Asymptotic stability of semigroups associated to linear weak dissipative systems (Q1764989)

The authors study the stability of the \(C_0\)-semigroups associated with the initial problem \(C u_{tt} + Au + Bu_t=0, u(0)=u_0, u_t(0)= u_1\), where \(A, B, C\) are self-adjoint positive operators such that their domains \(D(A)\subset D(C) \subset D(B)\) are dense in a Hilbert space \(H\). Supposing that \(\widetilde {A} = C^{-1}A\) and \(\widetilde {B} = C^{-1}B\) are self-adjoint positive defined operators, the authors consider a new Cauchy problem on \(D(\widetilde {A}^{1/2}) \times H\), generated by a block operator \(A_B\) defined in terms of \(\widetilde {A}\) and \(\widetilde {B}\). The first main result of the paper states that \(A_B\) generates a contraction semigroup \(\{S_B(t)\}_{t\geq 0}\) in \(D(\widetilde {A}^{1/2}) \times H\), if \(\widetilde {A}\) is bijective. Under certain assumptions, the authors prove that the initial problem is dissipative, but the semigroup \(S_B\) is not exponentially stable. The paper is nicely written and the results are applied in three interesting examples.