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

A general stability result in a memory-type Timoshenko system

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.
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, 347 (2009), 867. Google Scholar

[2]

F. Alabau-Bousosuira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonl. Differ. Eqns. Appl.,, \textbf{14} (2007), 14 (2007), 643. 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. doi: 10.1016/S0022-0396(03)00185-2. Google Scholar

[4]

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

[7]

A. Guesmia and S. A. Messaoudi, On the control of solutions of a viscoelastic equation,, Appl. Math. Comput., 206 (2008), 589. 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. 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. 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. 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: 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[21]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068. 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. 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. 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. 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. 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, 347 (2009), 867. Google Scholar

[2]

F. Alabau-Bousosuira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonl. Differ. Eqns. Appl.,, \textbf{14} (2007), 14 (2007), 643. 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. doi: 10.1016/S0022-0396(03)00185-2. Google Scholar

[4]

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

[7]

A. Guesmia and S. A. Messaoudi, On the control of solutions of a viscoelastic equation,, Appl. Math. Comput., 206 (2008), 589. 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. 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. 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. 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: 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[21]

J. E. Muñoz Rivera and R. Racke, Timoshenko systems with indefinite damping,, J. Math. Anal. Appl., 341 (2008), 1068. 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. 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. 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. 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. doi: 10.1080/14786442108636264. Google Scholar

[1]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021

[2]

Donatella Donatelli, Corrado Lattanzio. On the diffusive stress relaxation for multidimensional viscoelasticity. Communications on Pure & Applied Analysis, 2009, 8 (2) : 645-654. doi: 10.3934/cpaa.2009.8.645

[3]

Luci H. Fatori, Marcio A. Jorge Silva, Vando Narciso. Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6117-6132. doi: 10.3934/dcds.2016067

[4]

Baowei Feng. On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4729-4751. doi: 10.3934/dcds.2017203

[5]

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

[6]

Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure & Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667

[7]

J.E. Muñoz Rivera, Reinhard Racke. Global stability for damped Timoshenko systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1625-1639. doi: 10.3934/dcds.2003.9.1625

[8]

Ammar Khemmoudj, Taklit Hamadouche. General decay of solutions of a Bresse system with viscoelastic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4857-4876. doi: 10.3934/dcds.2017209

[9]

Salim A. Messaoudi, Jamilu Hashim Hassan. New general decay results in a finite-memory bresse system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1637-1662. doi: 10.3934/cpaa.2019078

[10]

Junxiong Jia, Jigen Peng, Kexue Li. On the decay and stability of global solutions to the 3D inhomogeneous MHD system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 745-780. doi: 10.3934/cpaa.2017036

[11]

Marta Lewicka, Piotr B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations & Control Theory, 2013, 2 (2) : 337-353. doi: 10.3934/eect.2013.2.337

[12]

Hualin Zheng. Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system. Kinetic & Related Models, 2015, 8 (3) : 559-585. doi: 10.3934/krm.2015.8.559

[13]

Salah Drabla, Salim A. Messaoudi, Fairouz Boulanouar. A general decay result for a multi-dimensional weakly damped thermoelastic system with second sound. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1329-1339. doi: 10.3934/dcdsb.2017064

[14]

Ammar Khemmoudj, Yacine Mokhtari. General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3839-3866. doi: 10.3934/dcds.2019155

[15]

Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure & Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721

[16]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423

[17]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197

[18]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541

[19]

Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511

[20]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]