Stabilization in a high‐dimensional chemotaxis system involving arbitrary superlinear degradation

The chemotaxis‐consumption system with generalized logistic sourceu t = Δ u − ∇ · ( u S ( x , u , v ) · ∇ v ) + λ u − μ u α , x ∈ Ω , t > 0 ,v t = Δ v − u v , x ∈ Ω , t > 0 ,is considered under homogeneous Neumann boundary conditions in a bounded smooth domain Ω ⊂ ℝ n ( n ≥ 1 ) with suitably regular positive initial data. Here λ ,  μ  > 0, α  > 1 and S ∈ C 2 ( Ω ¯× [ 0 , ∞ )2 ; ℝ n × n ) is a given matrix‐valued function. We construct globally defined solutions in an appropriately generalized sense and prove that these solutions converge to the spatially homogeneous equilibriumλ μ1 α − 1, 0in the large time limit.