@article{53323,
abstract = {{AbstractIn a ball $$\Omega =B_R(0)\subset \mathbb {R}^n$$
Ω
=
B
R
(
0
)
⊂
R
n
, $$n\ge 2$$
n
≥
2
, the chemotaxis system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \nabla \cdot \big ( D(u) \nabla u \big ) - \nabla \cdot \big ( uS(u)\nabla v\big ), \\ 0 = \Delta v - \mu + u, \qquad \mu =\frac{1}{|\Omega |} \int _\Omega u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
u
t
=
∇
·
(
D
(
u
)
∇
u
)
-
∇
·
(
u
S
(
u
)
∇
v
)
,
0
=
Δ
v
-
μ
+
u
,
μ
=
1
|
Ω
|
∫
Ω
u
,
(
⋆
)
is considered under no-flux boundary conditions, with a focus on nonlinearities $$S\in C^2([0,\infty ))$$
S
∈
C
2
(
[
0
,
∞
)
)
which exhibit super-algebraically fast decay in the sense that with some $$K_S>0, \beta \in [0,1)$$
K
S
>
0
,
β
∈
[
0
,
1
)
and $$\xi _0>0$$
ξ
0
>
0
, $$\begin{aligned} S(\xi )>0 \quad \text{ and } \quad S'(\xi ) \le -K_S\xi ^{-\beta } S(\xi ) \qquad \text{ for } \text{ all } \xi \ge \xi _0. \end{aligned}$$
S
(
ξ
)
>
0
and
S
′
(
ξ
)
≤
-
K
S
ξ
-
β
S
(
ξ
)
for
all
ξ
≥
ξ
0
.
It is, inter alia, shown that if furthermore $$D\in C^2((0,\infty ))$$
D
∈
C
2
(
(
0
,
∞
)
)
is positive and suitably small in relation to S by satisfying $$\begin{aligned} \frac{\xi S(\xi )}{D(\xi )} \ge K_{SD}\xi ^\lambda \qquad \text{ for } \text{ all } \xi \ge \xi _0 \end{aligned}$$
ξ
S
(
ξ
)
D
(
ξ
)
≥
K
SD
ξ
λ
for
all
ξ
≥
ξ
0
with some $$K_{SD}>0$$
K
SD
>
0
and $$\lambda >\frac{2}{n}$$
λ
>
2
n
, then throughout a considerably large set of initial data, ($$\star $$
⋆
) admits global classical solutions (u, v) fulfilling $$\begin{aligned} \frac{z(t)}{C} \le \Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )} \le Cz(t) \qquad \text{ for } \text{ all } t>0, \end{aligned}$$
z
(
t
)
C
≤
‖
u
(
·
,
t
)
‖
L
∞
(
Ω
)
≤
C
z
(
t
)
for
all
t
>
0
,
with some $$C=C^{(u,v)}\ge 1$$
C
=
C
(
u
,
v
)
≥
1
, where z denotes the solution of $$\begin{aligned} \left\{ \begin{array}{l}z'(t) = z^2(t) \cdot S\big ( z(t)\big ), \qquad t>0, \\ z(0)=\xi _0, \end{array} \right. \end{aligned}$$
z
′
(
t
)
=
z
2
(
t
)
·
S
(
z
(
t
)
)
,
t
>
0
,
z
(
0
)
=
ξ
0
,
which is seen to exist globally, and to satisfy $$z(t)\rightarrow +\infty $$
z
(
t
)
→
+
∞
as $$t\rightarrow \infty $$
t
→
∞
. As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in ($$\star $$
⋆
) at logarithmic and doubly logarithmic rates, respectively.}},
author = {{Winkler, Michael}},
issn = {{1040-7294}},
journal = {{Journal of Dynamics and Differential Equations}},
keywords = {{Analysis}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Slow Grow-up in a Quasilinear Keller–Segel System}}},
doi = {{10.1007/s10884-022-10167-w}},
year = {{2022}},
}
@article{63266,
abstract = {{AbstractIn a ball $$\Omega =B_R(0)\subset \mathbb {R}^n$$
Ω
=
B
R
(
0
)
⊂
R
n
, $$n\ge 2$$
n
≥
2
, the chemotaxis system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \nabla \cdot \big ( D(u) \nabla u \big ) - \nabla \cdot \big ( uS(u)\nabla v\big ), \\ 0 = \Delta v - \mu + u, \qquad \mu =\frac{1}{|\Omega |} \int _\Omega u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
u
t
=
∇
·
(
D
(
u
)
∇
u
)
-
∇
·
(
u
S
(
u
)
∇
v
)
,
0
=
Δ
v
-
μ
+
u
,
μ
=
1
|
Ω
|
∫
Ω
u
,
(
⋆
)
is considered under no-flux boundary conditions, with a focus on nonlinearities $$S\in C^2([0,\infty ))$$
S
∈
C
2
(
[
0
,
∞
)
)
which exhibit super-algebraically fast decay in the sense that with some $$K_S>0, \beta \in [0,1)$$
K
S
>
0
,
β
∈
[
0
,
1
)
and $$\xi _0>0$$
ξ
0
>
0
, $$\begin{aligned} S(\xi )>0 \quad \text{ and } \quad S'(\xi ) \le -K_S\xi ^{-\beta } S(\xi ) \qquad \text{ for } \text{ all } \xi \ge \xi _0. \end{aligned}$$
S
(
ξ
)
>
0
and
S
′
(
ξ
)
≤
-
K
S
ξ
-
β
S
(
ξ
)
for
all
ξ
≥
ξ
0
.
It is, inter alia, shown that if furthermore $$D\in C^2((0,\infty ))$$
D
∈
C
2
(
(
0
,
∞
)
)
is positive and suitably small in relation to S by satisfying $$\begin{aligned} \frac{\xi S(\xi )}{D(\xi )} \ge K_{SD}\xi ^\lambda \qquad \text{ for } \text{ all } \xi \ge \xi _0 \end{aligned}$$
ξ
S
(
ξ
)
D
(
ξ
)
≥
K
SD
ξ
λ
for
all
ξ
≥
ξ
0
with some $$K_{SD}>0$$
K
SD
>
0
and $$\lambda >\frac{2}{n}$$
λ
>
2
n
, then throughout a considerably large set of initial data, ($$\star $$
⋆
) admits global classical solutions (u, v) fulfilling $$\begin{aligned} \frac{z(t)}{C} \le \Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )} \le Cz(t) \qquad \text{ for } \text{ all } t>0, \end{aligned}$$
z
(
t
)
C
≤
‖
u
(
·
,
t
)
‖
L
∞
(
Ω
)
≤
C
z
(
t
)
for
all
t
>
0
,
with some $$C=C^{(u,v)}\ge 1$$
C
=
C
(
u
,
v
)
≥
1
, where z denotes the solution of $$\begin{aligned} \left\{ \begin{array}{l}z'(t) = z^2(t) \cdot S\big ( z(t)\big ), \qquad t>0, \\ z(0)=\xi _0, \end{array} \right. \end{aligned}$$
z
′
(
t
)
=
z
2
(
t
)
·
S
(
z
(
t
)
)
,
t
>
0
,
z
(
0
)
=
ξ
0
,
which is seen to exist globally, and to satisfy $$z(t)\rightarrow +\infty $$
z
(
t
)
→
+
∞
as $$t\rightarrow \infty $$
t
→
∞
. As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in ($$\star $$
⋆
) at logarithmic and doubly logarithmic rates, respectively.}},
author = {{Winkler, Michael}},
issn = {{1040-7294}},
journal = {{Journal of Dynamics and Differential Equations}},
number = {{2}},
pages = {{1677--1702}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Slow Grow-up in a Quasilinear Keller–Segel System}}},
doi = {{10.1007/s10884-022-10167-w}},
volume = {{36}},
year = {{2022}},
}
@article{53322,
abstract = {{AbstractThe Cauchy problem in $${\mathbb {R}}^n$$
R
n
, $$n\ge 1$$
n
≥
1
, for the parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$
u
t
=
u
p
Δ
u
(
⋆
)
is considered in the strongly degenerate regime $$p\ge 1$$
p
≥
1
. 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 $$\begin{aligned} t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \rightarrow \infty \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
→
∞
as
t
→
∞
.
The first result of this study complements this by asserting that given any positive $$f\in C^0([0,\infty ))$$
f
∈
C
0
(
[
0
,
∞
)
)
fulfilling $$f(t)\rightarrow +\infty $$
f
(
t
)
→
+
∞
as $$t\rightarrow \infty $$
t
→
∞
one can find a positive nondecreasing function $$\phi \in C^0([0,\infty ))$$
ϕ
∈
C
0
(
[
0
,
∞
)
)
such that whenever $$u_0\in C^0({\mathbb {R}}^n)$$
u
0
∈
C
0
(
R
n
)
is radially symmetric with $$0< u_0 < \phi (|\cdot |)$$
0
<
u
0
<
ϕ
(
|
·
|
)
, the corresponding minimal solution u satisfies $$\begin{aligned} \frac{t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}}{f(t)} \rightarrow 0 \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
f
(
t
)
→
0
as
t
→
∞
.
Secondly, ($$\star $$
⋆
) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data $$u_0$$
u
0
. It is shown that if the connected components of $$\{u_0>0\}$$
{
u
0
>
0
}
comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to ($$\star $$
⋆
) satisfies $$\begin{aligned} 0< \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} \le \limsup _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} <\infty . \end{aligned}$$
0
<
lim inf
t
→
∞
{
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
}
≤
lim sup
t
→
∞
{
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
}
<
∞
.
Under a somewhat complementary hypothesis, particularly fulfilled if $$\{u_0>0\}$$
{
u
0
>
0
}
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 = {{Winkler, Michael}},
issn = {{1040-7294}},
journal = {{Journal of Dynamics and Differential Equations}},
keywords = {{Analysis}},
number = {{S1}},
pages = {{3--23}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Approaching Critical Decay in a Strongly Degenerate Parabolic Equation}}},
doi = {{10.1007/s10884-020-09892-x}},
volume = {{36}},
year = {{2020}},
}
@article{63265,
abstract = {{AbstractThe Cauchy problem in $${\mathbb {R}}^n$$
R
n
, $$n\ge 1$$
n
≥
1
, for the parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$
u
t
=
u
p
Δ
u
(
⋆
)
is considered in the strongly degenerate regime $$p\ge 1$$
p
≥
1
. 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 $$\begin{aligned} t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \rightarrow \infty \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
→
∞
as
t
→
∞
.
The first result of this study complements this by asserting that given any positive $$f\in C^0([0,\infty ))$$
f
∈
C
0
(
[
0
,
∞
)
)
fulfilling $$f(t)\rightarrow +\infty $$
f
(
t
)
→
+
∞
as $$t\rightarrow \infty $$
t
→
∞
one can find a positive nondecreasing function $$\phi \in C^0([0,\infty ))$$
ϕ
∈
C
0
(
[
0
,
∞
)
)
such that whenever $$u_0\in C^0({\mathbb {R}}^n)$$
u
0
∈
C
0
(
R
n
)
is radially symmetric with $$0< u_0 < \phi (|\cdot |)$$
0
<
u
0
<
ϕ
(
|
·
|
)
, the corresponding minimal solution u satisfies $$\begin{aligned} \frac{t^\frac{1}{p}\Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)}}{f(t)} \rightarrow 0 \quad \hbox {as } t\rightarrow \infty . \end{aligned}$$
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
f
(
t
)
→
0
as
t
→
∞
.
Secondly, ($$\star $$
⋆
) is considered along with initial conditions involving nonnegative but not necessarily strictly positive bounded and continuous initial data $$u_0$$
u
0
. It is shown that if the connected components of $$\{u_0>0\}$$
{
u
0
>
0
}
comply with a condition reflecting some uniform boundedness property, then a corresponding uniquely determined continuous weak solution to ($$\star $$
⋆
) satisfies $$\begin{aligned} 0< \liminf _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} \le \limsup _{t\rightarrow \infty } \Big \{ t^\frac{1}{p} \Vert u(\cdot ,t)\Vert _{L^\infty ({\mathbb {R}}^n)} \Big \} <\infty . \end{aligned}$$
0
<
lim inf
t
→
∞
{
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
}
≤
lim sup
t
→
∞
{
t
1
p
‖
u
(
·
,
t
)
‖
L
∞
(
R
n
)
}
<
∞
.
Under a somewhat complementary hypothesis, particularly fulfilled if $$\{u_0>0\}$$
{
u
0
>
0
}
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 = {{Winkler, Michael}},
issn = {{1040-7294}},
journal = {{Journal of Dynamics and Differential Equations}},
number = {{S1}},
pages = {{3--23}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Approaching Critical Decay in a Strongly Degenerate Parabolic Equation}}},
doi = {{10.1007/s10884-020-09892-x}},
volume = {{36}},
year = {{2020}},
}
@article{63378,
author = {{Winkler, Michael}},
issn = {{1040-7294}},
journal = {{Journal of Dynamics and Differential Equations}},
number = {{1}},
pages = {{331--358}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{One-Dimensional Super-Fast Diffusion: Persistence Versus Extinction Revisited—Extinction at Spatial Infinity}}},
doi = {{10.1007/s10884-017-9577-3}},
volume = {{30}},
year = {{2017}},
}