Abstract
This paper is concerned with blowup phenomena and global existence for the periodic twocomponent DullinGottwaldHolm system. We first obtain several blowup results and the blowup rate of strong solutions to the system. We then present a global existence result for strong solutions to the system.
MSC: 35G25, 35L05.
Keywords:
periodic twocomponent DullinGottwaldHolm system; blowup; blowup rate; global existence1 Introduction
In this paper, we consider the following periodic twocomponent DullinGottwaldHolm (DGH) system:
where
System (1.1) has been recently derived by Zhu et al. in [1] by following Ivanov’s approach [2]. It was shown in [1] that the DGH system is completely integrable and can be written as a compatibility condition of two linear systems
and
where ξ is a spectral parameter. Moreover, this system has the following two Hamiltonians:
and
For
where A and α are two positive constants, modeling unidirectional propagation of surface waves
on a shallow layer of water which is at rest at infinity,
For
where
The Cauchy problem (1.1) has been discussed in [1]. Therein Zhu and Xu established the local wellposedness to system (1.1), derived the precise blowup scenario and investigated the wave breaking for it. The aim of this paper is to further study the blowup phenomena for strong solutions to (1.1) and to present a global existence result.
Our paper is organized as follows. In Section 2, we briefly state some needed results including the local well posedness of system (1.1), the precise blowup scenario and some useful lemmas to study blowup phenomena and global existence. In Section 3, we give several new blowup results and the precise blowup rate. In Section 4, we present a new global existence result of strong solutions to (1.1).
Notation Given a Banach space Z, we denote its norm by
2 Preliminaries
In this section, we will briefly give some needed results in order to pursue our goal.
With
Note that if
The local wellposedness of the Cauchy problem (2.1) can be obtained by applying Kato’s theorem. As a result, we have the following wellposedness result.
Lemma 2.1[1]
Given the initial data
of (2.1). Moreover, the solution
Consider now the following initial value problem:
whereudenotes the first component of the solution
Lemma 2.2[1]
Let
Lemma 2.3[1]
Let
Moreover, if there exists an
Next, we give two useful conservation laws of strong solutions to (2.1).
Lemma 2.4[1]
Let
Lemma 2.5Let
Proof By the first equation in (2.1), we have
This completes the proof of the lemma. □
Then we state the following precise blowup mechanism of (2.1).
Lemma 2.6[1]
Let
Lemma 2.7[23]
Let
and the functionmis almost everywhere differentiable on
Lemma 2.8[24]
(i) For every
where the constant
(ii) For every
with the best possible constantclying within the range
Lemma 2.9[25]
If
Moreover,
Lemma 2.10[26]
Assume that a differentiable function
whereC, Kare positive constants. If the initial datum
3 Blowup phenomena
In this section, we discuss the blowup phenomena of system (2.1). Firstly, we prove that there exist strong solutions to (2.1) which do not exist globally in time.
Theorem 3.1Let
then the corresponding solution to (2.1) blows up in finite time.
Proof Applying Lemma 2.1 and a simple density argument, we only need to show that the above
theorem holds for some
Define now
By Lemma 2.7, we let
Differentiating the first equation in (2.2) with respect to x and using the identity
Since the map
Thus
Valuating (3.1) at
here we used the relations
and
It follows that
where
Applying Lemma 2.6, the solution blows up in finite time. □
Theorem 3.2Let
then the corresponding solution to (2.1) blows up in finite time.
Proof By Lemma 2.5, we have
and
Following a similar proof in Theorem 3.1, we have
where
Letting
Corollary 3.1Let
then the corresponding solution to (2.1) blows up in finite time.
Remark 3.1 Note that system (2.1) is variational under the transformation
Next, we will give more insight into the blowup mechanism for the wavebreaking solution to system (2.1), that is, the blowup rate for strong solutions to (2.1).
Theorem 3.3Let
Proof As mentioned earlier, here we only need to show that the above theorem holds for
Define now
By the proof of Theorem 3.1, there exists a positive constant
Let
A combination of (3.5) and (3.6) enables us to infer
Since m is locally Lipschitz on
with
that is,
By the arbitrariness of
4 Global existence
In this section, we will present a global existence result.
Theorem 4.1Let
Proof Define
By Lemma 2.7, we let
Since the map
Set
where
By Lemmas 2.22.3, we know that
Letting
Differentiating (4.2) with respect to t and using (4.1), we obtain
By Gronwall’s inequality, we have
for all
Thus,
for all
This completes the proof by using Lemma 2.6. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This research is partially supported by the Doctoral Research Foundation of Zhengzhou University of Light Industry.
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