The problem considered in this paper is the one-dimensional free boundary problem for the heat equation \(u_{xx}-u_ t=0\) with Cauchy data on the free boundary, namely \[ u(s(t),t)=r(t),\quad u_ x(s(t),t)=h(t), \] r and h being given continuous functions, in the the domain \(-\infty <x<s(t)\), \(t>0\), with \(s(0)=0\) and zero initial data.
Clearly the authors ignore that such a problem has been studied in the literature, also in much more general situations [see e.g. the reviewer and M. Primicerio, Riv. Mat. Univ. Parma, IV. Ser. 5, 615-634 (1979)], and that in the simple case considered it is generally reducible to a Stefan problem, thus guaranteeing existence and uniqueness, under suitable assumptions on the data.
In order to find a solution they write down a system of Volterra integral equations, using a representation formula for u in terms of thermal potentials and imposing the free boundary conditions. Of course one of the resulting equations contains an integral of the form \[ \int^{t}_{0}p(\tau)(t-\tau)^{-3/2} \exp [-(s(t)-s(\tau))^ 2/4(t- \tau)] d\tau, \] which the authors claim to be convergent (note that p(t) represents the derivative of s(t)), substituting it in the existence proof with a “regularized” approximation. The justification of such a procedure is explained in appendix 2, where implicit use is made of some higher regularity of the free boundary (e.g. bounded second derivative), which would be hard (if not impossible) to be proved, particularly under the sole assumption of continuity of the free boundary data, which are not even assumed (at least explicitly) to match the condition \(u(x,0)=0\).