We consider the following dissipative 2-component Degasperis-Procesi system:
(1.1)
where λ, are nonnegative constants, , ().
In system (1.1), if , we get the classical Degasperis-Procesi equation [1]
(1.2)
where represents the fluid velocity at time t in x direction and . The nonlinear convection term causes the steepening of wave form. The nonlinear dispersion effect term makes the wave form spread. The Degasperis-Procesi equation has been studied in many works [2–8]. Escher et al. [2] demonstrated that there exists a unique solution to (1.2) with initial value (). Liu and Yin [3] obtained the global existence of solutions to (1.2). They derived several wave breaking mechanisms in Sobolev space with . Yin [4] established the local well-posedness for the Degasperis-Procesi equation with initial value () on the line. In [5], the author obtained the global existence of solutions to the Degasperis-Procesi equation on the circle. The precise blow-up scenario was also derived. The global existence of strong solutions and global weak solutions to the Degasperis-Procesi equation were shown in [6, 7]. Guo et al. [9] studied the dissipative Degasperis-Procesi equation,
(1.3)
where () is the dissipative term. They obtained the global weak solutions to (1.3). Guo [10] established the local well-posedness for (1.3), and also obtained the global existence, persistence properties and propagation speed of solutions. Wu and Yin [8] obtained the local well-posedness for (1.3), and also studied the blow-up scenarios of solutions in periodic case.
On the other hand, many researchers have studied the integrable multi-component generalizations of the Degasperis-Procesi equation [11–16]. Yan and Yin [11] investigated the 2-component Degasperis-Procesi system
(1.4)
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. [12] investigated the traveling wave solutions to the 2-component Degasperis-Procesi system. Manwai [16] studied the self-similar solutions to the 2-component Degasperis-Procesi system. Fu and Qu [13] obtained the persistence properties of solutions to the 2-component Degasperis-Procesi system in Sobolev space with . For system (1.4), Jin and Guo [14] studied the blow-up mechanisms and persistence properties of strong solutions.
Recently, a large amount of literature has been devoted to the study of the 2-component Camassa-Holm system [17–28]. Hu [18] studied the dissipative periodic 2-component Camassa-Holm system
(1.5)
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 [19] considered the persistence properties of solutions to the modified 2-component Camassa-Holm system. Zhu [20] 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 [29] 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 [35].
Now we rewrite system (1.1) as
(1.6)
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 .
Theorem 1.2 Let and with . is the corresponding solution to system (1.1). If there exists such that
then
uniformly on the interval .
Theorem 1.3 Let and with . is the corresponding solution to system (1.1). Assume the constant .
-
(1)
For ,
and there exists such that as , then
-
(2)
For ,
and there exists such that as , then
Theorem 1.4 Let in system (1.1). Assume and with . is the corresponding solution to system (1.1). If there exists such that
then
uniformly on the interval .
Theorem 1.5 Assume and with . is the corresponding solution to system (1.1). If there exists such that
then
Theorem 1.6 Assume and with . is the corresponding solution to system (1.1). If there exists such that
and there exists
such that
then
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.
Notation We denote the norm of Lebesgue space , by , the norm in Sobolev space , by and the norm in Besov space , by . For , we denote