American Institute of Mathematical Sciences

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2015, 2015(special): 678-685. doi: 10.3934/proc.2015.0678

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, France
2. 

Laboratoire de Mécanique et dAcoustique, UPR 7051 CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France

Received  September 2014 Revised  August 2015 Published  November 2015

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: linear elasticity, stability, nonlinear cracks, Neutral delay differential equations, energy method..
Mathematics Subject Classification: Primary: 34K40, 34K20; Secondary: 74J20, 74K1.
Citation: Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678
References:
[1]

R. Bellman and K. Cooke, Differential Difference Equations, Academic Press, 1963.

[2]

J. Cermák and J. Hrabalová, Delay-dependent stability criteria for neutral delay differential and difference equations, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 4577-4588.

[3]

H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcial. Ekvac., 34 (1991), 187-209.

[4]

P. S. Gromova, Stability of solutions of nonlinear equations in the asymptotically critical case, Mathematical Notes, 1-6 (1967), 472-479.

[5]

J. K. Hale, S. M. Verduyn Lunel, and M. Sjoerd, Introduction to functional-differential equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[6]

S. Junca, Oscillating Waves for Nonlinear Conservation Laws, Habilitation dissertation, Université de Nice, http://tel.archives-ouvertes.fr/tel-00845827, 2013.

[7]

S. Junca and B. Lombard, Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave, SIAM J. Appl. Math., 70 (2009), 735-761.

[8]

S. Junca and B. Lombard, Interaction between periodic elastic waves and two contact nonlinearities, Math. Models Methods Appl. Sci., 22 (2012), 41 pp.

[9]

S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation, J. Differential Equations, 256 (2014), 2368-2391.

[10]

B. Lombard, Modélisation numérique de la propagation et de la diffraction d'ondes mécaniques, [in French], Habilitation dissertation, Université de la Mediterranée, http://tel.archives-ouvertes.fr/docs/00/44/88/97/PDF/Hdr.pdf, 2010.

[11]

R. Rabah, G. M. Sklyar, and A. V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations, 214 (2005), 391-428.

[12]

X. F. Zhou, J. Liang, and T. J. Xiao, Asymptotic stability of the zero solution for degenerate retarded differential equations, Nonlinear Analysis, 70 (2009), 1415-1421.

show all references

References:
[1]

R. Bellman and K. Cooke, Differential Difference Equations, Academic Press, 1963.

[2]

J. Cermák and J. Hrabalová, Delay-dependent stability criteria for neutral delay differential and difference equations, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 4577-4588.

[3]

H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcial. Ekvac., 34 (1991), 187-209.

[4]

P. S. Gromova, Stability of solutions of nonlinear equations in the asymptotically critical case, Mathematical Notes, 1-6 (1967), 472-479.

[5]

J. K. Hale, S. M. Verduyn Lunel, and M. Sjoerd, Introduction to functional-differential equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[6]

S. Junca, Oscillating Waves for Nonlinear Conservation Laws, Habilitation dissertation, Université de Nice, http://tel.archives-ouvertes.fr/tel-00845827, 2013.

[7]

S. Junca and B. Lombard, Dilatation of a one-dimensional nonlinear crack impacted by a periodic elastic wave, SIAM J. Appl. Math., 70 (2009), 735-761.

[8]

S. Junca and B. Lombard, Interaction between periodic elastic waves and two contact nonlinearities, Math. Models Methods Appl. Sci., 22 (2012), 41 pp.

[9]

S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation, J. Differential Equations, 256 (2014), 2368-2391.

[10]

B. Lombard, Modélisation numérique de la propagation et de la diffraction d'ondes mécaniques, [in French], Habilitation dissertation, Université de la Mediterranée, http://tel.archives-ouvertes.fr/docs/00/44/88/97/PDF/Hdr.pdf, 2010.

[11]

R. Rabah, G. M. Sklyar, and A. V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations, 214 (2005), 391-428.

[12]

X. F. Zhou, J. Liang, and T. J. Xiao, Asymptotic stability of the zero solution for degenerate retarded differential equations, Nonlinear Analysis, 70 (2009), 1415-1421.

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