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Energy decay for the damped wave equation on an unbounded network

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

    * Corresponding author: Rachid Assel, rachid.assel@fsm.rnu.tn 
The authors are grateful to Professor Kais Ammari for his precious remarks, suggestions and support.
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  • 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.

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


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  • Figure 1.  The network and the prescribed conditions

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