# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [4] T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory, Commun. Pure Appl. Anal., 6 (2007), 83-101. Google Scholar [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. Google Scholar [6] G. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, New York, 1985. Google Scholar [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.  Google Scholar [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.  Google Scholar [9] J. Lagnese, "Boundary Stabilization of Thing Plates," Philadelphia: SIAM, 1989. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [14] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New-York, 1983. Google Scholar [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.  Google Scholar [16] P. Schiavone and R. J.Tait, Thermal effects in Mindlin-type plates, Q. Jl. Mech. appl. Math., 46 (1993), 27-39. Google Scholar [17] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., Ser.4, 148 (1987), 65-96. Google Scholar

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##### 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [4] T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory, Commun. Pure Appl. Anal., 6 (2007), 83-101. Google Scholar [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. Google Scholar [6] G. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, New York, 1985. Google Scholar [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.  Google Scholar [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.  Google Scholar [9] J. Lagnese, "Boundary Stabilization of Thing Plates," Philadelphia: SIAM, 1989. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [14] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New-York, 1983. Google Scholar [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.  Google Scholar [16] P. Schiavone and R. J.Tait, Thermal effects in Mindlin-type plates, Q. Jl. Mech. appl. Math., 46 (1993), 27-39. Google Scholar [17] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., Ser.4, 148 (1987), 65-96. Google Scholar
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