Perturbation de la dynamique des équations des ondes amorties
Joly, Romain
Journées équations aux dérivées partielles, (2006), p. 1-16 / Harvested from Numdam
Publié le : 2006-01-01
DOI : https://doi.org/10.5802/jedp.33
@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},
     year = {2006},
     pages = {1-16},
     doi = {10.5802/jedp.33},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/JEDP_2006____A6_0}
}
Joly, Romain. Perturbation de la dynamique des équations des ondes amorties. Journées équations aux dérivées partielles,  (2006), pp. 1-16. doi : 10.5802/jedp.33. http://gdmltest.u-ga.fr/item/JEDP_2006____A6_0/

[1] S.B. Angenent, The Morse-Smale property for a semilinear parabolic equation, Journal of Differential Equations no 62 (1986), pp 427-442. | MR 837763 | Zbl 0581.58026

[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 1836048 | Zbl 0992.93039

[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 1106116 | Zbl 0736.35073

[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 1046760 | Zbl 0717.35009

[5] A.V. Babin et M.I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications no 25 (1992), North-Holland. | MR 1156492 | Zbl 0778.58002

[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 1155634 | Zbl 0850.93719

[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 1178650 | Zbl 0786.93009

[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 1434918 | Zbl 0868.35062

[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 2046732 | Zbl 1053.35099

[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 1941662 | Zbl 1021.35020

[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 1355412 | Zbl 0847.35078

[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 1607535 | Zbl 0897.35053

[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 1249036 | Zbl 0792.65066

[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 1295205 | Zbl 0814.35066

[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 1039237 | Zbl 0699.65055

[16] J.K. Hale, Asymptotic behavior of dissipative systems, Mathematical Survey no 25 (1988), American Mathematical Society. | MR 941371 | Zbl 0642.58013

[17] J.K. Hale, Numerical dynamics, chaotic numerics (Geelong, 1993), Contemporary Mathematics no 172, American Mathematical Society (1994). | MR 1294406 | Zbl 0808.34061

[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 917820 | Zbl 0666.35013

[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 1914080

[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 1043076 | Zbl 0712.47053

[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 1242226 | Zbl 0938.34536

[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 1151557 | Zbl 0840.35044

[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 1331562 | Zbl 0828.35055

[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 1021188 | Zbl 0679.93063

[25] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics no 840 (1981), Springer-Verlag. | MR 610244 | Zbl 0456.35001

[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 804887 | Zbl 0572.58012

[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 1700042 | Zbl 0947.65101

[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 2155898 | Zbl 1082.35109

[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 05072902

[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 1281205 | Zbl 0795.35134

[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 1054123 | Zbl 0636.93064

[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 1202555 | Zbl 0803.35088

[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 1804797 | Zbl 1096.93503

[35] Z. Liu et S. Zheng, Semigroups associated with dissipative systems, Chapman and Hall/CRC Resarch Notes in Mathematics no 398 (1999). | MR 1681343 | Zbl 0924.73003

[36] J. Palis, On Morse-Smale dynamical systems, Topology no 8 (1968), pp. 385-404. | MR 246316 | Zbl 0189.23902

[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 267603 | Zbl 0214.50702

[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 2016907 | Zbl 1036.35042

[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 404864 | Zbl 0335.35062

[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 1374110 | Zbl 0851.58038

[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 1191585 | Zbl 0832.35084

[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 1628345 | Zbl 0884.65069

[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 1402909 | Zbl 0869.65043

[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 0831.35022

[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 2012934 | Zbl 1033.65080

[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 1048959 | Zbl 0709.35044

[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 1032629 | Zbl 0716.35010

[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 1040470 | Zbl 0695.93090

[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 1697041 | Zbl 0939.93016