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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
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