We prove local and global energy decay for the wave equation in a wave guide with damping at infinity. More precisely, the absorption index is assumed to converge slowly to a positive constant, and we obtain the diffusive phenomenon typical for the contribution of low frequencies when the damping is effective at infinity. On the other hand, the usual Geometric Control Condition is not necessarily satisfied so we may have a loss of regularity for the contribution of high frequencies. Since our results are new even in the Euclidean space, we also state a similar result in this case.
Mots-clés : Local and global energy decay, dissipative wave equation, wave guides, diffusive phenomenon, semiclassical analysis, low frequency resolvent estimates
@article{COCV_2018__24_2_519_0,
author = {Malloug, Mohamed and Royer, Julien},
title = {Energy decay in a wave guide with dissipation at infinity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {519--549},
publisher = {EDP-Sciences},
volume = {24},
number = {2},
year = {2018},
doi = {10.1051/cocv/2017054},
mrnumber = {3816403},
zbl = {1409.35032},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017054/}
}
TY - JOUR AU - Malloug, Mohamed AU - Royer, Julien TI - Energy decay in a wave guide with dissipation at infinity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 519 EP - 549 VL - 24 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017054/ DO - 10.1051/cocv/2017054 LA - en ID - COCV_2018__24_2_519_0 ER -
%0 Journal Article %A Malloug, Mohamed %A Royer, Julien %T Energy decay in a wave guide with dissipation at infinity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 519-549 %V 24 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017054/ %R 10.1051/cocv/2017054 %G en %F COCV_2018__24_2_519_0
Malloug, Mohamed; Royer, Julien. Energy decay in a wave guide with dissipation at infinity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 519-549. doi : 10.1051/cocv/2017054. http://www.numdam.org/articles/10.1051/cocv/2017054/
[1] , and , Energy decay for linear dissipative wave equations in exterior domains. J. Differ. Equ. 259 (2015) 2061–2079 | DOI | MR | Zbl
[2] and , Stabilisation pour l’équation des ondes dans un domaine extérieur. Rev. Math. Iberoamericana 18 (2002) 1–16 | DOI | MR | Zbl
[3] , and , C0-groups Commutator Methods and Spectral theory of N-body Hamiltonians, volume 135 of Progress in mathematics. Birkhäuser Verlag (1996) | MR | Zbl
[4] and , Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE 7 (2014) 159–214 | DOI | MR | Zbl
[5] and , Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306 (1988) 837–852 | DOI | MR | Zbl
[6] , and , Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optimiz. 30 (1992) 1024–1065 | DOI | MR | Zbl
[7] , and , Fine scales of decay of operator semigroups. J. Eur. Math. Soc. (JEMS) 18 (2016) 853–929 | DOI | MR | Zbl
[8] and , Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008) 765–780 | DOI | MR | Zbl
[9] and , Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds. Ann. Sci. l’ École Normale Supérieure 45 (2012) 311–335 | DOI | Numdam | MR | Zbl
[10] and , Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455–478 | DOI | MR | Zbl
[11] and rík, PT-symmectric waveguides. Integral Equ. Oper. Theory 68 (2008) 489–515 | DOI | MR | Zbl
[12] , Low frequency estimates and local energy decay for asymptotically Euclidean laplacians. Commun. Part. Diff. Equ. 36 (2011) 1239–1286 | DOI | MR | Zbl
[13] and , Local energy decay for the damped wave equation. Jour. Func. Anal. 266 (2014) 4538–4615 | DOI | MR | Zbl
[14] and , Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies. Commun. Math. Phys. 299 (2010) 677–708 | DOI | MR | Zbl
[15] ,Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180 (1998) 1–29 | DOI | MR | Zbl
[16] ,Semi-classical estimates for the resolvent in nontrapping geometries. Int. Math. Res. Not. 5 (2002) 221–241 | DOI | MR | Zbl
[17] and , Energy decay for damped wave equations on partially rectangular domains. Math. Res. Lett. 14 (2007) 35–47 | DOI | MR | Zbl
[18] and , Exponential decay for the damped wave equation in unbounded domains. Commun. Contemporary Math. 18(2016) | DOI | MR | Zbl
[19] and , An optimal estimate for the time singular limit of an abstract wave equation. Funkc. Ekvacioj, Ser. Int. 47 (2004) 277–290 | DOI | MR | Zbl
[20] and , Spectral Theory and Differential Operators. Oxford University Press, New York (1987) | MR | Zbl
[21] and , Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 (1993) 559–607 | DOI | MR | Zbl
[22] and , On the essential spectrum. J. Math. Anal. Appl. 25 (1969) 121–127 | DOI | MR | Zbl
[23] and , Large time behavior and Lp-Lq estimate of solutions of 2-dimensional nonlinear damped wave equations. J. Differ. Equ. 203 (2004) 82–118 | DOI | MR | Zbl
[24] , Diffusion phenomenon for linear dissipative wave equations in an exterior domain. J. Differ. Equ. 186 (2002) 633–651 | DOI | MR | Zbl
[25] , and , Optimal decay rate of the energy for wave equations with critical potential. J. Math. Soc. Jpn 65 (2013) 183–236 | DOI | MR | Zbl
[26] , From classical to semiclassical non-trapping behaviour. C. R., Math., Acad. Sci. Paris 338 (2004) 545–548 | DOI | MR | Zbl
[27] , Équation des ondes amorties dans un domaine extérieur. Bull. Soc. Math. France 131 (2003) 211–228 | DOI | Numdam | MR | Zbl
[28] M. Khenissi and J. Royer, Local energy decay and smoothing effect for the damped Schrödinger equation. Anal. PDE | MR
[29] and , On the spectrum of curved planar waveguides. Publ. Res. Inst. Math. Sci. 41 (2005) 757–791 | DOI | MR | Zbl
[30] and , Magnetic effects in curved quantum waveguides. Ann. Henri Poincaré 15 (2014) 1993–2024 | DOI | MR | Zbl
[31] M. Léautaud and N. Lerner, Energy decay for a locally undamped wave equation. Ann. Fac. Sci. Toulouse. Math. To appear. | MR
[32] , Équation des ondes amorties. In Algebraic and geometric methods in mathematical physics, edited by and Kluwer Academic Publishers (1996) 73–109 | DOI | MR | Zbl
[33] and , Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86 (1997) 465–491 | DOI | MR | Zbl
[34] , Local energy decay for the damped Klein−Gordon equation in exterior domain. Appl. Anal. (2016) | MR
[35] and , The Lp–Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media. J. Differ. Equ. 191 (2003) 445–469 | DOI | MR | Zbl
[36] , On the asymptotic behavior of solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12 (1976) 169–189 | DOI | MR | Zbl
[37] , Singularities and energy decay in acoustical scattering. Duke Math. J. 46 (1979) 43–59 | DOI | MR | Zbl
[38] , Scattering theory for wave equations with dissipative terms. Publ. Res. Inst. Math. Sci. 12 (1976) 383–390 | DOI | MR | Zbl
[39] , and , Decay of the solution of the wave equation outside non-trapping obstacles. Commun. Pure Appl. Math. 30 (1977) 447–508 | DOI | MR | Zbl
[40] , Opérateurs conjugués et propriétés de propagation. Comm. Math. Phys. 91 (1983) 279–300 | DOI | MR | Zbl
[41] , Lp-Lq estimatesfor damped wave equations and their applications to semi-linear problem. J. Math. Soc. Jpn 56 (2004) 585–626 | DOI | MR | Zbl
[42] , Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application. Math. Z. 244 (2003) 631–649 | DOI | MR | Zbl
[43] H. Nishiyama, Remarks on the asymptotic behavior of the solution of an abstract damped wave equation.
[44] , Polynomial decay for damped wave equations on partially rectangular domains. Math. Res. Lett. 16 (2009) 881–894 | DOI | MR | Zbl
[45] and , Quantum decay rates in chaotic scattering. Acta. Math. 203 (2009) | DOI | MR | Zbl
[46] , and , Asymptotic expansion for damped wave equations with periodic coefficients. Math. Models Methods Appl. Sci. 11 (2001) 1285–1310 | DOI | MR | Zbl
[47] , and , Decay estimates for wave equations with variable coefficients. Trans. Amer. Math. Soc. 362 (2010) 2279–2299 | DOI | MR | Zbl
[48] , and , The generalized diffusion phenomenon and applications. SIAM J. Math. Anal. 48 (2016) 174–203 | DOI | MR | Zbl
[49] , Solution of the wave equation with localized energy. Commun. Pure Appl. Math. 22 (1969) 807–823 | DOI | MR | Zbl
[50] and , Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974) 79–86 | DOI | MR | Zbl
[51] and , Method of Modern Math. Phys., volume IV, Analysis of Operator. Academic Press (1979) | MR
[52] , Local energy decay and diffusive phenomenon in a dissipative wave guide. [] | arXiv | Zbl
[53] , Limiting absorption principle for the dissipative Helmholtz equation. Commun. Part. Diff. Equ. 35 (2010) 1458–1489 | DOI | MR | Zbl
[54] , Uniform resolvent estimates for a non-dissipative Helmholtz equation. Bulletin de la S.M.F. 142 (2014) 591–633 | MR | Zbl
[55] , Exponential decay for the Schrödinger equation on a dissipative wave guide. Ann. Henri Poincaré 16 (2015) 1807–1836. | DOI | MR | Zbl
[56] , Local decay for the damped wave equation in the energy space. J. Institute Math. Jussieu (2016). Available at: . | DOI | MR
[57] , Mourre’s commutators method for a dissipative form perturbation. J. Operator Theory 76 (2016) 351–385 | DOI | MR | Zbl
[58] , On the essential spectrum of an arbitrary operator. I. J. Math. Anal. Appl. 13 (1966) 205–215 | DOI | MR | Zbl
[59] , Exponential stabilization without geometric control. Math. Res. Lett. 18 (2011) 379–388 | DOI | MR | Zbl
[60] and , Weighted L2-estimates for dissipative wave equations with variable coefficients. J. Differ. Equ. 246 (2009) 4497–4518 | DOI | MR | Zbl
[61] , On diffusion phenomena for the linear wave equation with space-dependent damping. J. Hyperbolic Differ. Equ. 11 (2014) 795–819 | DOI | MR | Zbl
[62] , Periodic damping gives polynomial energy decay. Math. Res. Lett. 24 (2017) 571–580 | DOI | MR | Zbl
[63] , Semiclassical Analysis, volume 138 of Graduate Studies in Mathematics. Amer. Math. Soc. (2012) | Zbl
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