First, we note that if *A* is reducible, then the full system (1.1) can be reduced to several sub-systems, independent of each other. Therefore, in the following, we assume that *A* is irreducible. In addition, we suppose that *k*_{1} - *m*_{1} ≤ *k*_{2} - *m*_{2} ≤ · · · *k*_{
n
} - *m*_{
n
} .

Let

be the first eigenfunction of

with the first eigenvalue
, normalized by
, then
,
in Ω and
and
on ∂Ω (see [14–16]).

Thus, there exist some positive constants

,

,

, and

such that

We also have
provided
with
and some positive constant
. For the fixed
, there exists a positive constant
such that
if
.

**Proof of the sufficiency**. We divide this proof into three different cases.

Case 1. (

*k*_{
i
} <

*m*_{
i
} (

*i* = 1,...,

*n*)). Let

where

*Q*_{
i
} satisfies

, and constants

*P*_{
i
} ,

*α*_{
i
} (

*i* = 1,...,

*n*) remain to be determined. Since

, by performing direct calculations, we have

in Ω × ℝ

^{+}. By setting

if

*m*_{
i
} ≥ 1,

if

*m*_{
i
} < 1, we have one the boundary that

Note that

*k*_{
i
} <

*m*_{
i
} (

*i* = 1,...,

*n*). From Lemmas 2.2 and 2.3, we know that inequalities (3.4) and (3.5) hold for suitable choices of

*P*_{
i
} ,

*α*_{
i
} (

*i* = 1,...,

*n*). Moreover, if we choose

*P*_{
i
} ,

*α*_{
i
} to be large enough such that

then
,
. Therefore, we have proved that
is a global upper solution of the system (1.1). The global existence of solutions to the problem (1.1) follows from the comparison principle.

Case 2. (

*k*_{
i
} ≥

*m*_{
i
} (

*i* = 1,...,

*n*)). Let

where

if

*m*_{
i
} ≥ 1,

if

*m*_{
i
} < 1,

,

,

,

are defined in (3.1) and (3.2),

*α*_{
i
} (

*i* = 1,...,

*n*) are positive constants that remain to be determined, and

Since -

*ye*^{-y}≥ -

*e*^{-1} for any

*y* > 0, we know that

. Thus, for (

*x*,

*t*) ∈ Ω × ℝ

^{+}, a simple computation shows that

Noting

on ∂Ω, we have on the boundary that

From Lemma 2.2, we know that inequalities (3.7) hold for suitable choices of

*α*_{
i
} (

*i* = 1,...,

*n*). Moreover, if we choose ∞

_{
i
} to be large enough such that

then
. Therefore, we have shown that
is an upper solution of (1.1) and exists globally. Therefore,
, and hence the solution (*u*_{1},..., *u*_{
n
} ) of (1.1) exists globally.

Case 3. (

*k*_{
i
}<

*m*_{
i
}(

*i* = 1,...,

*s*);

*k*_{
i
}≥

*m*_{
i
}(

*i* =

*s* + 1,...,

*n*)). Let

be as in (3.3) and

where

, and

*A*_{
i
}are as in case 2. By Lemma 2.3, we choose

*P*_{
i
}≥ (log

*Q*_{
i
})

^{-1}||

*u*_{i 0}||

_{∞} (

*i* = 1,...,

*s*) and

*M*_{
i
}≥ max{1, ||

*u*_{i 0}||

_{∞}} (

*i* =

*s* + 1,...,

*n*) such that

By similar arguments, in cases 1 and 2, we have on the boundary that

Therefore employing (3.8), we see that

We deduce from Lemma 2.2 that (3.9) holds for suitable choices of

*α*_{
i
} (

*i* = 1,...,

*n*). Moreover, we can choose

*α*_{
i
} large enough to assure that

Then, as in the calculations of cases 1 and 2, we have
. We prove that
is an upper solution of (1.1), so (*u*_{1},..., *u*_{
n
}) exists globally.

**Proof of the necessity**.

Without loss of generality, we first assume that all the lower-order principal minor determinants of A are non-negative, and |

*A*| < 0, for, if not, there exists some

*l* th-order (1 ≤

*l* <

*n*) principal minor determinant det

*A*_{l × l}of

*A* = (

*a*_{
ij
})

_{n×n}which is negative. Without loss of generality, we may consider that

and all of the

*sth*-order (1 ≤

*s* ≤

*l* - 1) principal minor determinants det

*A*_{s × s}of

*A*_{l × l}are non-negative. Then, we consider the following problem:

Note that
. If we can prove that the solution (*w*_{1},..., *w*_{
l
} ) of (3.10) blows up in finite time, then (*w*_{1},... *w*_{
l
} , *δ*,..., *δ*) is a lower solution of (1.1) that blows up in finite time. Therefore, the solution of (1.1) blows up in finite time.

We will complete the proof of the necessity of our theorem in three different cases.

Case 1. (

*k*_{
i
} <

*m*_{
i
} (

*i* = 1,...,

*n*)). Let

where

,

,

,

, the

*α*_{
i
} are as given in Lemma 2.4 and satisfy

,

By direct computation for

, we have

For

, we have

Thus, by (3.12) and Lemma 2.4, we have

We confirm that (u_{1},..., u_{
n
}) is a lower solution of (1.1), which blows up in finite time. We know by the comparison principle that the solution (*u*_{1},..., *u*_{
n
} ) blows up in finite time.

Case 2. (

*k*_{
i
} ≥

*m*_{
i
} (

*i* = 1,...,

*n*)). Let

if

*m*_{
i
} < 1,

if

*m*_{
i
} ≥ 1. for

*k*_{
i
} ≥

*m*_{
i
} (

*i* = 1,...,

*n*), set

where

*α*_{
i
} (

*i* = 1,...,

*n*) are to determined later and

By a direct computation, for

*x* ∈ Ω, 0 <

*t* <

*c*/

*b*, we obtain that

If

, we have

, and thus

On the other hand, since -

*ye*^{-y}≥ -

*e*^{-1} for any

*y* > 0, we have

We have by (3.16), (3.18), and (3.19) that
.

If

, then

, and then

It follows from (3.16), (3.17), and (3.20) that
.

We have on the boundary that

Moreover, by (3.14) and Lemma 2.4, we have that

(3.15), (3.21), and (3.22) imply that
. Therefore, (u_{1},..., u_{1}) is a lower solution of (1.1).

For

*k*_{
i
} =

*m*_{
i
} (

*i* = 1,...,

*n*), let

For *k*_{
i
} = *m*_{
i
} (*i* = 1,..., *s*) and *k*_{
i
} > *m*_{
i
} (*i* = *s* + 1,..., *n*), let
as in (3.13) and (3.23). Using similar arguments as above, we can prove that (u_{1},..., u_{
n
}) is a lower solution of (1.1). Therefore, (u_{1},..., u_{
n
}) ≤ (*u*_{1},..., *u*_{
n
} ). Consequently, (*u*_{1},..., *u*_{
n
} ) blows up in finite time.

Case 3. (

*k*_{
i
}<

*m*_{
i
}(

*i* = 1,...,

*s*);

*k*_{
i
}≥

*m*_{
i
}(

*i* =

*s* + 1,...,

*n*)). Let

be as in (3.11) and

where

*α*_{
i
} 's are to determined later and

Based on arguments in cases 1 and 2, we have

for

. Furthermore, for

, we have

From Lemma 2.4, we know that inequalities (3.24) hold for suitable choices of *α*_{
i
} (*i* = 1,..., *n*). We show that (u_{1},..., u_{
n
}) is a lower solution of (1.1). Since (u_{1},..., u_{
n
}) blows up in finite time, it follows that the solution of (1.1) blows up in finite time.