\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Instability of coupled systems with delay

Abstract Related Papers Cited by
  • We consider linear initial-boundary value problems that are a coupling like second-order thermoelasticity, or the thermoelastic plate equation or its generalization (the $\alpha$-$\beta$-system introduced in [1, 26]). Now, there is a delay term given in part of the coupled system, and we demonstrate that the expected inherent damping will not prevent the system from not being stable; indeed, the systems will shown to be ill-posed: a sequence of bounded initial data may lead to exploding solutions (at any fixed time).
    Mathematics Subject Classification: Primary: 35 B, 35K20, 35K55, 35L20, 35Q, 35R25; Secondary: 80A20.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    F. Ammar Khodja and A. Benabdallah, Sufficient conditions for uniform stabilization of second order equations by dynamical controllers, Dyn. Contin. Discrete Impulsive Syst., 7 (2000), 207-222.

    [2]

    K. Ammari, S. Nicaise and C. PignottiFeedback boundary stabilization of wave equations with interior delay, preprint, arXiv:math/1005.2547v1.

    [3]

    G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Instit. Mat. Univ. Trieste Suppl., 28 (1997), 1-28.

    [4]

    A. Bátkai and S. Piazzera, "Semigroups for Delay Equations,'' Research Notes in Mathematics, 10, A.K. Peters, Wellesley MA, 2005.

    [5]

    D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729.doi: 10.1115/1.3098984.

    [6]

    R. Denk and R. Racke, $L^p$ resolvent estimates and time decay for generalized thermoelastic plate equations, Electronic J. Differential Equations, 48 (2006), 1-16.

    [7]

    R. Denk, R. Racke and Y. Shibata, $L_p$ theory for the linear thermoelastic plate equations in bounded and exterior domains, Advances Differential Equations, 14 (2009), 685-715.

    [8]

    R. Denk, R. Racke and Y. Shibata, Local energy decay estimate of solutions to the thermoelastic plate equations in two- and three-dimensional exterior domains, J. Analysis Appl., 29 (2010), 21-62.

    [9]

    M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Letters, 22 (2009), 1374-1379.doi: 10.1016/j.aml.2009.03.010.

    [10]

    E. Fridman, S. Nicaise and J. Valein, Stabilization of second order evoluion equations with unbounded feedback with time-dependent delay, SIAM J. Control Optim., 48 (2010), 5028-5052.doi: 10.1137/090762105.

    [11]

    S. Jiang and R. Racke, "Evolution Equations in Thermoelasticity,'' $\pi$ Monographs Surveys Pure Appl. Math. 112, Chapman & Hall/CRC, Boca Raton, 2000.

    [12]

    P. M. Jordan, W. Dai and R. E. Mickens, A note on the delayed heat equation: Instability with respect to initial data, Mech. Research Comm., 35 (2008), 414-420.doi: 10.1016/j.mechrescom.2008.04.001.

    [13]

    J. U. Kim, On th energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889-899.doi: 10.1137/0523047.

    [14]

    M. Kirane, B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko systme with a time-varying delay term in the internal feedbacks, Comm. Pure Appl. Anal., 10 (2011), 667-686.doi: 10.3934/cpaa.2011.10.667.

    [15]

    J. Lagnese, "Boundary Stabilization of Thin Plates,'' SIAM Studies Appl. Math., 10, SIAM, Philadelphia, 1989.

    [16]

    I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the S.C. semigroup arising in abstract thermoelastic equations, Adv. Differential Equations, 3 (1998), 387-416.

    [17]

    I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups, ESAIM, Proc., 4 (1998), 199-222.

    [18]

    I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann boundary conditions, Abstract Appl. Anal., 3 (1998), 153-169.doi: 10.1155/S1085337598000487.

    [19]

    I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Annali Scuola Norm. Sup. Pisa, 27 (1998), 457-482.

    [20]

    Z. Liu and M. Renardy, A note on the equation of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.doi: 10.1016/0893-9659(95)00020-Q.

    [21]

    K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. angew. Math. Phys., 48 (1997), 885-904.doi: 10.1007/s000330050071.

    [22]

    Z. Liu and J. Yong, Qualitative properties of certain $C_0$ semigroups arising in elastic systems with various dampings, Adv. Differential Equations, 3 (1998), 643-686.

    [23]

    Z. Liu and S. Zheng, Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 53 (1997), 551-564.

    [24]

    Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems,'' $\pi$ Research Notes Math., 398, Chapman & Hall/ CRC, Boca Raton, 1999.

    [25]

    J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547-1563.doi: 10.1137/S0036142993255058.

    [26]

    J.E. Muñoz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems, J. Differential Equations, 127 (1996), 454-483.doi: 10.1006/jdeq.1996.0078.

    [27]

    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), 1561-1585.doi: 10.1137/060648891.

    [28]

    S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary of internal distributed delay, Diff. Integral Equations, 21 (2008), 935-958.

    [29]

    J. Prüß, "Evolutionary Integral Equations and Applications,'' Monographs Math., 87, Birkhäuser, Basel, 1993.

    [30]

    R. Quintanilla, A well posed problem for the dual-phase-lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269.doi: 10.1080/01495730701738272.

    [31]

    R. Quintanilla, A well posed problem for the three dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.doi: 10.1080/01495730903310599.

    [32]

    R. Quintanilla, Some solutions for a family of exact phase-lag heat conduction problems, Mech. Research Communications, 38 (2011), 355-360.doi: 10.1016/j.mechrescom.2011.04.008.

    [33]

    R. Racke, Thermoelasticity with second sound -- exponential stability in linear and nonlinear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441.doi: 10.1002/mma.298.

    [34]

    R. Racke, Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound, Quart. Appl. Math., 61 (2003), 315-328.

    [35]

    R. Racke, "Thermoelasticity,'' Chapter in: Handbook of Differential Equations. Evolutionary Equations, Vol. 5. Eds.: C.M. Dafermos, M. Pokorn\'y, Elsevier B.V., Amsterdam, 2009.

    [36]

    D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.doi: 10.1006/jmaa.1993.1071.

    [37]

    B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Appl. Math. Comp., 217 (2010), 2857-2869.doi: 10.1006/jmaa.1993.1071.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(217) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return