Expansions and eigenfrequencies for damped wave equations
Journées équations aux dérivées partielles (2001), article no. 6, 10 p.

We study eigenfrequencies and propagator expansions for damped wave equations on compact manifolds. In the strongly damped case, the propagator is shown to admit an expansion in terms of the finitely many eigenmodes near the real axis, with an error exponentially decaying in time. In the presence of an elliptic closed geodesic not meeting the support of the damping coefficient, we show that there exists a sequence of eigenfrequencies converging rapidly to the real axis. In the case of Zoll manifolds, the set of all eigenfrequencies is shown to exhibit a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. We then show that the propagator can be expanded in terms of the clusters of eigenfrequencies in the entire spectral band.

@article{JEDP_2001____A6_0,
     author = {Hitrik, Michael},
     title = {Expansions and eigenfrequencies for damped wave equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.590},
     zbl = {01808682},
     mrnumber = {1843407},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2001____A6_0}
}
Hitrik, Michael. Expansions and eigenfrequencies for damped wave equations. Journées équations aux dérivées partielles (2001), article  no. 6, 10 p. doi : 10.5802/jedp.590. http://www.numdam.org/item/JEDP_2001____A6_0/

[AschLebeau]M. Asch and G. Lebeau The spectrum of the damped wave operator for a bounded domain in 2 , preprint, 2000. | MR 2016708

[BLR]C. Bardos, G. Lebeau, J. Rauch Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control and Optimization, 30, 1992, 1024-1065. | MR 1178650 | Zbl 0786.93009

[Burq2]N. Burq Mesures semi-classiques et mesures de défaut, Sém. Bourbaki, Asterisque 245, 1997, 167-195. | Numdam | MR 1627111 | Zbl 0954.35102

[Burq3]N. Burq Semi-classical estimates for the resolvent in non-trapping geometries, preprint, 2000. | MR 1876933

[BurqZworski]N. Burq and M. Zworski Resonance expansions in semi-classical propagation, Comm. Math. Phys., to appear. | MR 1860756 | Zbl 1042.81582

[PopovCardoso]F. Cardoso and G. Popov Quasimodes with exponentially small errors associated with elliptic periodic rays, preprint, 2001. | MR 1932033

[Hitrik1]M. Hitrik Eigenfrequencies for damped wave equations on Zoll manifolds, preprint, 2001. | MR 1937840

[Hitrik2]M. Hitrik Propagator expansions for damped wave equations, in preparation.

[HormIV]L. Hörmander The analysis of linear partial differential operators IV, Springer Verlag 1985. | MR 781537 | Zbl 0612.35001

[Lebeau]G. Lebeau Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli 1993), 73-109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. | MR 1385677 | Zbl 0863.58068

[Markus] A. S. Markus Introduction to the spectral theory of polynomial operator pencils, Stiintsa, Kishinev 1986 (Russian). Engl. transl. in Transl. Math. Monographs 71, Amer. Math. Soc., Providence 1988. | MR 971506 | Zbl 0678.47005

[Ralston] J. Ralston On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys. 51 1976, 219-242. | MR 426057 | Zbl 0333.35066

[RT1]J. Rauch and M. Taylor Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J. 24, 1974, 79-86. | MR 361461 | Zbl 0281.35012

[RT2] J. Rauch and M. Taylor Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math. 28, 1975, 501-523. | MR 397184 | Zbl 0295.35048

[Sjostrand1] J. Sjöstrand Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. R.I.M.S., 36 (2000), 573-611. | MR 1798488 | Zbl 0984.35121

[Stefanov]P. Stefanov Quasimodes and resonances : sharp lower bounds, Duke Math. J., 99, 1999, 75-92. | MR 1700740 | Zbl 0952.47013

[StefanovVodev] P. Stefanov and G. Vodev Neumann resonances in linear elasticity for an arbitrary body, Comm. Math. Phys., 176 1996, 645-659. | MR 1376435 | Zbl 0851.35032

[TZ]S. H. Tang and M. Zworski From quasimodes to resonances, Math. Res. Lett., 5, 1998, 261-272. | MR 1637824 | Zbl 0913.35101

[Weinstein] A. Weinstein Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 1977, 883-892. | MR 482878 | Zbl 0385.58013

[Zworski]M. Zworski Resonance expansions in wave propagation, Séminaire E.D.P., 1999-2000, École Polytechnique, XXII-1-XXII-9. | Numdam | MR 1813184