@article{53343,
abstract = {{Abstract
The Cauchy problem in
R
n
{{\mathbb{R}}}^{n}
,
n
≥
2
n\ge 2
, for
u
t
=
Δ
u
−
∇
⋅
(
u
S
⋅
∇
v
)
,
0
=
Δ
v
+
u
,
(
⋆
)
\begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array}
is considered for general matrices
S
∈
R
n
×
n
S\in {{\mathbb{R}}}^{n\times n}
. A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to
BUC
(
R
n
)
∩
L
p
(
R
n
)
{\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n})
with some
p
∈
[
1
,
n
)
p\in \left[1,n)
, there exist
T
max
∈
(
0
,
∞
]
{T}_{\max }\in \left(0,\infty ]
and a uniquely determined
u
∈
C
0
(
[
0
,
T
max
)
;
BUC
(
R
n
)
)
∩
C
0
(
[
0
,
T
max
)
;
L
p
(
R
n
)
)
∩
C
∞
(
R
n
×
(
0
,
T
max
)
)
u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }))
such that with
v
≔
Γ
⋆
u
v:= \Gamma \star u
, and with
Γ
\Gamma
denoting the Newtonian kernel on
R
n
{{\mathbb{R}}}^{n}
, the pair
(
u
,
v
)
\left(u,v)
forms a classical solution of (
⋆
\star
) in
R
n
×
(
0
,
T
max
)
{{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })
, which has the property that
if
T
max
<
∞
,
then both
limsup
t
↗
T
max
‖
u
(
⋅
,
t
)
‖
L
∞
(
R
n
)
=
∞
and
limsup
t
↗
T
max
‖
∇
v
(
⋅
,
t
)
‖
L
∞
(
R
n
)
=
∞
.
\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty .
An exemplary application of this provides a result on global classical solvability in cases when
∣
S
+
1
∣
| S+{\bf{1}}|
is sufficiently small, where
1
=
diag
(
1
,
…
,
1
)
{\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1)
.}},
author = {{Winkler, Michael}},
issn = {{2391-5455}},
journal = {{Open Mathematics}},
keywords = {{General Mathematics}},
number = {{1}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type}}},
doi = {{10.1515/math-2022-0578}},
volume = {{21}},
year = {{2023}},
}
@article{63288,
abstract = {{Abstract
The Cauchy problem in
R
n
{{\mathbb{R}}}^{n}
,
n
≥
2
n\ge 2
, for
u
t
=
Δ
u
−
∇
⋅
(
u
S
⋅
∇
v
)
,
0
=
Δ
v
+
u
,
(
⋆
)
\begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array}
is considered for general matrices
S
∈
R
n
×
n
S\in {{\mathbb{R}}}^{n\times n}
. A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to
BUC
(
R
n
)
∩
L
p
(
R
n
)
{\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n})
with some
p
∈
[
1
,
n
)
p\in \left[1,n)
, there exist
T
max
∈
(
0
,
∞
]
{T}_{\max }\in \left(0,\infty ]
and a uniquely determined
u
∈
C
0
(
[
0
,
T
max
)
;
BUC
(
R
n
)
)
∩
C
0
(
[
0
,
T
max
)
;
L
p
(
R
n
)
)
∩
C
∞
(
R
n
×
(
0
,
T
max
)
)
u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }))
such that with
v
≔
Γ
⋆
u
v:= \Gamma \star u
, and with
Γ
\Gamma
denoting the Newtonian kernel on
R
n
{{\mathbb{R}}}^{n}
, the pair
(
u
,
v
)
\left(u,v)
forms a classical solution of (
⋆
\star
) in
R
n
×
(
0
,
T
max
)
{{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })
, which has the property that
if
T
max
<
∞
,
then both
limsup
t
↗
T
max
‖
u
(
⋅
,
t
)
‖
L
∞
(
R
n
)
=
∞
and
limsup
t
↗
T
max
‖
∇
v
(
⋅
,
t
)
‖
L
∞
(
R
n
)
=
∞
.
\hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty .
An exemplary application of this provides a result on global classical solvability in cases when
∣
S
+
1
∣
| S+{\bf{1}}|
is sufficiently small, where
1
=
diag
(
1
,
…
,
1
)
{\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1)
.}},
author = {{Winkler, Michael}},
issn = {{2391-5455}},
journal = {{Open Mathematics}},
number = {{1}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type}}},
doi = {{10.1515/math-2022-0578}},
volume = {{21}},
year = {{2023}},
}