Global existence and stability of solution for a p−Kirchhoff type hyperbolic equation with damping and source terms
DOI:
https://doi.org/10.24193/subbmath.2022.4.11Keywords:
p−Kirchhoff type hyperbolic equation, global existence, source term, Komornik’s integral inequality.Abstract
In this paper, we consider a nonlinear p−Kirchhoff type hyperbolic equation with damping and source terms f utt − M Ω |∇u|p dx ∆pu + |ut| m−2 ut = |u| r−2 u. Under suitable assumptions and positive initial energy, we prove the global existence of solution by using the potential energy and Nehari’s functionals. Finally, the stability of equation is established based on Komornik’s integral inequality.
Mathematics Subject Classification (2010): 35L70, 35L05, 35B40, 93D20.
Received 30 December 2019; Accepted 03 February 2020.
References
Autuori, G., Pucci, P., Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal. Theory, Methods Appl., 73(2010), no. 7, 1952-1965.
Autuori, G., Pucci, P., Salvatori, MC., Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196(2010), no. 2, 489-516.
Benaissa, A., Messaoudi, S.A., Blow-up of solutions for Kirchhoff equation of q-Laplacian type with nonlinear dissipation, Colloq. Math., 94(2002), no. 1, 103-109.
Gao, Q., Wang, Y., Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. Eur. J. Math., 9(2011), no. 3, 686-698.
Georgiev, V., Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source term, J. Dfferential Equations., 109(1994), no. 2, 295-308.
Ghisi, M., Gobbino, M., Hyperbolic-parabolic singular perturbation for middly degenerate Kirchhoff equations: time-decay estimates, J. Differential Equations, 245(2008), no. 10, 2979-3007.
Kirchhoff, G., Mechanik, Teubner, 1883.
Komornik, V., Exact Controllability and Stabilization the Multiplier Method, Paris: Masson – John Wiley, 1994.
Li, F., Global existence and blow-up of solutions for a higher-order Kirchhoff-type equation with nonlinear dissipation, Appl. Math. Lett., 17(2004), no. 12, 1409-1414.
Messaoudi, S.A., Said Houari, B., A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation, Appl. Math. Lett., 20(2007), no. 8, 866-871.
Messaoudi, S.A., Talahmeh, A., Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci., 40(2017), no. 18, 6976-6986.
Ono, K., Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation, Nonlinear Anal. Theory, Methods Appl., 30(1997), no. 2, 4449-4457.
Ono, K., Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations., 137(1997), no. 2, 273-301.
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), no. 2, 151-177.
Wu, S.T., Tsai, L.Y., Blow-up of solutions for some nonlinear wave equations of Kirchhoff type with some dissipation, Nonlinear Anal. Theory Methods Appl., 65(2006), no. 2, 243-264.
Zeng, R., Mu, C.L., Zhou, S.M., A blow-up result for Kirchhoff-type equations with high energy, Math. Methods Appl. Sci., 34(2011), no. 4, 479-486.
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