Uniform decay in a Timoshenko-type system with past history
The authors deal with the initial-boundary value problem for a Timoshenko beam in a form \[ \begin{cases} \rho_1\varphi_{tt}(x,t)-K(\varphi_x+\psi)_x=0,\\ \rho_2\psi_{tt}(x,t)-b\psi_{xx}+\int_0^\infty g(s)\psi_{xx}(t-s,.)\,ds+ K(\varphi_x+\psi)=0,\\ \varphi(0,t)=\varphi(1,t)=\psi(0,t)=\psi(1,t)=0,\;t>0,\\ \varphi(.,0)=\varphi_0,\;\varphi_t(.,0)=\varphi_1,\;\psi(.,0)=\psi_0,\;\psi_t(.,0)=\psi_1, \end{cases} \] where \(\rho_1, \rho_2, K, b\) are positive constants and \(g\) is a differentiable function satisfying for a positive \(k_0\) and \(1\leq p\leq 3/2\) the conditions \(g(t)>0\), \(\hat b=b-\int_0^\infty g(s)\,ds>0\), \(g'(t)\leq -k_0g^p(t)\). They show that, for \(\frac {\rho_1}K=\frac {\rho_2}b\), the first energy decays exponentially if \(p=1\) and polynomially if \(p>1\). For \(\frac {\rho_1}K\neq\frac {\rho_2}b\), the decay is in the rate of \(1/t^p\) provided that the initial data are regular enough.
scientific article; zbMATH DE number 5626922
scientific article