We set

The corresponding energy to the problem (1.1)-(1.3) is given by

and one can find that
easily from

whence

We note that from (1.6) and (1.7), we have

and by Sobolev inequality (1.8),
,
, where

Note that
has the maximum value
at
which are given in (2.1).

Adapting the idea of Zhou [10], we have the following lemma.

Lemma 2.1.

Suppose that

and

. Then

for all
.

Theorem 2.2.

For

, suppose that

and

satisfy

If

, then the global solution of the problem (1.1)–(1.3) blows up in finite time and the lifespan

Proof.

To prove the theorem, it suffices to show that the function

satisfies the hypotheses of the Lemma 1.1, where

,

and

to be determined later. To achieve this goal let us observe

Let us compute the derivatives

and

. Thus one has

for all

. In the above assumption (1.7), the definition of energy functionals (2.2) and (2.4) has been used. Then, due to (2.1) and (2.7) and taking

,

Hence

for all

and by assumption (2.8) we have

Therefore

for all

and by the construction of

, it is clearly that

whence,

. Thus for all

, from (2.13), (2.15), and (2.17) we obtain

Then using Lemma 1.1, one obtain that

as

Now, we are in a position to choose suitable

and

. Let

be a number that depends on

,

,

, and

as

To choose

, we may fix

as

Thus, for

the lifespan

is estimated by

which completes the proof.

Theorem 2.3.

Assume that

and the following conditions are valid:

Then the corresponding solution to (1.1)–(1.3) blows up in finite time.

Proof.

where the left-hand side of assumption (1.7) and the energy functional (2.2) have been used. Taking the inequality (2.27) and integrating this, we obtain

By using Poincare-Friedrich's inequality

where

. Using (2.30) and (2.31), we find from (2.28) that

Since

as

so, there must be a

such that

and by virtue of (2.33) and using (2.32), we get

Therefore, there exits a positive constant

This completes the proof.