---
_id: '63311'
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 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 "
article_number: '47'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
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
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
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.'
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).
date_created: 2025-12-18T19:30:04Z
date_updated: 2025-12-18T20:05:38Z
doi: 10.1007/s42985-022-00186-z
intvolume: ' 3'
issue: '4'
language:
- iso: eng
publication: Partial Differential Equations and Applications
publication_identifier:
issn:
- 2662-2963
- 2662-2971
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Oscillatory decay in a degenerate parabolic equation
type: journal_article
user_id: '31496'
volume: 3
year: '2022'
...