# American Institute of Mathematical Sciences

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

## Energy decay for the damped wave equation on an unbounded network

 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.

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
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The network and the prescribed conditions
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