We study the decay of the global energy for the damped Klein–Gordon equation on non-compact manifolds with finitely many cylindrical and subconic ends up to bounded perturbation. We prove that under the Geometric Control Condition, the decay is exponential, and that under the weaker Network Control Condition, the decay is logarithmic, by developing the global Carleman estimate with multiple weights.
Nous étudions la décroissance de l’énergie globale pour l’équation de Klein–Gordon amortie sur des variétés non compactes avec un nombre fini des bouts cylindriques et subconiques jusqu’à une perturbation bornée. Nous prouvons que sous la condition de contrôle géométrique, la décroissance est exponentielle, et que sous la condition de contrôle de réseau plus faible, la décroissance est logarithmique, en développant l’estimation globale de Carleman avec des poids multiples.
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Keywords: Damped waves, damped Klein–Gordon, exponential decay, non-compact manifolds, Carleman estimates.
Mot clés : Ondes amorties, Klein–Gordon amorti, décroissance exponentielle, variétés non-compactes, estimations de Carleman.
Wang, Ruoyu P. T. 1
@article{AIF_2024__74_6_2623_0, author = {Wang, Ruoyu P. T.}, title = {Exponential decay for damped {Klein{\textendash}Gordon} equations on asymptotically cylindrical and conic manifolds}, journal = {Annales de l'Institut Fourier}, pages = {2623--2666}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {74}, number = {6}, year = {2024}, doi = {10.5802/aif.3623}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3623/} }
TY - JOUR AU - Wang, Ruoyu P. T. TI - Exponential decay for damped Klein–Gordon equations on asymptotically cylindrical and conic manifolds JO - Annales de l'Institut Fourier PY - 2024 SP - 2623 EP - 2666 VL - 74 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3623/ DO - 10.5802/aif.3623 LA - en ID - AIF_2024__74_6_2623_0 ER -
%0 Journal Article %A Wang, Ruoyu P. T. %T Exponential decay for damped Klein–Gordon equations on asymptotically cylindrical and conic manifolds %J Annales de l'Institut Fourier %D 2024 %P 2623-2666 %V 74 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3623/ %R 10.5802/aif.3623 %G en %F AIF_2024__74_6_2623_0
Wang, Ruoyu P. T. Exponential decay for damped Klein–Gordon equations on asymptotically cylindrical and conic manifolds. Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2623-2666. doi : 10.5802/aif.3623. https://aif.centre-mersenne.org/articles/10.5802/aif.3623/
[1] Sharp polynomial decay rates for the damped wave equation on the torus, Anal. PDE, Volume 7 (2014) no. 1, pp. 159-214 | DOI | MR | Zbl
[2] Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. Nonlinear hyperbolic equations in applied sciences, Rend. Semin. Mat., Torino (1988) no. Special Issue, pp. 11-31 | MR | Zbl
[3] Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992) no. 5, pp. 1024-1065 | DOI | MR | Zbl
[4] Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math., Volume 180 (1998) no. 1, pp. 1-29 | DOI | MR | Zbl
[5] Imperfect geometric control and overdamping for the damped wave equation, Commun. Math. Phys., Volume 336 (2015) no. 1, pp. 101-130 | DOI | MR | Zbl
[6] Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris, Volume 325 (1997) no. 7, pp. 749-752 | DOI | MR | Zbl
[7] Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett., Volume 14 (2007) no. 1, pp. 35-47 | DOI | MR | Zbl
[8] Exponential decay for the damped wave equation in unbounded domains, Commun. Contemp. Math., Volume 18 (2016) no. 6, 1650012, 27 pages | DOI | MR | Zbl
[9] Nonuniform stability of damped contraction semigroups, Anal. PDE, Volume 16 (2023) no. 5, pp. 1089-1132 | DOI | MR
[10] From resolvent estimates to damped waves, J. Anal. Math., Volume 122 (2014), pp. 143-162 | DOI | MR | Zbl
[11] Mathematical theory of scattering resonances, Graduate Studies in Mathematics, 200, American Mathematical Society, 2019, xi+634 pages | DOI | MR | Zbl
[12] Global analysis on open manifolds, Nova Science Publishers, 2007, x+644 pages | MR | Zbl
[13] Spectral theory for contraction semigroups on Hilbert space, Trans. Am. Math. Soc., Volume 236 (1978), pp. 385-394 | DOI | MR | Zbl
[14] On the energy decay rate of the fractional wave equation on with relatively dense damping, Proc. Am. Math. Soc., Volume 148 (2020) no. 11, pp. 4745-4753 | DOI | MR | Zbl
[15] Uncertainty principles associated to sets satisfying the geometric control condition, J. Geom. Anal., Volume 32 (2022) no. 3, 80, 16 pages | DOI | MR | Zbl
[16] Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equations, Volume 1 (1985) no. 1, pp. 43-56 | MR | Zbl
[17] 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
[18] -theory of elliptic differential operators on manifolds of bounded geometry, Acta Appl. Math., Volume 23 (1991) no. 3, pp. 223-260 | DOI | MR | Zbl
[19] Null-controllability of the Kolmogorov equation in the whole phase space, J. Differ. Equations, Volume 260 (2016) no. 4, pp. 3193-3233 | DOI | MR | Zbl
[20] Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993) (Mathematical Physics Studies), Volume 19, Kluwer Academic Publishers, 1996, pp. 73-109 | MR | Zbl
[21] Stabilisation de l’équation des ondes par le bord, Duke Math. J., Volume 86 (1997) no. 3, pp. 465-491 | DOI | MR | Zbl
[22] On the energy decay rates for the 1D damped fractional Klein–Gordon equation, Math. Nachr., Volume 293 (2020) no. 2, pp. 363-375 | DOI | MR | Zbl
[23] Energy decay in a wave guide with dissipation at infinity, ESAIM, Control Optim. Calc. Var., Volume 24 (2018) no. 2, pp. 519-549 | DOI | MR | Zbl
[24] Geometric scattering theory, Stanford Lectures, Cambridge University Press, 1995, xii+116 pages | MR | Zbl
[25] On the spectrum of -semigroups, Trans. Am. Math. Soc., Volume 284 (1984) no. 2, pp. 847-857 | DOI | MR | Zbl
[26] Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., Volume 24 (1974), pp. 79-86
[27] Energy decay for the Klein–Gordon equation with highly oscillating damping, Ann. Henri Lebesgue, Volume 1 (2018), pp. 297-312 | DOI | MR | Zbl
[28] Spectral theory of elliptic operators on noncompact manifolds, Méthodes semi-classiques, Vol. 1 (Nantes, 1991) (Astérisque), Société Mathématique de France, 1992 no. 207, pp. 5, 35-108 | MR | Zbl
[29] Theory of function spaces II, Modern Birkhäuser Classics, Birkhäuser, 2010 | DOI | Zbl
[30] Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains, J. Differ. Equations, Volume 23 (1977) no. 3, pp. 459-471 | DOI | MR | Zbl
[31] Periodic damping gives polynomial energy decay, Math. Res. Lett., Volume 24 (2017) no. 2, pp. 571-580 | DOI | MR | Zbl
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