Dynamical systems method (DSM) for unbounded operators

Let $L$ be an unbounded linear operator in a real Hilbert space $H$, a generator of $C_0$ semigroup, and $g:H\to H$ be a $C_{loc}^2$ nonlinear map. The DSM (dynamical systems method) for solving equ$ $F(v):=Lv+gv=0$ consists of solving the Cauchy problem $\dot{u}=\mathrm{\Phi }(t,u)$, $u(0)=u_0$, where $\mathrm{\Phi }$ is a suitable operator, and proving that i) $\mathrm{\exists }u(t)\mathrm{\forall }t>0$, ii) $\mathrm{\exists }u(\mathrm{\infty })$, and iii) $F(u(\mathrm{\infty }))=0$.