Decay of global solutions, stability and blowup for a reaction-diffusion problem with free boundary

We consider a one-phase Stefan problem for the heat equation with a nonline ar reaction term. We first exhibit an energy condition, involving the initi al data, under which the solution blows up in finite time in L-infinity nor m. We next prove that all global solutions are bounded and decay uniformly to 0, and that either: (i) the free boundary converges to a finite limit an d the solution decays at an exponential rate, or (ii) the free boundary gro ws up to infinity and the decay rate is at most polynomial. Finally, we sho w that small data solutions behave like (i).