Volume 1 (2018)
Royer, Julien
Energy decay for the Klein–Gordon equation with highly oscillating damping
Annales Henri Lebesgue, Volume 1 (2018), pp. 297-312.

Metadata

DOI10.5802/ahl.9
Keywords Damped wave equation, energy decay, resolvent estimates, oscillating damping.

Abstract

We consider the free Klein–Gordon equation with periodic damping. We show on this simple model that if the usual geometric condition holds then the decay of the energy is uniform with respect to the oscillations of the damping, and in particular the sizes of the derivatives do not play any role. We also show that without geometric condition the polynomial decay of the energy is even slightly better for a highly oscillating damping. To prove these estimates we provide a parameter dependent version of well known results of semigroup theory.


References

[ALM16] Anantharaman, Nalini; Léautaud, Matthieu; Macià, Fabricio Wigner measures and observability for the Schrödinger equation on the disk, Invent. Math., Volume 206 (2016) no. 2, pp. 485-599 | DOI | Zbl

[BEPS06] Bátkai, András; Engel, Klaus-Jochen; Prüss, Jan; Schnaubelt, Roland Polynomial stability of operator semigroups, Math. Nachr., Volume 279 (2006) no. 13-14, pp. 1425-1440 | DOI | MR | Zbl

[BJ16] Burq, Nicolas; Joly, Romain Exponential decay for the damped wave equation in unbounded domains, Commun. Contemp. Math., Volume 18 (2016) no. 6, 1650012, 27 pages (Art. ID 1650012, 27 pages) | MR | Zbl

[BT10] Borichev, Alexander; Tomilov, Yuri Optimal polynomial decay of functions and operator semigroups, Math. Ann., Volume 347 (2010) no. 2, pp. 455-478 | DOI | MR | Zbl

[BZ12] Burq, Nicolas; Zworski, Maciej Control for Schrödinger operators on tori, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 309-324 | DOI | Zbl

[EN00] Engel, Klaus-Jochen; Nagel, Rainer One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194, Springer, 2000, xxi+586 pages | MR | Zbl

[Gea78] Gearhart, Larry Spectral theory for contraction semigroups on Hilbert space, Trans. Am. Math. Soc., Volume 236 (1978), pp. 385-394 | DOI | MR | Zbl

[Hua85] Huang, Falun Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equations, Volume 1 (1985), pp. 43-56 | MR | Zbl

[Jaf90] Jaffard, Stéphane Contrôle interne exact des vibrations d’une plaque rectangulaire., Port. Math., Volume 47 (1990) no. 4, pp. 423-429 | Zbl

[JR18] Joly, Romain; Royer, Julien Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation, J. Math. Soc. Japan, Volume 70 (2018) no. 4, pp. 1375-1418 | DOI | MR | Zbl

[Leb96] Lebeau, Gilles Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Mathematical Physics Studies), Volume 19, Kluwer Academic Publishers, 1996, pp. 73-109 | DOI | Zbl

[Prü84] Prüss, Jan On the spectrum of C 0 -semigroups., Trans. Am. Math. Soc., Volume 284 (1984), pp. 847-857 | Zbl

[Roy10] Royer, Julien Limiting absorption principle for the dissipative Helmholtz equation, Commun. Partial Differ. Equations, Volume 35 (2010) no. 8, pp. 1458-1489 | DOI | MR | Zbl

[Wun17] Wunsch, Jared Periodic damping gives polynomial energy decay, Math. Res. Lett., Volume 24 (2017) no. 2, pp. 571-580 | DOI | MR | Zbl

[Zwo12] Zworski, Maciej Semiclassical Analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012, xii+431 pages | MR | Zbl