Stability of vibrations for some Kirchhoff equation with dissipation

Prasanta Kumar Nandi; Ganesh Chandra Gorain; Samarjit Kar

Applications of Mathematics (2014)

  • Volume: 59, Issue: 2, page 205-215
  • ISSN: 0862-7940

Abstract

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In this paper we consider the boundary value problem of some nonlinear Kirchhoff-type equation with dissipation. We also estimate the total energy of the system over any time interval with a tolerance level . The amplitude of such vibrations is bounded subject to some restrictions on the uncertain disturbing force . After constructing suitable Lyapunov functional, uniform decay of solutions is established by means of an exponential energy decay estimate when the uncertain disturbances are insignificant.

How to cite

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Nandi, Prasanta Kumar, Gorain, Ganesh Chandra, and Kar, Samarjit. "Stability of vibrations for some Kirchhoff equation with dissipation." Applications of Mathematics 59.2 (2014): 205-215. <http://eudml.org/doc/261075>.

@article{Nandi2014,
abstract = {In this paper we consider the boundary value problem of some nonlinear Kirchhoff-type equation with dissipation. We also estimate the total energy of the system over any time interval $[0,T]$ with a tolerance level $\gamma $. The amplitude of such vibrations is bounded subject to some restrictions on the uncertain disturbing force $f$. After constructing suitable Lyapunov functional, uniform decay of solutions is established by means of an exponential energy decay estimate when the uncertain disturbances are insignificant.},
author = {Nandi, Prasanta Kumar, Gorain, Ganesh Chandra, Kar, Samarjit},
journal = {Applications of Mathematics},
keywords = {Kirchhoff equation; dissipation; vibration; stabilization; energy decay estimate; Kirchhoff equation; dissipation; vibration; stabilization; energy decay estimate},
language = {eng},
number = {2},
pages = {205-215},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stability of vibrations for some Kirchhoff equation with dissipation},
url = {http://eudml.org/doc/261075},
volume = {59},
year = {2014},
}

TY - JOUR
AU - Nandi, Prasanta Kumar
AU - Gorain, Ganesh Chandra
AU - Kar, Samarjit
TI - Stability of vibrations for some Kirchhoff equation with dissipation
JO - Applications of Mathematics
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 2
SP - 205
EP - 215
AB - In this paper we consider the boundary value problem of some nonlinear Kirchhoff-type equation with dissipation. We also estimate the total energy of the system over any time interval $[0,T]$ with a tolerance level $\gamma $. The amplitude of such vibrations is bounded subject to some restrictions on the uncertain disturbing force $f$. After constructing suitable Lyapunov functional, uniform decay of solutions is established by means of an exponential energy decay estimate when the uncertain disturbances are insignificant.
LA - eng
KW - Kirchhoff equation; dissipation; vibration; stabilization; energy decay estimate; Kirchhoff equation; dissipation; vibration; stabilization; energy decay estimate
UR - http://eudml.org/doc/261075
ER -

References

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  1. Aassila, M., Benaissa, A., Global existence and asymptotic behavior of solutions of mildly degenerate Kirchhoff equations with nonlinear dissipative term, Funkc. Ekvacioj, Ser. Int. 44 (2001), 309-333 French. (2001) Zbl1145.35432MR1865394
  2. Autuori, G., Pucci, P., Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces, Complex Var. Elliptic Equ. 56 (2011), 715-753. (2011) Zbl1230.35018MR2832211
  3. Autuori, G., Pucci, P., 10.1080/00036811.2010.483433, Appl. Anal. 90 (2011), 493-514. (2011) Zbl1223.35051MR2780908DOI10.1080/00036811.2010.483433
  4. Autuori, G., Pucci, P., Salvatori, M. C., 10.1016/j.jmaa.2008.04.066, J. Math. Anal. Appl. 352 (2009), 149-165. (2009) Zbl1175.35013MR2499894DOI10.1016/j.jmaa.2008.04.066
  5. Autuori, G., Pucci, P., Salvatori, M. C., 10.1016/j.nonrwa.2007.11.011, Nonlinear Anal., Real World Appl. 10 (2009), 889-909. (2009) Zbl1167.35314MR2474268DOI10.1016/j.nonrwa.2007.11.011
  6. D'Ancona, P., Spagnolo, S., 10.1002/cpa.3160470705, Commun. Pure Appl. Math. 47 (1994), 1005-1029. (1994) Zbl0807.35093MR1283880DOI10.1002/cpa.3160470705
  7. Gorain, G. C., 10.1016/j.jmaa.2005.06.031, J. Math. Anal. Appl. 319 (2006), 635-650. (2006) MR2227928DOI10.1016/j.jmaa.2005.06.031
  8. Gorain, G. C., 10.1016/j.amc.2005.11.003, Appl. Math. Comput. 177 (2006), 235-242. (2006) Zbl1098.74024MR2234515DOI10.1016/j.amc.2005.11.003
  9. Komornik, V., Zuazua, E., A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., IX. Sér. 69 (1990), 33-54. (1990) Zbl0636.93064MR1054123
  10. Lasiecka, I., Ong, J., 10.1080/03605309908821495, Commun. Partial Differ. Equations 24 (1999), 2069-2107. (1999) Zbl0936.35031MR1720766DOI10.1080/03605309908821495
  11. Menzala, G. P., 10.1016/0362-546X(79)90090-7, Nonlinear Anal., Theory Methods Appl. 3 (1979), 613-627. (1979) Zbl0419.35062MR0541872DOI10.1016/0362-546X(79)90090-7
  12. Mitrinović, D. S., Pečarić, J. E., Fink, A. M., Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications: East European Series 53 Kluwer Academic Publishers, Dordrecht (1991). (1991) Zbl0744.26011MR1190927
  13. Nandi, P. K., Gorain, G. C., Kar, S., 10.4236/am.2011.26087, Appl. Math. (Irvine) 2 (2011), 661-665. (2011) MR2910175DOI10.4236/am.2011.26087
  14. Narasimha, R., 10.1016/0022-460X(68)90200-9, J. Sound Vib. 8 (1968), 134-146. (1968) Zbl0164.26701DOI10.1016/0022-460X(68)90200-9
  15. Nayfeh, A. H., Mook, D. T., Nonlinear Oscillations, Pure and Applied Mathematics. A Wiley-Interscience Publication John Wiley & Sons, New York (1979). (1979) Zbl0418.70001MR0549322
  16. Newman, W. G., 10.1006/jmaa.1995.1198, J. Math. Anal. Appl. 192 (1995), 689-704. (1995) Zbl0837.35095MR1336472DOI10.1006/jmaa.1995.1198
  17. Nishihara, K., 10.3836/tjm/1270151737, Tokyo J. Math. 7 (1984), 437-459. (1984) Zbl0586.35059MR0776949DOI10.3836/tjm/1270151737
  18. Nishihara, K., Yamada, Y., On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkc. Ekvacioj, Ser. Int. 33 (1990), 151-159. (1990) Zbl0715.35053MR1065473
  19. Ono, K., 10.1006/jdeq.1997.3263, J. Differ. Equations 137 (1997), 273-301. (1997) Zbl0879.35110MR1456598DOI10.1006/jdeq.1997.3263
  20. Ono, K., Nishihara, K., On a nonlinear degenerate integro-differential equation of hyperbolic type with a strong dissipation, Adv. Math. Sci. Appl. 5 (1995), 457-476. (1995) Zbl0842.45005MR1361000
  21. Shahruz, S. M., 10.1109/9.533679, IEEE Trans. Autom. Control 41 (1996), 1179-1182. (1996) Zbl0863.93076MR1407204DOI10.1109/9.533679
  22. Yamazaki, T., 10.1002/mma.530, Math. Methods Appl. Sci. 27 (2004), 1893-1916. (2004) Zbl1072.35559MR2092828DOI10.1002/mma.530
  23. Yamazaki, T., 10.1016/j.jde.2004.10.012, J. Differ. Equations 210 (2005), 290-316. (2005) Zbl1062.35045MR2119986DOI10.1016/j.jde.2004.10.012
  24. Ye, Y., 10.4236/am.2010.16070, Applied Mathematics 1 (2010), 529-533. (2010) DOI10.4236/am.2010.16070

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