Consider the initial value problem
u
t
t
=
Δ
u
+
|
v
|
p
,
a
m
p
;
v
t
t
=
Δ
v
+
|
u
|
q
,
a
m
p
;
x
∈
R
n
,
a
m
p
;
t
>
0
,
u
(
x
,
0
)
=
f
(
x
)
,
a
m
p
;
v
(
x
,
0
)
=
h
(
x
)
,
a
m
p
;
a
m
p
;
u
t
(
x
,
0
)
=
g
(
x
)
,
a
m
p
;
v
t
(
x
,
0
)
=
k
(
x
)
,
\begin{equation*} \begin {array}{llll} u_{tt} = \Delta u+\vert v\vert ^{p}, & v_{tt} = \Delta v +\vert u\vert ^{q}, &x\in \mathbb {R}^{n},&t>0,\\ u(x,0)=f(x),&v(x,0)=h(x),&{}&{}\\ u_{t}(x,0) = g(x), &v_{t}(x,0) = k(x), \end{array} \end{equation*}
with
1
≤
n
≤
3
1\le n\le 3
and
p
,
q
>
0
p,q>0
. We show that there exists a bound
B
(
n
)
(
≤
∞
)
B(n) (\le \infty )
such that if
1
>
p
q
>
B
(
n
)
1>pq>B(n)
all nontrivial solutions with compact support blow up in finite time.