American Institute of Mathematical Sciences

Advanced
September  2018, 7(3): 335-351. doi: 10.3934/eect.2018017

Energy decay for the damped wave equation on an unbounded network

Rachid Assel , and  Mohamed Ghazel

UR13ES64, Analysis and Control of PDEs, Faculty of Sciences of Monastir, University of Monastir, Monastir, 5019, Tunisia

* Corresponding author: Rachid Assel, rachid.assel@fsm.rnu.tn

Received  January 2018 Revised  April 2018 Published  July 2018

Fund Project: The authors are grateful to Professor Kais Ammari for his precious remarks, suggestions and support

We study the wave equation on an unbounded network of $N, N∈\mathbb{N}^*$, finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges. The dissipation is given by a damping constant $α>0$ via the condition $\sum_{j = 0}^N\partial_xu_j(0, t) = α \partial_tu_0(0, t)$. We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for $α\not = N+1$ we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.

Keywords: Damping, network of strings, energy decay, resolvent, semigroup expansion.
Mathematics Subject Classification: Primary: 35B37, 35B40, 35L05; Secondary: 35B35.
Citation: Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017
References:
[1]

F. Alabau-Boussouira, V. Perrolaz and L. Rosier, Finite-time stabilization of a network of strings, Mathematical Control and Related Fields, AIMS, 5 (2015), 721-742. doi: 10.3934/mcrf.2015.5.721. Google Scholar

[2]

L. Aloui and M. Khenissi, Stabilisation de l'équation des ondes dans un domaine extérieur, Rev. Math. Iberoamericana, 18 (2002), 1-16. doi: 10.4171/RMI/309. Google Scholar

[3]

K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Contol Syst., 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7. Google Scholar

[4]

K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2015), 327-343. doi: 10.1007/s10492-007-0018-1. Google Scholar

[5]

K. Ammari and M. Jellouli, Méthode numérique pour la déroissance de l'éergie d'un réseau de cordes, Bulletin of the Belgian Mathematical Society, 4 (2010), 717-735. Google Scholar

[6]

R. Assel, M. Jellouli and M. Khenissi, Optimal decay rate for the local energy of a unbounded network, J. Differential Equations, 261 (2016), 4030-4054. doi: 10.1016/j.jde.2016.06.016. Google Scholar

[7]

R. Assel, M. Khenissi and J. Royer, Energy Decay for a Unbounded Degenerate Network, work in progress.Google Scholar

[8]

R. Dager and H. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Mathématiques & Applications, 50 (2006), x+221 pp. doi: 10.1007/3-540-37726-3. Google Scholar

[9]

K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. Google Scholar

[10]

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2. Google Scholar

[11]

M. Jellouli, Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662. doi: 10.1080/00207179.2015.1012652. Google Scholar

[12]

M. Khenissi, Equation des ondes amorties dans un domaine extérieur, Bull. Soc. math. France, 131 (2003), 211-228. doi: 10.24033/bsmf.2440. Google Scholar

[13]

P. Kuchment, Quantum graphs: Ⅰ. Some basic structures, Ⅱ. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen, 14 (2004), S107-S128. doi: 10.1088/0959-7174/14/1/014. Google Scholar

[14]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Multi-Link Flexible Structures, Systems and Control: Foundations and Applications, Birkhauser, Basel, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[15]

N. E. Mastorakis and G. Q. Xu, Spectral distribution of star-shaped coupled network, WSEAS Transactions on Applied and Theoretical Mechanics, 4 (2008), 739-746. Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[17]

H. Reinhard, Equations Différentielles, Fondements et Applications, Dunod, Paris, 1982. Google Scholar

[18]

S.-H. Tang and M. Zworski, Resonance expansions of scattered waves, Comm. Pure. Appl. Math., 53 (2000), 1305-1334. Google Scholar

[19]

B. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989. Google Scholar

[20]

B. Vainberg, On the short wave asymptotic behavior of solutions of stationary problems and asymptotic behavior as t→+∞ of solutions of non-stationary problems, Uspehi Mat. Nauk, 30 (1975), 3-55. Google Scholar

[21]

G. Vodev, On the uniform decay of the local energy, Serdica Math. J., 25 (1999), 191-206. Google Scholar

[22]

Y. N. Xu and G. Q. Xu, Exponential stabilization of a tree- shaped network of strings with variable coefficients, Glasgow Math. J., 53 (2011), 481-499. doi: 10.1017/S0017089511000085. Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, V. Perrolaz and L. Rosier, Finite-time stabilization of a network of strings, Mathematical Control and Related Fields, AIMS, 5 (2015), 721-742. doi: 10.3934/mcrf.2015.5.721. Google Scholar

[2]

L. Aloui and M. Khenissi, Stabilisation de l'équation des ondes dans un domaine extérieur, Rev. Math. Iberoamericana, 18 (2002), 1-16. doi: 10.4171/RMI/309. Google Scholar

[3]

K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Contol Syst., 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7. Google Scholar

[4]

K. Ammari and M. Jellouli, Remark on stabilization of tree-shaped networks of strings, Applications of Mathematics, 52 (2015), 327-343. doi: 10.1007/s10492-007-0018-1. Google Scholar

[5]

K. Ammari and M. Jellouli, Méthode numérique pour la déroissance de l'éergie d'un réseau de cordes, Bulletin of the Belgian Mathematical Society, 4 (2010), 717-735. Google Scholar

[6]

R. Assel, M. Jellouli and M. Khenissi, Optimal decay rate for the local energy of a unbounded network, J. Differential Equations, 261 (2016), 4030-4054. doi: 10.1016/j.jde.2016.06.016. Google Scholar

[7]

R. Assel, M. Khenissi and J. Royer, Energy Decay for a Unbounded Degenerate Network, work in progress.Google Scholar

[8]

R. Dager and H. Zuazua, Wave propagation, observation and control in 1-d flexible multi-structures, Mathématiques & Applications, 50 (2006), x+221 pp. doi: 10.1007/3-540-37726-3. Google Scholar

[9]

K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. Google Scholar

[10]

P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0741-2. Google Scholar

[11]

M. Jellouli, Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662. doi: 10.1080/00207179.2015.1012652. Google Scholar

[12]

M. Khenissi, Equation des ondes amorties dans un domaine extérieur, Bull. Soc. math. France, 131 (2003), 211-228. doi: 10.24033/bsmf.2440. Google Scholar

[13]

P. Kuchment, Quantum graphs: Ⅰ. Some basic structures, Ⅱ. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen, 14 (2004), S107-S128. doi: 10.1088/0959-7174/14/1/014. Google Scholar

[14]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Multi-Link Flexible Structures, Systems and Control: Foundations and Applications, Birkhauser, Basel, 1994. doi: 10.1007/978-1-4612-0273-8. Google Scholar

[15]

N. E. Mastorakis and G. Q. Xu, Spectral distribution of star-shaped coupled network, WSEAS Transactions on Applied and Theoretical Mechanics, 4 (2008), 739-746. Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[17]

H. Reinhard, Equations Différentielles, Fondements et Applications, Dunod, Paris, 1982. Google Scholar

[18]

S.-H. Tang and M. Zworski, Resonance expansions of scattered waves, Comm. Pure. Appl. Math., 53 (2000), 1305-1334. Google Scholar

[19]

B. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989. Google Scholar

[20]

B. Vainberg, On the short wave asymptotic behavior of solutions of stationary problems and asymptotic behavior as t→+∞ of solutions of non-stationary problems, Uspehi Mat. Nauk, 30 (1975), 3-55. Google Scholar

[21]

G. Vodev, On the uniform decay of the local energy, Serdica Math. J., 25 (1999), 191-206. Google Scholar

[22]

Y. N. Xu and G. Q. Xu, Exponential stabilization of a tree- shaped network of strings with variable coefficients, Glasgow Math. J., 53 (2011), 481-499. doi: 10.1017/S0017089511000085. Google Scholar

Figure 1.  The network and the prescribed conditions
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[2]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[3]

Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040
[4]

Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control & Related Fields, 2015, 5 (4) : 721-742. doi: 10.3934/mcrf.2015.5.721
[5]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
[6]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43
[7]

Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407
[8]

Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic & Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605
[9]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021
[10]

Zhong-Jie Han, Gen-Qi Xu. Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks. Networks & Heterogeneous Media, 2010, 5 (2) : 315-334. doi: 10.3934/nhm.2010.5.315
[11]

Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic & Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79
[12]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273
[13]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851
[14]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721
[15]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[16]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425
[17]

Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713
[18]

Marcio A. Jorge Silva, Vando Narciso, André Vicente. On a beam model related to flight structures with nonlocal energy damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3281-3298. doi: 10.3934/dcdsb.2018320
[19]

Vincent Giovangigli, Wen-An Yong. Erratum: ``Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion''. Kinetic & Related Models, 2016, 9 (4) : 813-813. doi: 10.3934/krm.2016018
[20]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (151)
  • HTML views (283)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences