Structural decomposition of thermo-elastic semigroups with rotational forces
\textit{J. E. Lagnese} in his monograph [Boundary stabilization of thin plates. SIAM Studies in Applied Mathematics, 10. Philadelphia: Society for Industrial and Applied Mathematics (1989; Zbl 0696.73034)] derived partial differential equations of linear thermoelastic plate in a bounded domain \(\Omega\) in \(\mathbb{R}^2\). If \(w\) is the vertical displacement and \(\theta\) is the relative temperature, in the linear homogeneous case these equations are \(w_{tt}- \gamma\Delta w_{tt}+ \Delta^2 w+\Delta\theta= 0\) in \((0,T]\times \Omega\equiv Q\), \(\theta_t- \Delta\theta- \Delta\omega_t=0\) in \(Q\), \(w(0,.)= w_0\), \(w_t(0,.)= w_1\), \(\theta(0,.)= \theta_0\) in \(\Omega\) (non-essential constants and lower-order terms are omitted). The equations are supplemented by boundary conditions on \(\partial\Omega\) where \(\gamma> 0\). In the present paper the above equations are associated with some canonical boundary conditions. At first, it is considered a challenging case of coupled boundary conditions: hinged mechanical boundary conditions and \(w|_\Sigma\equiv 0\), \([\Delta w+\theta]_\Sigma\equiv 0\), \([{\partial\theta\over\partial \nu}+ b\theta]_\Sigma\equiv 0\), \(b\geq 0\), \(\Sigma= (0,T]\times \Gamma\), where \(\nu\) is unit outward normal. This case is typical with the coupling between \(w\) and \(\theta\) in the second boundary conditions. For \(\gamma= 0\) it has been recently shown that, under all canonical boundary conditions, the above equations define an analytic semicontinuous contraction semigroup \([w_0, w_1,\theta_0)\to [w(t), w_t(t),\theta(t)]\) on a natural energy space. The present paper is entirely devoted to the case \(\gamma> 0\); here, the corresponding semicontinuous semigroup on a differential natural energy space displays radically different structural properties; nevertheless, the property of uniform exponential stability is preserved.
analytic semicontinuous contraction semigroup
scientific article; zbMATH DE number 1394455