## Journal of Integral Equations and Applications

### General and optimal decay in a memory-type Timoshenko system

#### Abstract

This paper is concerned with the following memory-type Timoshenko system $\rho _1\varphi _{tt}-K(\varphi _x+\psi )_x=0$ $\rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+ \displaystyle \int _0^tg(t-s)\psi _{xx}(s)\,ds=0,$ $(x,t)\in (0,L)\times (0,\infty )$, with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some constant $1\leq p\lt {3}/{2}$, $g'(t)\leq -\xi (t)g^p(t),\quad \mbox {for all }t\geq 0.$ We prove some decay results which generalize and improve many earlier results in the literature. In particular, our result gives the optimal decay for the case of polynomial stability.

#### Article information

Source
J. Integral Equations Applications, Volume 30, Number 1 (2018), 117-145.

Dates
First available in Project Euclid: 10 April 2018

https://projecteuclid.org/euclid.jiea/1523347341

Digital Object Identifier
doi:10.1216/JIE-2018-30-1-117

Mathematical Reviews number (MathSciNet)
MR3784885

Zentralblatt MATH identifier
06873401

#### Citation

Messaoudi, Salim A.; Hassan, Jamilu Hashim. General and optimal decay in a memory-type Timoshenko system. J. Integral Equations Applications 30 (2018), no. 1, 117--145. doi:10.1216/JIE-2018-30-1-117. https://projecteuclid.org/euclid.jiea/1523347341

#### References

• F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonlin. Diff. Eq. Appl. 14 (2007), 643–669.
• ––––, Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations, Nonlin. Diff. Eq. Appl. 18 (2011), 571–597.
• D.S. Almeida Júnior, M.L. Santos and J.E. Muñoz Rivera, Stability to weakly dissipative Timoshenko systems, Math. Meth. Appl. Sci. 36 (2013), 1965–1976.
• F. Ammar-Khodja, A. Benabdallah, J. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Diff. Eqs. 194 (2003), 82–115.
• T.A. Apalara, S.A. Messaoudi and A.A. Keddi, On the decay rates of Timoshenko system with second sound, Math. Meth. Appl. Sci. 39 (2015), 2671–2684.
• M.M. Cavalcanti, V.N. Domingos Cavalcanti, F.A. Falcão Nascimento, I. Lasiecka and J.H. Rodrigues, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. angew. Math. Phys. 65 (2014), 1189–1206.
• D. Feng, D. Shi and W. Zhang, Boundary feedback stabilization of Timoshenko beam with boundary dissipation, Sci. China Math. 41 (1998), 483–490.
• H.D. Fernández Sare and J.E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (2008), 482–502.
• A. Guesmia and S. Messaoudi, Some stability results for Timoshenko systems with cooperative frictional and infinite-memory dampings in the displacement, Acta Math. Sci. 36 (2016), 1–33.
• ––––, On the control of a viscoelastic damped Timoshenko-type system, Appl. Math. Comp. 206 (2008), 589–597.
• ––––, On the stabilization of Timoshenko systems with memory and different speeds of wave propagation, Appl. Math. Comp. 219 (2013), 9424–9437.
• ––––, A general stability result in a Timoshenko system with infinite memory: A new approach, Math. Meth. Appl. Sci. 37 (2014), 384–392.
• A. Guesmia, S.A. Messaoudi and A. Soufyane, Stabilization of a linear Timoshenko system with infinite history and applications to Timoshenko-heat systems, Electr. J. Differ. Eqs. 2012 (2012), 1–45.
• J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Contr. Optim. 25 (1987), 1417–1429.
• W. Liu, Y. Sun and G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term, Topol. Meth. Nonlin. Anal. (2016).
• Z. Liu and S. Zheng, Semigroups associated with dissipative systems, CRC Press, Boca Raton, 1999.
• S.A. Messaoudi, On the control of solutions of a viscoelastic equation, J. Franklin Inst. 344 (2007), 765–776.
• S.A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett. 66 (2017), 16–22.
• S.A. Messaoudi and M.I. Mustafa, On the internal and boundary stabilization of Timoshenko beams Nonlin. Diff. Eqs. Appl. 15 (2008), 655–671.
• ––––, A stability result in a memory-type Timoshenko system, Dynam. Syst. Appl. 18 (2009), 457–468.
• 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.
• J.E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systemsGlobal existence and exponential stability, J. Math. Anal. Appl. 276 (2002), 248–278.
• C. Raposo, J. Ferreira, M. Santos and N. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett. 18 (2005), 535–541.
• M. Santos, D, Almeida Júnior and J. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Diff. Eqs. 253 (2012), 2715–2733.
• A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electr. J. Diff. Eqs. 2003 (2003), 1–14.
• P.S. Timoshenko,On the correction for shear of the differential equation for transverse vibrations of prismatic bars, The London, Edinburgh, and Dublin Philos. Mag. J. Sci. 41 (1921), 744–746.