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

<jats:p>The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given by</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0308210518000057_equ01" /></jats:disp-formula></jats:p><jats:p>under Neumann boundary conditions in a bounded domain <jats:italic>Ω ⊂</jats:italic> ℝ<jats:sup><jats:italic>n</jats:italic></jats:sup>, where <jats:italic>n</jats:italic> ≥ 1 is arbitrary. It is shown that whenever <jats:italic>U</jats:italic> : <jats:italic>Ω ×</jats:italic> (0,<jats:italic>∞</jats:italic>) <jats:italic>→</jats:italic> ℝ<jats:sup><jats:italic>n</jats:italic></jats:sup> is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (<jats:italic>⋆</jats:italic>) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces <jats:italic>L</jats:italic><jats:sup>1</jats:sup>(<jats:italic>Ω</jats:italic>) and <jats:italic>L<jats:sup>∞</jats:sup></jats:italic>(<jats:italic>Ω</jats:italic>), can be controlled from above and below by appropriate multiples of 1<jats:italic>/</jats:italic>(<jats:italic>t</jats:italic> + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (<jats:italic>⋆</jats:italic>) 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 <jats:italic>n</jats:italic> = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial <jats:italic>L</jats:italic><jats:sup>1</jats:sup> norms of solutions.</jats:p>