September  2019, 39(9): 5125-5147. doi: 10.3934/dcds.2019208

Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary

1. 

Grupo de Dinámica No Lineal, Universidad Pontificia Comillas de Madrid, C/Alberto Aguilera 23, 28015 Madrid, Spain

2. 

Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040, Madrid

3. 

ICMAT§, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Spain

* Corresponding author: A. Rodríguez–Bernal

The first author is supported by Projects MTM2016-75465, MICINN and GR58/08 Grupo 920894, UCM, Spain and FIS2016-78883-C2-2-P(AEI/FEDER, U.E.), Spain.
The second author is supported by Projects MTM2016-75465, MICINN and GR58/08 Grupo 920894, UCM, Spain and EPSRC grant EP/R023778/1, UK.
ICMAT§ is Partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO)

Received  June 2018 Revised  February 2019 Published  May 2019

In this paper we show how solutions of a wave equation with distributed damping near the boundary converge to solutions of a wave equation with boundary feedback damping. Sufficient conditions are given for the convergence of solutions to occur in the natural energy space.

Citation: Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5125-5147. doi: 10.3934/dcds.2019208
References:
[1]

G. S. Aragão and F. D. M. Bezerra, Upper semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary, J. Math. Anal. Appl., 462 (2018), 871-899.  doi: 10.1016/j.jmaa.2017.12.047.  Google Scholar

[2]

G. S. Aragão and S. M. Bruschi, Concentrated terms and varying domains in elliptic equations: Lipschitz case, Math. Methods Appl. Sci., 39 (2016), 3450-3460.  doi: 10.1002/mma.3791.  Google Scholar

[3]

G. S. AragãoA. L. Pereira and M. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differential Equations, 26 (2014), 871-888.  doi: 10.1007/s10884-014-9412-z.  Google Scholar

[4]

G. S. Aragão and S. M. Oliva, Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary, J. Differential Equations, 253 (2012), 2573-2592.  doi: 10.1016/j.jde.2012.07.008.  Google Scholar

[5]

G. S. AragãoA. L. Pereira and M. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Methods Appl. Sci., 35 (2012), 1110-1116.  doi: 10.1002/mma.2525.  Google Scholar

[6]

J. M. ArrietaA. Jiménez-Casas and A. Rodríguez-Bernal, Nonhomogeneous flux condition as lim of concentrated reactions, Revista Iberoamericana de Matematicas, 24 (2008), 183-211.   Google Scholar

[7]

J. M. ArrietaA. Rodríguez–Bernal and J. Rossi, The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary, Proceedings of The Royal Society of Edinburgh, 138A (2008), 223-237.  doi: 10.1017/S0308210506000813.  Google Scholar

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J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. American Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.  Google Scholar

[9]

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., 30 (1992), 1024 -1065. doi: 10.1137/0330055.  Google Scholar

[10]

G. A. ChechkinD. CioranescuA. Damlamian and A. L. Piatnitski, On boundary value problem with singular inhomogeneity concentrated on the boundary, J. Math. Pures Appl., 98 (2012), 115-138.  doi: 10.1016/j.matpur.2011.11.002.  Google Scholar

[11]

P. CornilleauJ. P. Loheác and A. Osses, Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers, J. of Dynamical and Control Systems, 16 (2010), 163-188.  doi: 10.1007/s10883-010-9088-6.  Google Scholar

[12]

Y. D. GolovatyD. GómezM. Lobo and E. Pérez, On vibrating membranes with very heavy thin inclusions, Math. Models Methods Appl. Sci., 14 (2004), 987-1034.  doi: 10.1142/S0218202504003520.  Google Scholar

[13]

D. GómezM. LoboS. A. Nazarov and E. Pérez, Spectral stiff problems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues, J. Math. Pures Appl., 85 (2006), 598-632.  doi: 10.1016/j.matpur.2005.10.013.  Google Scholar

[14]

Á. Jiménez-Casas and A. Rodríguez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. and Appl., 379 (2011), 567-588.  doi: 10.1016/j.jmaa.2011.01.051.  Google Scholar

[15]

A. Jiménez-Casas and A. Rodríguez-Bernal, Dynamic boundary conditions as a singular limit of parabolic problems with terms concentrating at the boundary, Dynamics of Partial Differential Equations, 9 (2012), 341-368.  doi: 10.4310/DPDE.2012.v9.n4.a3.  Google Scholar

[16]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54.   Google Scholar

[17]

J. Lagnese, Note on the boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.  doi: 10.1137/0326068.  Google Scholar

[18]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Appl. Math., vol. 10, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[19]

P. D. Lamberti, Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems, Complex Var. Elliptic Equ., 59 (2014), 309-323.  doi: 10.1080/17476933.2011.557155.  Google Scholar

[20] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Cambridge University Press, 2000.   Google Scholar
[21]

J. L. Lions, Quelques Méthodes de Rèsolution des Problèmes aux Limes non Lineaires, Dunod, 1969.  Google Scholar

[22]

J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, RMA 8, 1988.  Google Scholar

[23]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[24]

M. Nakao, Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation, J. of Differential Equations, 148 (1998), 388-406.  doi: 10.1006/jdeq.1998.3468.  Google Scholar

[25]

O. A. OleinikJ. Sanchez-Hubert and G. A. Yosifian, On the vibration of membranes with concentrated masses, Bull. Sci. Math., 15 (1991), 1-27.   Google Scholar

[26]

M. C. Pereira, Remarks on semilinear parabolic systems with terms concentrating in the boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1921-1930.  doi: 10.1016/j.nonrwa.2013.01.003.  Google Scholar

[27]

A. Rodríguez-Bernal, A singular perturbation in a linear parabolic equation with terms concentrating on the boundary, Revista Matemática Complutense, 25 (2012), 165-197.  doi: 10.1007/s13163-011-0064-9.  Google Scholar

[28]

A. Rodríguez-Bernal and E. Zuazua, Parabolic singular lim of a wave equation with localized boundary damping, Dis. Cont. Dyn. Sys., 1 (1995), 303-346.  doi: 10.3934/dcds.1995.1.303.  Google Scholar

[29]

A. Rodríguez-Bernal and E. Zuazua, Parabolic singular limit of a wave equation with localized interior damping, Comm. Contemp. Math., 3 (2001), 215-257.  doi: 10.1142/S0219199701000330.  Google Scholar

[30]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equa- tions. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.  Google Scholar

show all references

References:
[1]

G. S. Aragão and F. D. M. Bezerra, Upper semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary, J. Math. Anal. Appl., 462 (2018), 871-899.  doi: 10.1016/j.jmaa.2017.12.047.  Google Scholar

[2]

G. S. Aragão and S. M. Bruschi, Concentrated terms and varying domains in elliptic equations: Lipschitz case, Math. Methods Appl. Sci., 39 (2016), 3450-3460.  doi: 10.1002/mma.3791.  Google Scholar

[3]

G. S. AragãoA. L. Pereira and M. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differential Equations, 26 (2014), 871-888.  doi: 10.1007/s10884-014-9412-z.  Google Scholar

[4]

G. S. Aragão and S. M. Oliva, Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary, J. Differential Equations, 253 (2012), 2573-2592.  doi: 10.1016/j.jde.2012.07.008.  Google Scholar

[5]

G. S. AragãoA. L. Pereira and M. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Methods Appl. Sci., 35 (2012), 1110-1116.  doi: 10.1002/mma.2525.  Google Scholar

[6]

J. M. ArrietaA. Jiménez-Casas and A. Rodríguez-Bernal, Nonhomogeneous flux condition as lim of concentrated reactions, Revista Iberoamericana de Matematicas, 24 (2008), 183-211.   Google Scholar

[7]

J. M. ArrietaA. Rodríguez–Bernal and J. Rossi, The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary, Proceedings of The Royal Society of Edinburgh, 138A (2008), 223-237.  doi: 10.1017/S0308210506000813.  Google Scholar

[8]

J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. American Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.  Google Scholar

[9]

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., 30 (1992), 1024 -1065. doi: 10.1137/0330055.  Google Scholar

[10]

G. A. ChechkinD. CioranescuA. Damlamian and A. L. Piatnitski, On boundary value problem with singular inhomogeneity concentrated on the boundary, J. Math. Pures Appl., 98 (2012), 115-138.  doi: 10.1016/j.matpur.2011.11.002.  Google Scholar

[11]

P. CornilleauJ. P. Loheác and A. Osses, Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers, J. of Dynamical and Control Systems, 16 (2010), 163-188.  doi: 10.1007/s10883-010-9088-6.  Google Scholar

[12]

Y. D. GolovatyD. GómezM. Lobo and E. Pérez, On vibrating membranes with very heavy thin inclusions, Math. Models Methods Appl. Sci., 14 (2004), 987-1034.  doi: 10.1142/S0218202504003520.  Google Scholar

[13]

D. GómezM. LoboS. A. Nazarov and E. Pérez, Spectral stiff problems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues, J. Math. Pures Appl., 85 (2006), 598-632.  doi: 10.1016/j.matpur.2005.10.013.  Google Scholar

[14]

Á. Jiménez-Casas and A. Rodríguez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. and Appl., 379 (2011), 567-588.  doi: 10.1016/j.jmaa.2011.01.051.  Google Scholar

[15]

A. Jiménez-Casas and A. Rodríguez-Bernal, Dynamic boundary conditions as a singular limit of parabolic problems with terms concentrating at the boundary, Dynamics of Partial Differential Equations, 9 (2012), 341-368.  doi: 10.4310/DPDE.2012.v9.n4.a3.  Google Scholar

[16]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54.   Google Scholar

[17]

J. Lagnese, Note on the boundary stabilization of wave equations, SIAM J. Control Optim., 26 (1988), 1250-1256.  doi: 10.1137/0326068.  Google Scholar

[18]

J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Appl. Math., vol. 10, 1989. doi: 10.1137/1.9781611970821.  Google Scholar

[19]

P. D. Lamberti, Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems, Complex Var. Elliptic Equ., 59 (2014), 309-323.  doi: 10.1080/17476933.2011.557155.  Google Scholar

[20] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Cambridge University Press, 2000.   Google Scholar
[21]

J. L. Lions, Quelques Méthodes de Rèsolution des Problèmes aux Limes non Lineaires, Dunod, 1969.  Google Scholar

[22]

J. L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Masson, Paris, RMA 8, 1988.  Google Scholar

[23]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[24]

M. Nakao, Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation, J. of Differential Equations, 148 (1998), 388-406.  doi: 10.1006/jdeq.1998.3468.  Google Scholar

[25]

O. A. OleinikJ. Sanchez-Hubert and G. A. Yosifian, On the vibration of membranes with concentrated masses, Bull. Sci. Math., 15 (1991), 1-27.   Google Scholar

[26]

M. C. Pereira, Remarks on semilinear parabolic systems with terms concentrating in the boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1921-1930.  doi: 10.1016/j.nonrwa.2013.01.003.  Google Scholar

[27]

A. Rodríguez-Bernal, A singular perturbation in a linear parabolic equation with terms concentrating on the boundary, Revista Matemática Complutense, 25 (2012), 165-197.  doi: 10.1007/s13163-011-0064-9.  Google Scholar

[28]

A. Rodríguez-Bernal and E. Zuazua, Parabolic singular lim of a wave equation with localized boundary damping, Dis. Cont. Dyn. Sys., 1 (1995), 303-346.  doi: 10.3934/dcds.1995.1.303.  Google Scholar

[29]

A. Rodríguez-Bernal and E. Zuazua, Parabolic singular limit of a wave equation with localized interior damping, Comm. Contemp. Math., 3 (2001), 215-257.  doi: 10.1142/S0219199701000330.  Google Scholar

[30]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equa- tions. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[31]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.  Google Scholar

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