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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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