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Keywords:
Kirchhoff equation; dissipation; vibration; stabilization; energy decay estimate
Kirchhoff equation; dissipation; vibration; stabilization; energy decay estimate
Summary:
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.
References:
[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. MR 1865394 | Zbl 1145.35432
[2] Autuori, G., Pucci, P.: Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces. Complex Var. Elliptic Equ. 56 (2011), 715-753. MR 2832211 | Zbl 1230.35018
[3] Autuori, G., Pucci, P.: Local asymptotic stability for polyharmonic Kirchhoff systems. Appl. Anal. 90 (2011), 493-514. DOI 10.1080/00036811.2010.483433 | MR 2780908 | Zbl 1223.35051
[4] Autuori, G., Pucci, P., Salvatori, M. C.: Asymptotic stability for anisotropic Kirchhoff systems. J. Math. Anal. Appl. 352 (2009), 149-165. DOI 10.1016/j.jmaa.2008.04.066 | MR 2499894 | Zbl 1175.35013
[5] Autuori, G., Pucci, P., Salvatori, M. C.: Asymptotic stability for nonlinear Kirchhoff systems. Nonlinear Anal., Real World Appl. 10 (2009), 889-909. DOI 10.1016/j.nonrwa.2007.11.011 | MR 2474268 | Zbl 1167.35314
[6] D'Ancona, P., Spagnolo, S.: Nonlinear perturbations of the Kirchhoff equation. Commun. Pure Appl. Math. 47 (1994), 1005-1029. DOI 10.1002/cpa.3160470705 | MR 1283880 | Zbl 0807.35093
[7] Gorain, G. C.: Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in $R^n$. J. Math. Anal. Appl. 319 (2006), 635-650. DOI 10.1016/j.jmaa.2005.06.031 | MR 2227928
[8] Gorain, G. C.: Exponential energy decay estimates for the solutions of $n$-dimensional Kirchhoff type wave equation. Appl. Math. Comput. 177 (2006), 235-242. DOI 10.1016/j.amc.2005.11.003 | MR 2234515 | Zbl 1098.74024
[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. MR 1054123 | Zbl 0636.93064
[10] Lasiecka, I., Ong, J.: Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation. Commun. Partial Differ. Equations 24 (1999), 2069-2107. DOI 10.1080/03605309908821495 | MR 1720766 | Zbl 0936.35031
[11] Menzala, G. P.: On classical solutions of a quasilinear hyperbolic equation. Nonlinear Anal., Theory Methods Appl. 3 (1979), 613-627. DOI 10.1016/0362-546X(79)90090-7 | MR 0541872 | Zbl 0419.35062
[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). MR 1190927 | Zbl 0744.26011
[13] Nandi, P. K., Gorain, G. C., Kar, S.: Uniform exponential stabilization for flexural vibrations of a solar panel. Appl. Math. (Irvine) 2 (2011), 661-665. DOI 10.4236/am.2011.26087 | MR 2910175
[14] Narasimha, R.: Non-linear vibration of an elastic string. J. Sound Vib. 8 (1968), 134-146. DOI 10.1016/0022-460X(68)90200-9 | Zbl 0164.26701
[15] Nayfeh, A. H., Mook, D. T.: Nonlinear Oscillations. Pure and Applied Mathematics. A Wiley-Interscience Publication John Wiley & Sons, New York (1979). MR 0549322 | Zbl 0418.70001
[16] Newman, W. G.: Global solution of a nonlinear string equation. J. Math. Anal. Appl. 192 (1995), 689-704. DOI 10.1006/jmaa.1995.1198 | MR 1336472 | Zbl 0837.35095
[17] Nishihara, K.: On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 7 (1984), 437-459. DOI 10.3836/tjm/1270151737 | MR 0776949 | Zbl 0586.35059
[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. MR 1065473 | Zbl 0715.35053
[19] Ono, K.: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differ. Equations 137 (1997), 273-301. DOI 10.1006/jdeq.1997.3263 | MR 1456598 | Zbl 0879.35110
[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. MR 1361000 | Zbl 0842.45005
[21] Shahruz, S. M.: Bounded-input bounded-output stability of a damped nonlinear string. IEEE Trans. Autom. Control 41 (1996), 1179-1182. DOI 10.1109/9.533679 | MR 1407204 | Zbl 0863.93076
[22] Yamazaki, T.: Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three. Math. Methods Appl. Sci. 27 (2004), 1893-1916. DOI 10.1002/mma.530 | MR 2092828 | Zbl 1072.35559
[23] Yamazaki, T.: Global solvability for the Kirchhoff equations in exterior domains of dimension three. J. Differ. Equations 210 (2005), 290-316. DOI 10.1016/j.jde.2004.10.012 | MR 2119986 | Zbl 1062.35045
[24] Ye, Y.: On the exponential decay of solutions for some Kirchhoff-type modelling equations with strong dissipation. Applied Mathematics 1 (2010), 529-533. DOI 10.4236/am.2010.16070
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