[{"_id":"63311","user_id":"31496","article_number":"47","language":[{"iso":"eng"}],"publication":"Partial Differential Equations and Applications","type":"journal_article","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 degenerate 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 for $$p\\ge 1$$\r\n \r\n p\r\n ≥\r\n 1\r\n \r\n . It is shown 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 and $$g\\in C^0([0,\\infty ))$$\r\n \r\n g\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 satisfying $$\\begin{aligned} f(t)\\rightarrow + \\infty \\quad \\text{ and } \\quad g(t)\\rightarrow 0 \\qquad \\text{ as } t\\rightarrow \\infty , \\end{aligned}$$\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 \r\n \r\n and\r\n \r\n \r\n g\r\n (\r\n t\r\n )\r\n →\r\n 0\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 one can find positive and radially symmetric continuous initial data with the property that the initial value problem for ($$\\star $$\r\n ⋆\r\n ) admits a positive classical solution such that $$\\begin{aligned} t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)} \\rightarrow \\infty \\qquad \\text{ and } \\qquad \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)} \\rightarrow 0 \\qquad \\text{ as } t\\rightarrow \\infty , \\end{aligned}$$\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 \r\n and\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 0\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 but that $$\\begin{aligned} \\liminf _{t\\rightarrow \\infty } \\frac{t^\\frac{1}{p} \\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)}}{f(t)} =0 \\end{aligned}$$\r\n \r\n \r\n \r\n \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 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 \r\n \r\n \r\n \r\n and $$\\begin{aligned} \\limsup _{t\\rightarrow \\infty } \\frac{\\Vert u(\\cdot ,t)\\Vert _{L^\\infty (\\mathbb {R}^n)}}{g(t)} =\\infty . \\end{aligned}$$\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 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 g\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 \r\n ","lang":"eng"}],"status":"public","publisher":"Springer Science and Business Media LLC","date_updated":"2025-12-18T20:05:38Z","volume":3,"author":[{"full_name":"Winkler, Michael","id":"31496","last_name":"Winkler","first_name":"Michael"}],"date_created":"2025-12-18T19:30:04Z","title":"Oscillatory decay in a degenerate parabolic equation","doi":"10.1007/s42985-022-00186-z","publication_identifier":{"issn":["2662-2963","2662-2971"]},"publication_status":"published","issue":"4","year":"2022","intvolume":" 3","citation":{"apa":"Winkler, M. (2022). Oscillatory decay in a degenerate parabolic equation. Partial Differential Equations and Applications, 3(4), Article 47. https://doi.org/10.1007/s42985-022-00186-z","mla":"Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.” Partial Differential Equations and Applications, vol. 3, no. 4, 47, Springer Science and Business Media LLC, 2022, doi:10.1007/s42985-022-00186-z.","short":"M. Winkler, Partial Differential Equations and Applications 3 (2022).","bibtex":"@article{Winkler_2022, title={Oscillatory decay in a degenerate parabolic equation}, volume={3}, DOI={10.1007/s42985-022-00186-z}, number={447}, journal={Partial Differential Equations and Applications}, publisher={Springer Science and Business Media LLC}, author={Winkler, Michael}, year={2022} }","chicago":"Winkler, Michael. “Oscillatory Decay in a Degenerate Parabolic Equation.” Partial Differential Equations and Applications 3, no. 4 (2022). https://doi.org/10.1007/s42985-022-00186-z.","ieee":"M. Winkler, “Oscillatory decay in a degenerate parabolic equation,” Partial Differential Equations and Applications, vol. 3, no. 4, Art. no. 47, 2022, doi: 10.1007/s42985-022-00186-z.","ama":"Winkler M. Oscillatory decay in a degenerate parabolic equation. Partial Differential Equations and Applications. 2022;3(4). doi:10.1007/s42985-022-00186-z"}}]