In this section, we are able to derive an import estimate for the -norm of strong solutions. This enables us to establish precise blow-up scenario and several blow-up results for equation (1.1).
Lemma 3.1 Let , be given and assume the T is the maximal existence time of the corresponding solution u to equation (1.1) with the initial data . Then we have
(3.1)
Proof The first equation of the Cauchy problem (1.1) is
In view of equation (1.5), we have
A direct computation implies that
It follows that
(3.2)
So we have
In view of equation (2.1) we have
Combing the above relations, we arrive at
Integrating the above inequality with respect to on yields
Thus
In view of the diffeomorphism property of , we can obtain
This completes the proof of Lemma 3.1. □
Theorem 3.2 Let , be given and assume that T is the maximal existence time of the corresponding solution to the Cauchy problem (1.1) with the initial data . If there exists such that
then the -norm of does not blow up on .
Proof We assume that c is a generic positive constant depending only on s. Let . Applying the operator to the first one in equation (1.3), multiplying by , and integrating over , we obtain
(3.3)
Let us estimate the first term of the above equation,
(3.4)
where we used Lemma 2.3 with . Furthermore, we estimate the second term of the right hand side of equation (3.3) in the following way:
(3.5)
Combing equations (3.4) and (3.5) with equation (3.3) we arrive at
An application of Gronwall’s inequality and the assumption of the theorem yield
This completes the proof of the theorem. □
The following result describes the precise blow-up scenario. Although the result which is proved in [16], our method is new, concise, and direct.
Theorem 3.3 Let , be given and assume that T is the maximal existence time of the corresponding solution to the Cauchy problem (1.1) with the initial data . Then the corresponding solution blows up in finite time if and only if
Proof Since the maximal existence time T is independent of the choice of s by Theorem 2.1, applying a simple density argument, we only need to consider the case . Multiplying the first one in equation (1.2) by y and integrating over with respect to x yield
If is bounded from below on , then there exists such that
then
Applying Gronwall’s inequality then yields for
Note that
Since and , Lemma 2.2 implies that
Theorem 3.1 ensures that the solution u does not blow up in finite time. On the other hand, by the Sobolev embedding theorem it is clear that if
then . This completes the proof of the theorem. □
We now give first sufficient conditions to guarantee wave breaking.
Theorem 3.4 Let , and T be the maximal time of the solution to equation (1.1) with the initial data . If
then the corresponding solution to equation (1.1) blow up in finite time in the following sense: there exists satisfying
where , such that
Proof As mentioned early, we only need to consider the case . Let
and let be a point where this minimum is attained by using Lemma 2.4. It follows that
Differentiating the first one in equation (1.3) with respect to x, we have
From equation (1.6) we deduce that
(3.6)
Obviously and . Substituting into equation (3.6), we get
Set
Then we obtain
Note that if , then for all . From the above inequality we obtain
Since
then there exists ,
such that . Theorem 3.3 implies that the solution u blows up in finite time. □
We give another blow-up result for the solutions of equation (1.1).
Theorem 3.5 Let , and T be the maximal time of the solution to equation (1.1) with the initial data . If is odd satisfies , then the corresponding solution to equation (1.1) blows up in finite time.
Proof By , we can check the function
is also a solution of equation (1.1), therefore is odd for any . By continuity with respect to x of u and , we get
Define for . From equation (3.6), we obtain
Note that if , then for all . From the above inequality we obtain
Since
there exists ,
such that . Theorem 3.3 implies that the solution u blows up in finite time. □