Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Balakrishnan-Taylor damping; polynomial decay; memory term; viscoelasticity
Balakrishnan-Taylor damping; polynomial decay; memory term; viscoelasticity
Summary:
A viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping is considered. Using integral inequalities and multiplier techniques we establish polynomial decay estimates for the energy of the problem. The results obtained in this paper extend previous results by Tatar and Zaraï [25].
References:
[1] Alabau-Boussouira, F., Cannarsa, P., Sforza, D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254 (5) (2008), 1342–1372. DOI 10.1016/j.jfa.2007.09.012 | MR 2386941 | Zbl 1145.35025
[2] Appleby, J. A. D., Fabrizio, M., Lazzari, B., Reynolds, D. W.: On exponential asymptotic stability in linear viscoelasticity. Math. Models Methods Appl. Sci. 16 (2006), 1677–1694. DOI 10.1142/S0218202506001674 | MR 2264556 | Zbl 1114.45003
[3] Balakrishnan, A. V., Taylor, L. W.: Distributed parameter nonlinear damping models for flight structures. Damping 89', Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989.
[4] Ball, J.: Remarks on blow up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. Oxford 28 (1977), 473–486. DOI 10.1093/qmath/28.4.473 | MR 0473484 | Zbl 0377.35037
[5] Bass, R. W., Zes, D.: Spillover, nonlinearity and flexible structures. The Fourth NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conference Publication 10065 (L.W.Taylor, ed.), 1991, pp. 1–14.
[6] Berrimi, S., Messaoudi, S.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64 (2006), 2314–2331. MR 2213903 | Zbl 1094.35070
[7] Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Filho, J. S. Prates, Soriano, J. A.: Existence and uniform decay rates for viscoelastic problems with nonlocal boundary damping. Differential Integral Equations 14 (1) (2001), 85–116. MR 1797934
[8] Cavalcanti, M. M., Oquendo, H. P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. (electronic) 42 (4) (2003), 1310–1324. DOI 10.1137/S0363012902408010 | MR 2044797 | Zbl 1053.35101
[9] Clark, H. R.: Elastic membrane equation in bounded and unbounded domains. Electron. J. Qual. Theory Differ. Equ. 11 (2002), 1–21. MR 1918508 | Zbl 1032.35039
[10] Glassey, R. T.: Blow-up theorems for nonlinear wave equations. Math. Z. 132 (1973), 183–203. DOI 10.1007/BF01213863 | MR 0340799 | Zbl 0247.35083
[11] Haraux, A., Zuazua, E.: Decay estimates for some semilinear damped hyperbolic problems. Arch. Rational Mech. Anal. 100 (1988), 191–206. DOI 10.1007/BF00282203 | MR 0913963 | Zbl 0654.35070
[12] Kalantarov, V. K., Ladyzhenskaya, O. A.: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type. J. Soviet Math. 10 (1978), 53–70. DOI 10.1007/BF01109723
[13] Kirchhoff, G.: Vorlesungen über Mechanik. Tauber, Leipzig, 1883.
[14] Komornik, V.: Exact controllability and stabilization. The multiplier method. RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. MR 1359765 | Zbl 0937.93003
[15] Kopackova, M.: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation. Comment. Math. Univ. Carolin. 30 (1989), 713–719. MR 1045899 | Zbl 0707.35026
[16] Li, M. R., Tsai, L. Y.: Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Anal. 54 (2003), 1397–1415. MR 1997226
[17] Matsuyama, T., Ikehata, R.: On global solutions and energy decay for the wave equation of Kirchhoff type with nonlinear damping terms. J. Math. Anal. Appl. 204 (3) (1996), 729–753. DOI 10.1006/jmaa.1996.0464 | MR 1422769
[18] Medjden, M., e. Tatar, N.: On the wave equation with a temporal nonlocal term. Dynam. Systems Appl. 16 (2007), 665–672. MR 2370149
[19] Nakao, M.: Decay of solutions of some nonlinear evolution equation. J. Math. Anal. Appl. 60 (1977), 542–549. DOI 10.1016/0022-247X(77)90040-3 | MR 0499564
[20] Nishihara, K., Yamada, Y.: On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkcial. Ekvac. 33 (1990), 151–159. MR 1065473 | Zbl 0715.35053
[21] Ono, K.: Global existence, decay and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differential Equations 137 (2) (1997), 273–301. DOI 10.1006/jdeq.1997.3263 | MR 1456598 | Zbl 0879.35110
[22] Ono, K.: On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation. Math. Methods Appl. Sci. 20 (1997), 151–177. DOI 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0 | MR 1430038 | Zbl 0878.35081
[23] Ono, K.: On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation. J. Math. Anal. Appl. 216 (1997), 321–342. DOI 10.1006/jmaa.1997.5697 | MR 1487267 | Zbl 0893.35078
[24] Pata, V.: Exponential stability in linear viscoelasticity. Quart. Appl. Math. 64 (3) (2006), 499–513. MR 2259051 | Zbl 1117.35052
[25] Tatar, N-e. and Zaraï, A., : Exponential stability and blow up for a problem with Balakrishnan-Taylor damping. to appear in Demonstratio Math.
[26] Tatar, N-e. and Zaraï, A., : On a Kirchhoff equation with Balakrishnan-Taylor damping and source term. to apper in DCDIS.
[27] You, Y.: Inertial manifolds and stabilization of nonlinear beam equations with Balakrishnan-Taylor damping. Abstr. Appl. Anal. 1 (1) (1996), 83–102. DOI 10.1155/S1085337596000048 | MR 1390561 | Zbl 0940.35036
Search
Partner of
