Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity

We prove convergence of positive solutions to $$u_t=u\mathrm{\Delta }u+u{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2,u{\mid }_{\mathrm{\partial }\mathrm{\Omega }}=0,u(\cdot ,0)=u_0$$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 1$, with smooth boundary in the case of ${\int }_{\mathrm{\Omega }}{u}_0=1$ and identify the $W_0^{1,2}(\mathrm{\Omega })$-limit of $u(t)$ as $t\to \mathrm{\infty }$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of ${\int }_{\mathrm{\Omega }}{u}_0<1$ and ${\int }_{\mathrm{\Omega }}{u}_0>1$ 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 ${\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$ along trajectories and the analysis of an associated constrained minimization problem. Keywords: degenerate diffusion, nonlocal nonlinearity, long-term behaviour