Non-homogeneous thermoelastic Timoshenko systems
The authors consider the problem \[ \begin{cases} \rho_1\varphi _{tt}-\left( k(\varphi _{x}+\psi )\right) _x+\left(m\theta \right) _{x}=0,\;\text{ in }(0,l)\times \mathbb{R}^{+}, \\ \rho _{2}\psi _{tt}-\left( b\psi _{x}\right) _{x}+k(\varphi _{x}+\psi )-m\theta =0,\;\text{ in }(0,l)\times \mathbb{R}^{+}, \\ \rho _{3}\theta _{t}-\left( c\theta _{x}\right) _{x}+m(\varphi _{xt}+\psi_{t})=0,\;\text{ in }(0,l)\times \mathbb{R}^{+},\end{cases} \] with non-homogeneous coefficients and with initial conditions and Dirichlet boundary conditions. They proved that under the condition \[ \frac{k(x)}{\rho_1(x)}=\frac{b(x)}{\rho _2(x)} \] for \(x\) in some open subinterval of \((0,l),\) the system is exponentially stable. In case this condition is violated, then the stability is of polynomial type. To this end, they use the resolvent characterization of polynomial and exponential stability of linear semigroups methods. These results improve and extend considerably several existing results in the literature.
scientific article; zbMATH DE number 6816650
scientific article