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 is a timedependent function on the unite circle and denotes its mean. The μDP equation can be formally described as an evolution equation on the space of tensor densities over the Lie algebra of smooth vector fields on the circle . In [2], the authors verified that the periodic μDP equation describes the geodesic flows of a rightinvariant affine connection on the Fréchet Lie group of all smooth and orientationpreserving diffeomorphisms of the circle .
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 , is an isomorphism between and . Moreover, it is easy to see that is also a conserved quantity for the μDP equation.
Obviously, under the constraint of , the μDP equation is reduced to the μBurgers equation [7].
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 models energy dissipation. Obviously, if then the equation reduces to the μDP equation. we can rewrite the system (1.1) as follows:
Let , be the associated Green’s function of the operator , then the operator can be expressed by its associated Green’s function,
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 has the inverse
Since and commute, the following identities hold:
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]
Letand; then there is a maximal timeTand a unique solution
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 in Theorem 2.1 is independent of the Sobolev index .
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]
Lemma 2.3[19]
wherecis a constant depending only onr.
Lemma 2.4[20]
Letand, then for everythere exists at least one pointwith
and the functionmis almost everywhere differentiable onwith
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 is the solution of the CamassaHolm equation and satisfies the following equation:
where T is the maximal existence time of solution, then is a diffeomorphism of the line. Taking the derivative with respect to x, we have
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 norm of strong solutions. This enables us to establish precise blowup scenario and several blowup results for equation (1.1).
Lemma 3.1Let, be given and assume the T is the maximal existence time of the corresponding solutionuto equation (1.1) with the initial data. Then we have
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 on yields
Thus
In view of the diffeomorphism property of , we can obtain
This completes the proof of Lemma 3.1. □
Theorem 3.2Let, be given and assume thatTis the maximal existence time of the corresponding solutionto the Cauchy problem (1.1) with the initial data. If there existssuch that
then thenorm ofdoes 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
Let us estimate the first term of the above equation,
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:
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, be given and assume thatTis the maximal existence time of the corresponding solutionto 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 3.4Let, andTbe the maximal time of the solutionto 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 existssatisfying
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
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
such that . Theorem 3.3 implies that the solution u blows up in finite time. □
We give another blowup result for the solutions of equation (1.1).
Theorem 3.5Let, andTbe the maximal time of the solutionto equation (1.1) with the initial data. Ifis 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
such that . Theorem 3.3 implies that the solution u blows up in finite time. □
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, andTbe the maximal time of the solutionto equation (1.1) with the initial data. IfTis finite, then
Proof It is inferred from Lemma 2.4 that the function
is locally Lipschitz with , . Note that , a.e. . Then we deduce that
It follows that
Thus,
Now fix any . In view of Theorem 3.1, there exists such that . Being locally Lipschitz, the function is absolutely continuous on . It then follows from the above inequality that is decreasing on and satisfies
Since is decreasing on , it follows that
It is found from equation (4.1) that
Integrating both sides of equation (4.2) on , we obtain
that is,
By the arbitrariness of , we have
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, be given and assume thatis the maximal existence time of the corresponding solutionto the Cauchy problem (1.1). Letbe the unique solution of equation (2.1). Then we have
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 have the same sign.
Theorem 5.2Let, . Ifdoes not change sign, then the corresponding solutionto equation (1.1) with the initial dataexists globally in time.
Proof By equation (2.1), we know that is diffeomorphism of the line and the periodicity of u with respect to spatial variable x, given , there exists a such that .
We first consider the case that on , in which case Lemma 5.1 ensures that . For , we have
On the other hand, if on , then Lemma 5.1 ensures that . Therefore, for , we have
It follows that . By using Theorem 3.2, we immediately conclude that the solution is global. This completes the proof of the theorem. □
Corollary 5.3If the initial valuesuch that
then the corresponding solutionuof the initial valueexists globally in time.
Proof Since , by Lemma 2.2, we obtain
□
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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