# American Institute of Mathematical Sciences

September  2012, 11(5): 1753-1773. doi: 10.3934/cpaa.2012.11.1753

## Instability of coupled systems with delay

 1 Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz

Received  January 2011 Revised  July 2011 Published  March 2012

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).
Citation: Reinhard Racke. Instability of coupled systems with delay. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753
##### References:
 [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. Google Scholar [2] K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay,, preprint, (). Google Scholar [3] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation,, Rend. Instit. Mat. Univ. Trieste Suppl., 28 (1997), 1. Google Scholar [4] A. Bátkai and S. Piazzera, "Semigroups for Delay Equations,'', Research Notes in Mathematics, 10 (2005). Google Scholar [5] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature,, Appl. Mech. Rev., 51 (1998), 705. doi: 10.1115/1.3098984. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Letters, 22 (2009), 1374. doi: 10.1016/j.aml.2009.03.010. Google Scholar [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. doi: 10.1137/090762105. Google Scholar [11] S. Jiang and R. Racke, "Evolution Equations in Thermoelasticity,'', $\pi$ Monographs Surveys Pure Appl. Math. \textbf{112}, 112 (2000). Google Scholar [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. doi: 10.1016/j.mechrescom.2008.04.001. Google Scholar [13] J. U. Kim, On th energy decay of a linear thermoelastic bar and plate,, SIAM J. Math. Anal., 23 (1992), 889. doi: 10.1137/0523047. Google Scholar [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. doi: 10.3934/cpaa.2011.10.667. Google Scholar [15] J. Lagnese, "Boundary Stabilization of Thin Plates,'', SIAM Studies Appl. Math., 10 (1989). Google Scholar [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. Google Scholar [17] I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups,, ESAIM, 4 (1998), 199. Google Scholar [18] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann boundary conditions,, Abstract Appl. Anal., 3 (1998), 153. doi: 10.1155/S1085337598000487. Google Scholar [19] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions,, Annali Scuola Norm. Sup. Pisa, 27 (1998), 457. Google Scholar [20] Z. Liu and M. Renardy, A note on the equation of a thermoelastic plate,, Appl. Math. Lett., 8 (1995), 1. doi: 10.1016/0893-9659(95)00020-Q. Google Scholar [21] K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems,, Z. angew. Math. Phys., 48 (1997), 885. doi: 10.1007/s000330050071. Google Scholar [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. Google Scholar [23] Z. Liu and S. Zheng, Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping,, Quart. Appl. Math., 53 (1997), 551. Google Scholar [24] Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems,'', $\pi$ Research Notes Math., 398 (1999). Google Scholar [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. doi: 10.1137/S0036142993255058. Google Scholar [26] J.E. Muñoz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems,, J. Differential Equations, 127 (1996), 454. doi: 10.1006/jdeq.1996.0078. Google Scholar [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. doi: 10.1137/060648891. Google Scholar [28] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary of internal distributed delay,, Diff. Integral Equations, 21 (2008), 935. Google Scholar [29] J. Prüß, "Evolutionary Integral Equations and Applications,'', Monographs Math., 87 (1993). Google Scholar [30] R. Quintanilla, A well posed problem for the dual-phase-lag heat conduction,, J. Thermal Stresses, 31 (2008), 260. doi: 10.1080/01495730701738272. Google Scholar [31] R. Quintanilla, A well posed problem for the three dual-phase-lag heat conduction,, J. Thermal Stresses, 32 (2009), 1270. doi: 10.1080/01495730903310599. Google Scholar [32] R. Quintanilla, Some solutions for a family of exact phase-lag heat conduction problems,, Mech. Research Communications, 38 (2011), 355. doi: 10.1016/j.mechrescom.2011.04.008. Google Scholar [33] R. Racke, Thermoelasticity with second sound -- exponential stability in linear and nonlinear 1-d,, Math. Meth. Appl. Sci., 25 (2002), 409. doi: 10.1002/mma.298. Google Scholar [34] R. Racke, Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound,, Quart. Appl. Math., 61 (2003), 315. Google Scholar [35] R. Racke, "Thermoelasticity,'', Chapter in: Handbook of Differential Equations. Evolutionary Equations, 5 (2009). Google Scholar [36] D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems,, J. Math. Anal. Appl., 173 (1993), 339. doi: 10.1006/jmaa.1993.1071. Google Scholar [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. doi: 10.1006/jmaa.1993.1071. Google Scholar

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##### References:
 [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. Google Scholar [2] K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay,, preprint, (). Google Scholar [3] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation,, Rend. Instit. Mat. Univ. Trieste Suppl., 28 (1997), 1. Google Scholar [4] A. Bátkai and S. Piazzera, "Semigroups for Delay Equations,'', Research Notes in Mathematics, 10 (2005). Google Scholar [5] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature,, Appl. Mech. Rev., 51 (1998), 705. doi: 10.1115/1.3098984. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Letters, 22 (2009), 1374. doi: 10.1016/j.aml.2009.03.010. Google Scholar [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. doi: 10.1137/090762105. Google Scholar [11] S. Jiang and R. Racke, "Evolution Equations in Thermoelasticity,'', $\pi$ Monographs Surveys Pure Appl. Math. \textbf{112}, 112 (2000). Google Scholar [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. doi: 10.1016/j.mechrescom.2008.04.001. Google Scholar [13] J. U. Kim, On th energy decay of a linear thermoelastic bar and plate,, SIAM J. Math. Anal., 23 (1992), 889. doi: 10.1137/0523047. Google Scholar [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. doi: 10.3934/cpaa.2011.10.667. Google Scholar [15] J. Lagnese, "Boundary Stabilization of Thin Plates,'', SIAM Studies Appl. Math., 10 (1989). Google Scholar [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. Google Scholar [17] I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, of thermo-elastic semigroups,, ESAIM, 4 (1998), 199. Google Scholar [18] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann boundary conditions,, Abstract Appl. Anal., 3 (1998), 153. doi: 10.1155/S1085337598000487. Google Scholar [19] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions,, Annali Scuola Norm. Sup. Pisa, 27 (1998), 457. Google Scholar [20] Z. Liu and M. Renardy, A note on the equation of a thermoelastic plate,, Appl. Math. Lett., 8 (1995), 1. doi: 10.1016/0893-9659(95)00020-Q. Google Scholar [21] K. Liu and Z. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems,, Z. angew. Math. Phys., 48 (1997), 885. doi: 10.1007/s000330050071. Google Scholar [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. Google Scholar [23] Z. Liu and S. Zheng, Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping,, Quart. Appl. Math., 53 (1997), 551. Google Scholar [24] Z. Liu and S. Zheng, "Semigroups Associated with Dissipative Systems,'', $\pi$ Research Notes Math., 398 (1999). Google Scholar [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. doi: 10.1137/S0036142993255058. Google Scholar [26] J.E. Muñoz Rivera and R. Racke, Large solutions and smoothing properties for nonlinear thermoelastic systems,, J. Differential Equations, 127 (1996), 454. doi: 10.1006/jdeq.1996.0078. Google Scholar [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. doi: 10.1137/060648891. Google Scholar [28] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary of internal distributed delay,, Diff. Integral Equations, 21 (2008), 935. Google Scholar [29] J. Prüß, "Evolutionary Integral Equations and Applications,'', Monographs Math., 87 (1993). Google Scholar [30] R. Quintanilla, A well posed problem for the dual-phase-lag heat conduction,, J. Thermal Stresses, 31 (2008), 260. doi: 10.1080/01495730701738272. Google Scholar [31] R. Quintanilla, A well posed problem for the three dual-phase-lag heat conduction,, J. Thermal Stresses, 32 (2009), 1270. doi: 10.1080/01495730903310599. Google Scholar [32] R. Quintanilla, Some solutions for a family of exact phase-lag heat conduction problems,, Mech. Research Communications, 38 (2011), 355. doi: 10.1016/j.mechrescom.2011.04.008. Google Scholar [33] R. Racke, Thermoelasticity with second sound -- exponential stability in linear and nonlinear 1-d,, Math. Meth. Appl. Sci., 25 (2002), 409. doi: 10.1002/mma.298. Google Scholar [34] R. Racke, Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound,, Quart. Appl. Math., 61 (2003), 315. Google Scholar [35] R. Racke, "Thermoelasticity,'', Chapter in: Handbook of Differential Equations. Evolutionary Equations, 5 (2009). Google Scholar [36] D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems,, J. Math. Anal. Appl., 173 (1993), 339. doi: 10.1006/jmaa.1993.1071. Google Scholar [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. doi: 10.1006/jmaa.1993.1071. Google Scholar
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