  The purpose of this paper is to study the asymptotic behavior of the positive
solutions of the problem $$ \partial_t u-\Delta u=a u-b(x) u^p \text{in}
\Omega\times \R^+, u(0)=u_0, u(t)|_{\partial \Omega}=0 $$ as $p\to +\infty$,
where $\Omega$ is a bounded domain and $b(x)$ is a nonnegative function. We
deduce that the limiting configuration solves a parabolic obstacle problem, and
afterwards we fully describe its long time behavior.
