Persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system

Sen Ming; Han Yang; Ls Yong

doi:10.1186/1687-2770-2014-108

Research
Open Access

Persistence properties of solutions to the dissipative 2-component Degasperis-Procesi system

Sen Ming¹,
Han Yang¹Email author and
Ls Yong²

Boundary Value Problems20142014:108

https://doi.org/10.1186/1687-2770-2014-108

Received: 29 March 2014
Accepted: 28 April 2014
Published: 9 May 2014

Abstract

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.

Keywords

2-component
Degasperis-Procesi system
dissipative
persistence properties

1 Introduction

We consider the following dissipative 2-component Degasperis-Procesi system:

{\begin{cases} u_{t} - u_{x x t} + 4 u u_{x} + λ_{1} (u - u_{x x}) + c ρ ρ_{x} = 3 u_{x} u_{x x} + u u_{x x x}, & t > 0, x \in R, \\ ρ_{t} + u ρ_{x} + 2 u_{x} ρ + λ ρ = 0, & t > 0, x \in R, \\ u (0, x) = u_{0} (x), ρ (0, x) = ρ_{0} (x), & x \in R, \end{cases}

(1.1)

where λ, $λ_{1}$ are nonnegative constants, $c \in R$ , $(u_{0}, ρ_{0}) \in H^{s} (R) \times H^{s - 1} (R)$ ( $s > \frac{3}{2}$ ).

In system (1.1), if

λ_{1} = ρ = 0

, we get the classical Degasperis-Procesi equation [1]

u_{t} - u_{x x t} + 4 u u_{x} = 3 u_{x} u_{x x} + u u_{x x x},

(1.2)

where

u (t, x)

represents the fluid velocity at time t in x direction and

k \in R

. The nonlinear convection term

u u_{x}

causes the steepening of wave form. The nonlinear dispersion effect term

3 u_{x} u_{x x} + u u_{x x x}

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

u (t, x) \in C ([0, T], H^{s} (R)) \cap C^{1} ([0, T], H^{s - 1} (R))

to (1.2) with initial value

u_{0} (x) \in H^{s} (R)

(

s > \frac{3}{2}

). Liu and Yin [3] obtained the global existence of solutions to (1.2). They derived several wave breaking mechanisms in Sobolev space

H^{s} (R)

with

s > \frac{3}{2}

. Yin [4] established the local well-posedness for the Degasperis-Procesi equation with initial value

u_{0} \in H^{s} (R)

(

s > \frac{3}{2}

) 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,

u_{t} - u_{x x t} + 4 u u_{x} + λ (u - u_{x x}) = 3 u_{x} u_{x x} + u u_{x x x},

(1.3)

where $λ (u - u_{x x})$ ( $λ > 0$ ) 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

{\begin{cases} u_{t} - u_{x x t} + 4 u u_{x} + c ρ ρ_{x} = 3 u_{x} u_{x x} + u u_{x x x}, & t > 0, x \in R, \\ ρ_{t} + u ρ_{x} + 2 u_{x} ρ = 0, & t > 0, x \in R, \\ u (0, x) = u_{0} (x), ρ (0, x) = ρ_{0} (x), & x \in R, \end{cases}

(1.4)

where $c \in R$ . They established the local well-posedness for system (1.4) in Besov space $B_{p, r}^{s} (R) \times B_{p, r}^{s - 1} (R)$ with $s > max (1 + \frac{1}{p}, \frac{3}{2})$ , and also derived the precise blow-up scenarios of strong solutions in Sobolev space $H^{s} (R) \times H^{s - 1} (R)$ with $s > \frac{3}{2}$ . 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 $H^{s} (R) \times H^{s - 1} (R)$ with $s \geq 2$ . 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

{\begin{cases} u_{t} - u_{x x t} + 3 u u_{x} + λ (u - u_{x x}) + ρ ρ_{x} = 2 u_{x} u_{x x} + u u_{x x x}, & t > 0, x \in S, \\ ρ_{t} + u ρ_{x} + u_{x} ρ + λ ρ = 0, & t > 0, x \in S, \\ u (t, x + 1) = u (t, x), ρ (t, x + 1) = ρ (t, x), & t > 0, x \in S, \\ u (0, x) = u_{0} (x), ρ (0, x) = ρ_{0} (x), & x \in S, \end{cases}

(1.5)

where $λ > 0$ . The author not only established the local well-posedness for system (1.5) in Besov space $B_{p, r}^{s} (S) \times B_{p, r}^{s - 1} (S)$ with $s > max (1 + \frac{1}{p}, \frac{3}{2})$ , but she also presented global existence results and the exact blow-up scenarios of strong solutions in Sobolev space $H^{s} (S) \times H^{s - 1} (S)$ with $s > \frac{3}{2}$ . For $λ = 0$ 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 $λ = 0$ . 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 $L^{\infty} (R)$ . The main idea of this work comes from [35].

Now we rewrite system (1.1) as

{\begin{cases} u_{t} + u u_{x} = P (D) (\frac{3}{2} u^{2} + \frac{1}{2} c ρ^{2}) - λ_{1} u, & t > 0, x \in R, \\ ρ_{t} + u ρ_{x} = - 2 u_{x} ρ - λ ρ, & t > 0, x \in R, \\ u (0, x) = u_{0} (x), ρ (0, x) = ρ_{0} (x), & x \in R, \end{cases}

(1.6)

where the operator $P (D) = - \partial_{x} {(1 - \partial_{x}^{2})}^{- 1}$ .

The main results are presented as follows.

Theorem 1.1 Assume $T > 0$ and $(u_{0}, ρ_{0}) \in H^{s} (R) \times H^{s - 1} (R)$ with $s > \frac{3}{2}$ . Then the Cauchy problem (1.1) has a unique solution $(u, ρ) \in C ([0, T]; H^{s} (R)) \cap C^{1} ([0, T]; H^{s - 1} (R)) \times C ([0, T]; H^{s - 1} (R)) \cap C^{1} ([0, T]; H^{s - 2} (R))$ .

Theorem 1.2 Let

T > 0

and

(u_{0}, ρ_{0}) \in H^{s} (R) \times H^{s - 1} (R)

with

s > \frac{3}{2}

.

(u, ρ) \in C ([0, T]; H^{s} (R)) \cap C^{1} ([0, T]; H^{s - 1} (R)) \times C ([0, T]; H^{s - 1} (R)) \cap C^{1} ([0, T]; H^{s - 2} (R))

is the corresponding solution to system (1.1). If there exists

θ \in (0, 1)

such that

| u_{0} (x) |, | \partial_{x} u_{0} (x) |, | ρ_{0} (x) | \sim O (e^{- θ x}) as x \to \infty,

then

| u (t, x) |, | \partial_{x} u (t, x) |, | ρ (t, x) | \sim O (e^{- θ x}) as x \to \infty

uniformly on the interval $[0, T]$ .

Theorem 1.3 Let

T > 0

and

(u_{0}, ρ_{0}) \in H^{s} (R) \times H^{s - 1} (R)

with

s > \frac{3}{2}

.

(u, ρ) \in C ([0, T]; H^{s} (R)) \cap C^{1} ([0, T]; H^{s - 1} (R)) \times C ([0, T]; H^{s - 1} (R)) \cap C^{1} ([0, T]; H^{s - 2} (R))

is the corresponding solution to system (1.1). Assume the constant

θ \in (\frac{1}{2}, 1)

.

(1)

For $c > 0$ ,
$| u_{0} (x) | \sim o (e^{- x}), | \partial_{x} u_{0} (x) | \sim O (e^{- θ x}), | ρ_{0} (x) | \sim O (e^{- θ x}) as x \to \infty,$

and there exists

t_{1} \in (0, T)

such that

| u (t_{1}, x) | \sim o (e^{- x})

as

x \to \infty

, then

u (t, x) = 0, ρ (t, x) = 0 .

(2)

For $c = 0$ ,
$| u_{0} (x) | \sim o (e^{- x}), | \partial_{x} u_{0} (x) | \sim O (e^{- θ x}) as x \to \infty,$

and there exists

t_{1} \in (0, T)

such that

| u (t_{1}, x) | \sim o (e^{- x})

as

x \to \infty

, then

u (t, x) = 0, ρ (t, x) = ρ_{0} (x) e^{- λ t} .

Theorem 1.4 Let

λ_{1} = 0

in system (1.1). Assume

T > 0

and

(u_{0}, ρ_{0}) \in H^{s} (R) \times H^{s - 1} (R)

with

s > \frac{3}{2}

.

(u, ρ) \in C ([0, T]; H^{s} (R)) \cap C^{1} ([0, T]; H^{s - 1} (R)) \times C ([0, T]; H^{s - 1} (R)) \cap C^{1} ([0, T]; H^{s - 2} (R))

is the corresponding solution to system (1.1). If there exists

θ \in (\frac{1}{2}, 1)

such that

| u_{0} (x) | \sim O (e^{- x}), | \partial_{x} u_{0} (x) | \sim O (e^{- θ x}), | ρ_{0} (x) | \sim O (e^{- θ x}) as x \to \infty,

then

| u (t, x) | \sim O (e^{- x}), | ρ (t, x) | \sim O (e^{- θ x}) as x \to \infty

uniformly on the interval $[0, T]$ .

Theorem 1.5 Assume

T > 0

and

(u_{0}, ρ_{0}) \in H^{s} (R) \times H^{s - 1} (R)

with

s > \frac{5}{2}

.

(u, ρ) \in C ([0, T]; H^{s} (R)) \cap C^{1} ([0, T]; H^{s - 1} (R)) \times C ([0, T]; H^{s - 1} (R)) \cap C^{1} ([0, T]; H^{s - 2} (R))

is the corresponding solution to system (1.1). If there exists

θ \in (0, 1)

such that

| u_{0} (x) |, | ρ_{0} (x) |, | u_{0 x} (x) |, | ρ_{0 x} (x) | \sim O (e^{- θ x}) as x \to \infty,

then

| u (t, x) |, | ρ (t, x) |, | u_{x} (t, x) |, | ρ_{x} (t, x) | \sim O (e^{- θ x}) as x \to \infty .

Theorem 1.6 Assume

T > 0

and

(u_{0}, ρ_{0}) \in H^{s} (R) \times H^{s - 1} (R)

with

s > \frac{5}{2}

.

(u, ρ) \in C ([0, T]; H^{s} (R)) \cap C^{1} ([0, T]; H^{s - 1} (R)) \times C ([0, T]; H^{s - 1} (R)) \cap C^{1} ([0, T]; H^{s - 2} (R))

is the corresponding solution to system (1.1). If there exists

θ \in (\frac{1}{2}, 1)

such that

| u_{0} (x) |, | ρ_{0} (x) | \sim o (e^{- x}) as x \to \infty, | u_{0 x} (x) |, | ρ_{0 x} (x) | \sim O (e^{- θ x}) as x \to \infty,

and there exists

t_{1} \in [0, T]

such that

| u (t_{1}, x) |, | ρ (t_{1}, x) | \sim o (e^{- x}) as x \to \infty,

then

u (t, x) = 0, ρ (t, x) = 0 .

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

L^{p} (R)

,

1 \leq p \leq \infty

by

{∥ \cdot ∥}_{L^{p}}

, the norm in Sobolev space

H^{s} (R)

,

s \in R

by

{∥ \cdot ∥}_{H^{s}}

and the norm in Besov space

B_{p, r}^{s} (R)

,

s \in R

by

{∥ \cdot ∥}_{B_{p, r}^{s}}

. For

θ \in R

, we denote

\begin{matrix} | f (x) | \sim O (e^{θ x}) as x \to \infty, if lim_{x \to \infty} \frac{| f (x) |}{e^{θ x}} = L (L \neq 0); \\ | f (x) | \sim o (e^{θ x}) as x \to \infty, if lim_{x \to \infty} \frac{| f (x) |}{e^{θ x}} = 0 . \end{matrix}

2 Proofs of Theorems 1.1 and 1.2

We write the definition of Besov space. One may check [36–39] for more details.

Proposition 2.1 [39]

Let

s \in R

and

1 \leq p, r \leq \infty

. The nonhomogeneous Besov space is defined by

B_{p, r}^{s} (R) = {f \in S^{'} (R) | {∥ f ∥}_{B_{p, r}^{s} (R)} < \infty}

, where

{∥ f ∥}_{B_{p, r}^{s} (R)} = {\begin{cases} {(\sum_{j = - 1}^{\infty} 2^{j r s} {∥ Δ_{j} f ∥}_{L^{p}}^{r})}^{\frac{1}{r}}, & r < \infty, \\ {sup}_{j \geq - 1} 2^{j s} {∥ Δ_{j} f ∥}_{L^{p}}, & r = \infty . \end{cases}

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 [11] 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 $(u_{0}, ρ_{0}) \in B_{p, r}^{s} \times B_{p, r}^{s - 1}$ ( $s > max (1 + \frac{1}{p}, \frac{3}{2})$ ), we see that the corresponding solution $(u, ρ) \in C ([0, T]; B_{p, r}^{s}) \cap C^{1} ([0, T]; B_{p, r}^{s - 1}) \times C ([0, T]; B_{p, r}^{s - 1}) \cap C^{1} ([0, T]; B_{p, r}^{s - 2})$ . Thus we complete the proof of Theorem 1.1.

2.2 Proof of Theorem 1.2

We denote

F (u, ρ) = \frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}, M = sup_{t \in [0, T]} {∥ (u, ρ) ∥}_{H^{s} \times H^{s - 1}} .

Multiplying the second equation in (1.6) by

ρ^{2 n - 1}

with

n \in N^{*}

and integrating the resultant equation with respect to x yield

\int_{R} ρ_{t} ρ^{2 n - 1} d x + \int_{R} u ρ_{x} ρ^{2 n - 1} d x + 2 \int_{R} u_{x} ρ ρ^{2 n - 1} d x + λ \int_{R} ρ ρ^{2 n - 1} d x = 0 .

(2.1)

We have

\begin{matrix} \int_{R} ρ_{t} ρ^{2 n - 1} d x = \frac{1}{2 n} \frac{d}{d t} {∥ ρ (t) ∥}_{L^{2 n}}^{2 n} = {∥ ρ (t) ∥}_{L^{2 n}}^{2 n - 1} \frac{d}{d t} {∥ ρ (t) ∥}_{L^{2 n}}, \\ \int_{R} u ρ_{x} ρ^{2 n - 1} d x + 2 \int_{R} u_{x} ρ ρ^{2 n - 1} d x = - \frac{1}{2 n} \int_{R} u_{x} ρ^{2 n} d x + 2 \int_{R} u_{x} ρ^{2 n} d x \\ \leq (2 - \frac{1}{2 n}) {∥ u_{x} (t) ∥}_{L^{\infty}} {∥ ρ (t) ∥}_{L^{2 n}}^{2 n}, \\ λ \int_{R} ρ ρ^{2 n - 1} d x = λ {∥ ρ (t) ∥}_{L^{2 n}}^{2 n} . \end{matrix}

Thus

\frac{d}{d t} {∥ ρ (t) ∥}_{L^{2 n}} \leq [(2 - \frac{1}{2 n}) {∥ u_{x} (t) ∥}_{L^{\infty}} + λ] {∥ ρ (t) ∥}_{L^{2 n}} .

(2.2)

If

s > \frac{3}{2}

, using the Sobolev embedding theorem, we have

{∥ u_{x} (t) ∥}_{L^{\infty}} \leq {∥ u (t) ∥}_{H^{s}} \leq M

. Applying the Gronwall inequality to (2.2) yields

{∥ ρ (t) ∥}_{L^{2 n}} \leq {∥ ρ_{0} ∥}_{L^{2 n}} e^{[(2 - \frac{1}{2 n}) M + λ] t} .

Noting

u \in L^{2} \cap L^{\infty}

gives rise to

lim_{p \to \infty} {∥ u ∥}_{L^{p}} = {∥ u ∥}_{L^{\infty}}

and, taking the limit as

n \to \infty

, we obtain

{∥ ρ (t) ∥}_{L^{\infty}} \leq {∥ ρ_{0} ∥}_{L^{\infty}} e^{(2 M + λ) t} .

Multiplying the first equation in system (1.6) by

u^{2 n - 1}

with

n \in N^{*}

and integrating the resultant equation with respect to x yield

\int_{R} u_{t} u^{2 n - 1} d x + \int_{R} u u_{x} u^{2 n - 1} d x + λ_{1} \int_{R} u u^{2 n - 1} d x + \int_{R} \partial_{x} g * F (u, ρ) u^{2 n - 1} d x = 0 .

(2.3)

Using the Holder inequality, we have

\int_{R} \partial_{x} g * F (u, ρ) u^{2 n - 1} d x \leq {∥ \partial_{x} g * F (u, ρ) ∥}_{L^{2 n}} {∥ u ∥}_{L^{2 n}}^{2 n - 1},

(2.4)

which in combination with (2.3) yields

\frac{d}{d t} {∥ u ∥}_{L^{2 n}} \leq λ_{1} {∥ u ∥}_{L^{2 n}} + {∥ \partial_{x} g * F (u, ρ) ∥}_{L^{2 n}} .

Using the Gronwall inequality, one derives

{∥ u (t) ∥}_{L^{2 n}} \leq ({∥ u_{0} ∥}_{L^{2 n}} + \int_{0}^{t} {∥ \partial_{x} g * F (u, ρ) (τ) ∥}_{L^{2 n}} d τ) e^{λ_{1} t} .

(2.5)

Taking the limit as

n \to \infty

in (2.5), one gets

{∥ u (t) ∥}_{L^{\infty}} \leq ({∥ u_{0} ∥}_{L^{\infty}} + \int_{0}^{t} {∥ \partial_{x} g * F (u, ρ) (τ) ∥}_{L^{\infty}} d τ) e^{λ_{1} t} .

(2.6)

Differentiating the first equation in (1.6) in the variable x yields

u_{x t} + u u_{x x} + u_{x}^{2} + λ_{1} u_{x} + \partial_{x}^{2} g * F (u, ρ) = 0 .

(2.7)

Multiplying (2.7) by

u_{x}^{2 n - 1}

with

n \in N^{*}

, integrating the resultant equation with respect to x and using

\int_{R} (u u_{x x} u_{x}^{2 n - 1} + u_{x}^{2} u_{x}^{2 n - 1}) d x = (1 - \frac{1}{2 n}) \int_{R} u_{x}^{2 n + 1} d x \leq {∥ u_{x} ∥}_{L^{\infty}} {∥ u_{x} ∥}_{L^{2 n}}^{2 n}

(2.8)

and

\int_{R} [\partial_{x}^{2} g * F (u, ρ)] u_{x}^{2 n - 1} d x \leq {∥ \partial_{x}^{2} g * F (u, ρ) ∥}_{L^{2 n}} {∥ u_{x} ∥}_{L^{2 n}}^{2 n - 1},

we have

\frac{d}{d t} {∥ u_{x} ∥}_{L^{2 n}} \leq ({∥ u_{x} ∥}_{L^{\infty}} + λ_{1}) {∥ u_{x} ∥}_{L^{2 n}} + {∥ \partial_{x}^{2} g * F (u, ρ) ∥}_{L^{2 n}} .

We obtain

{∥ u_{x} (t) ∥}_{L^{\infty}} \leq [{∥ u_{0 x} ∥}_{L^{\infty}} + \int_{0}^{t} {∥ \partial_{x}^{2} g * F (u, ρ) (τ) ∥}_{L^{\infty}} d τ] e^{(M + λ_{1}) t} .

(2.9)

We introduce the weight function

φ_{N} (x)

which is independent on t

φ_{N} (x) = {\begin{cases} 1, & x \leq 0, \\ e^{θ x}, & x \in (0, N), \\ e^{θ N}, & x \geq N, \end{cases}

where

N \in N^{*}

. It follows

0 \leq {(φ_{N} (x))}_{x} \leq φ_{N} (x)

a.e.

x \in R

. Multiplying the first equation in system (1.6) and (2.7) by

φ_{N} (x)

, we obtain

\partial_{t} (u φ_{N}) + u φ_{N} u_{x} + λ_{1} u φ_{N} + φ_{N} \partial_{x} g * F (u, ρ) = 0,

(2.10)

\partial_{t} (u_{x} φ_{N}) + u u_{x x} φ_{N} + u_{x} φ_{N} u_{x} + λ_{1} u_{x} φ_{N} + φ_{N} \partial_{x}^{2} g * F (u, ρ) = 0 .

(2.11)

Multiplying (2.10) by

{(u φ_{N})}^{2 n - 1}

and (2.11) by

{(u_{x} φ_{N})}^{2 n - 1}

, respectively, and integrating the resultant equation with respect to x, we also note

\begin{array}{rcl} | \int_{R} u u_{x x} φ_{N} {(u_{x} φ_{N})}^{2 n - 1} d x | & = & | \int_{R} u {(u_{x} φ_{N})}^{2 n - 1} [\partial_{x} (u_{x} φ_{N}) - u_{x} {(φ_{N})}_{x}] d x | \\ \leq & | \frac{1}{2 n} \int_{R} u_{x} {(u_{x} φ_{N})}^{2 n} d x | + | \int_{R} u u_{x} φ_{N} {(u_{x} φ_{N})}^{2 n - 1} d x | \\ \leq & ({∥ u (t) ∥}_{L^{\infty}} + {∥ u_{x} (t) ∥}_{L^{\infty}}) {∥ u_{x} φ_{N} ∥}_{L^{2 n}}^{2 n} . \end{array}

As in the weightless case, we estimate

u φ_{N}

and

u_{x} φ_{N}

step by step as the previous estimates for u and

u_{x}

. Thus

\begin{matrix} {∥ u φ_{N} ∥}_{L^{\infty}} + {∥ u_{x} φ_{N} ∥}_{L^{\infty}} \\ \leq e^{(4 M + 2 λ_{1}) t} ({∥ u_{0} φ_{N} ∥}_{L^{\infty}} + {∥ u_{0 x} φ_{N} ∥}_{L^{\infty}}) \\ + e^{(4 M + 2 λ_{1}) t} \int_{0}^{t} ({∥ φ_{N} \partial_{x} g * F (u, ρ) ∥}_{L^{\infty}} + {∥ φ_{N} \partial_{x}^{2} g * F (u, ρ) ∥}_{L^{\infty}}) d τ . \end{matrix}

(2.12)

Multiplying the second equation in system (1.6) by

φ_{N}

, one deduces

{(ρ φ_{N})}_{t} + u ρ_{x} φ_{N} + 2 u_{x} ρ φ_{N} + λ ρ φ_{N} = 0 .

(2.13)

Multiplying (2.13) by

{(ρ φ_{N})}^{2 n - 1}

, integrating the resultant equation with respect to x and using

\begin{matrix} | \int_{R} u ρ_{x} φ_{N} {(ρ φ_{N})}^{2 n - 1} d x | = | \int_{R} u {(ρ φ_{N})}^{2 n - 1} [\partial_{x} (ρ φ_{N}) - ρ {(φ_{N})}_{x}] d x | \\ \leq | \frac{1}{2 n} \int_{R} u_{x} {(ρ φ_{N})}^{2 n} d x | + | \int_{R} u ρ φ_{N} {(ρ φ_{N})}^{2 n - 1} d x | \\ \leq ({∥ u (t) ∥}_{L^{\infty}} + {∥ u_{x} (t) ∥}_{L^{\infty}}) {∥ ρ φ_{N} ∥}_{L^{2 n}}^{2 n}, \\ \int_{R} 2 u_{x} ρ φ_{N} {(ρ φ_{N})}^{2 n - 1} d x \leq 2 {∥ u_{x} (t) ∥}_{L^{\infty}} {∥ ρ φ_{N} ∥}_{L^{2 n}}^{2 n}, \end{matrix}

we have

\frac{d}{d t} {∥ ρ φ_{N} ∥}_{L^{2 n}} \leq (3 {∥ u_{x} ∥}_{L^{\infty}} + {∥ u (t) ∥}_{L^{\infty}} + λ) {∥ ρ φ_{N} ∥}_{L^{2 n}} .

Applying the Gronwall inequality and the Sobolev embedding theorem yields

{∥ ρ φ_{N} ∥}_{L^{2 n}} \leq e^{(4 M + λ) t} {∥ ρ_{0} φ_{N} ∥}_{L^{2 n}} .

Taking the limit as

n \to \infty

, one obtains

{∥ ρ φ_{N} ∥}_{L^{\infty}} \leq e^{(4 M + λ) t} {∥ ρ_{0} φ_{N} ∥}_{L^{\infty}} .

(2.14)

There exists

c_{1} > 0

which depends on

θ \in (0, 1)

, such that for all

N \in N^{*}

φ_{N} (x) \int_{R} e^{- | x - y |} \frac{1}{φ_{N} (y)} d y \leq c_{1} .

Thus

\begin{matrix} | φ_{N} \partial_{x} g * F (u, ρ) | \\ \leq | \frac{1}{2} φ_{N} (x) \int_{R} sgn (x - y) e^{- | x - y |} (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d y | \\ \leq \frac{1}{2} φ_{N} (x) | \int_{R} e^{- | x - y |} \frac{1}{φ_{N} (y)} φ_{N} (y) (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d y | \\ \leq \frac{(3 + c) c_{1}}{4} ({∥ u ∥}_{L^{\infty}} {∥ u φ_{N} ∥}_{L^{\infty}} + {∥ ρ ∥}_{L^{\infty}} {∥ ρ φ_{N} ∥}_{L^{\infty}}) . \end{matrix}

(2.15)

Using

\partial_{x}^{2} g * f = g * f - f

for all f, we have

\begin{matrix} | φ_{N} \partial_{x}^{2} g * F (u, ρ) | \\ = | \frac{1}{2} φ_{N} (x) \int_{R} e^{- | x - y |} (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d y - φ_{N} (x) (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) | \\ \leq \frac{1}{2} φ_{N} | \int_{R} e^{- | x - y |} \frac{1}{φ_{N} (y)} φ_{N} (y) (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d y | + | φ_{N} (x) (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) | \\ \leq \frac{(3 + c) (c_{1} + 2)}{4} ({∥ u ∥}_{L^{\infty}} {∥ u φ_{N} ∥}_{L^{\infty}} + {∥ ρ ∥}_{L^{\infty}} {∥ ρ φ_{N} ∥}_{L^{\infty}}) . \end{matrix}

(2.16)

Plugging (2.15), (2.16) into (2.12) and using (2.14), there exists

C_{1} > 0

such that

\begin{matrix} {∥ u φ_{N} ∥}_{L^{\infty}} + {∥ u_{x} φ_{N} ∥}_{L^{\infty}} + {∥ ρ φ_{N} ∥}_{L^{\infty}} \\ \leq C_{1} ({∥ u_{0} φ_{N} ∥}_{L^{\infty}} + {∥ u_{0 x} φ_{N} ∥}_{L^{\infty}} + {∥ ρ_{0} φ_{N} ∥}_{L^{\infty}}) \\ + C_{1} \int_{0}^{t} ({∥ u ∥}_{L^{\infty}} + {∥ u_{x} ∥}_{L^{\infty}} + {∥ ρ ∥}_{L^{\infty}}) \\ \times ({∥ u φ_{N} ∥}_{L^{\infty}} + {∥ u_{x} φ_{N} ∥}_{L^{\infty}} + {∥ ρ φ_{N} ∥}_{L^{\infty}}) d τ . \end{matrix}

(2.17)

Using the Gronwall inequality, one deduces that for all

N \in N^{*}

and

t \in [0, T]

\begin{matrix} {∥ u φ_{N} ∥}_{L^{\infty}} + {∥ u_{x} φ_{N} ∥}_{L^{\infty}} + {∥ ρ φ_{N} ∥}_{L^{\infty}} \\ \leq C ({∥ u_{0} φ_{N} ∥}_{L^{\infty}} + {∥ u_{0 x} φ_{N} ∥}_{L^{\infty}} + {∥ ρ_{0} φ_{N} ∥}_{L^{\infty}}) \\ \leq C ({∥ e^{θ x} u_{0} ∥}_{L^{\infty}} + {∥ e^{θ x} u_{0 x} ∥}_{L^{\infty}} + {∥ e^{θ x} ρ_{0} ∥}_{L^{\infty}}) . \end{matrix}

Finally, taking the limit as

N \to \infty

, one obtains

\begin{matrix} sup_{t \in [0, T]} ({∥ e^{θ x} u ∥}_{L^{\infty}} + {∥ e^{θ x} u_{x} ∥}_{L^{\infty}} + {∥ e^{θ x} ρ ∥}_{L^{\infty}}) \\ \leq C ({∥ e^{θ x} u_{0} ∥}_{L^{\infty}} + {∥ e^{θ x} u_{0 x} ∥}_{L^{\infty}} + {∥ e^{θ x} ρ_{0} ∥}_{L^{\infty}}) . \end{matrix}

(2.18)

Thus

| u (t, x) |, | \partial_{x} u (t, x) |, | ρ (t, x) | \sim O (e^{- θ x}) as x \to \infty

uniformly on the interval $[0, T]$ . This completes the proof of Theorem 1.2.

3 Proofs of Theorems 1.3 and 1.4

3.1 Proof of Theorem 1.3

(1) For

c > 0

, integrating the first equation in (1.6) over the interval

[0, t_{1}]

, one has

\begin{matrix} u (t_{1}, x) - u (0, x) + \int_{0}^{t_{1}} u u_{x} (τ, x) d τ + λ_{1} \int_{0}^{t_{1}} u (τ, x) d τ \\ = - \int_{0}^{t_{1}} \partial_{x} g * (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d τ . \end{matrix}

(3.1)

From the assumption in Theorem 1.3, one deduces

u (t_{1}, x) - u (0, x) \sim o (e^{- x}) as x \to \infty .

(3.2)

It follows from Theorem 1.2 that

\begin{matrix} \int_{0}^{t_{1}} u u_{x} (τ, x) d τ \sim O (e^{- 2 θ x}) as x \to \infty; \\ \int_{0}^{t_{1}} u (τ, x) d τ \sim O (e^{- θ x}) as x \to \infty . \end{matrix}

For

θ \in (\frac{1}{2}, 1)

, we have

\int_{0}^{t_{1}} u u_{x} (τ, x) d τ \sim o (e^{- x}) as x \to \infty .

For the right side in (3.1), we have

\int_{0}^{t_{1}} \partial_{x} g * (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d τ = \partial_{x} g * \int_{0}^{t_{1}} (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d τ = \partial_{x} g * p (x),

(3.3)

where

p (x) = \int_{0}^{t_{1}} (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d τ

. From Theorem 1.2, one has

0 \leq p (x) \sim O (e^{- 2 θ x}), thus p (x) \sim o (e^{- x}) as x \to \infty .

Then

\begin{array}{rcl} \partial_{x} g * p (x) & = & - \frac{1}{2} \int_{R} sgn (x - y) e^{- | x - y |} p (y) d y \\ = & - \frac{1}{2} e^{- x} \int_{- \infty}^{x} e^{y} p (y) d y + \frac{1}{2} e^{x} \int_{x}^{\infty} e^{- y} p (y) d y . \end{array}

(3.4)

Noting

e^{x} \int_{x}^{\infty} e^{- y} p (y) d y = o (1) e^{x} \int_{x}^{\infty} e^{- 2 y} d y = o (1) e^{- x} \sim o (e^{- x})

, if there is at least one of the equalities

u (t, x) \neq 0

and

ρ (t, x) \neq 0

is valid, we have

p (x) \neq 0

. Then there exists

c_{2} > 0

such that

\int_{- \infty}^{x} e^{y} p (y) d y \geq c_{2} for x ≫ 1 .

Thus

- \partial_{x} g * p (x) \geq \frac{c_{2}}{2} e^{- x} for x ≫ 1,

which combined with the above estimates yields a contradiction. We obtain $p (x) = 0$ . Consequently, $u = 0$ , $ρ = 0$ .

(2) For

c = 0

, similar to the case

c > 0

, one deduces

u = 0

. Inserting

u = 0

into the second equation in (1.1), one derives

ρ_{t} = - λ ρ .

(3.5)

From (3.5), we have $ρ (t, x) = ρ_{0} (x) e^{- λ t}$ . This completes the proof of case (2) in Theorem 1.3.

3.2 Proof of Theorem 1.4

For

λ_{1} = 0

, integrating the first equation in system (1.6) on the interval

[0, t]

, one obtains

u (t, x) - u (0, x) + \int_{0}^{t} u u_{x} (τ, x) d τ = - \int_{0}^{t} \partial_{x} g * (\frac{3}{2} u^{2} + \frac{c}{2} ρ^{2}) d τ .

(3.6)

From the assumption in Theorem 1.4

u (0, x) \sim O (e^{- x})

as

x \to \infty

and Theorem 1.2, one deduces

\int_{0}^{t} u u_{x} (τ, x) d τ \sim O (e^{- 2 θ x}) as x \to \infty .

(3.7)

For

θ \in (\frac{1}{2}, 1)

, then

\int_{0}^{t} u u_{x} (τ, x) d τ \sim o (e^{- x}) as x \to \infty .

For the right side in (3.6), firstly, we have

\frac{3}{2} \int_{0}^{t} \partial_{x} g * u^{2} (τ, x) d τ = \frac{3}{2} \partial_{x} g * \int_{0}^{t} u^{2} (τ, x) d τ = \frac{3}{2} \partial_{x} g * p_{1} (x),

(3.8)

where

p_{1} (x) = \int_{0}^{t} u^{2} (τ, x) d τ

. Using Theorem 1.2, we obtain

0 \leq p_{1} (x) \sim O (e^{- 2 θ x}) as x \to \infty .

Thus

\begin{array}{rcl} \partial_{x} g * p_{1} (x) & = & - \frac{1}{2} \int_{R} sgn (x - y) e^{- | x - y |} p_{1} (y) d y \\ = & - \frac{1}{2} e^{- x} \int_{- \infty}^{x} e^{y} p_{1} (y) d y + \frac{1}{2} e^{x} \int_{x}^{\infty} e^{- y} p_{1} (y) d y . \end{array}

(3.9)

Noting

e^{x} \int_{x}^{\infty} e^{- y} p_{1} (y) d y \sim O (e^{- 2 θ x}) \sim o (e^{- x}) as x \to \infty

and

- \frac{1}{2} e^{- x} \int_{- \infty}^{x} e^{y} p_{1} (y) d y \leq - \frac{c_{0}}{2} e^{- x} for x ≫ 1,

we have

\frac{3}{2} \int_{0}^{t} \partial_{x} g * u^{2} (τ, x) d τ \sim O (e^{- x}) as x \to \infty .

Similarly, we have

\frac{3}{2} \int_{0}^{t} \partial_{x} g * ρ^{2} (τ, x) d τ \sim O (e^{- x}) as x \to \infty .

Then $u (t, x) \sim O (e^{- x})$ as $x \to \infty$ . From Theorem 1.2, if $ρ_{0} (x) \sim O (e^{- θ x})$ as $x \to \infty$ , we have $ρ (t, x) \sim O (e^{θ x})$ as $x \to \infty$ . This completes the proof of Theorem 1.4.

4 Proofs of Theorems 1.5 and 1.6

4.1 Proof of Theorem 1.5

From the proof of Theorem 1.2, here we need to differentiate the second equation in (1.6) with variable x, and one has

ρ_{x t} + 3 u_{x} ρ_{x} + u ρ_{x x} + 2 u_{x x} ρ + λ ρ_{x} = 0 .

(4.1)

Multiplying (4.1) by

ρ_{x}^{2 n - 1}

and integrating the resultant equation with respect to x, we also note

\begin{aligned} \int_{R} ρ_{x t} ρ_{x}^{2 n - 1} d x = {∥ ρ_{x} ∥}_{L^{2 n}}^{2 n - 1} \frac{d}{d t} {∥ ρ_{x} ∥}_{L^{2 n}}, \\ \int_{R} (3 u_{x} ρ_{x} + u ρ_{x x}) ρ_{x}^{2 n - 1} d x = (3 - \frac{1}{2 n}) \int_{R} u_{x} ρ_{x}^{2 n} d x \leq 3 {∥ u_{x} ∥}_{L^{\infty}} {∥ ρ_{x} ∥}_{L^{2 n}}^{2 n}, \\ 2 \int_{R} u_{x x} ρ ρ_{x}^{2 n - 1} d x \leq 2 {∥ u_{x x} ρ ∥}_{L^{2 n}} {∥ ρ_{x} ∥}_{L^{2 n}}^{2 n - 1} \leq 2 {∥ u_{x x} ∥}_{L^{\infty}} {∥ ρ ∥}_{L^{2 n}} {∥ ρ_{x} ∥}_{L^{2 n}}^{2 n - 1}, \\ λ \int_{R} ρ_{x} ρ_{x}^{2 n - 1} d x = λ {∥ ρ_{x} ∥}_{L^{2 n}}^{2 n} . \end{aligned}

Thus, we obtain

\frac{d}{d t} {∥ ρ_{x} ∥}_{L^{2 n}} \leq (3 {∥ u_{x} ∥}_{L^{\infty}} + λ) {∥ ρ_{x} ∥}_{L^{2 n}} + 2 {∥ u_{x x} ∥}_{L^{\infty}} {∥ ρ ∥}_{L^{2 n}} .

(4.2)

Taking the limit as

n \to \infty

and applying the Gronwall inequality yield

{∥ ρ_{x} ∥}_{L^{\infty}} \leq [{∥ ρ_{0 x} ∥}_{L^{\infty}} + 2 M \int_{0}^{t} {∥ ρ ∥}_{L^{\infty}} d τ] e^{(3 M + λ) t} .

(4.3)

In order to obtain the estimates for

{∥ ρ_{x} φ_{N} ∥}_{L^{\infty}}

, we multiply (4.1) with the weight function

φ_{N} (x)

, then

ρ_{x t} φ_{N} + 3 u_{x} ρ_{x} φ_{N} + u ρ_{x x} φ_{N} + 2 u_{x x} ρ φ_{N} + λ ρ_{x} φ_{N} = 0 .

(4.4)

Multiplying (4.4) with

{(ρ_{x} φ_{N})}^{2 n - 1}

and integrating the resultant equation with respect to x, we note

\begin{matrix} \int_{R} ρ_{x t} φ_{N} {(ρ_{x} φ_{N})}^{2 n - 1} d x = {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}}^{2 n - 1} \frac{d}{d t} {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}}, \\ \int_{R} (3 u_{x} ρ_{x} φ_{N}) {(ρ_{x} φ_{N})}^{2 n - 1} d x \leq 3 {∥ u_{x} ∥}_{L^{\infty}} {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}}^{2 n}, \\ \int_{R} (u ρ_{x x} φ_{N}) {(ρ_{x} φ_{N})}^{2 n - 1} d x \leq \int_{R} u {(ρ_{x} φ_{N})}^{2 n - 1} [\partial_{x} (ρ_{x} φ_{N}) - ρ_{x} {(φ_{N})}_{x}] d x \\ \leq | \int_{R} u_{x} {(ρ_{x} φ_{N})}^{2 n} d x | + | \int_{R} u {(ρ_{x} φ_{N})}^{2 n} d x | \\ \leq ({∥ u ∥}_{L^{\infty}} + {∥ u_{x} ∥}_{L^{\infty}}) {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}}^{2 n}, \\ 2 \int_{R} u_{x x} ρ φ_{N} {(ρ_{x} φ_{N})}^{2 n - 1} d x \leq 2 {∥ u_{x x} ∥}_{L^{\infty}} {∥ ρ φ_{N} ∥}_{L^{2 n}} {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}}^{2 n - 1}, \\ λ \int_{R} ρ_{x} φ_{N} {(ρ_{x} φ_{N})}^{2 n - 1} d x = λ {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}}^{2 n} . \end{matrix}

Hence, we have

\begin{matrix} \frac{d}{d t} {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}} \\ \leq (4 {∥ u_{x} ∥}_{L^{\infty}} + {∥ u ∥}_{L^{\infty}} + λ) {∥ ρ_{x} φ_{N} ∥}_{L^{2 n}} + 2 {∥ u_{x x} ∥}_{L^{\infty}} {∥ ρ φ_{N} ∥}_{L^{2 n}} . \end{matrix}

(4.5)

Taking the limit as

n \to \infty

and using the Gronwall inequality, one obtains

{∥ ρ_{x} φ_{N} ∥}_{L^{\infty}} \leq ({∥ ρ_{0 x} φ_{N} ∥}_{L^{\infty}} + 2 M \int_{0}^{t} {∥ ρ φ_{N} ∥}_{L^{\infty}} d τ) e^{(5 M + λ) t} .

(4.6)

From (4.6) and (2.17), one deduces that there exists

C_{2}

such that

Y (t) \leq C_{2} Y_{0} + C_{2} \int_{0}^{t} y (τ) Y (τ) d τ,

(4.7)

where

\begin{matrix} Y (t) = {∥ u φ_{N} ∥}_{L^{\infty}} + {∥ u_{x} φ_{N} ∥}_{L^{\infty}} + {∥ ρ φ_{N} ∥}_{L^{\infty}} + {∥ ρ_{x} φ_{N} ∥}_{L^{\infty}}, \\ y (t) = {∥ u ∥}_{L^{\infty}} + {∥ u_{x} ∥}_{L^{\infty}} + {∥ u_{x x} ∥}_{L^{\infty}} + {∥ ρ ∥}_{L^{\infty}} + {∥ ρ_{x} ∥}_{L^{\infty}} + 1 . \end{matrix}

Applying the Gronwall inequality to (4.7), for all

N \in N^{*}

and

t \in [0, T]

, one has

Y (t) \leq C Y_{0} \leq C ({∥ u_{0} e^{θ x} ∥}_{L^{\infty}} + {∥ u_{0 x} e^{θ x} ∥}_{L^{\infty}} + {∥ ρ_{0} e^{θ x} ∥}_{L^{\infty}} + {∥ ρ_{0 x} e^{θ x} ∥}_{L^{\infty}}) .

(4.8)

Now taking the limit as

N \to \infty

in (4.8), one obtains

\begin{matrix} sup_{t \in [0, T]} ({∥ u (t, x) e^{θ x} ∥}_{L^{\infty}} + {∥ u_{x} (t, x) e^{θ x} ∥}_{L^{\infty}} + {∥ ρ (t, x) e^{θ x} ∥}_{L^{\infty}} + {∥ ρ_{x} (t, x) e^{θ x} ∥}_{L^{\infty}}) \\ \leq C ({∥ u_{0} e^{θ x} ∥}_{L^{\infty}} + {∥ u_{0 x} e^{θ x} ∥}_{L^{\infty}} + {∥ ρ_{0} e^{θ x} ∥}_{L^{\infty}} + {∥ ρ_{0 x} e^{θ x} ∥}_{L^{\infty}}) . \end{matrix}

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.

Declarations

Acknowledgements

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).

Authors’ Affiliations

(1)

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

(2)

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

References

Degasperis A, Procesi M: Asymptotic integrability. In Symmetry and Perturbation Theory. World Scientific, Singapore; 1999:23-37.Google Scholar
Escher J, Liu Y, Yin Z: Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J. Funct. Anal. 2006, 241: 457-485. 10.1016/j.jfa.2006.03.022MathSciNetView ArticleGoogle Scholar
Liu Y, Yin Z: Global existence and blow-up phenomena for the Degasperis-Procesi equation. Commun. Math. Phys. 2006, 267: 801-820. 10.1007/s00220-006-0082-5MathSciNetView ArticleGoogle Scholar
Yin Z: On the Cauchy problem for an integrable equation with peakon solutions. Ill. J. Math. 2003, 47: 649-666.Google Scholar
Yin Z: Global existence for a new periodic integrable equation. J. Math. Anal. Appl. 2003, 283: 129-139. 10.1016/S0022-247X(03)00250-6MathSciNetView ArticleGoogle Scholar
Yin Z: Global solutions to a new integrable equation with peakons. Indiana Univ. Math. J. 2004, 53: 1189-1210. 10.1512/iumj.2004.53.2479MathSciNetView ArticleGoogle Scholar
Yin Z: Global weak solutions to a new periodic integrable equation with peakon solutions. J. Funct. Anal. 2004, 212: 182-194. 10.1016/j.jfa.2003.07.010MathSciNetView ArticleGoogle Scholar
Wu S, Yin Z: Blow-up phenomena and decay for the periodic Degasperis-Procesi equation with weak dissipation. J. Nonlinear Math. Phys. 2008, 15: 28-49. 10.2991/jnmp.2008.15.s2.3MathSciNetView ArticleGoogle Scholar
Guo Y, Lai S, Wang Y: Global weak solutions to the weakly dissipative Degasperis-Procesi equation. Nonlinear Anal. 2011, 74: 4961-4973. 10.1016/j.na.2011.04.051MathSciNetView ArticleGoogle Scholar
Guo Z: Some properties of solutions to the weakly dissipative Degasperis-Procesi equation. J. Differ. Equ. 2009, 246: 4332-4344. 10.1016/j.jde.2009.01.032View ArticleGoogle Scholar
Yan K, Yin Z: On the Cauchy problem for a 2-component Degasperis-Procesi system. J. Differ. Equ. 2012, 252: 2131-2159. 10.1016/j.jde.2011.08.003MathSciNetView ArticleGoogle Scholar
Zhou J, Tian L, Fan X: Soliton, kink and antikink solutions of a 2-component of the Degasperis-Procesi equation. Nonlinear Anal., Real World Appl. 2012, 11: 2529-2536.MathSciNetView ArticleGoogle Scholar
Fu Y, Qu C: Unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations. Acta Math. Sci. 2012, 32: 652-662. 10.1016/S0252-9602(12)60046-0MathSciNetView ArticleGoogle Scholar
Jin L, Guo Z: On a 2-component Degasperis-Procesi shallow water system. Nonlinear Anal., Real World Appl. 2010, 11: 4164-4173. 10.1016/j.nonrwa.2010.05.003MathSciNetView ArticleGoogle Scholar
Yu L, Tian L: Loop solutions, breaking kink wave solutions, solitary wave solutions and periodic wave solutions for the 2-component Degasperis-Procesi equation. Nonlinear Anal., Real World Appl. 2014, 15: 140-148.MathSciNetView ArticleGoogle Scholar
Manwai Y: Self-similar blow-up solutions to the 2-component Degasperis-Procesi shallow water system. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 3463-3469. 10.1016/j.cnsns.2010.12.039MathSciNetView ArticleGoogle Scholar
Hu Q: Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system. Appl. Anal. 2013, 92: 398-410. 10.1080/00036811.2011.621893MathSciNetView ArticleGoogle Scholar
Hu Q: Global existence and blow-up phenomena for a weakly dissipative periodic 2-component Camassa-Holm system. J. Math. Phys. 2011., 52: Article ID 103701Google Scholar
Jin L, Guo Z: A note on a modified 2-component Camassa-Holm system. Nonlinear Anal., Real World Appl. 2012, 13: 887-892. 10.1016/j.nonrwa.2011.08.024MathSciNetView ArticleGoogle Scholar
Zhu M: Blow-up, global existence and persistence properties for the coupled Camassa-Holm equations. Math. Phys. Anal. Geom. 2011, 14: 197-209. 10.1007/s11040-011-9094-2MathSciNetView ArticleGoogle Scholar
Guo Z: Asymptotic profiles of solutions to the 2-component Camassa-Holm system. Nonlinear Anal. 2012, 75: 1-6. 10.1016/j.na.2011.01.030MathSciNetView ArticleGoogle Scholar
Guo Z, Ni L: Persistence properties and unique continuation of solutions to a 2-component Camassa-Holm equation. Math. Phys. Anal. Geom. 2011, 14: 101-114. 10.1007/s11040-011-9089-zMathSciNetView ArticleGoogle Scholar
Constantin A, Ivanov R: On an integrable 2-component Camassa-Holm shallow water system. Phys. Lett. A 2008, 372: 7129-7132. 10.1016/j.physleta.2008.10.050MathSciNetView ArticleGoogle Scholar
Gui G, Liu Y: On the global existence and wave-breaking criteria for the 2-component Camassa-Holm system. J. Funct. Anal. 2010, 258: 4251-4278. 10.1016/j.jfa.2010.02.008MathSciNetView ArticleGoogle Scholar
Gui G, Liu Y: On the Cauchy problem for the 2-component Camassa-Holm system. Math. Z. 2011, 268: 45-66. 10.1007/s00209-009-0660-2MathSciNetView ArticleGoogle Scholar
Ai X, Gui G: Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation. J. Math. Anal. Appl. 2010, 361: 457-465. 10.1016/j.jmaa.2009.07.031MathSciNetView ArticleGoogle Scholar
Chen W, Tian L, Deng X, Zhang J: Wave breaking for a generalized weakly dissipative 2-component Camassa-Holm system. J. Math. Anal. Appl. 2013, 400: 406-417. 10.1016/j.jmaa.2012.10.063MathSciNetView ArticleGoogle Scholar
Jin L, Jiang Z: Wave breaking of an integrable Camassa-Holm system with 2-component. Nonlinear Anal. 2014, 95: 107-116.MathSciNetView ArticleGoogle Scholar
Lenells J, Wunsch M: On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations. J. Differ. Equ. 2013, 255: 441-448. 10.1016/j.jde.2013.04.015MathSciNetView ArticleGoogle Scholar
Zhu M: On a shallow water equation perturbed in Schwartz class. Math. Phys. Anal. Geom. 2012. 10.1007/s11040-012-9112-zGoogle Scholar
Zhu M, Jiang Z: Some properties of solutions to the weakly dissipative b -family equation. Nonlinear Anal., Real World Appl. 2012, 13: 158-167. 10.1016/j.nonrwa.2011.07.020MathSciNetView ArticleGoogle Scholar
Ni L, Zhou Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 2011, 250: 3002-3021. 10.1016/j.jde.2011.01.030MathSciNetView ArticleGoogle Scholar
Zong X: Properties of the solutions to the 2-component b -family systems. Nonlinear Anal. 2012, 75: 6250-6259. 10.1016/j.na.2012.07.001MathSciNetView ArticleGoogle Scholar
Zhou S, Mu C: The properties of solutions for a generalized b -family equation with peakons. J. Nonlinear Sci. 2013, 23: 863-889. 10.1007/s00332-013-9171-8MathSciNetView ArticleGoogle Scholar
Himonas A, Misiolek G, Ponce G, Zhou Y: Persistence properties and unique continuation of solutions to the Camassa-Holm equation. Math. Phys. Anal. Geom. 2007, 271: 511-522.MathSciNetGoogle Scholar
Danchin R: A few remarks on the Camassa-Holm equation. Differ. Integral Equ. 2001, 14: 953-988.MathSciNetGoogle Scholar
Danchin R: A note on well-posedness for Camassa-Holm equation. J. Differ. Equ. 2003, 192: 429-444. 10.1016/S0022-0396(03)00096-2MathSciNetView ArticleGoogle Scholar
Danchin, R: Fourier analysis methods for PDEs. Lecture Notes, 14 Nov. Preprint (2005)Google Scholar
Bahouri H, Chemin J, Danchin R Grun. der Math. Wiss. In Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin; 2010.Google Scholar

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