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

We prove convergence of positive solutions to ut=uΔu+u∫Ω|∇u|2,u|∂Ω=0,u(⋅,0)=u0 in a bounded domain Ω⊂RnΩ⊂Rn, n≥1n≥1, with smooth boundary in the case of ∫Ωu0=1∫Ωu0=1 and identify the W01,2(Ω)-limit of u(t)u(t) as t→∞t→∞ as the solution of the corresponding stationary problem. This behaviour is different from the cases of ∫Ωu0<1∫Ωu0<1 and ∫Ωu0>1∫Ωu0>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 ∫Ω|∇u|2∫Ω|∇u|2 along trajectories and the analysis of an associated constrained minimization problem.