Stabilization of solutions of weakly singular quenching problems

In this paper we prove that if $0 < \beta < 1,\;D \subset {R^N}$ is bounded, and $\lambda > 0$, then every element of the $\omega$-limit set of weak solutions of

is a weak stationary solution of this problem. A consequence of this is that if $D$ is a ball, $\lambda$ is sufficiently small, and ${u_0}$ is a radial, then the set $\{ (x,t)\vert u = 0\}$ is a bounded subset of $D \times [0,\infty )$.