Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains

Abstract

The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given by

under Neumann boundary conditions in a bounded domain Ω ⊂ ℝⁿ, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝⁿ is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L¹(Ω) and L^∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L¹ norms of solutions.