In this paper, we prove the global existence and non-existence of solutions of the following problem: RDC{
u
t
=
u
xx
+
u
2
- ∫
u
2
(
x
) d
x
,
x
ϵ (0, 1),
t
> 0,
u
x
(0,
t
) =
u
x
(1,
t
) =
t
> 0,
u
(
x
, 0) =
u
0
(
x
),
x
ϵ (0, 1), ∫
1
0
u
(
x, t
) d
x
= 0,
t
> 0, Moreover, let
u
m
(
x
) be a stationary solution of problem RDC with
m
zeros in the interval (0, 1) for
m
ϵ
N
, and if we take
u
0
(
x
). Then we have that the solution exists globally if 0 < ϵ < 1, and blows up in finite time if ϵ > 1. This result verifies the numerical results of Budd
et al
. (1993,
SIAM Jl appl. Math
. 53, 718-742) that the non-zero stationary solutions are unstable.
Blow-up and global existence to a degenerate reaction–diffusion equation with nonlinear memory
