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 $C^2_{loc}$ nonlinear map. The DSM (dynamical systems method) for solving equ$$F(v):=Lv+gv=0$consists of solving the Cauchy problem$\dot {u}=\Phi(t,u)$,$u(0)=u_0$, where$\Phi$is a suitable operator, and proving that i)$\exists u(t) \quad \forall t>0$, ii)$\exists u(\infty)$, and iii)$F(u(\infty))=0$.