Stabilization of solutions of weakly singular quenching problems

In this paper we prove that if 0 0, then every element of the ω-limit set of weak solutions of u t − Δu + λu −β χu>0 = 0 in D × [0, ∞), u = 1 on ∂D × (0, ∞), u 0 > 0 on D × {0} is a weak stationary solution of this problem. A consequence of this is that if D is a ball, λ is sufficiently small, and u 0 is a radial, then the set {(x, t)|u = 0} is a bounded subset of D × [0, ∞)