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

we have

if

and

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

In addition, we have

Noting on ∂Ω, we have on the boundary that

Then, we have

if

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

Set

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

Therefore employing (3.8), we see that

if we knew

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

Thus,

holds if

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.