We consider the semilinear damped wave equation
In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that for some function with when . We provide general tools to deal with the semilinear stabilization problem in the case where has a sufficiently fast decay.
On considère l’équation des ondes amorties
Dans cet article, on obtient les premiers résultats concernant la stabilisation de cette équation dans des cas où ne vérifie pas la condition de contrôle géométrique. Quand certains des rayons géodésiques sont captés, la stabilisation du semigroupe est semi-uniforme dans le sens où pour une certaine fonction avec quand . On donne une méthode générale pour traiter le problème de la stabilisation semilinéaire dans le cas où décroit suffisamment.
Revised:
Accepted:
Published online:
Keywords: damped wave equations, stabilization, semi-uniform decay, unique continuation property, small trapped sets, weak attractors
@article{AHL_2020__3__1241_0, author = {Joly, Romain and Laurent, Camille}, title = {Decay of semilinear damped wave equations: cases without geometric control condition}, journal = {Annales Henri Lebesgue}, pages = {1241--1289}, publisher = {\'ENS Rennes}, volume = {3}, year = {2020}, doi = {10.5802/ahl.60}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.60/} }
TY - JOUR AU - Joly, Romain AU - Laurent, Camille TI - Decay of semilinear damped wave equations: cases without geometric control condition JO - Annales Henri Lebesgue PY - 2020 SP - 1241 EP - 1289 VL - 3 PB - ÉNS Rennes UR - http://archive.numdam.org/articles/10.5802/ahl.60/ DO - 10.5802/ahl.60 LA - en ID - AHL_2020__3__1241_0 ER -
%0 Journal Article %A Joly, Romain %A Laurent, Camille %T Decay of semilinear damped wave equations: cases without geometric control condition %J Annales Henri Lebesgue %D 2020 %P 1241-1289 %V 3 %I ÉNS Rennes %U http://archive.numdam.org/articles/10.5802/ahl.60/ %R 10.5802/ahl.60 %G en %F AHL_2020__3__1241_0
Joly, Romain; Laurent, Camille. Decay of semilinear damped wave equations: cases without geometric control condition. Annales Henri Lebesgue, Volume 3 (2020), pp. 1241-1289. doi : 10.5802/ahl.60. http://archive.numdam.org/articles/10.5802/ahl.60/
[AL14] Sharp polynomial decay rates for the damped wave equation on the torus, Anal. PDE, Volume 7 (2014) no. 1, pp. 159-214 | MR | Zbl
[Ana08] Entropy and the localization of eigenfunctions, Ann. Math., Volume 168 (2008) no. 2, pp. 435-475 | MR | Zbl
[BD08] Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., Volume 8 (2008) no. 4, pp. 765-780 | DOI | MR | Zbl
[BD18] Spectral gaps without the pressure condition, Ann. of Math., Volume 187 (2018) no. 3, pp. 825-867 | DOI | MR | Zbl
[BL13] Injections de Sobolev probabilistes et applications, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 6, pp. 917-962 | DOI | Numdam | MR | Zbl
[BLR92] 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
[BM04] Geometric control in the presence of a black box, J. Am. Math. Soc., Volume 17 (2004) no. 2, pp. 443-471 | DOI | MR | Zbl
[BT10] Optimal polynomial decay of functions and operator semigroups, Math. Ann., Volume 347 (2010) no. 2, pp. 455-478 | DOI | MR | Zbl
[Bur93] Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mém. Soc. Math. France (N.S.), Volume 55 (1993), pp. 3-126 | Numdam | MR | Zbl
[BV83] Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., Volume 62 (1983) no. 4, pp. 441-491 | MR | Zbl
[CKS + 10] Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math., Volume 181 (2010) no. 1, pp. 39-113 | Zbl
[CSVW14] From resolvent estimates to damped waves, J. Anal. Math., Volume 1222 (2014), pp. 143-160 | DOI | MR | Zbl
[Daf78] Asymptotic behavior of solutions of evolution equations, Nonlinear Evolution Equations. Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977) (Publication of the Mathematics Research Center, the University of Wisconsin), Volume 40, Academic Press, 1978, pp. 103-123 | MR | Zbl
[Del01] Stabilisation pour l’équation des ondes semi-linéaire, Asymptotic Anal., Volume 27 (2001) no. 2, pp. 171-181
[DLZ03] Stabilization and control for the subcritical semilinear wave equation, Ann. Sci. Éc. Norm. Sup., Volume 36 (2003) no. 4, pp. 525-551 | Numdam | MR | Zbl
[DV12] Propagation through trapped sets and semiclassical resolvent estimates, Ann. Inst. Fourier, Volume 62 (2012) no. 6, pp. 2347-2377 (Addendum ibid. pp. 2379–2384) | DOI | Numdam | MR | Zbl
[Gea78] Spectral theory for contraction semigroups on Hilbert spaces, Trans. Am. Math. Soc., Volume 236 (1978), pp. 385-394 | DOI | MR | Zbl
[GG12] Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, Volume 5 (2012) no. 5, pp. 1139-1155 | DOI | MR | Zbl
[Hal88] Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, 1988 | MR | Zbl
[Har85] Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differ. Equ., Volume 59 (1985) no. 2, pp. 145-154 | MR | Zbl
[HBTV15] Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi, Volume 3 (2015), e4, p. 63 | Zbl
[HR03] Regularity, determining modes and Galerkin methods, J. Math. Pures Appl., Volume 82 (2003) no. 9, pp. 1075-1136 | DOI | MR | Zbl
[Hua85] 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
[Hör85] The analysis of linear partial differential operators. IV: Fourier integral operators, Grundlehren der Mathematischen Wissenschaften, 275, Springer, 1985 | Zbl
[Hör97] On the uniqueness of the Cauchy problem under partial analyticity assumptions, Geometrical optics and related topics (Cortona, 1996 (Progress in Nonlinear Differential Equations and Their Applications), Volume 32, Birkhäuser (1997), pp. 179-219 | MR | Zbl
[Ika82] Decay of solutions of the wave equation in the exterior of two convex obstacles, Osaka J. Math., Volume 19 (1982), pp. 459-509 | MR | Zbl
[Ika88] Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, Volume 38 (1988) no. 2, pp. 113-146 | DOI | Numdam | MR | Zbl
[JL13] Stabilization for the semilinear wave equation with geometric control condition, Anal. PDE, Volume 6 (2013) no. 5, pp. 1089-1119 | DOI | MR | Zbl
[Jol07] New examples of damped wave equations with gradient-like structure, Asymptotic Anal., Volume 53 (2007) no. 4, pp. 237-253 | MR | Zbl
[KK93] Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities, Appl. Anal., Volume 50 (1993) no. 1-2, pp. 93-102 | DOI | MR | Zbl
[Leb93] Equation des ondes amorties, Algebraic and Geometric Methods in Mathematical Physics: proceedings of the Kaciveli Summer School, Crimea, Ukraine, 1993 (Mathematical Physics Studies), Volume 19, Springer (1993), pp. 73-109
[Lin20] Damped wave equations on compact hyperbolic surfaces, Commun. Math. Phys., Volume 373 (2020) no. 3, pp. 771-794 | MR
[LL15] Sharp polynomial energy decay for locally undamped waves, Sémin. Équ. Dériv. Partielles (2014-2015), 21, p. 13 | DOI | Zbl
[LR85] Unicité de Cauchy pour des opérateurs de type principal, J. Anal. Math., Volume 44 (1985), pp. 32-66 | DOI | Zbl
[LR97] Stabilisation de l’équation des ondes par le bord, Duke Math. J., Volume 86 (1997) no. 3, pp. 465-491 | DOI | Zbl
[LTZ00] Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability in one shot, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999) (Contemporary Mathematics), Volume 268, American Mathematical Society, 2000, pp. 227-325 | DOI | MR | Zbl
[Non18] Resonances in hyperbolic dynamics, Proceedings of the International Congress of Mathematicians (ICM 2018), Volume 2-4, World Scientific (2018), pp. 2495-2518 | DOI | MR | Zbl
[Prü84] On the spectrum of -semigroups, Trans. Am. Math. Soc., Volume 284 (1984) no. 2, pp. 847-857 | Zbl
[Riv14] Eigenmodes of the damped wave equation and small hyperbolic subsets, Ann. Inst. Fourier, Volume 64 (2014) no. 3, pp. 1229-1267 (With an appendix by Stéphane Nonnenmacher and Rivière) | DOI | Numdam | MR | Zbl
[Rui92] Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., Volume 71 (1992) no. 5, pp. 455-467 | MR | Zbl
[RZ98] Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., Volume 131 (1998) no. 3, pp. 493-539 | DOI | MR | Zbl
[Sch10] Energy Decay for the Damped Wave Equation Under a Pressure Condition, Comm. Math. Phys., Volume 300 (2010) no. 2, pp. 375-410 | DOI | MR | Zbl
[Sch11] Exponential stabilization without geometric control, Math. Res. Lett., Volume 18 (2011) no. 2, pp. 379-388 | DOI | MR | Zbl
[SU13] Recovery of a source term or a speed with one measurement and applications, Trans. Am. Math. Soc., Volume 365 (2013) no. 11, pp. 5737-5758 | DOI | MR | Zbl
[Tat95] Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem, Commun. Partial Differ. Equations, Volume 20 (1995) no. 5-6, pp. 855-884 | DOI | Zbl
[Tat96] Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl., Volume 75 (1996) no. 4, pp. 367-408 | MR | Zbl
[Tat99] Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl., Volume 78 (1999) no. 5, pp. 505-521 | DOI | MR | Zbl
[Zua90] Exponential decay for semilinear wave equations with localized damping, Commun. Partial Differ. Equations, Volume 15 (1990) no. 2, pp. 205-235
Cited by Sources: