Previous |  Up |  Next

Article

Keywords:
system of nonlinear wave equations of Kirchhoff-Carrier type; Balakrishnan-Taylor term; Faedo-Galerkin method; local existence; exponential decay
Summary:
We study a system of nonlinear wave equations of the Kirchhoff-Carrier type containing a variant of the Balakrishnan-Taylor damping in nonlinear terms. By the linearization method together with the Faedo-Galerkin method, we prove the local existence and uniqueness of a weak solution. On the other hand, by constructing a suitable Lyapunov functional, a sufficient condition is also established to obtain the exponential decay of weak solutions.
References:
[1] Balakrishnan, A. V., Taylor, L. W.: Distributed parameter nonlinear damping models for flight structures. Proceedings Damping 89. Report Number: WRDC-TR-89-3116 Volume II p. FDC-1 Flight Dynamics Laboratory, Chicago (1989), 9 pages.
[2] Bass, R. W., Zes, D.: Spillover, nonlinearity and flexible structures. 4th NASA Workshop on Computational Control of Flexible Aerospace Systems NASA Conference Publication 10065. NASA. Langley Research Center, Hampton (1991), 1-14.
[3] Boulaaras, S.: Polynomial decay rate for a new class of viscoelastic Kirchhoff equation related with Balakrishnan-Taylor dissipationand logarithmic source terms. Alexandria Eng. J. 59 (2020), 1059-1071. DOI 10.1016/j.aej.2019.12.013
[4] Boulaaras, S., Draifia, A., Zennir, K.: General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and logarithmic nonlinearity. Math. Methods Appl. Sci. 42 (2019), 4795-4814. DOI 10.1002/mma.5693 | MR 3992940 | Zbl 1428.35037
[5] Boulaaras, S., Ouchenane, D.: General decay for a coupled Lamé system of nonlinear viscoelastic equations. Math. Methods Appl. Sci. 43 (2020), 1717-1735. DOI 10.1002/mma.5998 | MR 4067018 | Zbl 1445.35054
[6] Boumaza, N., Boulaaras, S.: General decay for Kirchhoff type in viscoelasticity with not necessarily decreasing kernel. Math. Methods Appl. Sci. 41 (2018), 6050-6069. DOI 10.1002/mma.5117 | MR 3879228 | Zbl 1415.35038
[7] Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Filho, J. S. Prates, Soriano, J. A.: Existence and exponential decay for a Kirchhoff-Carrier model with viscosity. J. Math. Anal. Appl. 226 (1998), 40-60. DOI 10.1006/jmaa.1998.6057 | MR 1646453 | Zbl 0914.35081
[8] Emmrich, E., Thalhammer, M.: A class of integro-differential equations incorporating nonlinear and nonlocal damping with applications in nonlinear elastodynamics: Existence via time discretization. Nonlinearity 24 (2011), 2523-2546. DOI 10.1088/0951-7715/24/9/008 | MR 2819935 | Zbl 1222.74021
[9] Feng, B., Kang, Y. H.: Decay rates for a viscoelastic wave equation with Balakrishnan-Taylor and frictional dampings. Topol. Methods Nonlinear Anal. 54 (2019), 321-343. DOI 10.12775/tmna.2019.047 | MR 4018283 | Zbl 1437.35071
[10] Freitas, M. M.: Pullback attractors for non-autonomous porous elastic system with nonlinear damping and sources terms. Math. Methods Appl. Sci. 43 (2020), 658-681. DOI 10.1002/mma.5921 | MR 4056455 | Zbl 1445.35078
[11] Freitas, M. M., Santos, M. L., Langa, J. A.: Porous elastic system with nonlinear damping and sources terms. J. Differ. Equations 264 (2018), 2970-3051. DOI 10.1016/j.jde.2017.11.006 | MR 3737860 | Zbl 1394.35043
[12] Tavares, E. H. Gomes, Silva, M. A. Jorge, Narciso, V.: Long-time dynamics of Balakrishnan-Taylor extensible beams. J. Dyn. Differ. Equations 32 (2020), 1157-1175. DOI 10.1007/s10884-019-09766-x | MR 4126844 | Zbl 1445.35060
[13] Ha, T. G.: General decay rate estimates for viscoelastic wave equation with Balakrishnan-Taylor damping. Z. Angew. Math. Phys. 67 (2016), Article ID 32, 17 pages. DOI 10.1007/s00033-016-0625-3 | MR 3483881 | Zbl 1353.35064
[14] Ha, T. G.: Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping. Taiwanese J. Math. 21 (2017), 807-817. DOI 10.11650/tjm/7828 | MR 3684388 | Zbl 1394.35044
[15] Ha, T. G.: On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions. Evol. Equ. Control Theory 7 (2018), 281-291. DOI 10.3934/eect.2018014 | MR 3810197 | Zbl 1415.35042
[16] Hao, J., Hou, Y.: Stabilization for wave equation of variable coefficients with Balakrishnan-Taylor damping and source term. Comput. Math. Appl. 76 (2018), 2235-2245. DOI 10.1016/j.camwa.2018.08.023 | MR 3864576 | Zbl 1442.35267
[17] Hao, J., Wang, F.: General decay rate for weak viscoelastic wave equation with Balakrishnan-Taylor damping and time-varying delay. Comput. Math. Appl. 78 (2019), 2632-2640. DOI 10.1016/j.camwa.2019.04.010 | MR 4001729 | Zbl 1443.35095
[18] Shah, S. Hyder Ali Muttaqi: Some helical flows of a Burgers fluid with fractional derivative. Meccanica 45 (2010), 143-151. DOI 10.1007/s11012-009-9233-z | MR 2608341 | Zbl 1258.76026
[19] Jamil, M., Fetecau, C.: Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary. Nonlinear Anal., Real World Appl. 11 (2010), 4302-4311. DOI 10.1016/j.nonrwa.2010.05.016 | MR 2683877 | Zbl 1201.35159
[20] Kang, J.-R., Lee, M. J., Park, S. H.: Asymptotic stability for a viscoelastic problem with Balakrishnan-Taylor damping and time-varying delay. Comput. Math. Appl. 74 (2017), 1506-1515. DOI 10.1016/j.camwa.2017.06.033 | MR 3693349 | Zbl 1394.35280
[21] Lee, M. J., Kim, D., Park, J. Y.: General decay of solutions for Kirchhoff type containing Balakrishnan-Taylor damping with a delay and acoustic boundary conditions. Bound. Value Probl. 2016 (2016), Article ID 173, 21 pages. DOI 10.1186/s13661-016-0679-3 | MR 3550421 | Zbl 1350.35129
[22] Lee, M. J., Park, J. Y., Kang, Y. H.: Asymptotic stability of a problem with Balakrishnan-Taylor damping and a time delay. Comput. Math. Appl. 70 (2015), 478-487. DOI 10.1016/j.camwa.2015.05.004 | MR 3372039 | Zbl 1443.35098
[23] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Etudes mathematiques. Dunod, Gauthier-Villars, Paris (1969), French. MR 0259693 | Zbl 0189.40603
[24] Long, N. T., Ha, H. H., Ngoc, L. T. P., Triet, N. A.: Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Commun. Pure Appl. Anal. 19 (2020), 455-492. DOI 10.3934/cpaa.2020023 | MR 4025953 | Zbl 1437.35468
[25] Medeiros, L. A.: On some nonlinear perturbation of Kirchhoff-Carrier operator. Comput. Appl. Math. 13 (1994), 225-233. MR 1326759 | Zbl 0821.35100
[26] Mu, C., Ma, J.: On a system of nonlinear wave equations with Balakrishnan-Taylor damping. Z. Angew. Math. Phys. 65 (2014), 91-113. DOI 10.1007/s00033-013-0324-2 | MR 3160626 | Zbl 1295.35309
[27] Ngoc, L. T. P., Nhan, N. H., Nam, B. D., Long, N. T.: Existence and exponential decay of the Dirichlet problem for a nonlinear wave equation with the Balakrishnan-Taylor term. Lith. Math. J. 60 (2020), 225-247. DOI 10.1007/s10986-020-09469-7 | MR 4110669 | Zbl 1442.35243
[28] Qi, H., Jin, H.: Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative. Nonlinear Anal., Real World Appl. 10 (2009), 2700-2708. DOI 10.1016/j.nonrwa.2008.07.008 | MR 2523233 | Zbl 1162.76006
[29] Santos, M. L., Júnior, D. S. Almeida: On porous-elastic system with localized damping. Z. Angew. Math. Phys. 67 (2016), Article ID 63, 18 pages. DOI 10.1007/s00033-016-0622-6 | MR 3494484 | Zbl 1351.35217
[30] Showalter, R. E.: Hilbert space methods for partial differential equations. Electronic Journal of Differential Equations. Monograph 1. Southwest Texas State University, San Marcos (1994). MR 1302484 | Zbl 0991.35001
[31] Tatar, N.-e., Zaraï, A.: Exponential stability and blow up for a problem with Balakrishnan-Taylor damping. Demonstr. Math. 44 (2011), 67-90. DOI 10.1515/dema-2013-0297 | MR 2796763 | Zbl 1227.35074
[32] Tatar, N.-e., Zaraï, A.: On a Kirchhoff equation with Balakrishnan-Taylor damping and source term. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 18 (2011), 615-627. MR 2884753 | Zbl 1264.35244
[33] Tong, D., Zhang, X., Zhang, X.: Unsteady helical flows of a generalized Oldroyd-B fluid. J. Non-Newton. Fluid Mech. 156 (2009), 75-83. DOI 10.1016/j.nonrwa.2008.07.008 | MR 2523233 | Zbl 1274.76136
[34] Triet, N. A., Ngoc, L. T. P., Long, N. T.: On a nonlinear Kirchhoff-Carrier wave equation associated with Robin conditions. Nonlinear Anal., Real World Appl. 11 (2010), 3363-3388. DOI 10.1016/j.nonrwa.2009.11.028 | MR 2683795 | Zbl 1207.35208
[35] Zaraï, A., Tatar, N.-e.: Global existence and polynomial decay for a problem with Balakrishnan-Taylor damping. Arch. Math., Brno 46 (2010), 157-176. MR 2735903 | Zbl 1240.35330
Partner of
EuDML logo