@article{63372,
author = {{Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}},
issn = {{0025-5874}},
journal = {{Mathematische Zeitschrift}},
number = {{1-2}},
pages = {{71--108}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system}}},
doi = {{10.1007/s00209-017-1944-6}},
volume = {{289}},
year = {{2017}},
}
@article{63374,
author = {{Winkler, Michael}},
issn = {{0021-7824}},
journal = {{Journal de Mathématiques Pures et Appliquées}},
pages = {{118--169}},
publisher = {{Elsevier BV}},
title = {{{Singular structure formation in a degenerate haptotaxis model involving myopic diffusion}}},
doi = {{10.1016/j.matpur.2017.11.002}},
volume = {{112}},
year = {{2017}},
}
@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}},
}
@article{63379,
author = {{Winkler, Michael}},
issn = {{0022-0396}},
journal = {{Journal of Differential Equations}},
number = {{3}},
pages = {{2310--2350}},
publisher = {{Elsevier BV}},
title = {{{Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption}}},
doi = {{10.1016/j.jde.2017.10.029}},
volume = {{264}},
year = {{2017}},
}
@article{63383,
abstract = {{This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by
(\star)
{
u
t
=
∇
⋅
(
u
∇
u
u
2
+
|
∇
u
|
2
)
−
χ
∇
⋅
(
u
∇
v
1
+
|
∇
v
|
2
)
,
0
=
Δ
v
−
μ
+
u
,
\begin{equation}\tag {\star } \begin {cases} u_t=\nabla \cdot \Big (\frac {u\nabla u}{\sqrt {u^2+|\nabla u|^2}}\Big ) - \chi \, \nabla \cdot \Big (\frac {u\nabla v}{\sqrt {1+|\nabla v|^2}}\Big ), \\[3pt] 0=\Delta v - \mu + u, \end{cases} \end{equation}
under the initial condition
u
|
t
=
0
=
u
0
>
0
u|_{t=0}=u_0>0
and no-flux boundary conditions in a ball
Ω
⊂
R
n
\Omega \subset \mathbb {R}^n
, where
χ
>
0
\chi >0
and
μ
:=
1
|
Ω
|
∫
Ω
u
0
\mu :=\frac {1}{|\Omega |} \int _\Omega u_0
. A previous result of the authors [Comm. Partial Differential Equations 42 (2017), 436–473] has asserted global existence of bounded classical solutions for arbitrary positive radial initial data
u
0
∈
C
3
(
Ω
¯
)
u_0\in C^3(\bar \Omega )
when either
n
≥
2
n\ge 2
and
χ
>
1
\chi >1
, or
n
=
1
n=1
and
∫
Ω
u
0
>
1
(
χ
2
−
1
)
+
\int _\Omega u_0>\frac {1}{\sqrt {(\chi ^2-1)_+}}
.
This present paper shows that these conditions are essentially optimal: Indeed, it is shown that if the taxis coefficient satisfies
χ
>
1
\chi >1
, then for any choice of
\[
{
m
>
1
χ
2
−
1
a
m
p
;
if
n
=
1
,
m
>
0
is arbitrary
a
m
p
;
if
n
≥
2
,
\begin {cases} m>\frac {1}{\sqrt {\chi ^2-1}} & \text {if $n=1$}, \\ \text {$m>0$ is arbitrary} & \text {if $n\ge 2$}, \end {cases}
\]
there exist positive initial data
u
0
∈
C
3
(
Ω
¯
)
u_0\in C^3(\bar \Omega )
satisfying
∫
Ω
u
0
=
m
\int _\Omega u_0=m
which are such that for some
T
>
0
T>0
, (
⋆
\star
) possesses a uniquely determined classical solution
(
u
,
v
)
(u,v)
in
Ω
×
(
0
,
T
)
\Omega \times (0,T)
blowing up at time
T
T
in the sense that
lim sup
t
↗
T
‖
u
(
⋅
,
t
)
‖
L
∞
(
Ω
)
=
∞
\limsup _{t\nearrow T} \|u(\cdot ,t)\|_{L^\infty (\Omega )}=\infty
.
This result is derived by means of a comparison argument applied to the doubly degenerate scalar parabolic equation satisfied by the mass accumulation function associated with (
⋆
\star
).
}},
author = {{Bellomo, Nicola and Winkler, Michael}},
issn = {{2330-0000}},
journal = {{Transactions of the American Mathematical Society, Series B}},
number = {{2}},
pages = {{31--67}},
publisher = {{American Mathematical Society (AMS)}},
title = {{{Finite-time blow-up in a degenerate chemotaxis system with flux limitation}}},
doi = {{10.1090/btran/17}},
volume = {{4}},
year = {{2017}},
}