LAUSR

Abstract This paper deals with the nutrient taxis system { u t = Δ u − ∇ ⋅ ( u ∇ v ) , 0 = Δ v − u v − μ v + r ( x , t ) , in a bounded domain Ω ⊂ R n , n ≥ 1 , with smooth boundary, where μ ≥ 0 is a parameter and r ∈ C 1 ( Ω ‾ × [ 0 , ∞ ) ) is a given nonnegative function. It is shown that for any prescribed initial data u 0 ∈ W 1 , ∞ ( Ω ) with u 0 > 0 in Ω ‾ , the corresponding Neumann initial–boundary problem admits a global classical solution. With regard to qualitative aspects, it is moreover, inter alia, seen that if r additionally satisfies ∫ t t + 1 ∫ Ω | ∇ r | 2 → 0 as  t → ∞ , then in the large time limit the solution component u stabilizes toward the constant 1 | Ω | ∫ Ω u 0 with respect to the norm in L 1 ( Ω ) , and that if furthermore sup t > 0 ⁡ ‖ r ( ⋅ , t ) ‖ L q ( Ω ) ∞ for some q ≥ 1 fulfilling q > n 2 , then u is uniformly bounded.