Abstract
This paper is concerned with extensible beams with nonlocal frictional damping and polynomial nonlinearity. By using semigroup theory, potential well method, and energy method, the well-posedness and the conditions on global existence and finite time blow-up of solutions are studied. Moreover, the upper bound of blow-up time is also given by using ordinary differential inequalities.
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References
Antontsev, S., Ferreira, J.: A nonlinear viscoelastic plate equation with \(\overrightarrow{p}(x,t)\)-Laplace operator: blow up of solutions with negative initial energy. Nonlinear Anal. Real World Appl. 59, Paper No. 103240, 17 (2021)
Balakrishnan, A.V., Taylor, L.W.: Distributed parameter nonlinear damping models for flight structures. In: Proceedings “ Daming 89”, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB (1989)
Berger, H.M.: A new approach to the analysis of large deflections of plates. J. Appl. Mech. 22, 465–472 (1955)
Bociu, L., Rammaha, M., Toundykov, D.: On a wave equation with supercritical interior and boundary sources and damping terms. Math. Nachr. 284, 2032–2064 (2011)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Jorge Silva, M.A., Narciso, V.: Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan–Taylor type. J. Differ. Equ. 290, 197–222 (2021)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation. Commun. Contemp. Math. 6(5), 705–731 (2004)
Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations, Volume 13 of Oxford Lecture Series in Mathematics and Its Applications. Translated from the 1990 French original by Yvan Martel and revised by the authors. The Clarendon Press, Oxford University Press, New York (1998)
Chen, W.Y., Zhou, Y.: Global nonexistence for a semilinear Petrovsky equation. Nonlinear Anal. 70(9), 3203–3208 (2009)
Chueshov, I., Lasiecka, I.: Long-time behavior of second order evolution equations with nonlinear damping. Mem. Am. Math. Soc. 195(912), viii+183 (2008)
Chueshov, I., Lasiecka, I.: Von Karman evolution equations. Springer Monographs in Mathematics. Well-Posedness and Long-Time Dynamics. Springer, New York (2010)
Han, J.B., Xu, R.Z., Yang, Y.B.: Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation. Asymptot. Anal. 122(3–4), 349–369 (2021)
Horn, M.A., Lasiecka, I.: Asymptotic behavior with respect to thickness of boundary stabilizing feedback for the Kirchhoff plate. J. Differ. Equ. 114(2), 396–433 (1994)
Lagnese, J.E.: Boundary Stabilization of Thin Plates, Volume 10 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1989)
Lasiecka, I.: Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary. J. Differ. Equ. 79(2), 340–381 (1989)
Lasiecka, I., Wilke, M.: Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discret. Contin. Dyn. Syst. 33(11–12), 5189–5202 (2013)
Lian, W., Rădulescu, V.D., Xu, R.Z., Yang, Y.B., Zhao, N.: Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations. Adv. Calc. Var. 14(2), 589–611 (2021)
Liu, J.X., Wang, X.R., Zhou, J., Liu, X.: Dynamics of solutions to a semilinear plate equation with memory. Commun. Pure Appl. Anal. 20(11), 3911–3936 (2021)
Liu, Y., Chen, W.H.: Asymptotic profiles of solutions for regularity-loss-type generalized thermoelastic plate equations and their applications. Z. Angew. Math. Phys. 71(2), Paper No. 55, 26 (2020)
Liu, Y.Q.: Decay of solutions to an inertial model for a semilinear plate equation with memory. J. Math. Anal. Appl. 394(2), 616–632 (2012)
Liu, Y.Q., Kawashima, S.: Decay property for a plate equation with memory-type dissipation. Kinet. Relat. Models 4(2), 531–547 (2011)
Liu, Y.Q., Ueda, Y.: Decay estimate and asymptotic profile for a plate equation with memory. J. Differ. Equ. 268(5), 2435–2463 (2020)
Messaoudi, S.A.: Global existence and nonexistence in a system of Petrovsky. J. Math. Anal. Appl. 265(2), 296–308 (2002)
Nakao, M.: Decay of solutions of some nonlinear evolution equations. J. Math. Anal. Appl. 60(2), 542–549 (1977)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer-Verlag, New York (1983)
Pereira, D.C., Nguyen, H., Raposo, C.A., Maranhão, Celsa H.M.: On the solutions for an extensible beam equation with internal damping and source terms. Differ. Equ. Appl. 11(3), 367–377 (2019)
Ueda, Y.: Optimal decay estimates of a regularity-loss type system with constraint condition. J. Differ. Equ. 264(2), 679–701 (2018)
Woinowsky-Krieger, S.: The effect of an axial force on the vibration of hinged bars. J. Appl. Mech. 17, 35–36 (1950)
Xu, R.Z., Wang, X.C., Yang, Y.B., Chen, S.H.: Global solutions and finite time blow-up for fourth order nonlinear damped wave equation. J. Math. Phys. 59(6), 061503, 27 (2018)
Xu, R.Z., Chen, Y.X., Yang, Y.B., Chen, S.H., Shen, J.H., Yu, T., Xu, Z.S.: Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrödinger equations. Electron. J. Differ. Equ. 55, 52 (2018)
Yang, Z.J.: On an extensible beam equation with nonlinear damping and source terms. J. Differ. Equ. 254(9), 3903–3927 (2013)
Zheng, S.M.: Nonlinear evolution equations. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 133. Chapman & Hall/CRC, Boca Raton (2004)
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Chen, K., Zhou, J. Well-Posedness and Dynamical Properties for Extensible Beams with Nonlocal Frictional Damping and Polynomial Nonlinearity. Appl Math Optim 88, 92 (2023). https://doi.org/10.1007/s00245-023-10070-w
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DOI: https://doi.org/10.1007/s00245-023-10070-w