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March  2013, 12(2): 957-972. doi: 10.3934/cpaa.2013.12.957

A general stability result in a memory-type Timoshenko system

Salim A. Messaoudi 1, and  Muhammad I. Mustafa 1,

1. 

King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261, Saudi Arabia, Saudi Arabia

Received  March 2012 Revised  March 2012 Published  September 2012

In this paper we consider the following Timoshenko system \begin{eqnarray*} \varphi _{t t}-(\varphi _{x}+\psi )_{x}=0,\quad (0,1)\times R^+\\ \psi _{t t}-\psi _{x x}+\varphi _{x}+\psi +\int_0^t g(t-\tau )\psi_{x x}(\tau )d\tau =0,\quad (0,1)\times R^{+} \end{eqnarray*} with Dirichlet boundary conditions where $g$ is a positive nonincreasing function satisfying \begin{eqnarray*} g'(t)\leq -H(g(t)) \end{eqnarray*} and $H$ is a function satisfying some regularity and convexity conditions. We establish a general stability result for this system.
Keywords: viscoelasticity., relaxation function, Timoshenko system, Stability, general decay.
Mathematics Subject Classification: 35B37, 35L55, 74D05, 93D15, 93D2.
Citation: Salim A. Messaoudi, Muhammad I. Mustafa. A general stability result in a memory-type Timoshenko system. Communications on Pure and Applied Analysis, 2013, 12 (2) : 957-972. doi: 10.3934/cpaa.2013.12.957
References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 867-872.

[2]

F. Alabau-Bousosuira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonl. Differ. Eqns. Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.

[3]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differ. Eqns, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[4]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer-Verlag, New York, 1989.

[5]

H. D. Fernández Sare and J. E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. doi: 10.1016/j.jmaa.2007.07.012.

[6]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier's law, Arch. Rational Mech. Anal., 194 (2009), 221-251.

[7]

A. Guesmia and S. A. Messaoudi, On the control of solutions of a viscoelastic equation, Appl. Math. Comput., 206 (2008), 589-597. doi: 10.1016/j.amc.2008.05.122.

[8]

A. Guesmia and S. A. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122. doi: 10.1002/mma.1125.

[9]

K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. models Meth. Appl. Sci., 18 (2008), 647-667. doi: 10.1142/S0218202508002930.

[10]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[11]

Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, to be published in Math. models and Meth. Appl. Sci., 22 (2012), DOI No: 10.1142/S0218202511500126. doi: 10.1142/S0218202511500126.

[12]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, Nonlinear Differential Eqns. Appl., 15 (2008), 655-671. doi: 10.1007/s00030-008-7075-3.

[13]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307 doi: 10.1016/j.jmaa.2008.07.036.

[14]

S. A. Messaoudi and B. Said -Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III, Adv. Differ. Eqns, 14 (2009), 375-400.

[15]

S. A. Messaoudi and M. I. Mustafa, On the stabilization of the Timoshenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469. doi: 10.1002/mma.1047.

[16]

S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear Damped Timoshenko systems with second sound: Global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534. doi: 10.1002/mma.1049.

[17]

S. A. Messaoudi and B. Said -Houari, Uniform decay in a Timoshenko -type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475. doi: 10.1016/j.jmaa.2009.06.064.

[18]

S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dyn. Syst. Appl., 18 (2009), 457-468.

[19]

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

[20]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. syst., 9 (2003), 1625-1639.

[21]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: 10.1016/j.jmaa.2007.11.012.

[22]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyn. Control Syst., 16 (2010), 211-226 doi: 10.1007/s10883-010-9090-z.

[23]

C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-542. doi: 10.1016/j.aml.2004.03.017.

[24]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differ. Eqns, 29 (2003), 1-14.

[25]

S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 41 (1921), 744-746. doi: 10.1080/14786442108636264.

show all references

References:
[1]

F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 867-872.

[2]

F. Alabau-Bousosuira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonl. Differ. Eqns. Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.

[3]

F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differ. Eqns, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[4]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," Springer-Verlag, New York, 1989.

[5]

H. D. Fernández Sare and J. E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. doi: 10.1016/j.jmaa.2007.07.012.

[6]

H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier's law, Arch. Rational Mech. Anal., 194 (2009), 221-251.

[7]

A. Guesmia and S. A. Messaoudi, On the control of solutions of a viscoelastic equation, Appl. Math. Comput., 206 (2008), 589-597. doi: 10.1016/j.amc.2008.05.122.

[8]

A. Guesmia and S. A. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Meth. Appl. Sci., 32 (2009), 2102-2122. doi: 10.1002/mma.1125.

[9]

K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. models Meth. Appl. Sci., 18 (2008), 647-667. doi: 10.1142/S0218202508002930.

[10]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.

[11]

Y. Liu and S. Kawashima, Decay property for the Timoshenko system with memory-type dissipation, to be published in Math. models and Meth. Appl. Sci., 22 (2012), DOI No: 10.1142/S0218202511500126. doi: 10.1142/S0218202511500126.

[12]

S. A. Messaoudi and M. I. Mustafa, On the internal and boundary stabilization of Timoshenko beams, Nonlinear Differential Eqns. Appl., 15 (2008), 655-671. doi: 10.1007/s00030-008-7075-3.

[13]

S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307 doi: 10.1016/j.jmaa.2008.07.036.

[14]

S. A. Messaoudi and B. Said -Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III, Adv. Differ. Eqns, 14 (2009), 375-400.

[15]

S. A. Messaoudi and M. I. Mustafa, On the stabilization of the Timoshenko system by a weak nonlinear dissipation, Math. Meth. Appl. Sci., 32 (2009), 454-469. doi: 10.1002/mma.1047.

[16]

S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear Damped Timoshenko systems with second sound: Global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534. doi: 10.1002/mma.1049.

[17]

S. A. Messaoudi and B. Said -Houari, Uniform decay in a Timoshenko -type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475. doi: 10.1016/j.jmaa.2009.06.064.

[18]

S. A. Messaoudi and M. I. Mustafa, A stability result in a memory-type Timoshenko system, Dyn. Syst. Appl., 18 (2009), 457-468.

[19]

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

[20]

J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. syst., 9 (2003), 1625-1639.

[21]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping, J. Math. Anal. Appl., 341 (2008), 1068-1083. doi: 10.1016/j.jmaa.2007.11.012.

[22]

M. I. Mustafa and S. A. Messaoudi, General energy decay rates for a weakly damped Timoshenko system, J. Dyn. Control Syst., 16 (2010), 211-226 doi: 10.1007/s10883-010-9090-z.

[23]

C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-542. doi: 10.1016/j.aml.2004.03.017.

[24]

A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differ. Eqns, 29 (2003), 1-14.

[25]

S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 41 (1921), 744-746. doi: 10.1080/14786442108636264.

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