{"volume":20,"publication_identifier":{"issn":["1536-1365","2169-0375"]},"type":"journal_article","date_created":"2025-12-18T19:46:54Z","intvolume":" 20","date_updated":"2025-12-18T19:58:22Z","publication_status":"published","doi":"10.1515/ans-2020-2107","language":[{"iso":"eng"}],"abstract":[{"text":"Abstract\r\n The chemotaxis-growth system\r\n \r\n \r\n ($\\star$)\r\n \r\n \r\n \r\n {\r\n \r\n \r\n \r\n \r\n u\r\n t\r\n \r\n \r\n \r\n \r\n \r\n \r\n =\r\n \r\n \r\n \r\n \r\n D\r\n \r\n Δ\r\n \r\n u\r\n \r\n -\r\n \r\n \r\n χ\r\n \r\n ∇\r\n \r\n ⋅\r\n \r\n (\r\n \r\n u\r\n \r\n \r\n ∇\r\n \r\n v\r\n \r\n \r\n )\r\n \r\n \r\n \r\n +\r\n \r\n ρ\r\n \r\n u\r\n \r\n \r\n -\r\n \r\n μ\r\n \r\n \r\n u\r\n α\r\n \r\n \r\n \r\n \r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n v\r\n t\r\n \r\n \r\n \r\n \r\n \r\n =\r\n \r\n \r\n \r\n d\r\n \r\n Δ\r\n \r\n v\r\n \r\n -\r\n \r\n κ\r\n \r\n v\r\n \r\n \r\n +\r\n \r\n λ\r\n \r\n u\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n {}\\left\\{\\begin{aligned} \\displaystyle{}u_{t}&\\displaystyle=D\\Delta u-\\chi% \\nabla\\cdot(u\\nabla v)+\\rho u-\\mu u^{\\alpha},\\\\ \\displaystyle v_{t}&\\displaystyle=d\\Delta v-\\kappa v+\\lambda u\\end{aligned}\\right.\r\n \r\n \r\n \r\n is considered under homogeneous Neumann boundary conditions in smoothly bounded domains \r\n \r\n \r\n \r\n Ω\r\n ⊂\r\n \r\n ℝ\r\n n\r\n \r\n \r\n \r\n \r\n {\\Omega\\subset\\mathbb{R}^{n}}\r\n \r\n , \r\n \r\n \r\n \r\n n\r\n ≥\r\n 1\r\n \r\n \r\n \r\n {n\\geq 1}\r\n \r\n . For any choice of \r\n \r\n \r\n \r\n α\r\n >\r\n 1\r\n \r\n \r\n \r\n {\\alpha>1}\r\n \r\n , the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of ($\\star$), the present work shows that, whenever \r\n \r\n \r\n \r\n α\r\n ≥\r\n \r\n 2\r\n -\r\n \r\n 2\r\n n\r\n \r\n \r\n \r\n \r\n \r\n {\\alpha\\geq 2-\\frac{2}{n}}\r\n \r\n , under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state \r\n \r\n \r\n \r\n (\r\n \r\n \r\n (\r\n \r\n ρ\r\n μ\r\n \r\n )\r\n \r\n \r\n 1\r\n \r\n α\r\n -\r\n 1\r\n \r\n \r\n \r\n ,\r\n \r\n \r\n λ\r\n κ\r\n \r\n \r\n \r\n \r\n (\r\n \r\n ρ\r\n μ\r\n \r\n )\r\n \r\n \r\n 1\r\n \r\n α\r\n -\r\n 1\r\n \r\n \r\n \r\n \r\n )\r\n \r\n \r\n \r\n {\\bigl{(}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}},\\frac{\\lambda}{% \\kappa}\\bigl{(}\\frac{\\rho}{\\mu}\\bigr{)}^{\\frac{1}{\\alpha-1}}\\bigr{)}}\r\n \r\n in the large time limit.","lang":"eng"}],"status":"public","author":[{"id":"31496","first_name":"Michael","full_name":"Winkler, Michael","last_name":"Winkler"}],"publisher":"Walter de Gruyter GmbH","publication":"Advanced Nonlinear Studies","user_id":"31496","issue":"4","page":"795-817","year":"2020","_id":"63340","title":"Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions","citation":{"short":"M. Winkler, Advanced Nonlinear Studies 20 (2020) 795–817.","apa":"Winkler, M. (2020). Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions. Advanced Nonlinear Studies, 20(4), 795–817. https://doi.org/10.1515/ans-2020-2107","chicago":"Winkler, Michael. “Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions.” Advanced Nonlinear Studies 20, no. 4 (2020): 795–817. https://doi.org/10.1515/ans-2020-2107.","ama":"Winkler M. Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions. Advanced Nonlinear Studies. 2020;20(4):795-817. doi:10.1515/ans-2020-2107","mla":"Winkler, Michael. “Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions.” Advanced Nonlinear Studies, vol. 20, no. 4, Walter de Gruyter GmbH, 2020, pp. 795–817, doi:10.1515/ans-2020-2107.","ieee":"M. Winkler, “Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions,” Advanced Nonlinear Studies, vol. 20, no. 4, pp. 795–817, 2020, doi: 10.1515/ans-2020-2107.","bibtex":"@article{Winkler_2020, title={Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions}, volume={20}, DOI={10.1515/ans-2020-2107}, number={4}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter GmbH}, author={Winkler, Michael}, year={2020}, pages={795–817} }"}}