Stabilization in a chemotaxis system modelling T-cell dynamics with simultaneous production and consumption of signals

In a smoothly bounded domain Ω ⊂ Rⁿ , n≥ 1 , this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system {eqnarray*} \{ {array}{l} u_t = Δ u - ∇ · (u∇ v), \\[5pt] v_t = Δ v + u - α uv, {array} . {eqnarray*} with parameter α > 0 and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics. It is shown that there exists δ _ =δ _ (n)> 0 such that for any given α ≥ 1/δ _ and for any suitably regular initial data satisfying v(·, 0)≤ δ _ , this problem admits a unique classical solution that stabilizes to the constant equilibrium (1/|Ω |∫ _Ω u(·, 0), \, 1/α ) in the large time limit.