{"date_created":"2024-04-07T12:37:38Z","doi":"10.1007/s10884-020-09892-x","publication_status":"published","_id":"53322","status":"public","year":"2020","author":[{"full_name":"Winkler, Michael","last_name":"Winkler","first_name":"Michael"}],"user_id":"31496","publication_identifier":{"issn":["1040-7294","1572-9222"]},"publisher":"Springer Science and Business Media LLC","volume":36,"keyword":["Analysis"],"type":"journal_article","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.","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.","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","short":"M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23.","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.","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} }","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"},"language":[{"iso":"eng"}],"intvolume":" 36","issue":"S1","page":"3-23","date_updated":"2024-04-07T12:37:45Z","title":"Approaching Critical Decay in a Strongly Degenerate Parabolic Equation","publication":"Journal of Dynamics and Differential Equations","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"}]}