Energy decay for the damped wave equation on an unbounded network

Rachid Assel; Mohamed Ghazel

doi:10.3934/eect.2018017

Article Contents

2018, Volume 7, Issue 3: 335-351. Doi: 10.3934/eect.2018017

This issue Previous Article Next Article Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient

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

Received: December 31, 2017

Revised: March 31, 2018

Published: July 2018

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

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35B37, 35B40, 35L05; Secondary: 35B35.

Citation:

Full Text(HTML)

Figure 1. The network and the prescribed conditions

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[7]	R. Assel, M. Khenissi and J. Royer, Energy Decay for a Unbounded Degenerate Network, work in progress.
[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.
[9]	K. J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
[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.
[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.
[12]	M. Khenissi, Equation des ondes amorties dans un domaine extérieur, Bull. Soc. math. France, 131 (2003), 211-228. doi: 10.24033/bsmf.2440.
[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.
[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.
[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.
[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.
[17]	H. Reinhard, Equations Différentielles, Fondements et Applications, Dunod, Paris, 1982.
[18]	S.-H. Tang and M. Zworski, Resonance expansions of scattered waves, Comm. Pure. Appl. Math., 53 (2000), 1305-1334.
[19]	B. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989.
[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.
[21]	G. Vodev, On the uniform decay of the local energy, Serdica Math. J., 25 (1999), 191-206.
[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.