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

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- by Marcelo M. Cavalcanti, Irena Lasiecka and Daniel Toundykov PDF
- Trans. Amer. Math. Soc.
**364**(2012), 5693-5713 Request permission

## 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.

## References

- 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** - 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**, DOI 10.1088/0951-7715/23/9/011 - 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**, DOI 10.1007/s00028-009-0002-1 - 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**, DOI 10.1016/j.jde.2007.12.006 - 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**, DOI 10.1016/j.jde.2008.10.004 - 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**, DOI 10.1016/j.jmaa.2006.05.070 - 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. - 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**, DOI 10.3934/dcdss.2009.2.67 - 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**, DOI 10.1007/978-1-4757-2201-7 - 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**, DOI 10.1007/978-1-4684-9375-7_{5} - 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**, DOI 10.1051/cocv:2000123 - 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**, DOI 10.5209/rev_{R}EMA.2001.v14.n1.17061 - 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**, DOI 10.1090/conm/268/04314 - 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**, DOI 10.5269/bspm.v22i2.7486 - 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**, DOI 10.1007/s10483-008-0610-x - 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**, DOI 10.1090/conm/409/07706 - V. Komornik,
*Exact controllability and stabilization*, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR**1359765** - 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**, DOI 10.1137/1.9781611970821 - 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** - 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**142890**, DOI 10.1090/S0002-9904-1962-10865-9 - 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** - Keddour Lemrabet,
*Le problème de Ventcel pour le système de l’élasticité dans un domaine de $\textbf {R}^3$*, C. R. Acad. Sci. Paris Sér. I Math.**304**(1987), no. 6, 151–154 (French, with English summary). MR**880120** - 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**, DOI 10.1007/BF01766154 - 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**, DOI 10.1016/j.na.2005.07.024 - 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**, DOI 10.1090/conm/268/04315 - 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** - 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**473539**, DOI 10.1017/S0308210500018072 - A. A. Samarskiĭ and V. B. Andreev,
*Raznostnye metody dlya èllipticheskikh uravneniĭ*, Izdat. “Nauka”, Moscow, 1976 (Russian). MR**0502017** - Walter A. Strauss,
*Dispersal of waves vanishing on the boundary of an exterior domain*, Comm. Pure Appl. Math.**28**(1975), 265–278. MR**367461**, DOI 10.1002/cpa.3160280205 - D. Tataru. Private Communication.
- 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**, DOI 10.1016/j.na.2006.06.007 - 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**, DOI 10.1137/S0363012999354211 - Mahamadi Warma,
*The Robin and Wentzell-Robin Laplacians on Lipschitz domains*, Semigroup Forum**73**(2006), no. 1, 10–30. MR**2277314**, DOI 10.1007/s00233-006-0617-2 - Mahamadi Warma,
*Analyticity on $L_1$ of the heat semigroup with Wentzell boundary conditions*, Arch. Math. (Basel)**94**(2010), no. 1, 85–89. MR**2581338**, DOI 10.1007/s00013-009-0068-6 - A. D. Ventcel′,
*On boundary conditions for multi-dimensional diffusion processes*, Theor. Probability Appl.**4**(1959), 164–177. MR**121855**, DOI 10.1137/1104014

## Additional Information

**Marcelo M. Cavalcanti**- Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil.
- Email: mmcavalcanti@uem.br
**Irena Lasiecka**- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904 – and – Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahram, 31261, Saudi Arabia
- MR Author ID: 110465
- Email: il2v@virginia.edu
**Daniel Toundykov**- Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
- Email: dtoundykov@math.unl.edu
- Received by editor(s): July 31, 2010
- Published electronically: June 12, 2012
- Additional Notes: The research of the first author was partially supported by the CNPq under Grant 300631/2003-0.

The research of the second author was partially supported by the National Science Foundation under Grant DMS-0606682 and by AFOSR Grant FA9550-09-1-0459

The research of the third author was partially supported by the National Science Foundation under Grant DMS-0908270 - © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2946927