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

show all references

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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