---
_id: '63242'
abstract:
- lang: eng
text: "Abstract\r\n \r\n For\r\n
\ \r\n \r\n
\ $$p>2$$\r\n \r\n \r\n
\ p\r\n >\r\n
\ 2\r\n \r\n
\ \r\n \r\n
\ \r\n , the equation\r\n
\ \r\n \r\n
\ $$\\begin{aligned} u_t = u^p u_{xx}, \\qquad
x\\in \\mathbb {R}, \\ t\\in \\mathbb {R}, \\end{aligned}$$\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 u\r\n
\ p\r\n \r\n
\ \r\n u\r\n
\ \r\n xx\r\n
\ \r\n \r\n
\ ,\r\n \r\n
\ x\r\n ∈\r\n
\ R\r\n ,\r\n
\ \r\n t\r\n
\ ∈\r\n R\r\n
\ ,\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n is shown to admit
positive and spatially increasing smooth solutions on all of\r\n \r\n
\ \r\n $$\\mathbb
{R}\\times \\mathbb {R}$$\r\n \r\n \r\n
\ R\r\n ×\r\n
\ R\r\n \r\n
\ \r\n \r\n
\ \r\n which are precisely
of the form of an accelerating wave for\r\n \r\n
\ \r\n $$t<0$$\r\n
\ \r\n
\ \r\n t\r\n
\ <\r\n 0\r\n
\ \r\n \r\n
\ \r\n \r\n
\ , and of a wave slowing down for\r\n \r\n
\ \r\n $$t>0$$\r\n
\ \r\n
\ \r\n t\r\n
\ >\r\n 0\r\n
\ \r\n \r\n
\ \r\n \r\n
\ . These solutions satisfy\r\n \r\n
\ \r\n $$u(\\cdot
,t)\\rightarrow 0$$\r\n \r\n
\ \r\n u\r\n
\ (\r\n ·\r\n
\ ,\r\n t\r\n
\ )\r\n →\r\n
\ 0\r\n \r\n
\ \r\n \r\n
\ \r\n in\r\n \r\n
\ \r\n $$L^\\infty
_{loc}(\\mathbb {R})$$\r\n \r\n
\ \r\n \r\n
\ L\r\n \r\n
\ loc\r\n \r\n
\ ∞\r\n \r\n
\ \r\n (\r\n
\ R\r\n )\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n as\r\n \r\n
\ \r\n $$t\\rightarrow
+ \\infty $$\r\n \r\n
\ \r\n t\r\n
\ →\r\n +\r\n
\ ∞\r\n \r\n
\ \r\n \r\n
\ \r\n and as\r\n \r\n
\ \r\n $$t\\rightarrow
-\\infty $$\r\n \r\n
\ \r\n t\r\n
\ →\r\n -\r\n
\ ∞\r\n \r\n
\ \r\n \r\n
\ \r\n , and exhibit
a yet apparently undiscovered phenomenon of transient rapid spatial growth, in
the sense that\r\n \r\n \r\n
\ $$\\begin{aligned} \\lim _{x\\rightarrow
+\\infty } x^{-1} u(x,t) \\quad \\text{ exists } \\text{ for } \\text{ all } t<0,
\\end{aligned}$$\r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ lim\r\n \r\n
\ x\r\n →\r\n
\ +\r\n ∞\r\n
\ \r\n \r\n
\ \r\n x\r\n
\ \r\n -\r\n
\ 1\r\n \r\n
\ \r\n u\r\n
\ \r\n (\r\n
\ x\r\n ,\r\n
\ t\r\n )\r\n
\ \r\n \r\n
\ \r\n exists\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
\ \r\n \r\n
\ \r\n that\r\n \r\n
\ \r\n $$\\begin{aligned}
\\lim _{x\\rightarrow +\\infty } x^{-\\frac{2}{p}} u(x,t) \\quad \\text{ exists
} \\text{ for } \\text{ all } t>0, \\end{aligned}$$\r\n \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n lim\r\n
\ \r\n x\r\n
\ →\r\n +\r\n
\ ∞\r\n \r\n
\ \r\n \r\n
\ x\r\n \r\n
\ -\r\n \r\n
\ 2\r\n p\r\n
\ \r\n \r\n
\ \r\n u\r\n
\ \r\n (\r\n
\ x\r\n ,\r\n
\ t\r\n )\r\n
\ \r\n \r\n
\ \r\n exists\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
\ \r\n \r\n
\ \r\n but that\r\n \r\n
\ \r\n $$\\begin{aligned}
u(x,0)=K e^{\\alpha x} \\qquad \\text{ for } \\text{ all } x\\in \\mathbb {R}\\end{aligned}$$\r\n
\ \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n u\r\n
\ \r\n (\r\n
\ x\r\n ,\r\n
\ 0\r\n )\r\n
\ \r\n =\r\n
\ K\r\n \r\n
\ e\r\n \r\n
\ α\r\n x\r\n
\ \r\n \r\n
\ \r\n \r\n
\ for\r\n \r\n
\ \r\n all\r\n
\ \r\n x\r\n
\ ∈\r\n R\r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ \r\n \r\n
\ with some\r\n \r\n
\ \r\n $$K>0$$\r\n
\ \r\n
\ \r\n K\r\n
\ >\r\n 0\r\n
\ \r\n \r\n
\ \r\n \r\n
\ and\r\n \r\n \r\n
\ $$\\alpha >0$$\r\n \r\n \r\n
\ α\r\n >\r\n
\ 0\r\n \r\n
\ \r\n \r\n
\ \r\n .\r\n "
author:
- first_name: Celina
full_name: Hanfland, Celina
last_name: Hanfland
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Hanfland C, Winkler M. Exactly wave-type homoclinic orbits and emergence of
transient exponential growth in a super-fast diffusion equation. Journal of
Elliptic and Parabolic Equations. 2025;11(3):2041-2063. doi:10.1007/s41808-025-00316-9
apa: Hanfland, C., & Winkler, M. (2025). Exactly wave-type homoclinic orbits
and emergence of transient exponential growth in a super-fast diffusion equation.
Journal of Elliptic and Parabolic Equations, 11(3), 2041–2063. https://doi.org/10.1007/s41808-025-00316-9
bibtex: '@article{Hanfland_Winkler_2025, title={Exactly wave-type homoclinic orbits
and emergence of transient exponential growth in a super-fast diffusion equation},
volume={11}, DOI={10.1007/s41808-025-00316-9},
number={3}, journal={Journal of Elliptic and Parabolic Equations}, publisher={Springer
Science and Business Media LLC}, author={Hanfland, Celina and Winkler, Michael},
year={2025}, pages={2041–2063} }'
chicago: 'Hanfland, Celina, and Michael Winkler. “Exactly Wave-Type Homoclinic Orbits
and Emergence of Transient Exponential Growth in a Super-Fast Diffusion Equation.”
Journal of Elliptic and Parabolic Equations 11, no. 3 (2025): 2041–63.
https://doi.org/10.1007/s41808-025-00316-9.'
ieee: 'C. Hanfland and M. Winkler, “Exactly wave-type homoclinic orbits and emergence
of transient exponential growth in a super-fast diffusion equation,” Journal
of Elliptic and Parabolic Equations, vol. 11, no. 3, pp. 2041–2063, 2025,
doi: 10.1007/s41808-025-00316-9.'
mla: Hanfland, Celina, and Michael Winkler. “Exactly Wave-Type Homoclinic Orbits
and Emergence of Transient Exponential Growth in a Super-Fast Diffusion Equation.”
Journal of Elliptic and Parabolic Equations, vol. 11, no. 3, Springer Science
and Business Media LLC, 2025, pp. 2041–63, doi:10.1007/s41808-025-00316-9.
short: C. Hanfland, M. Winkler, Journal of Elliptic and Parabolic Equations 11 (2025)
2041–2063.
date_created: 2025-12-18T18:57:21Z
date_updated: 2025-12-18T20:16:49Z
doi: 10.1007/s41808-025-00316-9
intvolume: ' 11'
issue: '3'
language:
- iso: eng
page: 2041-2063
publication: Journal of Elliptic and Parabolic Equations
publication_identifier:
issn:
- 2296-9020
- 2296-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Exactly wave-type homoclinic orbits and emergence of transient exponential
growth in a super-fast diffusion equation
type: journal_article
user_id: '31496'
volume: 11
year: '2025'
...
---
_id: '53341'
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
is considered for the Keller–Segel system $$\\begin{aligned}
\\left\\{ \\begin{array}{l}u_t = \\Delta u - \\nabla \\cdot (u\\nabla v), \\\\
0 = \\Delta v + u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\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
\ Δ\r\n u\r\n
\ -\r\n ∇\r\n
\ ·\r\n \r\n
\ (\r\n u\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 0\r\n =\r\n
\ Δ\r\n v\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
\ with a focus
on a detailed description of behavior in the presence of nonnegative radially
symmetric initial data $$u_0$$\r\n \r\n
\ u\r\n 0\r\n
\ \r\n
with non-integrable behavior at spatial infinity. It is shown that if $$u_0$$\r\n \r\n
\ u\r\n 0\r\n
\ \r\n
is continuous and bounded, then ($$\\star
$$\r\n
\ ⋆\r\n )
admits a local-in-time classical solution, whereas if $$u_0(x)\\rightarrow
+\\infty $$\r\n
\ \r\n \r\n u\r\n
\ 0\r\n \r\n
\ \r\n (\r\n
\ x\r\n )\r\n
\ \r\n →\r\n
\ +\r\n ∞\r\n
\ \r\n
as $$|x|\\rightarrow \\infty
$$\r\n
\ \r\n |\r\n x\r\n
\ |\r\n →\r\n
\ ∞\r\n \r\n ,
then no such solution can be found. Furthermore, a collection of three sufficient
criteria for either global existence or global nonexistence indicates that with
respect to the occurrence of finite-time blow-up, spatial decay properties of
an explicit singular steady state plays a critical role. In particular, this underlines
that explosions in ($$\\star
$$\r\n
\ ⋆\r\n )
need not be enforced by initially high concentrations near finite points, but
can be exclusively due to large tails."
author:
- first_name: Michael
full_name: Winkler, Michael
last_name: Winkler
citation:
ama: Winkler M. Solutions to the Keller–Segel system with non-integrable behavior
at spatial infinity. Journal of Elliptic and Parabolic Equations. 2023;9(2):919-959.
doi:10.1007/s41808-023-00230-y
apa: Winkler, M. (2023). Solutions to the Keller–Segel system with non-integrable
behavior at spatial infinity. Journal of Elliptic and Parabolic Equations,
9(2), 919–959. https://doi.org/10.1007/s41808-023-00230-y
bibtex: '@article{Winkler_2023, title={Solutions to the Keller–Segel system with
non-integrable behavior at spatial infinity}, volume={9}, DOI={10.1007/s41808-023-00230-y},
number={2}, journal={Journal of Elliptic and Parabolic Equations}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2023}, pages={919–959}
}'
chicago: 'Winkler, Michael. “Solutions to the Keller–Segel System with Non-Integrable
Behavior at Spatial Infinity.” Journal of Elliptic and Parabolic Equations
9, no. 2 (2023): 919–59. https://doi.org/10.1007/s41808-023-00230-y.'
ieee: 'M. Winkler, “Solutions to the Keller–Segel system with non-integrable behavior
at spatial infinity,” Journal of Elliptic and Parabolic Equations, vol.
9, no. 2, pp. 919–959, 2023, doi: 10.1007/s41808-023-00230-y.'
mla: Winkler, Michael. “Solutions to the Keller–Segel System with Non-Integrable
Behavior at Spatial Infinity.” Journal of Elliptic and Parabolic Equations,
vol. 9, no. 2, Springer Science and Business Media LLC, 2023, pp. 919–59, doi:10.1007/s41808-023-00230-y.
short: M. Winkler, Journal of Elliptic and Parabolic Equations 9 (2023) 919–959.
date_created: 2024-04-07T12:52:52Z
date_updated: 2024-04-07T12:52:55Z
doi: 10.1007/s41808-023-00230-y
intvolume: ' 9'
issue: '2'
keyword:
- Applied Mathematics
- Numerical Analysis
- Analysis
language:
- iso: eng
page: 919-959
publication: Journal of Elliptic and Parabolic Equations
publication_identifier:
issn:
- 2296-9020
- 2296-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Solutions to the Keller–Segel system with non-integrable behavior at spatial
infinity
type: journal_article
user_id: '31496'
volume: 9
year: '2023'
...
---
_id: '63287'
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
is considered for the Keller–Segel system $$\\begin{aligned}
\\left\\{ \\begin{array}{l}u_t = \\Delta u - \\nabla \\cdot (u\\nabla v), \\\\
0 = \\Delta v + u, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\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
\ Δ\r\n u\r\n
\ -\r\n ∇\r\n
\ ·\r\n \r\n
\ (\r\n u\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
\ 0\r\n =\r\n
\ Δ\r\n v\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
\ with a focus
on a detailed description of behavior in the presence of nonnegative radially
symmetric initial data $$u_0$$\r\n \r\n
\ u\r\n 0\r\n
\ \r\n
with non-integrable behavior at spatial infinity. It is shown that if $$u_0$$\r\n \r\n
\ u\r\n 0\r\n
\ \r\n
is continuous and bounded, then ($$\\star
$$\r\n
\ ⋆\r\n )
admits a local-in-time classical solution, whereas if $$u_0(x)\\rightarrow
+\\infty $$\r\n
\ \r\n \r\n u\r\n
\ 0\r\n \r\n
\ \r\n (\r\n
\ x\r\n )\r\n
\ \r\n →\r\n
\ +\r\n ∞\r\n
\ \r\n
as $$|x|\\rightarrow \\infty
$$\r\n
\ \r\n |\r\n x\r\n
\ |\r\n →\r\n
\ ∞\r\n \r\n ,
then no such solution can be found. Furthermore, a collection of three sufficient
criteria for either global existence or global nonexistence indicates that with
respect to the occurrence of finite-time blow-up, spatial decay properties of
an explicit singular steady state plays a critical role. In particular, this underlines
that explosions in ($$\\star
$$\r\n
\ ⋆\r\n )
need not be enforced by initially high concentrations near finite points, but
can be exclusively due to large tails."
author:
- first_name: Michael
full_name: Winkler, Michael
id: '31496'
last_name: Winkler
citation:
ama: Winkler M. Solutions to the Keller–Segel system with non-integrable behavior
at spatial infinity. Journal of Elliptic and Parabolic Equations. 2023;9(2):919-959.
doi:10.1007/s41808-023-00230-y
apa: Winkler, M. (2023). Solutions to the Keller–Segel system with non-integrable
behavior at spatial infinity. Journal of Elliptic and Parabolic Equations,
9(2), 919–959. https://doi.org/10.1007/s41808-023-00230-y
bibtex: '@article{Winkler_2023, title={Solutions to the Keller–Segel system with
non-integrable behavior at spatial infinity}, volume={9}, DOI={10.1007/s41808-023-00230-y},
number={2}, journal={Journal of Elliptic and Parabolic Equations}, publisher={Springer
Science and Business Media LLC}, author={Winkler, Michael}, year={2023}, pages={919–959}
}'
chicago: 'Winkler, Michael. “Solutions to the Keller–Segel System with Non-Integrable
Behavior at Spatial Infinity.” Journal of Elliptic and Parabolic Equations
9, no. 2 (2023): 919–59. https://doi.org/10.1007/s41808-023-00230-y.'
ieee: 'M. Winkler, “Solutions to the Keller–Segel system with non-integrable behavior
at spatial infinity,” Journal of Elliptic and Parabolic Equations, vol.
9, no. 2, pp. 919–959, 2023, doi: 10.1007/s41808-023-00230-y.'
mla: Winkler, Michael. “Solutions to the Keller–Segel System with Non-Integrable
Behavior at Spatial Infinity.” Journal of Elliptic and Parabolic Equations,
vol. 9, no. 2, Springer Science and Business Media LLC, 2023, pp. 919–59, doi:10.1007/s41808-023-00230-y.
short: M. Winkler, Journal of Elliptic and Parabolic Equations 9 (2023) 919–959.
date_created: 2025-12-18T19:19:13Z
date_updated: 2025-12-18T20:07:25Z
doi: 10.1007/s41808-023-00230-y
intvolume: ' 9'
issue: '2'
language:
- iso: eng
page: 919-959
publication: Journal of Elliptic and Parabolic Equations
publication_identifier:
issn:
- 2296-9020
- 2296-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Solutions to the Keller–Segel system with non-integrable behavior at spatial
infinity
type: journal_article
user_id: '31496'
volume: 9
year: '2023'
...
---
_id: '53539'
abstract:
- lang: eng
text: "AbstractThe infinite Brownian loop on a
Riemannian manifold is the limit in distribution of the Brownian bridge of length
T around a fixed origin when $$T
\\rightarrow +\\infty $$\r\n
\ \r\n T\r\n →\r\n
\ +\r\n ∞\r\n
\ \r\n .
The aim of this note is to study its long-time asymptotics on Riemannian symmetric
spaces G/K of noncompact
type and of general rank. This amounts to the behavior of solutions to the heat
equation subject to the Doob transform induced by the ground spherical function.
Unlike the standard Brownian motion, we observe in this case phenomena which are
similar to the Euclidean setting, namely $$L^1$$\r\n \r\n
\ L\r\n 1\r\n
\ \r\n
asymptotic convergence without requiring bi-K-invariance
for initial data, and strong $$L^{\\infty
}$$\r\n
\ \r\n L\r\n ∞\r\n
\ \r\n
convergence."
author:
- first_name: Efthymia
full_name: Papageorgiou, Efthymia
id: '100325'
last_name: Papageorgiou
citation:
ama: Papageorgiou E. Asymptotics for the infinite Brownian loop on noncompact symmetric
spaces. Journal of Elliptic and Parabolic Equations. Published online 2023.
doi:10.1007/s41808-023-00250-8
apa: Papageorgiou, E. (2023). Asymptotics for the infinite Brownian loop on noncompact
symmetric spaces. Journal of Elliptic and Parabolic Equations. https://doi.org/10.1007/s41808-023-00250-8
bibtex: '@article{Papageorgiou_2023, title={Asymptotics for the infinite Brownian
loop on noncompact symmetric spaces}, DOI={10.1007/s41808-023-00250-8},
journal={Journal of Elliptic and Parabolic Equations}, publisher={Springer Science
and Business Media LLC}, author={Papageorgiou, Efthymia}, year={2023} }'
chicago: Papageorgiou, Efthymia. “Asymptotics for the Infinite Brownian Loop on
Noncompact Symmetric Spaces.” Journal of Elliptic and Parabolic Equations,
2023. https://doi.org/10.1007/s41808-023-00250-8.
ieee: 'E. Papageorgiou, “Asymptotics for the infinite Brownian loop on noncompact
symmetric spaces,” Journal of Elliptic and Parabolic Equations, 2023, doi:
10.1007/s41808-023-00250-8.'
mla: Papageorgiou, Efthymia. “Asymptotics for the Infinite Brownian Loop on Noncompact
Symmetric Spaces.” Journal of Elliptic and Parabolic Equations, Springer
Science and Business Media LLC, 2023, doi:10.1007/s41808-023-00250-8.
short: E. Papageorgiou, Journal of Elliptic and Parabolic Equations (2023).
date_created: 2024-04-17T13:16:39Z
date_updated: 2026-07-03T12:36:17Z
department:
- _id: '555'
doi: 10.1007/s41808-023-00250-8
keyword:
- Applied Mathematics
- Numerical Analysis
- Analysis
language:
- iso: eng
publication: Journal of Elliptic and Parabolic Equations
publication_identifier:
issn:
- 2296-9020
- 2296-9039
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Asymptotics for the infinite Brownian loop on noncompact symmetric spaces
type: journal_article
user_id: '100325'
year: '2023'
...