Abstract
Our concern in this paper is to prove blow-up results to the non-autonomous nonlinear system of wave equations
u
t
t
−
Δ
u
=
a
(
t
,
x
)
|
v
|
p
,
v
t
t
−
Δ
v
=
b
(
t
,
x
)
|
u
|
q
,
t
>
0
,
x
∈
R
N
in any space dimension. We show that a curve
F
˜
(
p
,
q
)
=
0
depending on the space dimension, on the exponents
p
,
q
and on the behavior of the functions
a
(
t
,
x
)
and
b
(
t
,
x
)
exists, such that all nontrivial solutions to the above system blow-up in a finite time whenever
F
˜
(
p
,
q
)
>
0
. Our method of proof uses some estimates developed by Galaktionov and Pohozaev in
[11] for a single non-autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived. Our result generalizes some important results such as the ones in Del Santo et al. (1996)
[12] and Galaktionov and Pohozaev (2003)
[11]. The advantage here is that our result applies to a wide variety of problems.