Analyticity of thermo-elastic semigroups with coupled hinged/Neumann boundary conditions (Q5926223)

The authors consider a thermo-elastic plate system where the elastic equation does not account for rotation forces. Let \({\mathcal A}\) be a positive, selfadjoint operator in \(L_2(\Omega)\): NEWLINE\[NEWLINE{\mathcal A}h=\Delta^2 h,\quad D({\mathcal A})= \{h\in H^4(\Omega)\cap H^1_0(\Omega):\Delta h+(1- \mu)B_1h= 0\text{ on }\Gamma\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\mathcal A}_Dh= -\Delta h,\quad D({\mathcal A}_D)= H^2(\Omega)\cap H^1_0(\Omega),NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\mathcal A}_Nh= -\Delta h,\quad D({\mathcal A}_N)= \Biggl\{h\in H^2(\Omega): \Biggl[{\partial h\over\partial v}+ bh\Biggr]_\Gamma= 0\Biggr\}.NEWLINE\]NEWLINE Theorem. Let the operator \(A\) be given by the formula NEWLINE\[NEWLINEA\left[\begin{matrix} w_1\\ w_2\\ \theta\end{matrix}\right]= \left[\begin{matrix} w_2\\ -{\mathcal A}[G(\theta|_\Gamma)+ w_1]+ {\mathcal A}_N\theta\\ -{\mathcal A}_D w_2- {\mathcal A}_N\theta\end{matrix}\right],NEWLINE\]NEWLINE NEWLINE\[NEWLINED(A)= \{w_1\in D({\mathcal A}_D)= D({\mathcal A}^{1/2}), w_2\in D({\mathcal A}_D), \theta\in D({\mathcal A}_N),\;w_1+ G(\theta/_\Gamma)\in D({\mathcal A})\}.NEWLINE\]NEWLINE Then the contraction semigroup \(e^{tA}\) is analytic on NEWLINE\[NEWLINEY- [H^2(\Omega)\cap H^1_0(\Omega]\times L_2(\Omega)\times L_2(\Omega)= D({\mathcal A}^{1/2})\times L_2(\Omega)\times L_2(\Omega).NEWLINE\]NEWLINE Moreover, there exist constants \(M\geq 1\) and \(\sigma>0\) such that \(\|e^{tA}\|_{L(y)}\leq Me^{-\sigma t}\), \(t\geq 0\).