Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay
The authors consider the following Timoshenko system of thermoelasticity of type III with discrete delay (for \(x \in (0,1)\) and \(t \geqslant 0\)) \[ \begin{cases}\rho_{1}\phi_{tt}-K(\phi_{x}+\psi)_{x}+\mu_{1}\phi_{t}(x,t)+\mu_{2}\phi _{t}(x,t-\tau)=0,\\ \rho_{2}\psi_{tt}-b\psi_{xx}+K(\phi_{x}+\psi)+\beta\theta_{tx}=0,\\ \rho_{3}\theta_{tt}-\delta\theta_{xx}+\gamma\psi_{tx}-k\theta_{txx} =0. \end{cases} \] These equations are complemented with initial data and history \(\phi _{t}(x,t-\tau)=f_{0}(x,t-\tau)\), \(t\in(0,\tau)\) together with Dirichlet boundary conditions for the first two components and Neumann boundary conditions for the third component. After proving the well-posedness in an appropriate space, the authors determined mainly two types of decay of the energy. Namely, an exponential decay in case of equal wave speeds and a polynomial decay in case of nonequal wave speeds. This is established in spite of the presence of the delay in the system.
scientific article; zbMATH DE number 6478439
scientific article