AbstractThis paper deals with the homogeneous Neumann boundary value problem for chemotaxis system $$\begin{aligned} \textstyle\begin{cases} u_{t} = \Delta u - \nabla \cdot (u\nabla v)+\kappa u-\mu u^{\alpha }, & x\in \Omega, t>0, \\ v_{t} = \Delta v - uv, & x\in \Omega, t>0, \end{cases}\displaystyle \end{aligned}$$
{
u
t
=
Δ
u
−
∇
⋅
(
u
∇
v
)
+
κ
u
−
μ
u
α
,
x
∈
Ω
,
t
>
0
,
v
t
=
Δ
v
−
u
v
,
x
∈
Ω
,
t
>
0
,
in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}(N\geq 2)$
Ω
⊂
R
N
(
N
≥
2
)
, where $\alpha >1$
α
>
1
and $\kappa \in \mathbb{R},\mu >0$
κ
∈
R
,
μ
>
0
for suitably regular positive initial data.When $\alpha \ge 2$
α
≥
2
, it has been proved in the existing literature that, for any $\mu >0$
μ
>
0
, there exists a weak solution to this system. We shall concentrate on the weaker degradation case: $\alpha <2$
α
<
2
. It will be shown that when $N<6$
N
<
6
, any sublinear degradation is enough to guarantee the global existence of weak solutions. In the case of $N\geq 6$
N
≥
6
, global solvability can be proved whenever $\alpha >\frac{4}{3}$
α
>
4
3
. It is interesting to see that once the space dimension $N\ge 6$
N
≥
6
, the qualified value of α no longer changes with the increase of N.
How far does logistic dampening influence the global solvability of a high-dimensional chemotaxis system?
