A stability theorem for a class of linear evolution systems (Q1301581)

The following abstract Cauchy problem \[ u_{t}(t)=Bu(t),\quad t>0,\quad u(0)=u_{0}, \] is considered, where the densely defined linear operator \(B\) is the infinitesimal generator of the semigroup of operators \(T(t,B)\) of class \(C_{0}\). The authors study the solutions \(u(t)\) extending to \(Pu_{0}\) when \(t\) tends to infinity, where \(P\) is the spectral eigenprojection onto the null space of \(B\). They derive the conditions guaranteeing stability of the solution to the considered problem.