On a system of nonlinear wave equations with the Kirchhoff-Carrier and Balakrishnan-Taylor terms

Nam, Bui Duc ; Nhan, Nguyen Huu ; Ngoc, Le Thi Phuong ; Long, Nguyen Thanh

Mathematica Bohemica, Tome 147 (2022) no. 2, pp. 237-270

Abstract (VO)

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.

Détail

DOI : 10.21136/MB.2021.0094-20

Classification : 35L20, 35L70, 35Q74, 37B25
Keywords: system of nonlinear wave equations of Kirchhoff-Carrier type; Balakrishnan-Taylor term; Faedo-Galerkin method; local existence; exponential decay

Bibliographie

[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

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[14] Ha, T. G.: Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping. Taiwanese J. Math. 21 (2017), 807-817. | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[18] Shah, S. Hyder Ali Muttaqi: Some helical flows of a Burgers fluid with fractional derivative. Meccanica 45 (2010), 143-151. | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[23] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Etudes mathematiques. Dunod, Gauthier-Villars, Paris (1969), French. | MR | JFM

[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 | MR | JFM

[25] Medeiros, L. A.: On some nonlinear perturbation of Kirchhoff-Carrier operator. Comput. Appl. Math. 13 (1994), 225-233. | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | JFM

[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 | MR | JFM

[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 | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | JFM