  We prove convergence of positive solutions to \[ u_t = u\Delta u +
u\int_{\Omega} |\nabla u|^2, \qquad u\rvert_{\partial\Omega} =0, \qquad
u(\cdot,0)=u_0 \] in a bounded domain $\Omega\subset \mathbb{R}^n$, $n\ge 1$,
with smooth boundary in the case of $\int_\Omega u_0=1$ and identify the
$W_0^{1,2}(\Omega)$-limit of $u(t)$ as $t\to \infty$ as the solution of the
corresponding stationary problem. This behaviour is different from the cases of
$\int_\Omega u_0<1$ and $\int_\Omega u_0>1$ which are known to result in
convergence to zero or blow-up in finite time, respectively.
  The proof is based on a monotonicity property of $\int_{\Omega} |\nabla u|^2$
along trajectories and the analysis of an associated constrained minimization
problem.
  Keywords: degenerate diffusion, nonlocal nonlinearity, long-term behaviour
