{"citation":{"bibtex":"@article{Cao_Winkler_2018, title={Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains}, volume={148}, DOI={<a href=\"https://doi.org/10.1017/s0308210518000057\">10.1017/s0308210518000057</a>}, number={5}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Cao, Xinru and Winkler, Michael}, year={2018}, pages={939–955} }","ieee":"X. Cao and M. Winkler, “Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains,” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>, vol. 148, no. 5, pp. 939–955, 2018, doi: <a href=\"https://doi.org/10.1017/s0308210518000057\">10.1017/s0308210518000057</a>.","mla":"Cao, Xinru, and Michael Winkler. “Sharp Decay Estimates in a Bioconvection Model with Quadratic Degradation in Bounded Domains.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>, vol. 148, no. 5, Cambridge University Press (CUP), 2018, pp. 939–55, doi:<a href=\"https://doi.org/10.1017/s0308210518000057\">10.1017/s0308210518000057</a>.","ama":"Cao X, Winkler M. Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>. 2018;148(5):939-955. doi:<a href=\"https://doi.org/10.1017/s0308210518000057\">10.1017/s0308210518000057</a>","chicago":"Cao, Xinru, and Michael Winkler. “Sharp Decay Estimates in a Bioconvection Model with Quadratic Degradation in Bounded Domains.” <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i> 148, no. 5 (2018): 939–55. <a href=\"https://doi.org/10.1017/s0308210518000057\">https://doi.org/10.1017/s0308210518000057</a>.","apa":"Cao, X., &#38; Winkler, M. (2018). Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains. <i>Proceedings of the Royal Society of Edinburgh: Section A Mathematics</i>, <i>148</i>(5), 939–955. <a href=\"https://doi.org/10.1017/s0308210518000057\">https://doi.org/10.1017/s0308210518000057</a>","short":"X. Cao, M. Winkler, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148 (2018) 939–955."},"title":"Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains","_id":"63369","year":"2018","page":"939-955","issue":"5","user_id":"31496","publication":"Proceedings of the Royal Society of Edinburgh: Section A Mathematics","publisher":"Cambridge University Press (CUP)","status":"public","author":[{"last_name":"Cao","full_name":"Cao, Xinru","first_name":"Xinru"},{"id":"31496","first_name":"Michael","full_name":"Winkler, Michael","last_name":"Winkler"}],"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"<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>"}],"doi":"10.1017/s0308210518000057","publication_status":"published","date_updated":"2025-12-19T11:03:03Z","date_created":"2025-12-19T11:02:55Z","intvolume":"       148","volume":148,"type":"journal_article","publication_identifier":{"issn":["0308-2105","1473-7124"]}}