In this section, we discuss the blow-up phenomena of Eq. (4) with . For Eq. (4), which describes shallow water waves, the blow-up occurs only in the form of wave-breaking, i.e. the solution remains bounded but its slope becomes unbounded in finite time.
Set , , then for all . Using this identity, we can rewrite Eq. (4) as follows:
(5)
or in the equivalent form:
(6)
where .
We now prove the following result.
Theorem 3.1 Let , , and , . If T is the existence time of the corresponding solution of initial data , then the -norm of to Eq. (4) (or (6)) blows up on if and only if
Proof Let be the solution of Eq. (4) with the initial data , , which is guaranteed by Theorem 2.1. If
by Sobolev’s embedding theorem, we obtain that the solution will blow up in finite time.
Next, applying the operator to Eq. (6), multiplying by , and integrating over R, we obtain
(7)
Here, . Assume that there exists such that
Similar to [18], using Lemma 2.1 with , we get
(8)
On the other hand, we estimate the second term of the right-hand side of Eq. (7)
(9)
Here, we applied Lemma 2.2 with and , Lemma 2.3 with . From (7)-(9), we obtain
Thus, by Gronwall’s inequality, we get
This completes the proof of Theorem 3.1. □
We have the following blow-up scenario for Eq. (4).
Theorem 3.2 Assume that , . Given , , the solution of Eq. (4) is uniformly bounded. Blow-up in finite time occurs if and only if
Proof is an invariant for Eq. (4). According to the inequalities
the invariance of ensures that the solution u is uniformly bounded as long as they exist.
If the slope of the solution becomes unbounded from below in finite time, then by Theorem 2.1 and Sobolev’s embedding theorem, we can see that the solution blows up in finite time.
Next, if the slope of the solution is bounded from below in finite time, then we deduce that the solution will not blow up in finite time. Differentiating Eq. (5) with respect to x, in view of the identity , we have
(10)
Note that and
By Young’s inequality, we get
and
Define . Since for all , it follows that a.e. on
Then
where
by
If , then and is decreasing. Otherwise, . Thus we get
By Theorem 3.1 and the above inequality, we have that if the slope of the solution is bounded from below in finite time, then the solution will not blow up in finite time. □
Next, we present the following blow-up result.
Theorem 3.3 Assume that , . Given , , , assume that we can find with
where
Then the corresponding solution to Eq. (4) for blows up in finite time. Moreover, the maximal time of existence T satisfies the inequality
Proof Now define by Lemma 2.4, and let be a point where this infimum is attained. From Eq. (10), we have
For , since , we arrive at
where
Note that
By Young’s inequality, we get
and
So, it follows that
(11)
where (note that )
The absolute continuity of the locally Lipschitz function allows us to perform an integration over and to have
We claim now that for all , where is fixed arbitrarily provided that . In fact, assuming the contrary, in view of being continuous, ensure the existence of such that in and . Then we deduce that
and a contradiction appears as we take . Using (11), we get
where . Since , and is locally Lipschitz, it follows that is locally Lipschitz as well. This gives
Integration of this inequality yields
Since , we obtain
In fact, as a consequence of these considerations, we obtain that the maximal existence time
An estimation from the above for T is obtained immediately, namely
The conclusion is reached by letting . □