Asymptotic stabilization for a class of chemotaxis-consumption systems with generalized logistic source. (April 2022) - Lyu, Wenbin
Abstract: This paper is concerned with a chemotaxis-consumption system u t = ∇ ⋅ ( ∇ u − u S ( x, u, v ) ⋅ ∇ v ) + ρ u − μ u l, v t = Δ v − u v, under no-flux boundary conditions in a smooth bounded domain Ω ⊂ R n ( n ⩾ 1 ), where the chemotactic sensitivity tensor S ∈ C 2 ( Ω ¯ × [ 0, + ∞ ) 2 ; R n × n ) fulfills that there exists some nondecreasing function S 0 on [ 0, + ∞ ) such that | S ( x, u, v ) | ⩽ S 0 ( v ) for all ( x, u, v ) ∈ Ω ¯ × [ 0, + ∞ ) × [ 0, + ∞ ) . We show that for any ρ, μ > 0 and l > 1, any generalized solution of the above system asymptotically approaches to the nontrivial spatially homogeneous steady state ρ μ 1 l − 1, 0 as t → + ∞ .