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11/12/2025
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Do, L., Pham, N. G., & Nguyen, T. L. (2025). The asymptotical stability of stationary solutions to three-dimensional kelvin-voigt equations with damping and unbounded delays. Journal of Water Resources & Environmental Engineering, 97, 55-59. https://dev.vojs.vn/index.php/jwree/article/view/14
The asymptotical stability of stationary solutions to three-dimensional kelvin-voigt equations with damping and unbounded delays
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Abstract
In this paper, we consider the three-dimensional Kelvin-Voigt equations involving unbounded delays in a bounded domain Ω⊂ℝ. We will study the asymptotical stability of stationary solutions via the construction of Lyapunov functionals.
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Keywords
Kelvin-voigt equation, stationary solutions: unbounded delays
References
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[KL09] V.K. Kalantarov, B. Levant and E.S. Titi, (2009), Gevrey regularity for the attractor of the 3D Navier-Stokes-Voigt equations, J. Nonlinear Sci. 19, 133-152.
[LC18] L. Liu, T. Caraballo and P. Marín-Rubio, (2018), Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265, 5685-5708.
[MM11] P. Marín-Rubio, A.M. Márquez-Durán and J. Real, (2011), Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. 31, 779-796.
[NS20] P.I. Naumkin, I. Sanchez-Suarez, (2020), KdV type asymptotics for solutions to higher-order nonlinear Schrodinger equations, Electron. J. Differential Equations, Vol. 2020 no. 77, 1-34.
[PS20] R. Pardo, A. Sanjuan, (2020), Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth, Electron. J. Differential Equations, Vol. 2020, no. 114, 1-17.
[YZ11] G. Yue and C.K. Zhong, (2011), Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voigt equations, Discrete Contin. Dyn. Syst. Ser. B 16, 985-1002.
[CJ08] X. Cai and Q. Jiu, (2008), Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., vol. 343, pp. 799-809, doi:10.1016/j.jmaa.2008.01.041.
[CR04] T. Caraballo and J. Real, (2004), Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations 205, pp. 271-297.
[KL09] V.K. Kalantarov, B. Levant and E.S. Titi, (2009), Gevrey regularity for the attractor of the 3D Navier-Stokes-Voigt equations, J. Nonlinear Sci. 19, 133-152.
[LC18] L. Liu, T. Caraballo and P. Marín-Rubio, (2018), Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265, 5685-5708.
[MM11] P. Marín-Rubio, A.M. Márquez-Durán and J. Real, (2011), Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. 31, 779-796.
[NS20] P.I. Naumkin, I. Sanchez-Suarez, (2020), KdV type asymptotics for solutions to higher-order nonlinear Schrodinger equations, Electron. J. Differential Equations, Vol. 2020 no. 77, 1-34.
[PS20] R. Pardo, A. Sanjuan, (2020), Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth, Electron. J. Differential Equations, Vol. 2020, no. 114, 1-17.
[YZ11] G. Yue and C.K. Zhong, (2011), Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voigt equations, Discrete Contin. Dyn. Syst. Ser. B 16, 985-1002.