= Finite-time extinction of a subcritical spherical seed
{title2=$t_e=\frac{Sa_c^2}{\kappa}[-\log(1-\eta_0)-\eta_0-\eta_0^2/2]$}
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, $a^3\sim3\kappa a_c(t_e-t)/S$. The time diverges logarithmically as the initial radius approaches the critical radius from below.
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