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.

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

Giuseppe, Viglialoro
;
2018-01-01

Abstract

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.
2018
Chemotaxis; Keller-Segel Model; Global Existence
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11584/234001
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