On the uniqueness of the classical solution of the fingering problem in a Hele–Shaw cell with surface tension

The existence of a classical solution was established for a one-phase radial viscous fingering problem in a Hele–Shaw cell under surface tension (original problem) by means of parabolic regularization for a certain subsequence $\{\varepsilon_n\}_{n \in \mathbb{N}}$, $\varepsilon_n>0$. In this paper, we prove the uniqueness of the classical solution to the original problem with the use of parabolic regularization for the full sequence of the parameter $\{\varepsilon\}$, $\varepsilon>0$.