Stability of neutral delay differential equations modeling wave propagation in cracked media

Stéphane  Junca; Bruno  Lombard

doi:10.3934/proc.2015.0678

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2015, Volume 2015: 678-685. Doi: 10.3934/proc.2015.0678 open access

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Stability of neutral delay differential equations modeling wave propagation in cracked media

Stéphane Junca^1, and
Bruno Lombard^2,

1.
Laboratoire J.A. Dieudonné, UMR 7351 CNRS, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02
2.
Laboratoire de Mécanique et dAcoustique, UPR 7051 CNRS, 31 chemin Joseph Aiguier, 13402 Marseille

Received: September 2014

Revised: August 2015

Published: November 2015

Abstract / Introduction Related Papers Cited by

Abstract

Propagation of elastic waves is studied in a 1D medium containing $N$ cracks modeled by nonlinear jump conditions. The case $N=1$ is fully understood. When $N>1$, the evolution equations are written as a system of nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. In the case $N=2$, some mathematical results about the existence, uniqueness and attractivity of periodic solutions have been obtained in 2012 by the authors, under the assumption of small sources. The difficulty of analysis follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. Here we propose a new result of stability in the homogeneous case, based on an energy method. One deduces the asymptotic stability of the zero steady-state. Extension to $N=3$ cracks is also considered, leading to new results in particular configurations.

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Mathematics Subject Classification: Primary: 34K40, 34K20; Secondary: 74J20, 74K10.

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References

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