Abstract
We consider the stability properties for thermoelastic Bresse system which describes the motion of a linear planar shearable thermoelastic beam. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. We use multiplier techniques to prove the exponential stability result when the wave speed of the vertical displacement coincides with the wave speed of the longitudinal or of the shear angle displacement. Moreover, the existence of the global attractor is firstly achieved.
Highlights
For the boundary conditions w3 x, t φ2 x, t θ3x x, t 0, at x 0, l, 1.9 they obtained exponential stability for the thermoelastic Timoshenko system 1.8 when E G; later, they proved energy decay for a Timoshenko-type system with history in thermoelasticity of type III 3, and this paper is similar to 2 with an extra damping that comes from the presence of a history term; it improves the result of 2 in the sense that the case of nonequal wave speed has been considered and the relaxation function g is allowed to decay exponentially or polynomially
We refer the reader to 4–10 for the Timoshenko system with other kinds of damping mechanisms such as viscous damping, viscoelastic damping of Boltzmann type acting on the motion equation of w3 or φ2
The transverse displacement w3 is only indirectly damped through the coupling, which can be observed from 1.2. The effectiveness of this damping depends on the type of coupling and the wave speeds
Summary
Copyright q 2010 Zhiyong Ma. Copyright q 2010 Zhiyong Ma We consider the stability properties for thermoelastic Bresse system which describes the motion of a linear planar shearable thermoelastic beam. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. We use multiplier techniques to prove the exponential stability result when the wave speed of the vertical displacement coincides with the wave speed of the longitudinal or of the shear angle displacement.
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