New decay rates for a Cauchy thermelastic laminated Timoshenko problem with interfacial slip under Fourier or Cattaneo laws

TL;DR

This paper investigates the decay rates of solutions for a laminated Timoshenko beam with interfacial slip under thermal effects modeled by Fourier or Cattaneo laws, revealing new stability conditions and polynomial decay estimates.

Contribution

It introduces new decay rate results and stability conditions for the Timoshenko system with thermal effects, considering different components affected by heat.

Findings

Both systems satisfy polynomial stability estimates in L2-norm.

Decay rate depends on initial data regularity.

New stability threshold between polynomial stability and convergence.

Abstract

The objective of the present paper is to investigate the decay of solutions for a laminated Timoshenko beam with interfacial slip in the whole space R subject to a thermal effect acting only on one component modelled by either Fourier or Cattaneo law. When the thermal effect is acting via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we obtain that both systems, Timoshenko-Fourier and Timoshenko-Cattaneo systems, satisfy the same polynomial stability estimates in the L2 -norm of the solution and its higher order derivatives with respect of the space variable. The decay rate depends on the regularity of the initial data. In addition, the presence and absence of the regularity-loss type property are determined by some relations between the parameters of systems. However, when the thermal effect is acting via the first comoponent of the system (transversal displacement), a new stability condition is introduced for both TimoshenkoFourier and Timoshenko-Cattaneo systems. This stability condition is in the form of threshold between polynomial stability and convergence to zero. To prove our results, we use the energy method in Fourier space combined with judicious choices of weight functions to build appropriate Lyapunov functionals.