Taiwanese Journal of Mathematics

Attractors for a Class of Kirchhoff Models with $p$-Laplacian and Time Delay

Sun-Hye Park

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This paper is concerned with a class of Kirchhoff models with time delay and perturbation of $p$-Laplacian type \[ u_{tt}(x,t) + \Delta^2 u(x,t) - \Delta_p u(x,t) - a_0 \Delta u_t(x,t) + a_1 u_t(x,t-\tau) + f(u(x,t)) = g(x), \] where $\Delta_p u = \operatorname{div}(|\nabla u|^{p-2} \nabla u)$ is the usual $p$-Laplacian operator. Many researchers have studied well-posedness and decay rates of energy for these equations without delay effects. But, there are not many studies on attractors for other delayed systems. Thus we establish the existence of global attractors and the finite dimensionality of the attractors by establishing some functionals which are related to the norm of the phase space to our problem.

Article information

Taiwanese J. Math., Volume 23, Number 4 (2019), 883-896.

Received: 11 September 2017
Revised: 4 October 2018
Accepted: 11 November 2018
First available in Project Euclid: 18 July 2019

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Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations 35B41: Attractors

attractor finite dimensionality time delay Kirchhoff model $p$-Laplacian


Park, Sun-Hye. Attractors for a Class of Kirchhoff Models with $p$-Laplacian and Time Delay. Taiwanese J. Math. 23 (2019), no. 4, 883--896. doi:10.11650/tjm/181105. https://projecteuclid.org/euclid.twjm/1563436873

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  • D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with $p$-Laplacian and memory terms, Math. Methods Appl. Sci. 35 (2012), no. 4, 417–426.
  • I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and long-time dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.
  • R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim. 26 (1988), no. 3, 697–713
  • M. A. Jorge Silva and T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math. 78 (2013), no. 6, 1130–1146.
  • ––––, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys. 54 (2013), no. 2, 021505, 15 pp.
  • M. A. Jorge Silva, J. E. Muñoz Rivera and R. Racke, On a class of nonlinear viscoelastic Kirchhoff plates: well-posedness and general decay rates, Appl. Math. Optim. 73 (2016), no. 1, 165–194.
  • M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys. 62 (2011), no. 6, 1065–1082.
  • T. F. Ma and J. A. Soriano, On weak solutions for an evolution equation with exponential nonlinearities, Nonlinear Anal. 37 (1999), no. 8, Ser. A: Theory Methods, 1029–1038.
  • M. I. Mustafa, Asymptotic behavior of second sound thermoelasticity with internal time-varying delay, Z. Angew. Math. Phys. 64 (2013), no. 4, 1353–1362.
  • S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006), no. 5, 1561–1585.
  • ––––, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential equations 2011 (2011), no. 41, 20 pp.
  • S. H. Park, Stability for a viscoelastic plate equation with $p$-Laplacian, Bull. Korean Math. Soc. 52 (2015), no. 3, 907–914.
  • ––––, Energy decay for a von Karman equation with time-varying delay, Appl. Math. Lett. 55 (2016), 10–17.
  • J. Y. Park, H. M. Kim and S. H. Park, On weak solutions for hyperbolic differential inclusion with discontinuous nonlinearities, Nonlinear Anal. 55 (2003), no. 1-2, 103–113.
  • Z. Yang, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Z. Angew. Math. Phys. 66 (2015), no. 3, 727–745.
  • Y. Zhijian, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Methods Appl. Sci. 32 (2009), no. 9, 1082–1104.
  • Y. Zhijian and J. Baoxia, Global attractor for a class of Kirchhoff models, J. Math. Phys. 50 (2009), no. 3, 032701, 29 pp.