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

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

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