Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension

Abstract

This paper deals with the nutrient taxis system{ut=Δu−∇⋅(u∇v),0=Δv−uv−μv+r(x,t), in a bounded domain Ω⊂Rn, n≥1, with smooth boundary, where μ≥0 is a parameter and r∈C1(Ω‾×[0,∞)) is a given nonnegative function.It is shown that for any prescribed initial data u0∈W1,∞(Ω) with u0>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∫tt+1∫Ω|∇r|2→0as t→∞, then in the large time limit the solution component u stabilizes toward the constant 1|Ω|∫Ωu0 with respect to the norm in L1(Ω), and that if furthermoresupt>0⁡‖r(⋅,t)‖Lq(Ω)<∞ for some q≥1 fulfilling q>n2, then u is uniformly bounded.