Elliptic equation of second order (Q2757055)

The author considers an abstract elliptic equation of second order NEWLINE\[NEWLINE\frac{\partial^2 u}{\partial t^2}+B\;\frac{du}{dt}+Cu=0,NEWLINE\]NEWLINE defined on \([0,+\infty)\), where the solution \(u(t)\) has values in a complex Hilbert space \(H\) and \(B\), \(C\) are two selfadjoint operators with dense domains on \(H\). Assuming some ellipticity condition and adding an initial condition \(u(0)=u_0\), the author proves the existence and the uniqueness of the solution of the above problem.