Energy decay for the damped wave equation on an unbounded network

Rachid Assel; Mohamed Ghazel

doi:10.3934/eect.2018017

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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: January 2018

Revised: April 2018

Published: July 2018

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

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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.

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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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