Abstract
In this paper, we consider a damped wave equation with mixed boundary conditions in a bounded domain. On one portion of the boundary, we have kinetic boundary condition: \(\partial _\nu y+m(x)y_{tt}-\Delta _{T} y=0\) with density function m(x), and on the other portion, we have homogeneous Neumann boundary condition: \(\partial _\nu y=0\). Based on the growth of the resolvent operator on the imaginary axis, solutions of the wave equations under consideration are proved to decay logarithmically. The proof of the resolvent estimation relies on the interpolation inequalities for an elliptic equation with Steklov type boundary conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ammari, K., Hassine, F., Robbiano, L.: Stabilization for vibration plate with singular structural damping, arXiv:1905.13089
Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Optim. 30, 1024–165 (1992)
Batty, C.J.K., Duyckaerts, T.: Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8, 765–780 (2008)
Bellassoued, M.: Decay of solutions of the elastic wave equation with a localized dissipation. Ann. Fac. Sci. Toulouse Math. 12, 267–301 (2003)
Buffe, R.: Stabilization of the wave equation with Ventcel boundary condition. J. Math. Pures Appl. 108, 207–259 (2007)
Burq, N., Hitrik, M.: Energy decay for damped wave equations on partially rectangular domains. Math. Res. Lett. 14, 35–47 (2007)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Toundykov, D.: Stabilization of the damped wave equation with Cauchy Ventcel boundary conditions. J. Evol. Equ. 9, 143–169 (2009)
Cavalcanti, M.M., Lasiecka, I., Toundykov, D.: Wave equation with damping affecting only a subset of static Wentzell boundary is uniform stable. Trans. Am. Math. Soc. 364, 5693–5713 (2012)
Cornilleau, P., Robbiano, L.: Carleman estimates for the Zaremba boundary condition and stabilization of waves. Am. J. Math. 136, 393–444 (2014)
Fu, X.: Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62, 667–680 (2011)
Fu, X., Liu, X., Zhu, X.Z.: Logarithmic decay of wave equations with Cauchy–Ventcel boundary conditions. Control Theory Appl. 36, 1879–1885 (2019)
Fu, X., Lü, Q., Zhang, X.: Carleman estimates for second order partial differential operators an applications, a unified approach. Springer, New York (2019)
Fursikov, A.V., Imanuvilov, OYu.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Research Institute of Mathematics, Seoul National University, Seoul (1994)
Gal, C.G.: The role of surface diffusion in dynamic boundary conditions: where do we stand? Milan J. Math. 83, 237–278 (2015)
Gal, C.G., Goldstein, G.R., Goldstein, J.A.: Oscillatory boundary conditions for acoustic wave equations. J. Evol. Equ. 3, 623–635 (2003)
Gal, C.G., Shomberg, J.L.: Coleman–Gurtin type equations with dynamic boundary conditions. Physica D 292–293, 29–45 (2015)
Gal, C.G., Tebou, L.: Carleman inequalities for wave equations with oscillatory boundary conditions and application. SIAM J. Control Optim. 5, 324–364 (2016)
Goldstein, G.R.: Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11, 457–480 (2006)
Heminna, A.: Stabilisation frontière de problèmes de Ventcel. ESAIM Control Optim. Calc. Var. 5, 591–622 (2000)
Komornik, V., Zuazua, E.: A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69, 33–54 (1990)
Lebeau, G.: Equation des ondes amorties. Algebraic and Geometric Methods in Mathematical Physics, pp. 73–109. Kluwer Acad. Publ., Dordrecht (1993)
Lebeau, G., Robbiano, L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491 (1997)
Li, C., Xiao, T.J.: Polynomial stability for wave equations with Wentzell boundary conditions. J. Nonlinear Convex Anal. 18, 1801–1814 (2017)
Liu, Z., Rao, B.: Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56, 630–644 (2005)
Nédélec, J.C.: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Applied Mathematical Sciences, vol. 144. Springer, New York (2001)
Nicaise, S., Laoubi, K.: Polynomial stabilization of the wave equation with Ventcel’s boundary conditions. New York Math. Nachr. 283, 1428–1438 (2010)
Phung, K.-D.: Polynomial decay rate for the dissipative wave equation. J. Differ. Equ. 240, 92–124 (2007)
Phung, K.-D., Zhang, X.: Time reversal focusing of the initial state for Kirchoff plate. SIAM J. Appl. Math. 68, 1535–1556 (2008)
Zhang, Z., Guo, D.: Uniform stabilization of semilinear wave equations with localized internal damping and dynamical Wentzell boundary conditions with a memory term. Z. Angew. Math. Phys. 70, 160 (2019)
Zhang, X., Zuazua, E.: Long time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Rat. Mech. Anal. 184, 49–120 (2007)
Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping. Communn. Partial Differ. Equ. 15, 205–235 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partially supported by the NSF of China under Grants 11971333, 11931011.
Rights and permissions
About this article
Cite this article
Fu, X., Kong, L. Longtime Behavior of Wave Equation with Kinetic Boundary Condition. Appl Math Optim 84, 2803–2817 (2021). https://doi.org/10.1007/s00245-020-09730-y
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-020-09730-y