In this paper, we investigate the Cauchy problem of a weakly dissipative generalized periodic Degasperis-Procesi equation. The precise blow-up scenarios of strong solutions to the equation are derived by a direct method. Several new criteria guaranteeing the blow-up of strong solutions are presented. The exact blow-up 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 Degasperis-Procesi equation; blow-up; global existence; blow-up rate
where is a time-dependent 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 , the authors verified that the periodic μDP equation describes the geodesic flows of a right-invariant affine connection on the Fréchet Lie group of all smooth and orientation-preserving diffeomorphisms of the circle .
Analogous to the generalized periodic Camassa-Holm (μCH) equation [4-6], μDP equation possesses bi-Hamiltonian form and infinitely many conservation laws. Here we list some of the simplest conserved quantities:
Obviously, under the constraint of , the μDP equation is reduced to the μBurgers equation .
which was derived by Degasperis and Procesi in  as a model for the motion of shallow water waves, and its asymptotic accuracy is the same as for the Camassa-Holm 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  considered the weakly dissipative Degasperis-Procesi equation. For related studies, we refer to  and . Liu and Yin  discussed the blow-up, global existence for the weakly dissipative μ-Hunter-Saxton equation.
In this paper, we investigate the Cauchy problem of the following weakly dissipative periodic Degasperis-Procesi equation :
where ∗ denotes the spatial convolution. Then equation (1.1) takes the equivalent form of a quasi-linear evolution equation of hyperbolic type:
The paper is organized as follows. In Section 2, we briefly give some needed results, including the local well-posedness of equation (1.1), and some useful lemmas and results which will be used in subsequent sections. In Section 3, we establish the precise blow-up scenarios and blow-up criteria of strong solutions. In Section 4, we give the blow-up rate of strong solutions. In Section 5, we give two global existence results of strong solutions.
Remark 1.1 Although blow-up criteria and global existence results of strong solutions to equation (1.1) are presented in , our blow-up results improve considerably earlier results.
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 well-posedness for equation (1.1) can be obtained by Kato’s theory , in  the authors gave a detailed description on well-posedness theorem.
of the Cauchy problems (1.1) which depends continuously on the initial data, i.e. the mapping
Next we present the Sobolev-type inequalities, which play a key role to obtain blow-up results for the Cauchy problem (1.1) in the sequel.
wherecis a constant depending only onr.
We also need to introduce the classical particle trajectory method which is motivated by McKean’s deep observation for the Camassa-Holm equation in . Suppose is the solution of the Camassa-Holm equation and satisfies the following equation:
which is always positive before the blow-up time.
In addition, integrating both sides of the first equation in equation (1.1) with respect to x on , we obtain
it follows that
3 Blow-up solutions
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
This completes the proof of Lemma 3.1. □
Let us estimate the first term of the above equation,
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 , 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
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
We now give first sufficient conditions to guarantee wave breaking.
Differentiating the first one in equation (1.3) with respect to x, we have
From equation (1.6) we deduce that
Then we obtain
We give another blow-up result for the solutions of equation (1.1).
4 Blow-up rate
In this section, we consider the blow-up profile; the blow-up rate of equation (1.1) with respect to time can be shown as follows.
Proof It is inferred from Lemma 2.4 that the function
It follows that
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
It is found from equation (4.1) that
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.
Proof By the first one in equation (1.2) and equation (2.1) we have
Thus the theorem is proved by using Theorem 5.2.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
This work is partially supported by the NSFC (Grant No. 11101376) the HiCi Project (Grant No. 27-130-35-HiCi).
Lenells, J, Misiolek, G, Tiğlay, F: Integrable evolution equations on spaces of tensor densities and their peakon solutions. Commun. Math. Phys.. 299, 129–161 (2010). Publisher Full Text
Fu, Y, Liu, Y, Qu, C: On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations. J. Funct. Anal.. 262, 3125–3158 (2012). Publisher Full Text
Chen, R, Lenells, J, Liu, Y: Stability of the μ-Camassa-Holm peakons. J. Nonlinear Sci.. 23, 97–112 (2013). Publisher Full Text
Christov, O: On the nonlocal symmetries of the μ-Camassa-Holm equation. J. Nonlinear Math. Phys.. 19, 411–427 (2012). Publisher Full Text
Zhou, Y: Blow up phenomena for the integrable Degasperis-Procesi equation. Phys. Lett. A. 328, 157–162 (2004). Publisher Full Text
Chen, W: On solutions to the Degasperis-Procesi equation. J. Math. Anal. Appl.. 379, 351–359 (2011). Publisher Full Text
Liu, Y, Yin, Z: Global existence and blow-up phenomena for the Degasperis-Procesi equation. Commun. Math. Phys.. 267, 801–820 (2006). Publisher Full Text
Wu, S, Yin, Z: Blow up, blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi equation. SIAM J. Math. Anal.. 40, 475–490 (2008). Publisher Full Text
Guo, Z: Some properties of solutions to the weakly dissipative Degasperis-Procesi equation. J. Differ. Equ.. 246, 4332–4344 (2009). Publisher Full Text
Kohlmann, M: Global existence and blow-up for a weakly dissipative μDP equation. Nonlinear Anal.. 74, 4746–4753 (2011). Publisher Full Text
Constantin, A: On the blow-up of solutions of a periodic shallow water equation. J. Nonlinear Sci.. 10, 391–399 (2000). Publisher Full Text
Kato, T, Ponce, G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math.. 41, 891–907 (1988). Publisher Full Text
Constantin, A, Escher, J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math.. 181, 229–243 (1998). Publisher Full Text