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 and 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), 207-222.

[2]

K. Ammari, S. Nicaise and C. Pignotti, Feedback 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.

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), 207-222.

[2]

K. Ammari, S. Nicaise and C. Pignotti, Feedback 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.

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