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

Pages: 678 - 685, Volume 2015, Issue special, November 2015      doi:10.3934/proc.2015.0678

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Stéphane Junca - Laboratoire J.A. Dieudonné, UMR 7351 CNRS, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France (email)
Bruno Lombard - Laboratoire de Mécanique et dAcoustique, UPR 7051 CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France (email)

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.

Keywords:  Neutral delay differential equations, linear elasticity, nonlinear cracks, stability, energy method.
Mathematics Subject Classification:  Primary: 34K40, 34K20; Secondary: 74J20, 74K10.

Received: September 2014;      Revised: August 2015;      Available Online: November 2015.