Heat-viscoelastic plate interaction via bending moment and shear forces operators: analyticity, spectral analysis, exponential decay (Q2198167)

The author considers a heat-plate interaction model where the 2D plate is subject to viscoelastic (strong) damping. He considers the bounded domain \(\Omega _{f}\subset \mathbb{R}^{2}\) which contains the plate domain \(\Omega_{s}\) and the problem \(u_{t}-\Delta u=0\) in \((0,T]\times \Omega _{f}\), \(w_{tt}+\Delta ^{2}w+\Delta ^{2}w_{t}+w=0\) in \((0,T]\times \Omega _{s}\). The boundary conditions \([\Delta (w+w_{t})+(1-\mu )B_{1}w_{t}]_{\Gamma _{s}}=0\), \(\frac{\partial (w+w_{t})}{\partial \nu }+(1-\mu )B_{2}w_{t}-w_{t}=-\frac{\partial u}{\partial \nu }\), \(u\mid _{\partial \Omega _{s}}=w_{t}\mid_{\partial \Omega _{s}}=0\), on \((0,T]\times \partial \Omega _{s}\), \(u\mid_{\partial \Omega _{f}}=0\), on \((0,T]\times \partial \Omega _{f}\) are imposed. Here, \(0<\mu <1\) is the Poisson modulus, \(B_{1}\) is the bending moments operator and \(B_{2}\) is the shear forces operator. The author introduces the Hilbert space \(\mathbf{H}\equiv H^{2}(\Omega _{s})\times L^{2}(\Omega _{s})\times L^{2}(\Omega _{f})\) and he rewrites this problem in an abstract form: \(\frac{d}{dt}\left[ \begin{array}{c} w \\ w_{t} \\ u \end{array} \right] =\mathcal{A}\left[ \begin{array}{c} w \\ w_{t} \\ u \end{array} \right] \), with the operator \(\mathcal{A}=\left[ \begin{array}{ccc} 0 & I & 0 \\ -\Delta ^{2}-I & -\Delta ^{2} & 0 \\ 0 & 0 & \Delta \end{array} \right] \). The main purpose of the paper consists to prove the maximal dissipative property of the operator \(\mathcal{A}\) whence properties of the associated semigroup. The author characterizes the domain of \(\mathcal{A}\) and the adjoint \(\mathcal{A}^{\ast }\) of \(\mathcal{A}\). He proves that \(\mathcal{A}\) and \(\mathcal{A}^{\ast }\) are dissipative, that \(\mathcal{A}\) is boundedly invertible, that \(\mathcal{A}\) is maximal dissipative on \(\mathbf{H}\), that the generator \(\mathcal{A}\) of the s.c. contraction semigroup \(e^{\mathcal{A}t}\) has no spectrum on the imaginary axis and satisfies a resolvent condition which implies that the s.c. contraction analytic semigroup is uniformly exponentially stable on \(\mathbf{H}\).