Definition 2.8. For ρ 0 ∈ Z E,p and ρ ∈ AC([0, ∞); P p (X)) a curve of maximal slope with ρ(0) = ρ 0 , we denote by ω ρ (ρ 0 ) the ω-limit set with initial **data** ρ 0 subordinate to ρ, defined by
ω ρ (ρ 0 ) := ρ ∈ P p (X) : ∃t n → ∞ s.t. ρ(t n ) σ →ρ as n → ∞ .
We now give a characterisation of stationary states in relation to the weak upper gradient G which we will use extensively in the sequel.