Decay Estimate for the Time-Delayed Fourth-Order Wave Equations
The objective of this article is to analyze the stability of solutions for the following fourth- order nonlinear wave equations with an internal delay term: \begin{equation*} u_{tt} + \Delta^2 u + u + \sigma_1(t) |u_{t}(x,t)|^{2m-2} u_t(x,t) + \sigma_2(t) |u_{t} (x,t-\tau)|^{2m-2} u_t(x,t-\tau) = 0. \end{equation*} We obtain appropriate conditions on $\sigma_1(t)$ and $\sigma_2(t)$ for the decay properties of the solutions. The multiplier technique and nonlinear integral inequalities are used in the proof.
Keywords:
fourth order wave, asymptotic behavior energy decay rate,
___
- [1] Ammari, K., Nicaise, S., Pignotti, C.: Feedback boundary stabilization of wave equations with interior delay. Systems and Control Letters. 59, 623–628 (2010).
- [2] Benaissa, A., Benaissa, A., Messaoudi, SA.: Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks. J. Math. Phys. 53, 123–514 (2012).
- [3] Benaissa, A., Messaoudi, SA.: Global Existence and Energy Decay of Solutions for a Nondissipative Wave Equation with a Time-Varying Delay Term. Progress in Partial Differential Equations. 1–26 (2013).
- [4] Benaissa, A., Benguessoum, A., Messaoudi, SA.: Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback. EJQTDE. 11, 1–13 (2014).
- [5] Çelebi, AO., Gür, S ̧., Kalantarov, VK.: Structural stability and decay estimate for marine riser equations. Math. Comput Model. 54, 3182-3188 (2011).
- [6] Gerbi, S., Said-Houari, B.: Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term. Applied Mathematics and Computation. 218 (24), 11900–11910 (2012).
- [7] Gür, Ş.: Global asymptotic stability of solutions to nonlinear marine riser equation. J Inequal Appl. (2010). doi:10.1155/2010/504670.
- [8] Kalantarov, VK., Kurt, A.: The Long-time behavior of solutions of a nonlinear fourth order wave equation Describing the dynamics of marine risers. ZAMM . 77 (3), 209–215 (1997).
- [9] Li, J., Chai, S.: Existence and energy decay rates of solutions to the variable-coefficient Euler–Bernoulli plate with a delay in localized nonlinear internal feedback. J. Math. Anal. Appl. 443 (2), 981–1006 (2016).
- [10] Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM:Control,Optimisation and Calculus of Variations. 4, 419–444 (1999).
- [11] Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedback. SIAM J. Control Optim. 45, 1561–1585 (2006).
- [12] Ning, Z., Shen, C., Zhao, X.: Stabilization of the wave equation with variable coefficients and a delay in dissipative internal feedback. J. Math. Anal. Appl. 405 (1), 148–155 (2013).
- [13] Park, S.: Energy decay for a von Karman equation with time-varying delay. Appl. Math. Lett. 55, 10—17 (2016).
- [14] Zhuang, Z., Zhang, Y.: Global existence and asymptotic behavior of solutions to a class of fourth-order wave equations. Boundary Value Problems. 2013:168 (2013).
- ISSN: 2147-6268
- Yayın Aralığı: Yılda 4 Sayı
- Başlangıç: 2013
- Yayıncı: -