Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial, and it was also proved that small data solutions behave like (i). Here we prove that there exist global solutions with slow decay and unbounded free boundary, i.e. of type (ii). Also, we establish uniform a priori estimates for all global solutions. Moreover, we provide a correction to an error in the proof of decay from Ghidouche, Souplet, & Tarzia [5].