Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable

Cavalcanti, Marcelo; Lasiecka, Irena; Toundykov, Daniel

doi:10.1090/S0002-9947-2012-05583-8

Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable

Authors: Marcelo M. Cavalcanti, Irena Lasiecka and Daniel Toundykov
Journal: Trans. Amer. Math. Soc. 364 (2012), 5693-5713
MSC (2010): Primary 35L05; Secondary 93B07, 93D15
DOI: https://doi.org/10.1090/S0002-9947-2012-05583-8
Published electronically: June 12, 2012
MathSciNet review: 2946927
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A stabilization/observability estimate and asymptotic energy decay rates are derived for a wave equation with nonlinear damping in a portion of the interior and Wentzell condition on the boundary: $\partial _{\nu } u + u = \Delta _{T}u$ . The dissipation does not affect a full collar of the boundary, thus leaving out a portion subjected to the high-order Wentzell condition.

Observability of wave equations with damping supported away from the Neumann boundary is known to be intrinsically more difficult than the corresponding Dirichlet problem because the uniform Lopatinskii condition is not satisfied by such a system. In the case of a Wentzell boundary, the situation is more difficult since the ``natural'' energy now includes the $H^{1}$ Sobolev norm of the solution on the boundary. To establish uniform stability it is necessary not only to overcome the presence of the Neumann boundary operator, but also to establish an inverse-type coercivity estimate on the $H^{1}$ trace norm of the solution. This goal is attained by constructing multipliers based on a refinement of nonradial vector fields employed for ``unobserved'' Neumann conditions. These multipliers, along with a suitable geometry (local convexity), allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.

[Bar93] Viorel Barbu, Analysis and control of nonlinear infinite-dimensional systems, Mathematics in Science and Engineering, vol. 190, Academic Press, Inc., Boston, MA, 1993. MR 1195128
[BT10] Francesca Bucci and Daniel Toundykov, Finite-dimensional attractor for a composite system of wave/plate equations with localized damping, Nonlinearity 23 (2010), no. 9, 2271–2306. MR 2672645, https://doi.org/10.1088/0951-7715/23/9/011
[CDCFT09] Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Ryuichi Fukuoka, and Daniel Toundykov, Stabilization of the damped wave equation with Cauchy-Ventcel boundary conditions, J. Evol. Equ. 9 (2009), no. 1, 143–169. MR 2501356, https://doi.org/10.1007/s00028-009-0002-1
[CGG08] Giuseppe M. Coclite, Gisèle R. Goldstein, and Jerome A. Goldstein, Stability estimates for parabolic problems with Wentzell boundary conditions, J. Differential Equations 245 (2008), no. 9, 2595–2626. MR 2455779, https://doi.org/10.1016/j.jde.2007.12.006
[CGG09] Giuseppe M. Coclite, Gisèle R. Goldstein, and Jerome A. Goldstein, Stability of parabolic problems with nonlinear Wentzell boundary conditions, J. Differential Equations 246 (2009), no. 6, 2434–2447. MR 2498847, https://doi.org/10.1016/j.jde.2008.10.004
[CKM07] Marcelo M. Cavalcanti, Ammar Khemmoudj, and Mohamed Medjden, Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl. 328 (2007), no. 2, 900–930. MR 2290021, https://doi.org/10.1016/j.jmaa.2006.05.070
[CLT] M. M. Cavalcanti, I. Lasiecka, and D. Toundykov. Geometrically constrained stabilization of wave equation with Wentzell boundary conditions. Appl. Anal., 2012. In press. DOI 10.1080/00036811.2011.647910.
[DLT09] Moez Daoulatli, Irena Lasiecka, and Daniel Toundykov, Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 1, 67–94. MR 2481581, https://doi.org/10.3934/dcdss.2009.2.67
[Doc] Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207
[GLLT04] R. Gulliver, I. Lasiecka, W. Littman, and R. Triggiani, The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber, Geometric methods in inverse problems and PDE control, IMA Vol. Math. Appl., vol. 137, Springer, New York, 2004, pp. 73–181. MR 2169903, https://doi.org/10.1007/978-1-4684-9375-7_5
[Hem00] Amar Heminna, Stabilisation frontière de problèmes de Ventcel, ESAIM Control Optim. Calc. Var. 5 (2000), 591–622 (French, with English and French summaries). MR 1799332, https://doi.org/10.1051/cocv:2000123
[Hem01] Amar Heminna, Contrôlabilité exacte d’un problème avec conditions de Ventcel évolutives pour le système linéaire de l’élasticité, Rev. Mat. Complut. 14 (2001), no. 1, 231–270 (French, with English and French summaries). MR 1851730, https://doi.org/10.5209/rev_REMA.2001.v14.n1.17061
[IY00] Victor Isakov and Masahiro Yamamoto, Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999) Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000, pp. 191–225. MR 1804796, https://doi.org/10.1090/conm/268/04314
[KM04] Ammar Khemmoudj and Mohamed Medjden, Exponential decay for the semilinear Cauchy-Ventcel problem with localized damping, Bol. Soc. Parana. Mat. (3) 22 (2004), no. 2, 97–116 (English, with French summary). MR 2190136, https://doi.org/10.5269/bspm.v22i2.7486
[KM08] A. Kanoune and N. Mehidi, Stabilization and control of subcritical semilinear wave equation in bounded domain with Cauchy-Ventcel boundary conditions, Appl. Math. Mech. (English Ed.) 29 (2008), no. 6, 787–800. MR 2423224, https://doi.org/10.1007/s10483-008-0610-x
[KZ06] Herbert Koch and Enrique Zuazua, A hybrid system of PDE’s arising in multi-structure interaction: coupling of wave equations in 𝑛 and 𝑛-1 space dimensions, Recent trends in partial differential equations, Contemp. Math., vol. 409, Amer. Math. Soc., Providence, RI, 2006, pp. 55–77. MR 2243949, https://doi.org/10.1090/conm/409/07706
[Kom94] V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
[Lag89] John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153
[LT93] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507–533. MR 1202555
[LMP62] Peter D. Lax, Cathleen S. Morawetz, and Ralph S. Phillips, The exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Bull. Amer. Math. Soc. 68 (1962), 593–595. MR 0142890, https://doi.org/10.1090/S0002-9904-1962-10865-9
[LL88] J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 6, Masson, Paris, 1988. MR 953313
[Lem87] Keddour Lemrabet, Le problème de Ventcel pour le système de l’élasticité dans un domaine de 𝑅³, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 6, 151–154 (French, with English summary). MR 880120
[LM88] W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl. (4) 152 (1988), 281–330. MR 980985, https://doi.org/10.1007/BF01766154
[LT06] Irena Lasiecka and Daniel Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms, Nonlinear Anal. 64 (2006), no. 8, 1757–1797. MR 2197360, https://doi.org/10.1016/j.na.2005.07.024
[LTZ00] I. Lasiecka, R. Triggiani, and X. Zhang, 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) Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000, pp. 227–325. MR 1804797, https://doi.org/10.1090/conm/268/04315
[Nic03] S. Nicaise, Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications, Rend. Mat. Appl. (7) 23 (2003), no. 1, 83–116. MR 2044992
[QR77] John P. Quinn and David L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 1-2, 97–127. MR 0473539, https://doi.org/10.1017/S0308210500018072
[SA76] A. A. Samarskiĭ and V. B. Andreev, \cyr Raznostnye metody dlya èllipticheskikh uravneniĭ, Izdat. ‘Nauka’, Moscow, 1976 (Russian). MR 0502017
[Str] Walter A. Strauss, Dispersal of waves vanishing on the boundary of an exterior domain, Comm. Pure Appl. Math. 28 (1975), 265–278. MR 0367461, https://doi.org/10.1002/cpa.3160280205
[Tat] D. Tataru. Private Communication.
[Tou07] Daniel Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions, Nonlinear Anal. 67 (2007), no. 2, 512–544. MR 2317185, https://doi.org/10.1016/j.na.2006.06.007
[VM00] Judith Vancostenoble and Patrick Martinez, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks, SIAM J. Control Optim. 39 (2000), no. 3, 776–797. MR 1786329, https://doi.org/10.1137/S0363012999354211
[War06] Mahamadi Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum 73 (2006), no. 1, 10–30. MR 2277314, https://doi.org/10.1007/s00233-006-0617-2
[War10] Mahamadi Warma, Analyticity on 𝐿₁ of the heat semigroup with Wentzell boundary conditions, Arch. Math. (Basel) 94 (2010), no. 1, 85–89. MR 2581338, https://doi.org/10.1007/s00013-009-0068-6
[Ven59] A. D. Ventcel′, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl. 4 (1959), 164–177. MR 0121855