November  2013, 12(6): 2645-2667. doi: 10.3934/cpaa.2013.12.2645

Decay rates for Kirchhoff-Timoshenko transmission problems

1. 

Department of Mathematics and Mechanics, Kharkov Karazin National University, 4, Svobody sq., Kharkov 61077, Ukraine

Received  August 2012 Revised  January 2013 Published  May 2013

A linear transmission problem for a thermoelastic Timoshenko beam model with Fourier low of heat conduction which has a Kirchhoff part with hereditary heat conduction of Gurtin-Pipkin type is considered. We prove that the system is exponentially stable under certain conditions on its parameters. The same result for the problem with purely elastic Kirchhoff part is obtained.
Citation: Tamara Fastovska. Decay rates for Kirchhoff-Timoshenko transmission problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2645-2667. doi: 10.3934/cpaa.2013.12.2645
References:
[1]

M. S. Alves, J. E. Muñoz Rivera, C. A. Raposo, M. Sepúlveda and O. P. Vera Villagrán, Uniform stabilization for transmission problem for Timoshenko's system with memory, J. Math. Anal.Appl., 369 (2010), 323-345. doi: 10.1016/j.jmaa.2010.02.045.

[2]

W. D. Bastos, C. A. Raposo and M. L. Santos, A transmission problem for the Timoshenko system, Comp. Appl. Math., 26 (2007), 215-234.

[3]

I. Chueshov and I. Lasiecka, Global attractors for Mindlin-Timoshenko plates and for their Kirchhoff limits, Milan J. Math., 74 (2006), 117-138. doi: 10.1007/s00032-006-0050-8.

[4]

T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory, Commun. Pure Appl. Anal., 6 (2007), 83-101.

[5]

T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermo-viscoelastic model with memory, Nonlin. Anal. TMA, Nonlinear Analysis TMA, 71 (2009), 4833-4851.

[6]

G. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, New York, 1985.

[7]

M. Grasselli, J. E. Muñoz Rivera and V. Pata, On the energy decay of the linear thermoelastic plate with memory, J. Math. Anal. Appl., 309 (2005), 1-14. doi: 10.1016/j.jmaa.2004.10.071.

[8]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.

[9]

J. Lagnese, "Boundary Stabilization of Thing Plates," Philadelphia: SIAM, 1989.

[10]

S. A. Messaoudi, M. Pokojovy and B. Said-Houary, Nonlinear damped Timoshenko systems with second sound - global existence and exponential stability, Math. Med. Appl. Sci., 32 (2009), 505-534.

[11]

J. E. Muñoz Rivera and H. Portillo Oquendo, The transmission problem for thermoelastic beams, J. Thermal Stresses, 24 (2001), 1137-1158. doi: 10.1080/014957301753251665.

[12]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems - global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.

[13]

J. E. Muñoz Rivera and J. C. Vila Bravo, The transmission problem to thermoelastic plate of hyperbolic type, IMA J. Appl. Math., 74 (2009), 950-962.

[14]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New-York, 1983.

[15]

R. Racke and B. Said-Houary, Decay rates and global existence for semilinear dissipative Timoshenko systems,, Quart. Appl. Math., ().  doi: 10.1090/S0033-569X-2012-01280-8.

[16]

P. Schiavone and R. J.Tait, Thermal effects in Mindlin-type plates, Q. Jl. Mech. appl. Math., 46 (1993), 27-39.

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., Ser.4, 148 (1987), 65-96.

show all references

References:
[1]

M. S. Alves, J. E. Muñoz Rivera, C. A. Raposo, M. Sepúlveda and O. P. Vera Villagrán, Uniform stabilization for transmission problem for Timoshenko's system with memory, J. Math. Anal.Appl., 369 (2010), 323-345. doi: 10.1016/j.jmaa.2010.02.045.

[2]

W. D. Bastos, C. A. Raposo and M. L. Santos, A transmission problem for the Timoshenko system, Comp. Appl. Math., 26 (2007), 215-234.

[3]

I. Chueshov and I. Lasiecka, Global attractors for Mindlin-Timoshenko plates and for their Kirchhoff limits, Milan J. Math., 74 (2006), 117-138. doi: 10.1007/s00032-006-0050-8.

[4]

T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory, Commun. Pure Appl. Anal., 6 (2007), 83-101.

[5]

T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermo-viscoelastic model with memory, Nonlin. Anal. TMA, Nonlinear Analysis TMA, 71 (2009), 4833-4851.

[6]

G. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, New York, 1985.

[7]

M. Grasselli, J. E. Muñoz Rivera and V. Pata, On the energy decay of the linear thermoelastic plate with memory, J. Math. Anal. Appl., 309 (2005), 1-14. doi: 10.1016/j.jmaa.2004.10.071.

[8]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.

[9]

J. Lagnese, "Boundary Stabilization of Thing Plates," Philadelphia: SIAM, 1989.

[10]

S. A. Messaoudi, M. Pokojovy and B. Said-Houary, Nonlinear damped Timoshenko systems with second sound - global existence and exponential stability, Math. Med. Appl. Sci., 32 (2009), 505-534.

[11]

J. E. Muñoz Rivera and H. Portillo Oquendo, The transmission problem for thermoelastic beams, J. Thermal Stresses, 24 (2001), 1137-1158. doi: 10.1080/014957301753251665.

[12]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems - global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.

[13]

J. E. Muñoz Rivera and J. C. Vila Bravo, The transmission problem to thermoelastic plate of hyperbolic type, IMA J. Appl. Math., 74 (2009), 950-962.

[14]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New-York, 1983.

[15]

R. Racke and B. Said-Houary, Decay rates and global existence for semilinear dissipative Timoshenko systems,, Quart. Appl. Math., ().  doi: 10.1090/S0033-569X-2012-01280-8.

[16]

P. Schiavone and R. J.Tait, Thermal effects in Mindlin-type plates, Q. Jl. Mech. appl. Math., 46 (1993), 27-39.

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., Ser.4, 148 (1987), 65-96.

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