On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

Laurent, Camille

Journées équations aux dérivées partielles (2011), article no. 6, 17 p.

On étudie la stabilisation et le contrôle interne de l’équation de Klein-Gordon critique sur des variétés de dimension 3. Sous des conditions géométriques légèrement plus fortes que la condition de contrôle géométrique classique, on prouve la décroissance exponentielle de solutions bornées dans l’espace d’énergie mais petites dans des normes plus faibles. La preuve combine la décomposition en profils et des arguments microlocaux. Cette décomposition, analogue à celle de Bahouri-Gérard [2] sur ${ℝ}^3$, nécessite l’analyse de certains effets dus à la géométrie. Elle utilise des résultats de S. Ibrahim [16] sur le comportement d’ondes de concentration sur les variétés.

We study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-Gérard [2] on ${ℝ}^3$, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim [16] on the behavior of concentrating waves on manifolds.

[1] L. Aloui, S. Ibrahim, and K. Nakanishi. Exponential energy decay for damped Klein-Gordon Equation with nonlinearities of arbitrary growth. Comm. Partial Diff. Equa., 36(5):797–818, 2011. | MR

[2] H. Bahouri and P. Gérard. High frequency approximation of critical nonlinear wave equations. American J. Math., 121:131–175, 1999. | MR | Zbl

[3] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim., 305:1024–1065, 1992. | MR | Zbl

[4] N. Burq. Mesures semi-classiques et mesures de defaut, Seminaire Bourbaki, Vol. 1996/97. Astérisque, 245:167–195, 1997. | Numdam | MR | Zbl

[5] N. Burq and P. Gérard. Condition nécéssaire et suffisante pour la contrôlabilite exacte des ondes. Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 325(7):749–752, 1997. | MR | Zbl

[6] M. Christ, J. Colliander, and T. Tao. Ill-posedness for nonlinear Schrödinger and wave equations. http://arxiv.org/ps/math.AP/0311048.pdf.

[7] H. Christianson. Semiclassical non-concentration near hyperbolic orbits. Journal of Functional Analysis, 246(2):145–195, 2007. | MR | Zbl

[8] M. Daoulatli, I. Lasiecka, and D. Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete Contin. Dyn. Syst. Ser, 2:67–94, 2009. | MR | Zbl

[9] B. Dehman and P. Gérard. Stabilization for the Nonlinear Klein Gordon Equation with critical Exponent. Prépublication de l’Université Paris-Sud, available at http://www.math.u-psud.fr/~biblio/saisie/fichiers/ppo_2002_35.ps, 2002.

[10] B. Dehman and G. Lebeau. Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time. SIAM Journal on Control and Optimization, 48(2):521–550, 2009. | MR | Zbl

[11] B. Dehman, G. Lebeau, and E. Zuazua. Stabilization and control for the subcritical semilinear wave equation. Annales scientifiques de l’Ecole normale supérieure, 36(4):525–551, 2003. | Numdam | MR | Zbl

[12] E. Fernández-Cara and E. Zuazua. Null and approximate controllability for weakly blowing up semilinear heat equations. Annales de l’Institut Henri Poincare/Analyse non lineaire, 17(5):583–616, 2000. | Numdam | MR | Zbl

[13] I. Gallagher and P. Gérard. Profile decomposition for the wave equation outside a convex obstacle. Journal de mathématiques pures et appliquées, 80(1):1–49, 2001. | MR | Zbl

[14] P. Gérard. Microlocal Defect Measures. Comm. Partial Diff. eq., 16:1762–1794, 1991. | MR | Zbl

[15] P. Gérard. Oscillations and Concentration Effects in Semilinear Dispersive Wave Equations. Journal of Functional Analysis, 141:60–98, 1996. | MR | Zbl

[16] S. Ibrahim. Geometric-Optics for Nonlinear Concentrating Waves in Focusing and Non-Focusing Two Geometries. Communications in Contemporary Mathematics, 6(1):1–24, 2004. | MR | Zbl

[17] S. Ibrahim and M. Majdoub. Solutions globales de l’equation des ondes semi-lineaire critique a coefficients variables. Bulletin de la Société Mathématique de France, 131(1):1–22, 2003. | Numdam | MR | Zbl

[18] R. Joly and C. Laurent. Stabilisation for the semilinear wave equation with geometric control condition. in preparation, 2011.

[19] L.V. Kapitanski. Some generalizations of the Strichartz-Brenner inequality. Leningrad Math. J., 1(10):693–726, 1990. | MR | Zbl

[20] H. Koch and D. Tataru. Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure and Appl. Math., 58(2):217–284, 2005. | MR | Zbl

[21] C. Laurent. Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3. SIAM Journal on Mathematical Analysis, 42(2):785–832, 2010. | MR | Zbl

[22] C. Laurent. On stabilization and control for the critical Klein Gordon equation on 3-D compact manifolds. Journal of Functional Analysis, 260(5):1304–1368, 2011. | MR

[23] G. Lebeau. Equation des ondes amorties. In Algebraic and geometric methods in mathematical physics: proceedings of the Kaciveli Summer School, Crimea, Ukraine, 1993, page 73. Springer, 1996. | MR | Zbl

[24] G. Lebeau. Non linear optic and supercritical wave equation. Bulletin-Société Royale des sciences de Liège, 70(4/6):267–306, 2001. | MR | Zbl

[25] G. Lebeau and L. Robbiano. Stabilisation de l’équation des ondes par le bord. Duke Mathematical Journal, 86(3):465–491, 1997. | MR | Zbl

[26] P.L. Lions. The Concentration-Compactness Principle in the Calculus of Variations.(The limit case, Part I.). Revista matemática iberoamericana, 1(1):145, 1985. | MR | Zbl

[27] J. Rauch and M. Taylor. Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains. Indiana Univ. Math. J., 24(1):79–86, 1975. | MR | Zbl

[28] L. Robbiano and C. Zuily. Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients. Invent. Math., 131:493–539, 1998. | MR | Zbl

[29] E. Schenck. Energy decay for the damped wave equation under a pressure condition. Communications in Mathematical Physics, pages 1–36, 2010. | MR | Zbl