@article{63242,
abstract = {{Abstract
For
$$p>2$$
p
>
2
, the equation
$$\begin{aligned} u_t = u^p u_{xx}, \qquad x\in \mathbb {R}, \ t\in \mathbb {R}, \end{aligned}$$
u
t
=
u
p
u
xx
,
x
∈
R
,
t
∈
R
,
is shown to admit positive and spatially increasing smooth solutions on all of
$$\mathbb {R}\times \mathbb {R}$$
R
×
R
which are precisely of the form of an accelerating wave for
$$t<0$$
t
<
0
, and of a wave slowing down for
$$t>0$$
t
>
0
. These solutions satisfy
$$u(\cdot ,t)\rightarrow 0$$
u
(
·
,
t
)
→
0
in
$$L^\infty _{loc}(\mathbb {R})$$
L
loc
∞
(
R
)
as
$$t\rightarrow + \infty $$
t
→
+
∞
and as
$$t\rightarrow -\infty $$
t
→
-
∞
, and exhibit a yet apparently undiscovered phenomenon of transient rapid spatial growth, in the sense that
$$\begin{aligned} \lim _{x\rightarrow +\infty } x^{-1} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t<0, \end{aligned}$$
lim
x
→
+
∞
x
-
1
u
(
x
,
t
)
exists
for
all
t
<
0
,
that
$$\begin{aligned} \lim _{x\rightarrow +\infty } x^{-\frac{2}{p}} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t>0, \end{aligned}$$
lim
x
→
+
∞
x
-
2
p
u
(
x
,
t
)
exists
for
all
t
>
0
,
but that
$$\begin{aligned} u(x,0)=K e^{\alpha x} \qquad \text{ for } \text{ all } x\in \mathbb {R}\end{aligned}$$
u
(
x
,
0
)
=
K
e
α
x
for
all
x
∈
R
with some
$$K>0$$
K
>
0
and
$$\alpha >0$$
α
>
0
.
}},
author = {{Hanfland, Celina and Winkler, Michael}},
issn = {{2296-9020}},
journal = {{Journal of Elliptic and Parabolic Equations}},
number = {{3}},
pages = {{2041--2063}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Exactly wave-type homoclinic orbits and emergence of transient exponential growth in a super-fast diffusion equation}}},
doi = {{10.1007/s41808-025-00316-9}},
volume = {{11}},
year = {{2025}},
}
@article{53341,
abstract = {{AbstractThe Cauchy problem in $$\mathbb {R}^n$$
R
n
is considered for the Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \Delta u - \nabla \cdot (u\nabla v), \\ 0 = \Delta v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
u
t
=
Δ
u
-
∇
·
(
u
∇
v
)
,
0
=
Δ
v
+
u
,
(
⋆
)
with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data $$u_0$$
u
0
with non-integrable behavior at spatial infinity. It is shown that if $$u_0$$
u
0
is continuous and bounded, then ($$\star $$
⋆
) admits a local-in-time classical solution, whereas if $$u_0(x)\rightarrow +\infty $$
u
0
(
x
)
→
+
∞
as $$|x|\rightarrow \infty $$
|
x
|
→
∞
, 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 ($$\star $$
⋆
) need not be enforced by initially high concentrations near finite points, but can be exclusively due to large tails.}},
author = {{Winkler, Michael}},
issn = {{2296-9020}},
journal = {{Journal of Elliptic and Parabolic Equations}},
keywords = {{Applied Mathematics, Numerical Analysis, Analysis}},
number = {{2}},
pages = {{919--959}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity}}},
doi = {{10.1007/s41808-023-00230-y}},
volume = {{9}},
year = {{2023}},
}
@article{63287,
abstract = {{AbstractThe Cauchy problem in $$\mathbb {R}^n$$
R
n
is considered for the Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \Delta u - \nabla \cdot (u\nabla v), \\ 0 = \Delta v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
u
t
=
Δ
u
-
∇
·
(
u
∇
v
)
,
0
=
Δ
v
+
u
,
(
⋆
)
with a focus on a detailed description of behavior in the presence of nonnegative radially symmetric initial data $$u_0$$
u
0
with non-integrable behavior at spatial infinity. It is shown that if $$u_0$$
u
0
is continuous and bounded, then ($$\star $$
⋆
) admits a local-in-time classical solution, whereas if $$u_0(x)\rightarrow +\infty $$
u
0
(
x
)
→
+
∞
as $$|x|\rightarrow \infty $$
|
x
|
→
∞
, 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 ($$\star $$
⋆
) need not be enforced by initially high concentrations near finite points, but can be exclusively due to large tails.}},
author = {{Winkler, Michael}},
issn = {{2296-9020}},
journal = {{Journal of Elliptic and Parabolic Equations}},
number = {{2}},
pages = {{919--959}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Solutions to the Keller–Segel system with non-integrable behavior at spatial infinity}}},
doi = {{10.1007/s41808-023-00230-y}},
volume = {{9}},
year = {{2023}},
}
@article{53539,
abstract = {{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 $$T \rightarrow +\infty $$
T
→
+
∞
. 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 $$L^1$$
L
1
asymptotic convergence without requiring bi-K-invariance for initial data, and strong $$L^{\infty }$$
L
∞
convergence.}},
author = {{Papageorgiou, Efthymia}},
issn = {{2296-9020}},
journal = {{Journal of Elliptic and Parabolic Equations}},
keywords = {{Applied Mathematics, Numerical Analysis, Analysis}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Asymptotics for the infinite Brownian loop on noncompact symmetric spaces}}},
doi = {{10.1007/s41808-023-00250-8}},
year = {{2023}},
}