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
Categories: math.AP