Persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system
© Ming et al.; licensee Springer. 2014
Received: 29 March 2014
Accepted: 28 April 2014
Published: 9 May 2014
The persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system are investigated. We find that if the initial data with their derivatives of the system exponentially decay at infinity, then the corresponding solution also exponentially decays at infinity.
MSC:35G25, 35L15, 35Q58.
Keywords2-component Degasperis-Procesi system dissipative persistence properties
where λ, are nonnegative constants, , ().
where () is the dissipative term. They obtained the global weak solutions to (1.3). Guo  established the local well-posedness for (1.3), and also obtained the global existence, persistence properties and propagation speed of solutions. Wu and Yin  obtained the local well-posedness for (1.3), and also studied the blow-up scenarios of solutions in periodic case.
where . They established the local well-posedness for system (1.4) in Besov space with , and also derived the precise blow-up scenarios of strong solutions in Sobolev space with . Zhou et al.  investigated the traveling wave solutions to the 2-component Degasperis-Procesi system. Manwai  studied the self-similar solutions to the 2-component Degasperis-Procesi system. Fu and Qu  obtained the persistence properties of solutions to the 2-component Degasperis-Procesi system in Sobolev space with . For system (1.4), Jin and Guo  studied the blow-up mechanisms and persistence properties of strong solutions.
where . The author not only established the local well-posedness for system (1.5) in Besov space with , but she also presented global existence results and the exact blow-up scenarios of strong solutions in Sobolev space with . For in system (1.5), Jin and Guo  considered the persistence properties of solutions to the modified 2-component Camassa-Holm system. Zhu  considered the persistence property of solutions to the coupled Camassa-Holm system, and also established the global existence and blow-up mechanisms of solutions. Guo [21, 22] studied the persistence properties and unique continuation of solutions to the 2-component Camassa-Holm system in the case . It was shown in  that the dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton and Novikov equations could be reduced to the non-dissipative versions by means of an exponentially time-dependent scaling. One may refer to [30–34] and the references therein for more details in this direction.
Motivated by the work in [13, 20, 35], we study the dissipative 2-component Degasperis-Procesi system (1.1). We note that the persistence properties of solutions to system (1.1) have not been discussed yet. The aim of this paper is to investigate the persistence properties of solutions in Sobolev space . The main idea of this work comes from .
where the operator .
The main results are presented as follows.
Theorem 1.1 Assume and with . Then the Cauchy problem (1.1) has a unique solution .
uniformly on the interval .
- (1)For ,
- (2)For ,
uniformly on the interval .
The remainder of this paper is organized as follows. In Section 2, the proofs of Theorems 1.1 and 1.2 are presented. Section 3 is devoted to the proofs of Theorems 1.3 and 1.4. The proofs of Theorems 1.5 and 1.6 are given in Section 4.
2 Proofs of Theorems 1.1 and 1.2
Proposition 2.1 
2.1 Proof of Theorem 1.1
Using the Littlewood-Paley theory and estimates for solutions to the transport equation, one may follow similar arguments as in  to establish the local well-posedness for system (1.1) with some modification. Here we omit the detailed proof. For system (1.1) with initial data (), we see that the corresponding solution . Thus we complete the proof of Theorem 1.1.
2.2 Proof of Theorem 1.2
uniformly on the interval . This completes the proof of Theorem 1.2.
3 Proofs of Theorems 1.3 and 1.4
3.1 Proof of Theorem 1.3
which combined with the above estimates yields a contradiction. We obtain . Consequently, , .
From (3.5), we have . This completes the proof of case (2) in Theorem 1.3.
3.2 Proof of Theorem 1.4
Then as . From Theorem 1.2, if as , we have as . This completes the proof of Theorem 1.4.
4 Proofs of Theorems 1.5 and 1.6
4.1 Proof of Theorem 1.5
Using the assumption in Theorem 1.5, we complete the proof.
4.2 Proof of Theorem 1.6
The proof of Theorem 1.6 is similar to the proof of Theorem 1.3, here we omit it.
The authors would like to express sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. This work was partially supported by National Natural Science Foundation of P.R. China (71003082) and Fundamental Research Funds for the Central Universities (SWJTU12CX061 and SWJTU09ZT36).
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