Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one-dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle—a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established. Copyright © 2008 John Wiley & Sons, Ltd.

[1]  Abdelaziz Soufyane,et al.  Stabilisation de la poutre de Timoshenko , 1999 .

[2]  R. Racke.  Lectures on nonlinear evolution equations , 1992 .

[3]  Mauro de Lima Santos,et al.  Exponential stability for the Timoshenko system with two weak dampings , 2005, Appl. Math. Lett..

[4]  R. Racke.  Asymptotic behavior of solutions in linear 2- or 3-d thermoelasticity with second sound , 2003 .

[5]  S. Messaoudi.  On the decay of solutions for a class of quasilinear hyperbolic equations with non‐linear damping and source terms , 2005 .

[6]  R. Racke,et al.  Energy decay for Timoshenko systems of memory type , 2003 .

[7]  R. Racke,et al.  Timoshenko systems with indefinite damping , 2008 .

[8]  R. Racke,et al.  Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability☆ , 2002 .

[9]  J. U. Kim,et al.  Boundary control of the Timoshenko beam , 1987 .

[10]  M. Slemrod.  Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity , 1981 .

[11]  Genqi Xu,et al.  Stabilization of Timoshenko Beam by Means of Pointwise Controls , 2003 .

[12]  Zhuangyi Liu,et al.  Semigroups Associated with Dissipative Systems , 1999 .

[13]  Song Jiang,et al.  Evolution Equations in Thermoelasticity , 2000 .

[14]  Stephen W. Taylor.  A smoothing property of a hyperbolic system and boundary controllability , 2000 .

[15]  S. Timoshenko,et al.  LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars , 1921 .

[16]  Decay rates for solutions of a Timoshenko system with a memory condition at the boundary , 2002 .

[17]  Dexing Feng,et al.  Boundary feedback stabilization of Timoshenko beam with boundary dissipation , 1998 .

[18]  Dexing Feng,et al.  Exponential decay of Timoshenko beam with locally distributed feedback , 2001 .

[19]  Michael A. Tarabek.  On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound , 1992 .

[20]  Abdelaziz Soufyane,et al.  UNIFORM STABILIZATION FOR THE TIMOSHENKO BEAM BY A LOCALLY DISTRIBUTED DAMPING , 2003 .

[21]  S. Messaoudi,et al.  Blowup of solutions with positive energy in nonlinear thermoelasticity with second sound , 2004 .

[22]  Dexing Feng,et al.  EXPONENTIAL DECAY RATE OF THE ENERGY OF A TIMOSHENKO BEAM WITH LOCALLY DISTRIBUTED FEEDBACK , 2002 .

[23]  S. Messaoudi,et al.  Exponential stability in one‐dimensional non‐linear thermoelasticity with second sound , 2004 .

[24]  Jaime E. Muñoz Rivera,et al.  Global stability for damped Timoshenko systems , 2003 .

[25]  Hugo D. Fernández Sare,et al.  On the Stability of Damped Timoshenko Systems: Cattaneo Versus Fourier Law , 2009 .

[26]  S. Messaoudi.  LOCAL EXISTENCE AND BLOW UP IN NONLINEAR THERMOELASTICITY WITH SECOND SOUND , 2002 .

[27]  R. Racke.  Thermoelasticity with second sound—exponential stability in linear and non‐linear 1‐d , 2002 .