction-diffusion equations in thin tubular domains, Applicable Analysis no 36 (1990), pp. 171-188. | MR | Zbl

[49] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping Communications in Partial Differential Equations no 15 (1990), pp. 205-235. | MR | Zbl

[50] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM Journal on Control and Optimization no 28 (1990), pp. 466-477. | MR | Zbl

[51] E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square, Journal de Mathématiques Pures et Appliquées no 78 (1999), pp. 523-563. | MR | Zbl

Cited by Sources: