Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth

In this paper we study the chemotaxis-system (Fourmula Prestend) defined in a convex smooth and bounded domain of Rn, n ≥ 1, with x > 0 and endowed with homogeneous Neumann boundary conditions. The source g behaves similarly to the logistic function and satisfies g(s) ≤ a - bsα, for s ≥ 0, with a ≥ 0, b > 0 and α > 1. Continuing the research initiated in [33], where for appropriate 1 < p < α < 2 and (u0; v0) ∈ C0 (ω) × C2 (ω) the global existence of very weak solutions (u; v) to the system (for any n ≥ 1) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when n = 3, we establish that - for all > 0 an upper bound for a b ; jju0jjL1(); jjv0jjW2;α(ω) can be prescribed in a such a way that (u; v) is bounded and Hölder continuous beyond - for all (u0; v0), and suficiently small ratio a b , there exists a T > 0 such that (u; v) is bounded and Hölder continuous beyond T. Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.