  In this paper, we consider the asymptotic behavior of the nonlocal parabolic
problem \[ u_{t}=\Delta u+\displaystyle\frac{\lambda
f(u)}{\big(\int_{\Omega}f(u)dx\big)^{p}}, x\in \Omega, t>0, \] with homogeneous
Dirichlet boundary condition, where $\lambda>0, p>0$, $f$ is nonincreasing. It
is found that: (a) For $0<p\leq1$, $u(x,t)$ is globally bounded and the unique
stationary solution is globally asymptotically stable for any $\lambda>0$; (b)
For $1<p<2$, $u(x,t)$ is globally bounded for any $\lambda>0$; (c) For $p=2$,
if $0<\lambda<2|\partial\Omega|^2$, then $u(x,t)$ is globally bounded, if
$\lambda=2|\partial\Omega|^2$, there is no stationary solution and $u(x,t)$ is
a global solution and $u(x,t)\to\infty$ as $t\to\infty$ for all $x\in\Omega$,
if $\lambda>2|\partial\Omega|^2$, there is no stationary solution and $u(x,t)$
blows up in finite time for all $x\in\Omega$; (d) For $p>2$, there exists a
$\lambda^*>0$ such that for $\lambda>\lambda^*$, or for
$0<\lambda\leq\lambda^*$ and $u_0(x)$ sufficiently large, $u(x,t)$ blows up in
finite time. Moreover, some formal asymptotic estimates for the behavior of
$u(x,t)$ as it blows up are obtained for $p\geq2$.
