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 \[ \begin {array}{*{20}{c}} {{u_t} - \Delta u + \lambda {u^{ - \beta }}{\chi _{u > 0}} = 0\quad {\text {in}}\;D \times [0,\infty ),} \\ {u = \left \{ \begin {gathered} 1\qquad \quad {\text {on}}\;\partial D \times (0,\infty ), \hfill \\ {u_0} > 0\quad {\text {on}}\;\bar D \times \{ 0\} \hfill \\ \end {gathered} \right .} \\ \end {array} \] 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 $\{ (x,t)|u = 0\}$ is a bounded subset of $D \times [0,\infty )$.