Volume 3 (2020)
Joly, Romain; Laurent, Camille
Decay of semilinear damped wave equations: cases without geometric control condition
Annales Henri Lebesgue, Volume 3 (2020), pp. 1241-1289.

Metadata

DOI10.5802/ahl.60
Keywords damped wave equations, stabilization, semi-uniform decay, unique continuation property, small trapped sets, weak attractors

Abstract

We consider the semilinear damped wave equation

∂tt2u(x,t)+γ(x)∂tu(x,t)=Δu(x,t)-αu(x,t)-f(x,u(x,t)).

In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where γ 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 ∥e At A -1 ∥≤h(t) for some function h with h(t)→0 when t→+∞. We provide general tools to deal with the semilinear stabilization problem in the case where h(t) has a sufficiently fast decay.


References

[AL14] Anantharaman, Nalini; Léautaud, Matthieu 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] Anantharaman, Nalini Entropy and the localization of eigenfunctions, Ann. Math., Volume 168 (2008) no. 2, pp. 435-475 | MR | Zbl

[BD08] Batty, Charles J. K.; Duyckaerts, Thomas 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] Bourgain, Jean; Dyatlov, Semyon Spectral gaps without the pressure condition, Ann. of Math., Volume 187 (2018) no. 3, pp. 825-867 | DOI | MR | Zbl

[BL13] Burq, Nicolas; Lebeau, Gilles Injections de Sobolev probabilistes et applications, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 6, pp. 917-962 | DOI | Numdam | MR | Zbl

[BLR92] Bardos, Claude; Lebeau, Gilles; Rauch, Jeffrey 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] Burq, Nicolas; Maciej, Zworski 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] Borichev, Alexander; Tomilov, Yury Optimal polynomial decay of functions and operator semigroups, Math. Ann., Volume 347 (2010) no. 2, pp. 455-478 | DOI | MR | Zbl

[Bur93] Burq, Nicolas 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] Babin, Anatolii V.; Vishik, Mark I. Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., Volume 62 (1983) no. 4, pp. 441-491 | MR | Zbl

[CKS + 10] Colliander, James E.; Keel, Markus; Staffilani, Gigliola; Takaoka, Hideo; Tao, Terence 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] Christianson, Hans; Schenck, Emmanuel; Vasy, András; Wunsch, Jared From resolvent estimates to damped waves, J. Anal. Math., Volume 1222 (2014), pp. 143-160 | DOI | MR | Zbl

[Daf78] Dafermos, Constantin M. 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] Delham, Belhassen Stabilisation pour l’équation des ondes semi-linéaire, Asymptotic Anal., Volume 27 (2001) no. 2, pp. 171-181

[DLZ03] Delham, Belhassen; Lebeau, Gilles; Zuazua, Enrique 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] Datchev, Kiril; Vasy, András 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] Gearhart, Larry Spectral theory for contraction semigroups on Hilbert spaces, Trans. Am. Math. Soc., Volume 236 (1978), pp. 385-394 | DOI | MR | Zbl

[GG12] Gérard, Patrick; Grellier, Sandrine Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, Volume 5 (2012) no. 5, pp. 1139-1155 | DOI | MR | Zbl

[Hal88] Hale, Jack K. Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, Volume 25, American Mathematical Society, 1988 | MR | Zbl

[Har85] Haraux, Alain Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differ. Equ., Volume 59 (1985) no. 2, pp. 145-154 | MR | Zbl

[HBTV15] Hani, Zaher; Benoît, Pausader; Tzvetkov, Nikolay; Visciglia, Nicola Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum Math. Pi, Volume 3 (2015), e4, p. 63 | Zbl

[HR03] Hale, Jack K.; Raugel, Geneviève Regularity, determining modes and Galerkin methods, J. Math. Pures Appl., Volume 82 (2003) no. 9, pp. 1075-1136 | DOI | MR | Zbl

[Hua85] Huang, Falun 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] Hörmander, Lars The analysis of linear partial differential operators. IV: Fourier integral operators, Grundlehren der Mathematischen Wissenschaften, Volume 275, Springer, 1985 | Zbl

[Hör97] Hörmander, Lars 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 (1997), pp. 179-219 | MR | Zbl

[Ika82] Ikawa, Mitsuru 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] Ikawa, Mitsuru 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] Joly, Romain; Laurent, Camille Stabilization for the semilinear wave equation with geometric control condition, Anal. PDE, Volume 6 (2013) no. 5, pp. 1089-1119 | DOI | MR | Zbl

[Jol07] Joly, Romain New examples of damped wave equations with gradient-like structure, Asymptotic Anal., Volume 53 (2007) no. 4, pp. 237-253 | MR | Zbl

[KK93] Kazemi, Mohammad; Klibanov, Michael V. 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] Lebeau, Gilles 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 (1993), pp. 73-109

[Lin20] Lin, Jong Damped wave equations on compact hyperbolic surfaces, Commun. Math. Phys., Volume 373 (2020) no. 3, pp. 771-794 | MR

[LL15] Léautaud, Matthieu; Lerner, Nicolas Sharp polynomial energy decay for locally undamped waves, Sémin. Équ. Dériv. Partielles (2014-2015), 21, p. 13 | DOI | Zbl

[LR85] Lerner, Nicolas; Robbiano, Luc Unicité de Cauchy pour des opérateurs de type principal, J. Anal. Math., Volume 44 (1985), pp. 32-66 | DOI | Zbl

[LR97] Lebeau, Gilles; Robbiano, Luc Stabilisation de l’équation des ondes par le bord, Duke Math. J., Volume 86 (1997) no. 3, pp. 465-491 | DOI | Zbl

[LTZ00] Lasiecka, Irena; Triggiani, Roberto; Zhang, Xu 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] Nonnenmacher, Stéphane Resonances in hyperbolic dynamics, Proceedings of the International Congress of Mathematicians (ICM 2018), Volume 2-4 (2018), pp. 2495-2518 | DOI | MR | Zbl

[Prü84] Prüss, Jan On the spectrum of C 0 -semigroups, Trans. Am. Math. Soc., Volume 284 (1984) no. 2, pp. 847-857 | Zbl

[Riv14] Rivière, Gabriel 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] Ruiz, Alberto 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] Robbiano, Luc; Zuily, Claude 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] Schenck, Emmanuel 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] Schenck, Emmanuel Exponential stabilization without geometric control, Math. Res. Lett., Volume 18 (2011) no. 2, pp. 379-388 | DOI | MR | Zbl

[SU13] Stefanov, Plamen; Ulhmann, Gunther 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] Tataru, Daniel 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] Tataru, Daniel 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] Tataru, Daniel Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl., Volume 78 (1999) no. 5, pp. 505-521 | DOI | MR | Zbl

[Zua90] Zuazua, Enrique Exponential decay for semilinear wave equations with localized damping, Commun. Partial Differ. Equations, Volume 15 (1990) no. 2, pp. 205-235