[{"publisher":"Walter de Gruyter GmbH","date_updated":"2025-12-18T20:10:00Z","date_created":"2025-12-18T19:09:41Z","author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"volume":24,"title":"A degenerate migration-consumption model in domains of arbitrary dimension","doi":"10.1515/ans-2023-0131","publication_status":"published","publication_identifier":{"issn":["2169-0375"]},"issue":"3","year":"2024","citation":{"mla":"Winkler, Michael. “A Degenerate Migration-Consumption Model in Domains of Arbitrary Dimension.” Advanced Nonlinear Studies, vol. 24, no. 3, Walter de Gruyter GmbH, 2024, pp. 592–615, doi:10.1515/ans-2023-0131.","short":"M. Winkler, Advanced Nonlinear Studies 24 (2024) 592–615.","bibtex":"@article{Winkler_2024, title={A degenerate migration-consumption model in domains of arbitrary dimension}, volume={24}, DOI={10.1515/ans-2023-0131}, number={3}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter GmbH}, author={Winkler, Michael}, year={2024}, pages={592–615} }","apa":"Winkler, M. (2024). A degenerate migration-consumption model in domains of arbitrary dimension. Advanced Nonlinear Studies, 24(3), 592–615. https://doi.org/10.1515/ans-2023-0131","ieee":"M. Winkler, “A degenerate migration-consumption model in domains of arbitrary dimension,” Advanced Nonlinear Studies, vol. 24, no. 3, pp. 592–615, 2024, doi: 10.1515/ans-2023-0131.","chicago":"Winkler, Michael. “A Degenerate Migration-Consumption Model in Domains of Arbitrary Dimension.” Advanced Nonlinear Studies 24, no. 3 (2024): 592–615. https://doi.org/10.1515/ans-2023-0131.","ama":"Winkler M. A degenerate migration-consumption model in domains of arbitrary dimension. Advanced Nonlinear Studies. 2024;24(3):592-615. doi:10.1515/ans-2023-0131"},"intvolume":" 24","page":"592-615","_id":"63264","user_id":"31496","language":[{"iso":"eng"}],"type":"journal_article","publication":"Advanced Nonlinear Studies","abstract":[{"lang":"eng","text":"Abstract\r\n In a smoothly bounded convex domain \r\n \r\n \r\n Ω\r\n ⊂\r\n \r\n \r\n R\r\n \r\n \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 with n ≥ 1, a no-flux initial-boundary value problem for\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n u\r\n \r\n \r\n t\r\n \r\n \r\n =\r\n Δ\r\n \r\n \r\n u\r\n ϕ\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 \r\n \r\n \r\n \r\n \r\n v\r\n \r\n \r\n t\r\n \r\n \r\n =\r\n Δ\r\n v\r\n −\r\n u\r\n v\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n$$\\begin{cases}_{t}={\\Delta}\\left(u\\phi \\left(v\\right)\\right),\\quad \\hfill \\\\ {v}_{t}={\\Delta}v-uv,\\quad \\hfill \\end{cases}$$\r\n\r\n \r\n \r\n is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by\r\n \r\n \r\n ϕ\r\n \r\n (\r\n \r\n ξ\r\n \r\n )\r\n \r\n =\r\n \r\n \r\n ξ\r\n \r\n \r\n α\r\n \r\n \r\n ,\r\n \r\n ξ\r\n ∈\r\n \r\n [\r\n \r\n 0\r\n ,\r\n \r\n \r\n ξ\r\n \r\n \r\n 0\r\n \r\n \r\n \r\n ]\r\n \r\n .\r\n \r\n \r\n$$\\phi \\left(\\xi \\right)={\\xi }^{\\alpha },\\qquad \\xi \\in \\left[0,{\\xi }_{0}\\right].$$\r\n\r\n \r\n \r\n By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy\r\n \r\n \r\n C\r\n \r\n (\r\n \r\n T\r\n \r\n )\r\n \r\n ≔\r\n \r\n \r\n ess sup\r\n \r\n \r\n t\r\n ∈\r\n \r\n (\r\n \r\n 0\r\n ,\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 Ω\r\n \r\n \r\n u\r\n \r\n (\r\n \r\n ⋅\r\n ,\r\n t\r\n \r\n )\r\n \r\n ln\r\n \r\n u\r\n \r\n (\r\n \r\n ⋅\r\n ,\r\n t\r\n \r\n )\r\n \r\n <\r\n ∞\r\n \r\n for all \r\n T\r\n >\r\n 0\r\n ,\r\n \r\n \r\n$$C\\left(T\\right){:=}\\underset{t\\in \\left(0,T\\right)}{\\text{ess\\,sup}}{\\int }_{{\\Omega}}u\\left(\\cdot ,t\\right)\\mathrm{ln}u\\left(\\cdot ,t\\right){< }\\infty \\qquad \\text{for\\,all\\,}T{ >}0,$$\r\n\r\n \r\n \r\n with sup\r\n T>0\r\n C(T) < ∞ if α ∈ [1, 2]."}],"status":"public"},{"publication":"Advanced Nonlinear Studies","type":"journal_article","abstract":[{"text":"AbstractThe chemotaxis–Stokes systemnt+u⋅∇n=∇⋅(D(n)∇n)−∇⋅(nS(x,n,c)⋅∇c),ct+u⋅∇c=Δc−nc,ut=Δu+∇P+n∇Φ,∇⋅u=0,\\left\\{\\begin{array}{l}{n}_{t}+u\\cdot \\nabla n=\\nabla \\cdot (D\\left(n)\\nabla n)-\\nabla \\cdot (nS\\left(x,n,c)\\cdot \\nabla c),\\\\ {c}_{t}+u\\cdot \\nabla c=\\Delta c-nc,\\\\ {u}_{t}=\\Delta u+\\nabla P+n\\nabla \\Phi ,\\hspace{1.0em}\\nabla \\cdot u=0,\\end{array}\\right.is considered in a smoothly bounded convex domainΩ⊂R3\\Omega \\subset {{\\mathbb{R}}}^{3}, with given suitably regular functionsD:[0,∞)→[0,∞)D:{[}0,\\infty )\\to {[}0,\\infty ),S:Ω¯×[0,∞)×(0,∞)→R3×3S:\\overline{\\Omega }\\times {[}0,\\infty )\\times \\left(0,\\infty )\\to {{\\mathbb{R}}}^{3\\times 3}andΦ:Ω¯→R\\Phi :\\overline{\\Omega }\\to {\\mathbb{R}}such thatD>0D\\gt 0on(0,∞)\\left(0,\\infty ). It is shown that if with some nondecreasingS0:(0,∞)→(0,∞){S}_{0}:\\left(0,\\infty )\\to \\left(0,\\infty )we have∣S(x,n,c)∣≤S0(c)c12for all(x,n,c)∈Ω¯×[0,∞)×(0,∞),| S\\left(x,n,c)| \\le \\frac{{S}_{0}\\left(c)}{{c}^{\\tfrac{1}{2}}}\\hspace{1.0em}\\hspace{0.1em}\\text{for all}\\hspace{0.1em}\\hspace{0.33em}\\left(x,n,c)\\in \\overline{\\Omega }\\times {[}0,\\infty )\\times \\left(0,\\infty ),then for allM>0M\\gt 0there existsL(M)>0L\\left(M)\\gt 0such that wheneverliminfn→∞D(n)>L(M)andliminfn↘0D(n)n>0,\\mathop{\\mathrm{liminf}}\\limits_{n\\to \\infty }D\\left(n)\\gt L\\left(M)\\hspace{1.0em}\\hspace{0.1em}\\text{and}\\hspace{0.1em}\\hspace{1.0em}\\mathop{\\mathrm{liminf}}\\limits_{n\\searrow 0}\\frac{D\\left(n)}{n}\\gt 0,for all sufficiently regular initial data(n0,c0,u0)\\left({n}_{0},{c}_{0},{u}_{0})fulfilling‖c0‖L∞(Ω)≤M\\Vert {c}_{0}{\\Vert }_{{L}^{\\infty }\\left(\\Omega )}\\le Man associated no-flux/no-flux/Dirichlet initial-boundary value problem admits a global bounded weak solution, classical if additionallyD(0)>0D\\left(0)\\gt 0. When combined with previously known results, this particularly implies global existence of bounded solutions whenD(n)=nm−1D\\left(n)={n}^{m-1},n≥0n\\ge 0, with arbitrarym>1m\\gt 1, but beyond this asserts global boundedness also in the presence of diffusivities which exhibit arbitrarily slow divergence to+∞+\\inftyat large densities and of possibly singular chemotactic sensitivities.","lang":"eng"}],"status":"public","_id":"63310","user_id":"31496","language":[{"iso":"eng"}],"publication_identifier":{"issn":["2169-0375"]},"publication_status":"published","issue":"1","year":"2022","intvolume":" 22","page":"88-117","citation":{"ieee":"M. Winkler, “Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings,” Advanced Nonlinear Studies, vol. 22, no. 1, pp. 88–117, 2022, doi: 10.1515/ans-2022-0004.","chicago":"Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures Involving Multiplicative Couplings.” Advanced Nonlinear Studies 22, no. 1 (2022): 88–117. https://doi.org/10.1515/ans-2022-0004.","ama":"Winkler M. Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings. Advanced Nonlinear Studies. 2022;22(1):88-117. doi:10.1515/ans-2022-0004","bibtex":"@article{Winkler_2022, title={Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings}, volume={22}, DOI={10.1515/ans-2022-0004}, number={1}, journal={Advanced Nonlinear Studies}, publisher={Walter de Gruyter GmbH}, author={Winkler, Michael}, year={2022}, pages={88–117} }","mla":"Winkler, Michael. “Chemotaxis-Stokes Interaction with Very Weak Diffusion Enhancement: Blow-up Exclusion via Detection of Absorption-Induced Entropy Structures Involving Multiplicative Couplings.” Advanced Nonlinear Studies, vol. 22, no. 1, Walter de Gruyter GmbH, 2022, pp. 88–117, doi:10.1515/ans-2022-0004.","short":"M. Winkler, Advanced Nonlinear Studies 22 (2022) 88–117.","apa":"Winkler, M. (2022). Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings. Advanced Nonlinear Studies, 22(1), 88–117. https://doi.org/10.1515/ans-2022-0004"},"publisher":"Walter de Gruyter GmbH","date_updated":"2025-12-18T20:05:30Z","volume":22,"date_created":"2025-12-18T19:29:40Z","author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael","id":"31496"}],"title":"Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings","doi":"10.1515/ans-2022-0004"},{"publication":"Advanced Nonlinear Studies","type":"journal_article","abstract":[{"lang":"eng","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."}],"status":"public","_id":"63340","user_id":"31496","language":[{"iso":"eng"}],"publication_identifier":{"issn":["1536-1365","2169-0375"]},"publication_status":"published","issue":"4","year":"2020","intvolume":" 20","page":"795-817","citation":{"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","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.","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.","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","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} }","short":"M. Winkler, Advanced Nonlinear Studies 20 (2020) 795–817.","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."},"date_updated":"2025-12-18T19:58:22Z","publisher":"Walter de Gruyter GmbH","volume":20,"author":[{"first_name":"Michael","last_name":"Winkler","id":"31496","full_name":"Winkler, Michael"}],"date_created":"2025-12-18T19:46:54Z","title":"Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions","doi":"10.1515/ans-2020-2107"}]