 Finite-time extinction of a subcritical spherical seed

In the quasistatic spherical model, ${\eta }_0=a_0/a_c\in (0,1)$. Separating the radius equation gives $t_e=(S/\kappa ){\int }_0^{a_0}{a}^2/(a_c-a)da$, which is finite and evaluates to the displayed expression. Near disappearance, $a^3\sim 3\kappa a_c(t_e-t)/S$. The time diverges logarithmically as the initial radius approaches the critical radius from below.