---
_id: '53323'
abstract:
- lang: eng
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."
author:
- first_name: Michael
full_name: Winkler, Michael
last_name: Winkler
citation:
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
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
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} }'
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.
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.'
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.
short: M. Winkler, Journal of Dynamics and Differential Equations (2022).
date_created: 2024-04-07T12:39:12Z
date_updated: 2024-04-07T12:39:17Z
doi: 10.1007/s10884-022-10167-w
keyword:
- Analysis
language:
- iso: eng
publication: Journal of Dynamics and Differential Equations
publication_identifier:
issn:
- 1040-7294
- 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Slow Grow-up in a Quasilinear Keller–Segel System
type: journal_article
user_id: '31496'
year: '2022'
...
---
_id: '63266'
abstract:
- lang: eng
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."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
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
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
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}
}'
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.'
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.'
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.
short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2022) 1677–1702.
date_created: 2025-12-18T19:10:32Z
date_updated: 2025-12-18T20:10:14Z
doi: 10.1007/s10884-022-10167-w
intvolume: ' 36'
issue: '2'
language:
- iso: eng
page: 1677-1702
publication: Journal of Dynamics and Differential Equations
publication_identifier:
issn:
- 1040-7294
- 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Slow Grow-up in a Quasilinear Keller–Segel System
type: journal_article
user_id: '31496'
volume: 36
year: '2022'
...
---
_id: '53322'
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."
author:
- first_name: Michael
full_name: Winkler, Michael
last_name: Winkler
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
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
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}
}'
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.'
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.'
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.
short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23.
date_created: 2024-04-07T12:37:38Z
date_updated: 2024-04-07T12:37:45Z
doi: 10.1007/s10884-020-09892-x
intvolume: ' 36'
issue: S1
keyword:
- Analysis
language:
- iso: eng
page: 3-23
publication: Journal of Dynamics and Differential Equations
publication_identifier:
issn:
- 1040-7294
- 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Approaching Critical Decay in a Strongly Degenerate Parabolic Equation
type: journal_article
user_id: '31496'
volume: 36
year: '2020'
...
---
_id: '63265'
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."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
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
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
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}
}'
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.'
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.'
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.
short: M. Winkler, Journal of Dynamics and Differential Equations 36 (2020) 3–23.
date_created: 2025-12-18T19:10:01Z
date_updated: 2025-12-18T20:10:07Z
doi: 10.1007/s10884-020-09892-x
intvolume: ' 36'
issue: S1
language:
- iso: eng
page: 3-23
publication: Journal of Dynamics and Differential Equations
publication_identifier:
issn:
- 1040-7294
- 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Approaching Critical Decay in a Strongly Degenerate Parabolic Equation
type: journal_article
user_id: '31496'
volume: 36
year: '2020'
...
---
_id: '63378'
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
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'
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'
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}
}'
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.'
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.
date_created: 2025-12-19T11:07:24Z
date_updated: 2025-12-19T11:07:30Z
doi: 10.1007/s10884-017-9577-3
intvolume: ' 30'
issue: '1'
language:
- iso: eng
page: 331-358
publication: Journal of Dynamics and Differential Equations
publication_identifier:
issn:
- 1040-7294
- 1572-9222
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: 'One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction
at Spatial Infinity'
type: journal_article
user_id: '31496'
volume: 30
year: '2017'
...