Finite-time blow-up in a degenerate chemotaxis system with flux limitation
Authors: Bellomo Nicola , Winkler Michael 
Published: June 2017

This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis reverse-solidus star right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row u Subscript t Baseline equals nabla dot left-parenthesis StartFraction u nabla u Over StartRoot u squared plus StartAbsoluteValue nabla u EndAbsoluteValue squared EndRoot EndFraction right-parenthesis minus chi nabla dot left-parenthesis StartFraction u nabla v Over StartRoot 1 plus StartAbsoluteValue nabla v EndAbsoluteValue squared EndRoot EndFraction right-parenthesis comma 2nd Row 0 equals normal upper Delta v minus mu plus u comma EndLayout EndLayout">

(\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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column m greater-than StartFraction 1 Over StartRoot chi squared minus 1 EndRoot EndFraction 2nd Column a m p semicolon if n equals 1 comma 2nd Row 1st Column m greater-than 0 is arbitrary 2nd Column a m p semicolon if n greater-than-or-equal-to 2 comma EndLayout">

{

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup double-vertical-bar u left-parenthesis dot comma t right-parenthesis double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals normal infinity">

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

).