Energy decay in a wave guide with dissipation at infinity
ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 2, pp. 519-549.

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.

DOI: 10.1051/cocv/2017054
Classification: 35L05, 35J10, 35J25, 35B40, 47A10, 47B44
Keywords: Local and global energy decay, dissipative wave equation, wave guides, diffusive phenomenon, semiclassical analysis, low frequency resolvent estimates
Malloug, Mohamed 1; Royer, Julien 1

1
@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://archive.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://archive.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://archive.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, Volume 24 (2018) no. 2, pp. 519-549. doi : 10.1051/cocv/2017054. http://archive.numdam.org/articles/10.1051/cocv/2017054/

[1] L. Aloui, S. Ibrahim and M. Khenissi, Energy decay for linear dissipative wave equations in exterior domains. J. Differ. Equ. 259 (2015) 2061–2079 | DOI | MR | Zbl

[2] L. Aloui and M. Khenissi, Stabilisation pour l’équation des ondes dans un domaine extérieur. Rev. Math. Iberoamericana 18 (2002) 1–16 | DOI | MR | Zbl

[3] W.O. Amrein, A. Boutet De Monvel and V. Georgescu, C0-groups Commutator Methods and Spectral theory of N-body Hamiltonians, volume 135 of Progress in mathematics. Birkhäuser Verlag (1996) | MR | Zbl

[4] N. Anantharaman and M. Léautaud, Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE 7 (2014) 159–214 | DOI | MR | Zbl

[5] W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306 (1988) 837–852 | DOI | MR | Zbl

[6] C. Bardos, G. Lebeau and J. Rauch, 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] C.J.K. Batty, R. Chill and Y. Tomilov, Fine scales of decay of operator semigroups. J. Eur. Math. Soc. (JEMS) 18 (2016) 853–929 | DOI | MR | Zbl

[8] C.J.K. Batty and Th. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008) 765–780 | DOI | MR | Zbl

[9] J.-F. Bony and D. Häfner, 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] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455–478 | DOI | MR | Zbl

[11] D. Borisov and D. Krejči rík, PT-symmectric waveguides. Integral Equ. Oper. Theory 68 (2008) 489–515 | DOI | MR | Zbl

[12] J.-M. Bouclet, Low frequency estimates and local energy decay for asymptotically Euclidean laplacians. Commun. Part. Diff. Equ. 36 (2011) 1239–1286 | DOI | MR | Zbl

[13] J.-M. Bouclet and J. Royer, Local energy decay for the damped wave equation. Jour. Func. Anal. 266 (2014) 4538–4615 | DOI | MR | Zbl

[14] N. Boussaid and S. Golénia, 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] N. Burq,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] N. Burq,Semi-classical estimates for the resolvent in nontrapping geometries. Int. Math. Res. Not. 5 (2002) 221–241 | DOI | MR | Zbl

[17] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains. Math. Res. Lett. 14 (2007) 35–47 | DOI | MR | Zbl

[18] N. Burq and R. Joly, Exponential decay for the damped wave equation in unbounded domains. Commun. Contemporary Math. 18(2016) | DOI | MR | Zbl

[19] R. Chill and A. Haraux, 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] D.E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators. Oxford University Press, New York (1987) | MR | Zbl

[21] P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71 (1993) 559–607 | DOI | MR | Zbl

[22] K. Gustafson and J. Weidmann, On the essential spectrum. J. Math. Anal. Appl. 25 (1969) 121–127 | DOI | MR | Zbl

[23] T. Hosono and T. Ogawa, 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] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain. J. Differ. Equ. 186 (2002) 633–651 | DOI | MR | Zbl

[25] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay rate of the energy for wave equations with critical potential. J. Math. Soc. Jpn 65 (2013) 183–236 | DOI | MR | Zbl

[26] Th. Jecko, From classical to semiclassical non-trapping behaviour. C. R., Math., Acad. Sci. Paris 338 (2004) 545–548 | DOI | MR | Zbl

[27] M. Khenissi, 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] D. Krejčiřík and J. Kříž, On the spectrum of curved planar waveguides. Publ. Res. Inst. Math. Sci. 41 (2005) 757–791 | DOI | MR | Zbl

[30] D. Krejčiřík and N. Raymond, 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] G. Lebeau, Équation des ondes amorties. In Algebraic and geometric methods in mathematical physics, edited by A. Boutet De Monvel and V. Marchenko. Kluwer Academic Publishers (1996) 73–109 | DOI | MR | Zbl

[33] G. Lebeau and L. Robbiano, Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86 (1997) 465–491 | DOI | MR | Zbl

[34] M. Malloug, Local energy decay for the damped Klein−Gordon equation in exterior domain. Appl. Anal. (2016) | MR

[35] P. Marcati and K. Nishihara, 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] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations. Publ. Res. Inst. Math. Sci. 12 (1976) 169–189 | DOI | MR | Zbl

[37] R. Melrose, Singularities and energy decay in acoustical scattering. Duke Math. J. 46 (1979) 43–59 | DOI | MR | Zbl

[38] K. Mochizuki, Scattering theory for wave equations with dissipative terms. Publ. Res. Inst. Math. Sci. 12 (1976) 383–390 | DOI | MR | Zbl

[39] C.S. Morawetz, J.V. Ralston and W.A. Strauss, Decay of the solution of the wave equation outside non-trapping obstacles. Commun. Pure Appl. Math. 30 (1977) 447–508 | DOI | MR | Zbl

[40] E. Mourre, Opérateurs conjugués et propriétés de propagation. Comm. Math. Phys. 91 (1983) 279–300 | DOI | MR | Zbl

[41] T. Narazaki, 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] K. Nishihara, 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] H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains. Math. Res. Lett. 16 (2009) 881–894 | DOI | MR | Zbl

[45] S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering. Acta. Math. 203 (2009) | DOI | MR | Zbl

[46] R. Orive, E. Zuazua and A.F. Pazoto, Asymptotic expansion for damped wave equations with periodic coefficients. Math. Models Methods Appl. Sci. 11 (2001) 1285–1310 | DOI | MR | Zbl

[47] P. Radu, G. Todorova and B. Yordanov, Decay estimates for wave equations with variable coefficients. Trans. Amer. Math. Soc. 362 (2010) 2279–2299 | DOI | MR | Zbl

[48] P. Radu,G. Todorova and B. Yordanov, The generalized diffusion phenomenon and applications. SIAM J. Math. Anal. 48 (2016) 174–203 | DOI | MR | Zbl

[49] J. Ralston, Solution of the wave equation with localized energy. Commun. Pure Appl. Math. 22 (1969) 807–823 | DOI | MR | Zbl

[50] J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24 (1974) 79–86 | DOI | MR | Zbl

[51] M. Reed and B. Simon, Method of Modern Math. Phys., volume IV, Analysis of Operator. Academic Press (1979) | MR

[52] J. Royer, Local energy decay and diffusive phenomenon in a dissipative wave guide. [] | arXiv | Zbl

[53] J. Royer, Limiting absorption principle for the dissipative Helmholtz equation. Commun. Part. Diff. Equ. 35 (2010) 1458–1489 | DOI | MR | Zbl

[54] J. Royer, Uniform resolvent estimates for a non-dissipative Helmholtz equation. Bulletin de la S.M.F. 142 (2014) 591–633 | MR | Zbl

[55] J. Royer, Exponential decay for the Schrödinger equation on a dissipative wave guide. Ann. Henri Poincaré 16 (2015) 1807–1836. | DOI | MR | Zbl

[56] J. Royer, Local decay for the damped wave equation in the energy space. J. Institute Math. Jussieu (2016). Available at: . | DOI | MR

[57] J. Royer, Mourre’s commutators method for a dissipative form perturbation. J. Operator Theory 76 (2016) 351–385 | DOI | MR | Zbl

[58] M. Schechter, On the essential spectrum of an arbitrary operator. I. J. Math. Anal. Appl. 13 (1966) 205–215 | DOI | MR | Zbl

[59] E. Schenck, Exponential stabilization without geometric control. Math. Res. Lett. 18 (2011) 379–388 | DOI | MR | Zbl

[60] G. Todorova and B. Yordanov, Weighted L2-estimates for dissipative wave equations with variable coefficients. J. Differ. Equ. 246 (2009) 4497–4518 | DOI | MR | Zbl

[61] Y. Wakasugi, On diffusion phenomena for the linear wave equation with space-dependent damping. J. Hyperbolic Differ. Equ. 11 (2014) 795–819 | DOI | MR | Zbl

[62] J. Wunsch, Periodic damping gives polynomial energy decay. Math. Res. Lett. 24 (2017) 571–580 | DOI | MR | Zbl

[63] M. Zworski, Semiclassical Analysis, volume 138 of Graduate Studies in Mathematics. Amer. Math. Soc. (2012) | Zbl

Cited by Sources: