[{"publisher":"Springer Science and Business Media LLC","_id":"53323","language":[{"iso":"eng"}],"doi":"10.1007/s10884-022-10167-w","user_id":"31496","author":[{"first_name":"Michael","last_name":"Winkler","full_name":"Winkler, Michael"}],"publication_identifier":{"issn":["1040-7294","1572-9222"]},"status":"public","year":"2022","title":"Slow Grow-up in a Quasilinear Keller–Segel System","date_updated":"2024-04-07T12:39:17Z","publication_status":"published","date_created":"2024-04-07T12:39:12Z","type":"journal_article","keyword":["Analysis"],"citation":{"mla":"Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations, Springer Science and Business Media LLC, 2022, doi:10.1007/s10884-022-10167-w.","bibtex":"@article{Winkler_2022, title={Slow Grow-up in a Quasilinear Keller–Segel System}, DOI={10.1007/s10884-022-10167-w}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022} }","ama":"Winkler M. Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations. Published online 2022. doi:10.1007/s10884-022-10167-w","ieee":"M. Winkler, “Slow Grow-up in a Quasilinear Keller–Segel System,” Journal of Dynamics and Differential Equations, 2022, doi: 10.1007/s10884-022-10167-w.","apa":"Winkler, M. (2022). Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations. https://doi.org/10.1007/s10884-022-10167-w","chicago":"Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations, 2022. https://doi.org/10.1007/s10884-022-10167-w.","short":"M. Winkler, Journal of Dynamics and Differential Equations (2022)."},"publication":"Journal of Dynamics and Differential Equations","abstract":[{"text":"AbstractIn a ball $$\\Omega =B_R(0)\\subset \\mathbb {R}^n$$\r\n \r\n Ω\r\n =\r\n \r\n B\r\n R\r\n \r\n \r\n (\r\n 0\r\n )\r\n \r\n ⊂\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n \r\n , $$n\\ge 2$$\r\n \r\n n\r\n ≥\r\n 2\r\n \r\n , the chemotaxis system $$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \\nabla \\cdot \\big ( D(u) \\nabla u \\big ) - \\nabla \\cdot \\big ( uS(u)\\nabla v\\big ), \\\\ 0 = \\Delta v - \\mu + u, \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\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 u\r\n t\r\n \r\n =\r\n ∇\r\n ·\r\n \r\n (\r\n \r\n D\r\n \r\n (\r\n u\r\n )\r\n \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 u\r\n S\r\n \r\n (\r\n u\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 0\r\n =\r\n Δ\r\n v\r\n -\r\n μ\r\n +\r\n u\r\n ,\r\n \r\n μ\r\n =\r\n \r\n 1\r\n \r\n |\r\n Ω\r\n |\r\n \r\n \r\n \r\n ∫\r\n Ω\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 \r\n (\r\n ⋆\r\n )\r\n \r\n \r\n \r\n \r\n \r\n \r\n is considered under no-flux boundary conditions, with a focus on nonlinearities $$S\\in C^2([0,\\infty ))$$\r\n \r\n S\r\n ∈\r\n \r\n C\r\n 2\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n which exhibit super-algebraically fast decay in the sense that with some $$K_S>0, \\beta \\in [0,1)$$\r\n \r\n \r\n K\r\n S\r\n \r\n >\r\n 0\r\n ,\r\n β\r\n ∈\r\n \r\n [\r\n 0\r\n ,\r\n 1\r\n )\r\n \r\n \r\n and $$\\xi _0>0$$\r\n \r\n \r\n ξ\r\n 0\r\n \r\n >\r\n 0\r\n \r\n , $$\\begin{aligned} S(\\xi )>0 \\quad \\text{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi ) \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n S\r\n \r\n (\r\n ξ\r\n )\r\n \r\n >\r\n 0\r\n \r\n \r\n and\r\n \r\n \r\n \r\n S\r\n ′\r\n \r\n \r\n (\r\n ξ\r\n )\r\n \r\n ≤\r\n -\r\n \r\n K\r\n S\r\n \r\n \r\n ξ\r\n \r\n -\r\n β\r\n \r\n \r\n S\r\n \r\n (\r\n ξ\r\n )\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n ξ\r\n ≥\r\n \r\n ξ\r\n 0\r\n \r\n .\r\n \r\n \r\n \r\n \r\n \r\n It is, inter alia, shown that if furthermore $$D\\in C^2((0,\\infty ))$$\r\n \r\n D\r\n ∈\r\n \r\n C\r\n 2\r\n \r\n \r\n (\r\n \r\n (\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n is positive and suitably small in relation to S by satisfying $$\\begin{aligned} \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0 \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n ξ\r\n S\r\n (\r\n ξ\r\n )\r\n \r\n \r\n D\r\n (\r\n ξ\r\n )\r\n \r\n \r\n ≥\r\n \r\n K\r\n \r\n SD\r\n \r\n \r\n \r\n ξ\r\n λ\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n ξ\r\n ≥\r\n \r\n ξ\r\n 0\r\n \r\n \r\n \r\n \r\n \r\n \r\n with some $$K_{SD}>0$$\r\n \r\n \r\n K\r\n \r\n SD\r\n \r\n \r\n >\r\n 0\r\n \r\n and $$\\lambda >\\frac{2}{n}$$\r\n \r\n λ\r\n >\r\n \r\n 2\r\n n\r\n \r\n \r\n , then throughout a considerably large set of initial data, ($$\\star $$\r\n ⋆\r\n ) admits global classical solutions (u, v) fulfilling $$\\begin{aligned} \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t) \\qquad \\text{ for } \\text{ all } t>0, \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n (\r\n t\r\n )\r\n \r\n C\r\n \r\n ≤\r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n \r\n ≤\r\n C\r\n z\r\n \r\n (\r\n t\r\n )\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n with some $$C=C^{(u,v)}\\ge 1$$\r\n \r\n C\r\n =\r\n \r\n C\r\n \r\n (\r\n u\r\n ,\r\n v\r\n )\r\n \r\n \r\n ≥\r\n 1\r\n \r\n , where z denotes the solution of $$\\begin{aligned} \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad t>0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n ′\r\n \r\n \r\n (\r\n t\r\n )\r\n \r\n =\r\n \r\n z\r\n 2\r\n \r\n \r\n (\r\n t\r\n )\r\n \r\n ·\r\n S\r\n \r\n (\r\n \r\n z\r\n \r\n (\r\n t\r\n )\r\n \r\n \r\n )\r\n \r\n ,\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n \r\n (\r\n 0\r\n )\r\n \r\n =\r\n \r\n ξ\r\n 0\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 which is seen to exist globally, and to satisfy $$z(t)\\rightarrow +\\infty $$\r\n \r\n z\r\n (\r\n t\r\n )\r\n →\r\n +\r\n ∞\r\n \r\n as $$t\\rightarrow \\infty $$\r\n \r\n t\r\n →\r\n ∞\r\n \r\n . As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in ($$\\star $$\r\n ⋆\r\n ) at logarithmic and doubly logarithmic rates, respectively.","lang":"eng"}]},{"volume":36,"user_id":"31496","_id":"63266","publisher":"Springer Science and Business Media LLC","page":"1677-1702","status":"public","citation":{"ieee":"M. Winkler, “Slow Grow-up in a Quasilinear Keller–Segel System,” Journal of Dynamics and Differential Equations, vol. 36, no. 2, pp. 1677–1702, 2022, doi: 10.1007/s10884-022-10167-w.","apa":"Winkler, M. (2022). Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations, 36(2), 1677–1702. https://doi.org/10.1007/s10884-022-10167-w","chicago":"Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations 36, no. 2 (2022): 1677–1702. https://doi.org/10.1007/s10884-022-10167-w.","short":"M. Winkler, Journal of Dynamics and Differential Equations 36 (2022) 1677–1702.","mla":"Winkler, Michael. “Slow Grow-up in a Quasilinear Keller–Segel System.” Journal of Dynamics and Differential Equations, vol. 36, no. 2, Springer Science and Business Media LLC, 2022, pp. 1677–702, doi:10.1007/s10884-022-10167-w.","bibtex":"@article{Winkler_2022, title={Slow Grow-up in a Quasilinear Keller–Segel System}, volume={36}, DOI={10.1007/s10884-022-10167-w}, number={2}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022}, pages={1677–1702} }","ama":"Winkler M. Slow Grow-up in a Quasilinear Keller–Segel System. Journal of Dynamics and Differential Equations. 2022;36(2):1677-1702. doi:10.1007/s10884-022-10167-w"},"doi":"10.1007/s10884-022-10167-w","language":[{"iso":"eng"}],"intvolume":" 36","publication_status":"published","date_updated":"2025-12-18T20:10:14Z","author":[{"id":"31496","full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"publication_identifier":{"issn":["1040-7294","1572-9222"]},"title":"Slow Grow-up in a Quasilinear Keller–Segel System","year":"2022","type":"journal_article","date_created":"2025-12-18T19:10:32Z","abstract":[{"text":"AbstractIn a ball $$\\Omega =B_R(0)\\subset \\mathbb {R}^n$$\r\n \r\n Ω\r\n =\r\n \r\n B\r\n R\r\n \r\n \r\n (\r\n 0\r\n )\r\n \r\n ⊂\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n \r\n , $$n\\ge 2$$\r\n \r\n n\r\n ≥\r\n 2\r\n \r\n , the chemotaxis system $$\\begin{aligned} \\left\\{ \\begin{array}{l}u_t = \\nabla \\cdot \\big ( D(u) \\nabla u \\big ) - \\nabla \\cdot \\big ( uS(u)\\nabla v\\big ), \\\\ 0 = \\Delta v - \\mu + u, \\qquad \\mu =\\frac{1}{|\\Omega |} \\int _\\Omega u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\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 u\r\n t\r\n \r\n =\r\n ∇\r\n ·\r\n \r\n (\r\n \r\n D\r\n \r\n (\r\n u\r\n )\r\n \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 u\r\n S\r\n \r\n (\r\n u\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 0\r\n =\r\n Δ\r\n v\r\n -\r\n μ\r\n +\r\n u\r\n ,\r\n \r\n μ\r\n =\r\n \r\n 1\r\n \r\n |\r\n Ω\r\n |\r\n \r\n \r\n \r\n ∫\r\n Ω\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 \r\n (\r\n ⋆\r\n )\r\n \r\n \r\n \r\n \r\n \r\n \r\n is considered under no-flux boundary conditions, with a focus on nonlinearities $$S\\in C^2([0,\\infty ))$$\r\n \r\n S\r\n ∈\r\n \r\n C\r\n 2\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n which exhibit super-algebraically fast decay in the sense that with some $$K_S>0, \\beta \\in [0,1)$$\r\n \r\n \r\n K\r\n S\r\n \r\n >\r\n 0\r\n ,\r\n β\r\n ∈\r\n \r\n [\r\n 0\r\n ,\r\n 1\r\n )\r\n \r\n \r\n and $$\\xi _0>0$$\r\n \r\n \r\n ξ\r\n 0\r\n \r\n >\r\n 0\r\n \r\n , $$\\begin{aligned} S(\\xi )>0 \\quad \\text{ and } \\quad S'(\\xi ) \\le -K_S\\xi ^{-\\beta } S(\\xi ) \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n S\r\n \r\n (\r\n ξ\r\n )\r\n \r\n >\r\n 0\r\n \r\n \r\n and\r\n \r\n \r\n \r\n S\r\n ′\r\n \r\n \r\n (\r\n ξ\r\n )\r\n \r\n ≤\r\n -\r\n \r\n K\r\n S\r\n \r\n \r\n ξ\r\n \r\n -\r\n β\r\n \r\n \r\n S\r\n \r\n (\r\n ξ\r\n )\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n ξ\r\n ≥\r\n \r\n ξ\r\n 0\r\n \r\n .\r\n \r\n \r\n \r\n \r\n \r\n It is, inter alia, shown that if furthermore $$D\\in C^2((0,\\infty ))$$\r\n \r\n D\r\n ∈\r\n \r\n C\r\n 2\r\n \r\n \r\n (\r\n \r\n (\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n is positive and suitably small in relation to S by satisfying $$\\begin{aligned} \\frac{\\xi S(\\xi )}{D(\\xi )} \\ge K_{SD}\\xi ^\\lambda \\qquad \\text{ for } \\text{ all } \\xi \\ge \\xi _0 \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n ξ\r\n S\r\n (\r\n ξ\r\n )\r\n \r\n \r\n D\r\n (\r\n ξ\r\n )\r\n \r\n \r\n ≥\r\n \r\n K\r\n \r\n SD\r\n \r\n \r\n \r\n ξ\r\n λ\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n ξ\r\n ≥\r\n \r\n ξ\r\n 0\r\n \r\n \r\n \r\n \r\n \r\n \r\n with some $$K_{SD}>0$$\r\n \r\n \r\n K\r\n \r\n SD\r\n \r\n \r\n >\r\n 0\r\n \r\n and $$\\lambda >\\frac{2}{n}$$\r\n \r\n λ\r\n >\r\n \r\n 2\r\n n\r\n \r\n \r\n , then throughout a considerably large set of initial data, ($$\\star $$\r\n ⋆\r\n ) admits global classical solutions (u, v) fulfilling $$\\begin{aligned} \\frac{z(t)}{C} \\le \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\Omega )} \\le Cz(t) \\qquad \\text{ for } \\text{ all } t>0, \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n (\r\n t\r\n )\r\n \r\n C\r\n \r\n ≤\r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n Ω\r\n )\r\n \r\n \r\n \r\n ≤\r\n C\r\n z\r\n \r\n (\r\n t\r\n )\r\n \r\n \r\n \r\n for\r\n \r\n \r\n all\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n with some $$C=C^{(u,v)}\\ge 1$$\r\n \r\n C\r\n =\r\n \r\n C\r\n \r\n (\r\n u\r\n ,\r\n v\r\n )\r\n \r\n \r\n ≥\r\n 1\r\n \r\n , where z denotes the solution of $$\\begin{aligned} \\left\\{ \\begin{array}{l}z'(t) = z^2(t) \\cdot S\\big ( z(t)\\big ), \\qquad t>0, \\\\ z(0)=\\xi _0, \\end{array} \\right. \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n ′\r\n \r\n \r\n (\r\n t\r\n )\r\n \r\n =\r\n \r\n z\r\n 2\r\n \r\n \r\n (\r\n t\r\n )\r\n \r\n ·\r\n S\r\n \r\n (\r\n \r\n z\r\n \r\n (\r\n t\r\n )\r\n \r\n \r\n )\r\n \r\n ,\r\n \r\n t\r\n >\r\n 0\r\n ,\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n z\r\n \r\n (\r\n 0\r\n )\r\n \r\n =\r\n \r\n ξ\r\n 0\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 which is seen to exist globally, and to satisfy $$z(t)\\rightarrow +\\infty $$\r\n \r\n z\r\n (\r\n t\r\n )\r\n →\r\n +\r\n ∞\r\n \r\n as $$t\\rightarrow \\infty $$\r\n \r\n t\r\n →\r\n ∞\r\n \r\n . As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in ($$\\star $$\r\n ⋆\r\n ) at logarithmic and doubly logarithmic rates, respectively.","lang":"eng"}],"issue":"2","publication":"Journal of Dynamics and Differential Equations"},{"publication_identifier":{"issn":["1040-7294","1572-9222"]},"author":[{"full_name":"Winkler, Michael","first_name":"Michael","last_name":"Winkler"}],"title":"Approaching Critical Decay in a Strongly Degenerate Parabolic Equation","year":"2020","intvolume":" 36","date_updated":"2024-04-07T12:37:45Z","publication_status":"published","language":[{"iso":"eng"}],"doi":"10.1007/s10884-020-09892-x","publication":"Journal of Dynamics and Differential Equations","issue":"S1","abstract":[{"text":"AbstractThe Cauchy problem in $${\\mathbb {R}}^n$$\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n , $$n\\ge 1$$\r\n \r\n n\r\n ≥\r\n 1\r\n \r\n , for the parabolic equation $$\\begin{aligned} u_t=u^p \\Delta u \\qquad \\qquad (\\star ) \\end{aligned}$$\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 u\r\n p\r\n \r\n Δ\r\n u\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 is considered in the strongly degenerate regime $$p\\ge 1$$\r\n \r\n p\r\n ≥\r\n 1\r\n \r\n . The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies $$\\begin{aligned} t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\rightarrow \\infty \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n →\r\n ∞\r\n \r\n as\r\n \r\n t\r\n →\r\n ∞\r\n .\r\n \r\n \r\n \r\n \r\n \r\n The first result of this study complements this by asserting that given any positive $$f\\in C^0([0,\\infty ))$$\r\n \r\n f\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n fulfilling $$f(t)\\rightarrow +\\infty $$\r\n \r\n f\r\n (\r\n t\r\n )\r\n →\r\n +\r\n ∞\r\n \r\n as $$t\\rightarrow \\infty $$\r\n \r\n t\r\n →\r\n ∞\r\n \r\n one can find a positive nondecreasing function $$\\phi \\in C^0([0,\\infty ))$$\r\n \r\n ϕ\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n such that whenever $$u_0\\in C^0({\\mathbb {R}}^n)$$\r\n \r\n \r\n u\r\n 0\r\n \r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n is radially symmetric with $$0< u_0 < \\phi (|\\cdot |)$$\r\n \r\n 0\r\n <\r\n \r\n u\r\n 0\r\n \r\n \r\n <\r\n ϕ\r\n (\r\n |\r\n \r\n ·\r\n \r\n |\r\n )\r\n \r\n \r\n , the corresponding minimal solution u satisfies $$\\begin{aligned} \\frac{t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)}}{f(t)} \\rightarrow 0 \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n \r\n \r\n f\r\n (\r\n t\r\n )\r\n \r\n \r\n →\r\n 0\r\n \r\n as\r\n \r\n t\r\n →\r\n ∞\r\n .\r\n \r\n \r\n \r\n \r\n \r\n Secondly, ($$\\star $$\r\n ⋆\r\n ) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data $$u_0$$\r\n \r\n u\r\n 0\r\n \r\n . It is shown that if the connected components of $$\\{u_0>0\\}$$\r\n \r\n {\r\n \r\n u\r\n 0\r\n \r\n >\r\n 0\r\n }\r\n \r\n comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to ($$\\star $$\r\n ⋆\r\n ) satisfies $$\\begin{aligned} 0< \\liminf _{t\\rightarrow \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\Big \\} \\le \\limsup _{t\\rightarrow \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\Big \\} <\\infty . \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n 0\r\n <\r\n \r\n lim inf\r\n \r\n t\r\n →\r\n ∞\r\n \r\n \r\n \r\n {\r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n \r\n }\r\n \r\n ≤\r\n \r\n lim sup\r\n \r\n t\r\n →\r\n ∞\r\n \r\n \r\n \r\n {\r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n 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 \r\n \r\n \r\n \r\n Under a somewhat complementary hypothesis, particularly fulfilled if $$\\{u_0>0\\}$$\r\n \r\n {\r\n \r\n u\r\n 0\r\n \r\n >\r\n 0\r\n }\r\n \r\n contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions.","lang":"eng"}],"date_created":"2024-04-07T12:37:38Z","type":"journal_article","keyword":["Analysis"],"status":"public","_id":"53322","publisher":"Springer Science and Business Media LLC","page":"3-23","volume":36,"user_id":"31496","citation":{"ama":"Winkler M. Approaching Critical Decay in a Strongly Degenerate Parabolic Equation. Journal of Dynamics and Differential Equations. 2020;36(S1):3-23. doi:10.1007/s10884-020-09892-x","bibtex":"@article{Winkler_2020, title={Approaching Critical Decay in a Strongly Degenerate Parabolic Equation}, volume={36}, DOI={10.1007/s10884-020-09892-x}, number={S1}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={3–23} }","mla":"Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation.” Journal of Dynamics and Differential Equations, vol. 36, no. S1, Springer Science and Business Media LLC, 2020, pp. 3–23, doi:10.1007/s10884-020-09892-x.","chicago":"Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation.” Journal of Dynamics and Differential Equations 36, no. S1 (2020): 3–23. https://doi.org/10.1007/s10884-020-09892-x.","short":"M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23.","apa":"Winkler, M. (2020). Approaching Critical Decay in a Strongly Degenerate Parabolic Equation. Journal of Dynamics and Differential Equations, 36(S1), 3–23. https://doi.org/10.1007/s10884-020-09892-x","ieee":"M. Winkler, “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation,” Journal of Dynamics and Differential Equations, vol. 36, no. S1, pp. 3–23, 2020, doi: 10.1007/s10884-020-09892-x."}},{"abstract":[{"lang":"eng","text":"AbstractThe Cauchy problem in $${\\mathbb {R}}^n$$\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n , $$n\\ge 1$$\r\n \r\n n\r\n ≥\r\n 1\r\n \r\n , for the parabolic equation $$\\begin{aligned} u_t=u^p \\Delta u \\qquad \\qquad (\\star ) \\end{aligned}$$\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 u\r\n p\r\n \r\n Δ\r\n u\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 is considered in the strongly degenerate regime $$p\\ge 1$$\r\n \r\n p\r\n ≥\r\n 1\r\n \r\n . The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies $$\\begin{aligned} t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\rightarrow \\infty \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n →\r\n ∞\r\n \r\n as\r\n \r\n t\r\n →\r\n ∞\r\n .\r\n \r\n \r\n \r\n \r\n \r\n The first result of this study complements this by asserting that given any positive $$f\\in C^0([0,\\infty ))$$\r\n \r\n f\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n fulfilling $$f(t)\\rightarrow +\\infty $$\r\n \r\n f\r\n (\r\n t\r\n )\r\n →\r\n +\r\n ∞\r\n \r\n as $$t\\rightarrow \\infty $$\r\n \r\n t\r\n →\r\n ∞\r\n \r\n one can find a positive nondecreasing function $$\\phi \\in C^0([0,\\infty ))$$\r\n \r\n ϕ\r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n [\r\n 0\r\n ,\r\n ∞\r\n )\r\n \r\n )\r\n \r\n \r\n such that whenever $$u_0\\in C^0({\\mathbb {R}}^n)$$\r\n \r\n \r\n u\r\n 0\r\n \r\n ∈\r\n \r\n C\r\n 0\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n is radially symmetric with $$0< u_0 < \\phi (|\\cdot |)$$\r\n \r\n 0\r\n <\r\n \r\n u\r\n 0\r\n \r\n \r\n <\r\n ϕ\r\n (\r\n |\r\n \r\n ·\r\n \r\n |\r\n )\r\n \r\n \r\n , the corresponding minimal solution u satisfies $$\\begin{aligned} \\frac{t^\\frac{1}{p}\\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)}}{f(t)} \\rightarrow 0 \\quad \\hbox {as } t\\rightarrow \\infty . \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n \r\n \r\n f\r\n (\r\n t\r\n )\r\n \r\n \r\n →\r\n 0\r\n \r\n as\r\n \r\n t\r\n →\r\n ∞\r\n .\r\n \r\n \r\n \r\n \r\n \r\n Secondly, ($$\\star $$\r\n ⋆\r\n ) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data $$u_0$$\r\n \r\n u\r\n 0\r\n \r\n . It is shown that if the connected components of $$\\{u_0>0\\}$$\r\n \r\n {\r\n \r\n u\r\n 0\r\n \r\n >\r\n 0\r\n }\r\n \r\n comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to ($$\\star $$\r\n ⋆\r\n ) satisfies $$\\begin{aligned} 0< \\liminf _{t\\rightarrow \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\Big \\} \\le \\limsup _{t\\rightarrow \\infty } \\Big \\{ t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty ({\\mathbb {R}}^n)} \\Big \\} <\\infty . \\end{aligned}$$\r\n \r\n \r\n \r\n \r\n \r\n 0\r\n <\r\n \r\n lim inf\r\n \r\n t\r\n →\r\n ∞\r\n \r\n \r\n \r\n {\r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n n\r\n \r\n )\r\n \r\n \r\n \r\n \r\n }\r\n \r\n ≤\r\n \r\n lim sup\r\n \r\n t\r\n →\r\n ∞\r\n \r\n \r\n \r\n {\r\n \r\n \r\n t\r\n \r\n 1\r\n p\r\n \r\n \r\n \r\n \r\n ‖\r\n u\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 L\r\n ∞\r\n \r\n \r\n (\r\n \r\n \r\n R\r\n \r\n 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 \r\n \r\n \r\n \r\n Under a somewhat complementary hypothesis, particularly fulfilled if $$\\{u_0>0\\}$$\r\n \r\n {\r\n \r\n u\r\n 0\r\n \r\n >\r\n 0\r\n }\r\n \r\n contains components with arbitrarily small principal eigenvalues of the associated Dirichlet Laplacian, it is finally seen that (0.1) continues to hold also for such not everywhere positive weak solutions."}],"issue":"S1","publication":"Journal of Dynamics and Differential Equations","type":"journal_article","date_created":"2025-12-18T19:10:01Z","publication_status":"published","date_updated":"2025-12-18T20:10:07Z","intvolume":" 36","year":"2020","title":"Approaching Critical Decay in a Strongly Degenerate Parabolic Equation","publication_identifier":{"issn":["1040-7294","1572-9222"]},"author":[{"id":"31496","last_name":"Winkler","first_name":"Michael","full_name":"Winkler, Michael"}],"doi":"10.1007/s10884-020-09892-x","language":[{"iso":"eng"}],"citation":{"chicago":"Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation.” Journal of Dynamics and Differential Equations 36, no. S1 (2020): 3–23. https://doi.org/10.1007/s10884-020-09892-x.","short":"M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23.","apa":"Winkler, M. (2020). Approaching Critical Decay in a Strongly Degenerate Parabolic Equation. Journal of Dynamics and Differential Equations, 36(S1), 3–23. https://doi.org/10.1007/s10884-020-09892-x","ieee":"M. Winkler, “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation,” Journal of Dynamics and Differential Equations, vol. 36, no. S1, pp. 3–23, 2020, doi: 10.1007/s10884-020-09892-x.","ama":"Winkler M. Approaching Critical Decay in a Strongly Degenerate Parabolic Equation. Journal of Dynamics and Differential Equations. 2020;36(S1):3-23. doi:10.1007/s10884-020-09892-x","bibtex":"@article{Winkler_2020, title={Approaching Critical Decay in a Strongly Degenerate Parabolic Equation}, volume={36}, DOI={10.1007/s10884-020-09892-x}, number={S1}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2020}, pages={3–23} }","mla":"Winkler, Michael. “Approaching Critical Decay in a Strongly Degenerate Parabolic Equation.” Journal of Dynamics and Differential Equations, vol. 36, no. S1, Springer Science and Business Media LLC, 2020, pp. 3–23, doi:10.1007/s10884-020-09892-x."},"status":"public","user_id":"31496","volume":36,"page":"3-23","_id":"63265","publisher":"Springer Science and Business Media LLC"},{"user_id":"31496","doi":"10.1007/s10884-017-9577-3","volume":30,"page":"331-358","publisher":"Springer Science and Business Media LLC","_id":"63378","language":[{"iso":"eng"}],"publication_status":"published","date_updated":"2025-12-19T11:07:30Z","intvolume":" 30","year":"2017","title":"One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity","status":"public","publication_identifier":{"issn":["1040-7294","1572-9222"]},"author":[{"full_name":"Winkler, Michael","first_name":"Michael","last_name":"Winkler","id":"31496"}],"type":"journal_article","date_created":"2025-12-19T11:07:24Z","issue":"1","publication":"Journal of Dynamics and Differential Equations","citation":{"bibtex":"@article{Winkler_2017, title={One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity}, volume={30}, DOI={10.1007/s10884-017-9577-3}, number={1}, journal={Journal of Dynamics and Differential Equations}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2017}, pages={331–358} }","ama":"Winkler M. One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity. Journal of Dynamics and Differential Equations. 2017;30(1):331-358. doi:10.1007/s10884-017-9577-3","mla":"Winkler, Michael. “One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity.” Journal of Dynamics and Differential Equations, vol. 30, no. 1, Springer Science and Business Media LLC, 2017, pp. 331–58, doi:10.1007/s10884-017-9577-3.","short":"M. Winkler, Journal of Dynamics and Differential Equations 30 (2017) 331–358.","chicago":"Winkler, Michael. “One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity.” Journal of Dynamics and Differential Equations 30, no. 1 (2017): 331–58. https://doi.org/10.1007/s10884-017-9577-3.","ieee":"M. Winkler, “One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity,” Journal of Dynamics and Differential Equations, vol. 30, no. 1, pp. 331–358, 2017, doi: 10.1007/s10884-017-9577-3.","apa":"Winkler, M. (2017). One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity. Journal of Dynamics and Differential Equations, 30(1), 331–358. https://doi.org/10.1007/s10884-017-9577-3"}}]