TY  JOUR
AB  AbstractThe quasilinear Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u)  \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta vv+u, \end{array}\right. \end{aligned}$$
u
t
=
∇
·
(
D
(
u
)
∇
u
)

∇
·
(
S
(
u
)
∇
v
)
,
v
t
=
Δ
v

v
+
u
,
endowed with homogeneous Neumann boundary conditions is considered in a bounded domain $$\Omega \subset {\mathbb {R}}^n$$
Ω
⊂
R
n
, $$n \ge 3$$
n
≥
3
, with smooth boundary for sufficiently regular functions D and S satisfying $$D>0$$
D
>
0
on $$[0,\infty )$$
[
0
,
∞
)
, $$S>0$$
S
>
0
on $$(0,\infty )$$
(
0
,
∞
)
and $$S(0)=0$$
S
(
0
)
=
0
. On the one hand, it is shown that if $$\frac{S}{D}$$
S
D
satisfies the subcritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$
S
(
s
)
D
(
s
)
≤
C
s
α
for
all
s
≥
1
with
some
α
<
2
n
and $$C>0$$
C
>
0
, then for any sufficiently regular initial data there exists a global weak energy solution such that $${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$
ess
sup
t
>
0
‖
u
(
t
)
‖
L
p
(
Ω
)
<
∞
for some $$p > \frac{2n}{n+2}$$
p
>
2
n
n
+
2
. On the other hand, if $$\frac{S}{D}$$
S
D
satisfies the supercritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$
S
(
s
)
D
(
s
)
≥
c
s
α
for
all
s
≥
1
with
some
α
>
2
n
and $$c>0$$
c
>
0
, then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value $$\alpha = \frac{2}{n}$$
α
=
2
n
for $$n \ge 3$$
n
≥
3
, without any additional assumption on the behavior of D(s) as $$s \rightarrow \infty $$
s
→
∞
, in particular without requiring any algebraic lower bound for D. When applied to the Keller–Segel system with volumefilling effect for probability distribution functions of the type $$Q(s) = \exp (s^\beta )$$
Q
(
s
)
=
exp
(

s
β
)
, $$s \ge 0$$
s
≥
0
, for global solvability the exponent $$\beta = \frac{n2}{n}$$
β
=
n

2
n
is seen to be critical.
AU  Stinner, Christian
AU  Winkler, Michael
ID  53316
IS  2
JF  Journal of Evolution Equations
KW  Mathematics (miscellaneous)
SN  14243199
TI  A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volumefilling effects
VL  24
ER 