Explicit solution of gravity-limited frost heave ($\eta =[1+({\eta }_0^{7/4}-1)e^{-7\tau /4}{]}^{4/7}$)

For ${\eta }^{\prime }={\eta }^{-3/4}-\eta$, set $u={\eta }^{7/4}$ to get $u^{\prime }=(7/4)(1-u)$. The displayed solution approaches the stable equilibrium $\eta =1$. From the formal zero start it behaves as $(7\tau /4{)}^{4/7}$ at early times and $1-(4/7)e^{-7\tau /4}$ at late times. The zero-start asymptote is an intermediate reduced-model regime, since the physical thin-film approximation fails at vanishing ice thickness.