Abstract
AbstractWe study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - \frac{1}{|\Omega |} \int \limits _\Omega u + u \end{array}\right. } \end{aligned}$$
u
t
=
∇
·
(
(
u
+
1
)
m
-
1
∇
u
-
u
∇
v
)
+
λ
u
-
μ
u
1
+
κ
,
0
=
Δ
v
-
1
|
Ω
|
∫
Ω
u
+
u
and $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u -u \nabla v) + \lambda u - \mu u^{1+\kappa }, \\ 0 = \Delta v - v + u, \end{array}\right. } \end{aligned}$$
u
t
=
∇
·
(
(
u
+
1
)
m
-
1
∇
u
-
u
∇
v
)
+
λ
u
-
μ
u
1
+
κ
,
0
=
Δ
v
-
v
+
u
,
where $$\lambda $$
λ
and $$\mu $$
μ
are given spatially radial nonnegative functions and $$m, \kappa > 0$$
m
,
κ
>
0
are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $$m,\kappa $$
m
,
κ
leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $$\lambda , \mu > 0$$
λ
,
μ
>
0
, we find that there are initial data which lead to blow-up in (JL) if $$\begin{aligned} 0 \le \kappa&< \min \left\{ \frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&\quad \text {if } m\in \left[ \frac{2}{n},\frac{2n-2}{n}\right) \\ \text { or }\quad 0 \le \kappa&<\min \left\{ \frac{1}{2},\frac{n-1}{n}-\frac{m}{2}\right\}&\quad \text {if } m\in \left( 0,\frac{2}{n}\right) , \end{aligned}$$
0
≤
κ
<
min
1
2
,
n
-
2
n
-
(
m
-
1
)
+
if
m
∈
2
n
,
2
n
-
2
n
or
0
≤
κ
<
min
1
2
,
n
-
1
n
-
m
2
if
m
∈
0
,
2
n
,
and in (PE) if $$m \in [1, \frac{2n-2}{n})$$
m
∈
[
1
,
2
n
-
2
n
)
and $$\begin{aligned} 0 \le \kappa < \min \left\{ \frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\} . \end{aligned}$$
0
≤
κ
<
min
(
m
-
1
)
n
+
1
2
(
n
-
1
)
,
n
-
2
-
(
m
-
1
)
n
n
(
n
-
1
)
.
Funder
Studienstiftung des Deutschen Volkes
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Physics and Astronomy,General Mathematics
Cited by
20 articles.
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