Abstract
In this paper, we investigate the Cauchy problem of a weakly dissipative generalized periodic DegasperisProcesi equation. The precise blowup scenarios of strong solutions to the equation are derived by a direct method. Several new criteria guaranteeing the blowup of strong solutions are presented. The exact blowup rates of strong solutions are also determined. Finally, we give a new global existence results to the equation.
MSC: 35G25, 35Q35, 58D05.
Keywords:
weakly dissipative; generalized periodic DegasperisProcesi equation; blowup; global existence; blowup rate1 Introduction
Recently, the following generalized periodic DegasperisProcesi equation (μDP) was introduced and studied in [13]
where
Analogous to the generalized periodic CamassaHolm (μCH) equation [46], μDP equation possesses biHamiltonian form and infinitely many conservation laws. Here we list some of the simplest conserved quantities:
where
Obviously, under the constraint of
It is clear that the closest relatives of the μDP equation are the DP equation [811]
which was derived by Degasperis and Procesi in [8] as a model for the motion of shallow water waves, and its asymptotic accuracy is the same as for the CamassaHolm equation.
Generally speaking, energy dissipation is a very common phenomenon in the real world. It is interesting for us to study this kind of equation. Recently, Wu and Yin [12] considered the weakly dissipative DegasperisProcesi equation. For related studies, we refer to [13] and [14]. Liu and Yin [15] discussed the blowup, global existence for the weakly dissipative μHunterSaxton equation.
In this paper, we investigate the Cauchy problem of the following weakly dissipative periodic DegasperisProcesi equation [16]:
the constant λ is a nonnegative dissipative parameter and the term
Let
where ∗ denotes the spatial convolution. Then equation (1.1) takes the equivalent form of a quasilinear evolution equation of hyperbolic type:
It is easy to check that the operator
Since
and
The paper is organized as follows. In Section 2, we briefly give some needed results, including the local wellposedness of equation (1.1), and some useful lemmas and results which will be used in subsequent sections. In Section 3, we establish the precise blowup scenarios and blowup criteria of strong solutions. In Section 4, we give the blowup rate of strong solutions. In Section 5, we give two global existence results of strong solutions.
Remark 1.1 Although blowup criteria and global existence results of strong solutions to equation (1.1) are presented in [16], our blowup results improve considerably earlier results.
2 Preliminaries
In this section we recall some elementary results which we want to use in this paper. We list them and skip their proofs for conciseness. Local wellposedness for equation (1.1) can be obtained by Kato’s theory [17], in [16] the authors gave a detailed description on wellposedness theorem.
Theorem 2.1[16]
Let
of the Cauchy problems (1.1) which depends continuously on the initial data, i.e. the mapping
is continuous.
Remark 2.1 The maximal time of existence
Next we present the Sobolevtype inequalities, which play a key role to obtain blowup results for the Cauchy problem (1.1) in the sequel.
Lemma 2.2[18]
If
Lemma 2.3[19]
If
wherecis a constant depending only onr.
Lemma 2.4[20]
Let
and the functionmis almost everywhere differentiable on
We also need to introduce the classical particle trajectory method which is motivated
by McKean’s deep observation for the CamassaHolm equation in [21]. Suppose
where T is the maximal existence time of solution, then
Hence
which is always positive before the blowup time.
In addition, integrating both sides of the first equation in equation (1.1) with respect to x on , we obtain
it follows that
where
3 Blowup solutions
In this section, we are able to derive an import estimate for the
Lemma 3.1Let
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
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
Thus
In view of the diffeomorphism property of
This completes the proof of Lemma 3.1. □
Theorem 3.2Let
then the
Proof We assume that c is a generic positive constant depending only on s. Let
Let us estimate the first term of the above equation,
where we used Lemma 2.3 with
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 blowup scenario. Although the result which is proved in [16], our method is new, concise, and direct.
Theorem 3.3Let
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
If
then
Applying Gronwall’s inequality then yields for
Note that
Since
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
We now give first sufficient conditions to guarantee wave breaking.
Theorem 3.4Let
then the corresponding solution to equation (1.1) blow up in finite time in the following sense: there exists
where
Proof As mentioned early, we only need to consider the case
and let
Differentiating the first one in equation (1.3) with respect to x, we have
From equation (1.6) we deduce that
Obviously
Set
Then we obtain
Note that if
Since
then there exists
such that
We give another blowup result for the solutions of equation (1.1).
Theorem 3.5Let
Proof By
is also a solution of equation (1.1), therefore
Define
Note that if
Since
there exists
such that
4 Blowup rate
In this section, we consider the blowup profile; the blowup rate of equation (1.1) with respect to time can be shown as follows.
Theorem 4.1Let
Proof It is inferred from Lemma 2.4 that the function
is locally Lipschitz with
It follows that
Thus,
Now fix any
Since
It is found from equation (4.1) that
Integrating both sides of equation (4.2) on
that is,
By the arbitrariness of
This completes the proof of the theorem. □
5 Global existence
In this section, we will present some global existence results. Let us now prove the following lemma.
Lemma 5.1Let
where
Proof By the first one in equation (1.2) and equation (2.1) we have
Therefore
□
Lemma 5.1 and equation (2.2) imply that y and
Theorem 5.2Let
Proof By equation (2.1), we know that
We first consider the case that
It follows that
On the other hand, if
It follows that
Corollary 5.3If the initial value
then the corresponding solutionuof the initial value
Proof Since
If
If
□
Thus the theorem is proved by using Theorem 5.2.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This work is partially supported by the NSFC (Grant No. 11101376) the HiCi Project (Grant No. 2713035HiCi).
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