We consider the nonlinear Stefan problem
| { | ut-dΔu = au − bu2 | for x Ω(t), t > 0, |
| | u = 0 and ut = μ|∇xu|2 | for x ∂Ω(t), t > 0, |
| | u(0,x) = u0(x) | for x Ω0, |
where Ω(0)=Ω0 is an unbounded Lipschitz domain in ℝN, u0 > 0 in Ω0 and u0 vanishes on ∂Ω0. When Ω0 is bounded, the long-time behavior of this problem has been rather well-understood by Du et al. (J Differ Equ 250:4336-4366, 2011; J Differ Equ 253:996-1035, 2012; J Ellip Par Eqn 2:297-321, 2016; Arch Ration Mech Anal 212:957-1010, 2014). Here we reveal some interesting different behavior for certain unbounded Ω0. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded Ω0.