Zhou et al. Boundary Value Problems 2014, 2014:9http://www.boundaryvalueproblems.com/content/2014/1/9

RESEARCH Open Access

Local well-posedness and persistenceproperties for a model containing bothCamassa-Holm and Novikov equationShouming Zhou1*, Chun Wu1 and Baoshuai Zhang2

*Correspondence:zhoushouming76@163.com1College of Mathematics Science,Chongqing Normal University,Chongqing, 401331, P.R. ChinaFull list of author information isavailable at the end of the article

AbstractThis paper deals with the Cauchy problem for a generalized Camassa-Holm equationwith high-order nonlinearities,

ut – uxxt + kux + aumux = (n + 2)unuxuxx + un+1uxxx ,

where k,a ∈R andm,n ∈ Z+. This equation is a generalization of the famous

equation of Camassa-Holm and the Novikov equation. The local well-posedness ofstrong solutions for this equation in Sobolev space Hs(R) with s > 3

2 is obtained, andpersistence properties of the strong solutions are studied. Furthermore, underappropriate hypotheses, the existence of its weak solutions in low order Sobolevspace Hs(R) with 1 < s≤ 3

2 is established.

Keywords: persistence properties; local well-posedness; weak solution

1 IntroductionThis work is concerned with the following one-dimensional nonlinear dispersive PDE:

{ut – uxxt + kux + aumux = (n + )unuxuxx + un+uxxx, t > ,x ∈R,u(x, ) = u(x), x ∈R,

(.)

where k,a ∈ R and m,n ∈ Z+.

Obviously, if n = ,m = , a = , equation (.) becomes the Camassa-Holm equation,

ut – uxxt + kux + uux – uxuxx – uuxxx = , (.)

where the variable u(t,x) represents the fluid velocity at time t and in the spatial direc-tion x, and k is a nonnegative parameter related to the critical shallow water speed [].The Camassa-Holm equation (.) is also a model for the propagation of axially symmet-ric waves in hyperelastic rods (cf. []). It is well known that equation (.) has also a bi-Hamiltonian structure [, ] and is completely integrable (see [, ] and the in-depth dis-cussion in [, ]). In [], Qiao has shown that the Camassa-Holm spectral problem yieldstwo different integrable hierarchies of nonlinear evolution equations, one is of negative or-der CH hierachy while the other one is of positive order CH hierarchy. Its solitary waves

©2014 Zhou et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

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are smooth if k > and peaked in the limiting case c = (cf. []). The orbital stability ofthe peaked solitons is proved in [], and the stability of the smooth solitons is consideredin []. It is worth pointing out that solutions of this type are not mere abstractions: thepeakons replicate a feature that is characteristic for the waves of great height - waves oflargest amplitude that are exact solutions of the governing equations for irrotational waterwaves (cf. [–]). The explicit interaction of the peaked solitons is given in [] and allpossible explicit single soliton solutions are shown in []. The Cauchy problem for theCamassa-Holm equation (.) has been studied extensively. It has been shown that thisproblem is locally well-posed for initial data u ∈ Hs(R) with s >

[–]. Moreover, ithas global strong solutions and also admits finite time blow-up solutions [, , , ].On the other hand, it also has global weak solutions in H(R) [–]. The advantage oftheCamassa-Holm equation in comparisonwith theKdV equation (.) lies in the fact thatthe Camassa-Holm equation has peaked solitons and models the peculiar wave breakingphenomena [, ].For n = ,m ∈ Z

+, a ∈R, equation (.) becomes a generalized Camassa-Holm equation,

ut – uxxt + kux + aumux = uxuxx + uuxxx, (.)

Wazwaz [, ] studied the solitary wave solutions for the generalized Camassa-Holmequation (.) with m = , a = , and the peakon wave solutions for this equation werestudied in [–], and the periodic blow-up solutions and limit forms for (.) were ob-tained in []. In [, ], the authors have given the traveling waves solution, peakedsolitary wave solutions for (.).On the other hand, taking m = , a = , k = in (.) we found the Novikov equation

[]:

ut – uxxt + uux = uuxuxx + uuxxx, t > ,x ∈R, (.)

The Novikov equation (.) possesses a matrix Lax pair, many conserved densities, a bi-Hamiltonian structure as well as peakon solutions []. These apparently exotic wavesreplicate a feature that is characteristic of the waves of great height-waves of largest am-plitude that are exact solutions of the governing equations for water waves, as far as the de-tails are concerned [, , ]. The Novikov equation possesses the explicit formulas formultipeakon solutions []. It has been shown that the Cauchy problem for the Novikovequation is locally well-posed in the Besov spaces and in Sobolev spaces and possesses thepersistence properties [, ]. In [, ], the authors showed that the data-to-solutionmap for equation (.) is not uniformly continuous on bounded subsets of Hs for s > /.Analogous to the Camassa-Holm equation, the Novikov equation shows the blow-up phe-nomenon [] and has global weak solutions []. Recently, Zhao andZhou [] discussedthe symbolic analysis and exact traveling wave solutions of a modified Novikov equation,which is new in that it has a nonlinear term uux instead of uux.Other integrable CH-type equations with cubic nonlinearity have been discovered:

{mt + (u – ux)mx + uxm + γux = , m = u – uxx, t > ,x ∈R,u(,x) = u(x), x ∈R,

(.)

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where γ is a constant. equation (.) was independently proposed by Fokas [], byFuchssteiner [], and Olver and Rosenau [] as a new generalization of integrable sys-tem by using the general method of tri-Hamiltonian duality to the bi-Hamiltonian rep-resentation of the modified Korteweg-de Vries equation. Later, it was obtained by Qiao[, ] from the two-dimensional Euler equations, where the variables u(t,x) andm(t,x)represent, respectively, the velocity of the fluid and its potential density. Ivanov and Lyons[] obtain a class of soliton solutions of the integrable hierarchy which has been put for-ward in a series of woks by Qiao [, ]. It was shown that equation (.) admits theLax-pair and the Cauchy problem (.) may be solved by the inverse scattering transformmethod. The formation of singularities and the existence of peaked traveling-wave solu-tions for equation (.) was investigated in []. The well-posedness, blow-upmechanism,and persistence properties are given in []. It was also found that equation (.) is relatedto the short-pulse equation derived by Schäfer and Wayne [].Applying the method of pseudoparabolic regularization, Lai and Wu [] investigated

the local well-posedness and existence of weak solutions for the following generalizedCamassa-Holm equation with dissipative term:

ut – uxxt + kux + aumux =(nun–ux + unuxx

)x+ β∂x

[(ux)N–], (.)

wherem,n,N ∈ Z+, and a, k, β are constants. Hakkaev and Kirchev [] studied the local

well-posedness and orbital stability of solitary wave solution for equation (.) with a =(m+)(m+)

, n =m and β = .Motivated by the results mentioned above, the goal of this paper is to establish the well-

posedness of strong solutions and weak solutions for problem (.). First, we use Kato’stheorem to obtain the existence and uniqueness of strong solutions for equation (.).

Theorem . Let u ∈ Hs(R) with s > /. Then there exists a maximal T = T(‖u‖Hs(R)),and a unique solution u(x, t) to the problem (.) such that

u = u(·,u) ∈ C([,T);Hs(R)

) ∩C([,T);Hs–(R)).

Moreover, the solution depends continuously on the initial data, i.e. the mapping

u → u(·,u) :Hs(R)→ C([,T);Hs(R)

) ∩C([,T);Hs–(R))

is continuous.

In [, , ], the spatial decay rates for the strong solution to the Camassa-HolmNovikov equation were established provided that the corresponding initial datum decaysat infinity. This kind of property is so-called the persistence property. Similarly, for equa-tion (.), we also have the following persistence properties for the strong solution.

Theorem . Assume that u ∈ C([,T);Hs(R)) with s > / satisfies

∣∣u(x)∣∣, ∣∣ux(x)∣∣ ∼O(e–θx) as x ↑ ∞(

respectively,∣∣u(x)∣∣, ∣∣ux(x)∣∣ ∼O

(( + x)–α

)as x ↑ ∞)

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for some θ ∈ (, ) (respectively, α ≥ max{ m+ ,

n }), then the corresponding strong solution

u ∈ C([,T);Hs(R)) to equation (.) satisfies for some T >

∣∣u(x, t)∣∣, ∣∣ux(x, t)∣∣ ∼O(e–θx) as x ↑ ∞(

respectively,∣∣u(x)∣∣ ∼O

(( + x)–α

)as x ↑ ∞)

uniformly in the time interval [,T].

Theorem . Assume that k = ,m = n, and u ∈ C([,T);Hs(R)) with s > / satisfies

∣∣u(x)∣∣ ∼O(e–x

),

∣∣ux(x)∣∣ ∼O(e–θx) as x ↑ ∞(

respectively,∣∣u(x)∣∣ ∼O

(( + x)–α

),∣∣ux(x)∣∣ ∼O

(( + x)–β

)as x ↑ ∞)

for some θ ∈ (/(m+), ) (respectively, α ≥ m+ , β ∈ ( α

m+ ,α)), then the corresponding strongsolution u ∈ C([,T);Hs(R)) to equation (.) satisfies for some T >

∣∣u(x, t)∣∣ ∼O(e–x

)as x ↑ ∞(

respectively,∣∣u(x)∣∣ ∼O

(( + x)–α

)as x ↑ ∞)

uniformly in the time interval [,T].

Remark . The notations mean that

∣∣f (x)∣∣ ∼O(e–θx) as x ↑ ∞ if lim

x→∞f (x)e–θx = L.

Finally, we have the following theorem for the existence of a weak solution for equation(.).

Theorem . Suppose that u(x) ∈ Hs(R) with < s ≤ and ‖ux‖L∞(R) < ∞. Then there

exists a life span T > such that problem (.) has aweak solution u(x, t) ∈ L([,T],Hs(R))in the sense of a distribution and ux ∈ L∞([,T]×R).

The plan of this paper is as follows. In the next section, the local well-posedness andpersistence properties of strong solutions for the problem (.) are established, and The-orems .-. are proved. The existence of weak solutions for the problem (.) is provedin Section , and this proves Theorem ..

2 Well-posedness and persistence properties of strong solutionsNotation The space of all infinitely differentiable functions f (x, t) with compact supportin R × [, +∞) is denoted by C∞

. Let p be any constant with ≤ p < ∞ and denote Lp =Lp(R) to be the space of all measurable functions f such that ‖f ‖pLp =

∫R

|f (x)|p dx < ∞.The space L∞ = L∞(R) with the standard norm ‖f ‖L∞ = infm(e)= supx∈R/e |f (x)|. For anyreal number s, let Hs =Hs(R) denote the Sobolev space with the norm defined by

‖f ‖Hs =(∫

R

( + |ξ |)s∣∣f (ξ , t)∣∣ dξ

) < ∞,

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where f (ξ , t) =∫Re–ixξ f (x, t)dx. Let C([,T];Hs(R)) denote the class of continuous func-

tions from [,T] to Hs(R) and � = ( – ∂x )

.

Proof of Theorem . To prove well-posedness we apply Kato’s semigroup approach [].For this, we rewrite the Cauchy problemof equation (.) as follows for the transport equa-tion:{

ut + um+ux + F(u) = ,u(x, ) = u(x),

(.)

where F(u) := P ∗ E(u). E(u) = kux + aumux – un+ux + n+ ∂x(unux) +

nu

n–ux and P(x) =e

–|x|. Let A(u) := um+∂x, Y = Hs, X = Hs– and Q = � = ( – ∂x )

. Following closely the

considerations made in [, , ], we obtain the statement of Theorem .. �

Proof of Theorem . We introduce the notationM = supt∈[,T] ‖u(t)‖Hs . The first step wewill give estimates on ‖u(x, t)‖L∞ . Integrating the both sides with respect to x variable bymultiplying the first equation of (.) by up– with p ∈ Z

+, we get

∫R

up–ut dx +∫R

up–(un+ux

)dx +

∫R

up–(P ∗ E(u)

)dx = . (.)

Note that the estimates∫R

up–ut dx =p

ddt

∥∥u(x, t)∥∥pLp =

∥∥u(x, t)∥∥p–Lp

ddt

∥∥u(x, t)∥∥Lp ,

and ∣∣∣∣∫R

up–(un+ux

)dx

∣∣∣∣ ≤ ∥∥ux(x, t)∥∥L∞∥∥u(x, t)∥∥p+n

Lp

are true. Moreover, using Hölder’s inequality

∣∣∣∣∫R

up–(P ∗ E(u)

)dx

∣∣∣∣ ≤ ∥∥u(x, t)∥∥p–Lp

∥∥P ∗ E(u)∥∥Lp .

From equation (.) we can obtain

ddt

∥∥u(x, t)∥∥Lp ≤ ∥∥ux(x, t)∥∥L∞∥∥u(x, t)∥∥n+

Lp +∥∥P ∗ E(u)

∥∥Lp .

Since ‖f ‖Lp → ‖f ‖L∞ as p → ∞ for any f ∈ L∞ ∩L. From the above inequality we deducethat

ddt

∥∥u(x, t)∥∥L∞ ≤Mn+∥∥u(x, t)∥∥L∞ +∥∥P ∗ E(u)

∥∥L∞ ,

where we use

∥∥ux(x, t)∥∥L∞∥∥u(x, t)∥∥n

L∞ ≤ ∥∥ux(x, t)∥∥H

+

∥∥u(x, t)∥∥n

H +

≤ ∥∥u(x, t)∥∥n+Hs ≤Mn+.

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Because of Gronwall’s inequality, we get

∥∥u(x, t)∥∥L∞ ≤ exp(Mn+t

)(∥∥u(x)∥∥L∞ +∫ t

∥∥(P ∗ E(u)

)(x, τ )

∥∥L∞ dτ

).

Next, we will give estimates on ‖ux(x, t)‖L∞ . Differentiating (.) with respect to thex-variable produces the equation

uxt + un+uxx + (n + )unux + ∂x(P ∗ E(u)

)= . (.)

Multiplying this equation by (ux)p– with p ∈ Z+, integrating the result in the x-variable,

and using integration by parts:∫R

(ux)p–uxt dx =p

ddt

∥∥ux(x, t)∥∥pLp =

∥∥ux(x, t)∥∥p–Lp

ddt

∥∥ux(x, t)∥∥Lp ,∣∣∣∣∫R

(ux)p–(unux

)dx

∣∣∣∣ ≤ ∥∥u(x, t)∥∥nL∞

∥∥ux(x, t)∥∥L∞∥∥ux(x, t)∥∥p

Lp ,∣∣∣∣∫R

(ux)p–(un+uxx

)dx

∣∣∣∣ = ∣∣∣∣n + p

∫R

unup+x dx∣∣∣∣

≤ n + p

∥∥u(x, t)∥∥nL∞

∥∥ux(x, t)∥∥L∞∥∥ux(x, t)∥∥p

Lp .

From the above inequalities, we also can get the following inequality:

ddt

∥∥ux(x, t)∥∥Lp ≤(n + +

n + p

)Mn+∥∥ux(x, t)∥∥Lp +

∥∥∂x(P ∗ E(u)

)∥∥Lp ,

where we use ‖ux(x, t)‖L∞‖u(t)‖nL∞ ≤Mn+. Then passing to the limit in this inequality andusing Gronwall’s inequality one can obtain

∥∥ux(x, t)∥∥L∞ ≤ exp((n + )Mn+t

)(∥∥ux(x)∥∥L∞ +∫ t

∥∥∂x(P ∗ E(u)

)(x, τ )

∥∥L∞ dτ

).

We shall now repeat the arguments using the weight

ϕN (x) =

⎧⎪⎨⎪⎩, x≤ ,eθx, < x <N ,eθx, x≥N ,

where N ∈ Z. Observe that for all N we have

≤ ϕ′N (x)≤ ϕN (x), for all x ∈ R. (.)

Using the notation E(u), from (.) we get

∂t(uϕN ) +(un+ϕN

)ux + ϕN

(P ∗ E(u)

)= ,

and from (.), we also obtain

∂t(ϕN∂xu) + un+ϕN∂x u + (n + )un(ϕN∂xu)∂xu + ϕN∂x

(P ∗ E(u)

)= .

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We need to eliminate the second derivatives in the second term in the above equality.Thus, combining integration by parts and equation (.) we find∣∣∣∣∫

R

un+ϕN∂x u(∂xuϕN )p–

∣∣∣∣=

∣∣∣∣∫R

un+(∂xuϕN )p–(∂x(ϕN∂xu) – ∂xuϕ′

N)dx

∣∣∣∣=

∣∣∣∣∫R

p

un+∂x((∂xuϕN )p

)– un+(∂xuϕN )p–∂xuϕ′

N dx∣∣∣∣

≤ (‖u‖L∞ + ‖∂xu‖L∞)‖u‖nL∞‖∂xuϕN‖pLp .

Hence, as in the weightless case, we have

‖uϕN‖L∞ + ‖∂xuϕN‖L∞

≤ exp((n + )Mn+t

)(∥∥u(x)ϕN∥∥L∞ +

∥∥ux(x)ϕN∥∥L∞

)+ exp

((n + )Mn+t

)∫ t

(∥∥ϕN∂x(E(u)

)∥∥L∞ +

∥∥ϕN(E(u)

)∥∥L∞

)dτ .

A simple calculation shows that there exists C > , depending only on θ ∈ (, ) such thatfor any N ∈ Z

+,

ϕN

∫R

ϕN (y)

dy≤ C =

– θ.

Thus, we have

∣∣ϕN( – ∂

x)–(un–∂

x u)∣∣ =

∣∣∣∣ϕN

∫R

e–|x–y|(un–∂x u

)(y)dy

∣∣∣∣=

∣∣∣∣ϕN

∫R

e–|x–y| ϕN (y)

(ϕN∂xu)(un–∂

x u)(y)dy

∣∣∣∣≤

(ϕN

∫R

e–|x–y| ϕN (y)

dy)

‖ϕN∂xu‖L∞∥∥un–∂

x u∥∥L∞

≤ c‖ϕN∂xu‖L∞∥∥un–∂

x u∥∥L∞ ,

and

∣∣ϕN( – ∂

x)–

∂x(un–∂

x u)∣∣ =

∣∣∣∣ϕN

∫R

sgn(x – y)e–|x–y|(un–∂x u

)(y)dy

∣∣∣∣≤ c‖ϕN∂xu‖L∞

∥∥un–∂x u

∥∥L∞ .

Using the same method, we can estimate the other terms:

∣∣ϕN( – ∂

x)–(kux + aumux – un+ux

)∣∣ ≤ c( + ‖u‖mL∞ + ‖u‖n+L∞

)‖ϕN∂xu‖L∞ ,∣∣ϕN( – ∂

x)–

∂x(kux + aumux – un+ux

)∣∣=

∣∣∣∣ϕN( – ∂

x)–

∂x

(ku +

am +

um+ –

n + un+

)∣∣∣∣

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≤∣∣∣∣ϕN

(ku +

am +

um+ –

n + un+

)∣∣∣∣+

∣∣∣∣ϕN( – ∂

x)–(ku +

am +

um+ –

n + un+

)∣∣∣∣≤ c

( + ‖u‖mL∞ + ‖u‖n+L∞

)‖ϕNu‖L∞ ,

and

∣∣ϕN( – ∂

x)–

∂x(unux

)∣∣ ≤ c‖ϕNu‖L∞∥∥un–∂

x u∥∥L∞ ,∣∣ϕN

( – ∂

x)–

∂x(unux

)∣∣ ≤ ∣∣ϕNunux∣∣ + ∣∣ϕN

( – ∂

x)–(unux)∣∣

≤ c‖ϕNu‖L∞∥∥un–∂

x u∥∥L∞ .

Thus, it follows that there exists a constant C > which depends only on M, m, n, k, a,and T , such that

‖uϕN‖L∞ + ‖∂xuϕN‖L∞

≤ C(‖uϕN‖L∞ + ‖uxϕN‖L∞

)+C

∫ t

(( + ‖u‖mL∞ + ‖u‖n+L∞ +

∥∥un–∂x u

∥∥L∞

)(‖ϕN∂xu‖L∞ + ‖ϕNu‖L∞))dτ

≤ C(‖uϕN‖L∞ + ‖uxϕN‖L∞

)+C

∫ t

(‖ϕN∂xu‖L∞ + ‖ϕNu‖L∞)dτ .

Hence, for any n ∈ Z and any t ∈ [,T] we have

‖uϕN‖L∞ + ‖∂xuϕN‖L∞ ≤ C(‖uϕN‖L∞ + ‖uxϕN‖L∞

)≤ C

(∥∥umax(, eθx)∥∥

L∞ +∥∥uxmax

(, eθx)∥∥

L∞).

Finally, taking the limit as N goes to infinity we find that for any t ∈ [,T],

∥∥ueθx∥∥L∞ +

∥∥∂xueθx∥∥L∞ ≤ C

(∥∥umax(, eθx)∥∥

L∞ +∥∥uxmax

(, eθx)∥∥

L∞),

which completes the proof of Theorem .. �

Next, we give a simple proof for Theorem ..

Proof of Theorem . We should use Theorem . to prove this theorem.For any t ∈ [,T], integrating equation (.) over the time interval [, t] we get

u(x, t) – u(x, ) +∫ t

(un+ux

)(x, τ )dτ +

∫ t

(P ∗ E(u)

)(x, τ )dτ = . (.)

From Theorem ., it follows that

∫ t

(un+ux

)(x, τ )dτ ∼O

(e–(n+)αx

)as x ↑ ∞

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and so∫ t

(un+ux

)(x, τ )dτ ∼O

(e–x

)as x ↑ ∞.

We shall show that the last term in equation (.) is O(e–x); thus we have

∫ t

(P ∗ E(u)

)(x, τ )dτ = P(x) ∗

∫ t

(E(u)

)(x, τ )dτ

.= P(x) ∗ ρ(x).

From the given condition and Theorem .. we know ρ(x)∼O(e–x) as x ↑ ∞. Since

P(x) ∗ ρ(x) =

∫R

e–|x–y|ρ(y)dy =e–x

∫ x

–∞eyρ(y)dy +

ex

∫ ∞

xe–yρ(y)dy

we have

e–x∫ x

–∞eyρ(y)dy =O()e–x

∫ x

–∞ey dy∼O()e–x ∼O

(e–x

)as x ↑ ∞,

ex∫ ∞

xe–y|ρ(y)dy =O()ex

∫ ∞

xe–y dy∼O()e–x ∼O

(e–x

)as x ↑ ∞.

Thus∫ t

(P ∗ E(u)

)(x, τ )dτ ∼O

(e–x

)as x ↑ ∞.

From equation (.) and |u(x)| ∼ O(e–x) as x ↑ ∞, we know

∣∣u(x, t)∣∣ ∼O(e–x

)as x ↑ ∞.

By the arbitrariness of t ∈ [,T], we get

∣∣u(x, t)∣∣ ∼O(e–x

)as x ↑ ∞

uniformly in the time interval [,T]. This completes the proof of Theorem .. �

3 Existence of solution of the regularized equationIn order to prove Theorem ., we consider the regularized problem for equation (.) inthe following form:

⎧⎪⎨⎪⎩ut – uxxt + εuxxxt = ∂x(–ku – a

m+um+) +

n+∂x (un+)

– n+ ∂x(unux) –

nu

n–ux ,u(x, ) = u(x),

(.)

where < ε < ,m ≥ , n≥ and a, k are constants. One can easily check that when ε = ,

equation (.) is equivalent to the IVP (.).Before giving the proof of Theorem ., we give several lemmas.

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Lemma . (See []) Let p and q be real numbers such that –p < q ≤ p. Then

‖fg‖Hq ≤ c‖f ‖Hp‖g‖Hq , if p >,

‖fg‖Hp+q–

≤ c‖f ‖Hp‖g‖Hq , if p <

.

Lemma . Let u(x) ∈ Hs(R) with s > /. Then the Cauchy problem (.) has a uniquesolution u(x, t) ∈ C([,T];Hs(R)) where T > depends on ‖u‖Hs(R). If s ≥ , the solutionu(x, t) ∈ C([,T];Hs(R)) exists for all time. In particular, when s ≥ , the correspondingsolution is a classical globally defined solution of problem (.).

Proof First, we note that, for any < ε < and any s, the integral operator

D =( – ∂

x + ε∂x)– :Hs →Hs+

defines a bounded linear operator on the indicated Sobolev spaces.To prove the existence of a solution to the problem (.), we apply the operator D to

both sides of equation (.) and then integrate the resulting equations with regard to t.This leads to the following equations:

u(x, t) = u(x) +∫ t

D

[∂x

(–ku –

am +

um+)+

n +

∂x(un+

)–n +

∂x(unux

)–nun–ux

]dτ . (.)

Suppose thatA is the operator in the right-hand side of equation (.). For fixed t ∈ [,T],we get

∥∥∥∥∫ t

D

[∂x

(–ku –

am +

um+)+ ∂

x(un+ux

)–n +

∂x(unux

)–nun–ux

]dτ

–∫ t

D

[∂x

(–kv –

am +

vm+)+ ∂

x(vn+vx

)–n +

∂x(vnvx

)–nvn–vx

]dτ

∥∥∥∥Hs

≤ CT(sup

≤t≤T‖u – v‖Hs + sup

≤t≤T

∥∥um+ – vm+∥∥Hs + sup

≤t≤T

∥∥un+ – vn+∥∥Hs

+ sup≤t≤T

∥∥D∂x[unux – vnvx

]∥∥Hs + sup

≤t≤T

∥∥D[un–ux – vn–vx

]∥∥Hs

).

Since Hs is an algebra for s ≥ , we have the inequalities

∥∥um+ – vm+∥∥Hs =

∥∥∥∥∥(u – v)m∑j=

(um–jvj

)∥∥∥∥∥Hs

≤ C

(‖u – v‖Hs

m∑j=

‖u‖m–jHs ‖v‖jHs

).

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 11 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

Since s > /, by Lemma ., we get

∥∥D∂x[unux – vnvx

]∥∥Hs

=

n + ∥∥D∂x

[∂x

(un+

)∂xu – ∂x

(vn+

)∂xv

]∥∥Hs

≤ C(∥∥D∂x

[∂x

(un+

)∂x(u – v)

]∥∥Hs +

∥∥D∂x[∂x

(un+ – vn+

)∂xv

]∥∥Hs

)≤ C

(∥∥∂x(un+

)∂x(u – v)

∥∥Hs– +

∥∥∂x(un+ – vn+

)∂xv

∥∥Hs–

)≤ C

(∥∥un+∥∥Hs‖u – v‖Hs +∥∥un+ – vn+

∥∥Hs‖v‖Hs

)≤ C

(‖u‖n+Hs ‖u – v‖Hs +

(‖u – v‖Hs

n∑j=

‖u‖n–jHs ‖v‖jHs

)‖v‖Hs

),

and

∥∥D[un–ux – vn–vx

]∥∥Hs

≤ ∥∥D[un–

(ux – vx

)]∥∥Hs +

∥∥D[(un– – vn–

)vx

]∥∥Hs

≤ C[∥∥un–(ux – vx

)∥∥Hs– +

∥∥(un– – vn–

)vx

∥∥Hs–

]≤ C

[∥∥un–∥∥Hs–

∥∥ux – vx∥∥Hs– +

∥∥un– – vn–∥∥Hs–

∥∥vx∥∥Hs–]

≤ C

[‖u‖n–Hs ‖u – v‖Hs

∑j=

‖u‖–jHs ‖v‖jHs +

(‖u – v‖Hs

n–∑j=

‖u‖n––jHs ‖v‖jHs

)‖v‖Hs

],

where C, C only depend on n. Suppose that both u and v are in the closed ball BR() ofradius R about the zero function in C([,T];Hs(R)); by the above inequalities, we obtain

‖Au –Av‖C([,T];Hs) ≤ θ‖u – v‖C([,T];Hs),

where θ = TC(Rm + Rn+) and C only depend on a, k, m, n. Choosing T sufficiently smallsuch that θ < , we know that A is a contraction. Applying the above inequality yields

‖Au‖C([,T];Hs) ≤ ‖u‖Hs + θ‖u‖C([,T];Hs).

Taking T sufficiently small so that θR+ ‖u‖Hs < R, we deduce that Amaps BR() to itself.It follows from the contraction-mapping principle that the mapping A has a unique fixedpoint u in BR().For s ≥ ,multiplying the first equation of the system (.) by u, integratingwith respect

to x, one derives

ddt

∫R

(u + ux + εuxx

)dx =

∫R

u(–kux – aumux + (n + )unuxuxx + un+uxxx

)dx

=∫R

((n + )un+uxuxx + un+uxxx

)dx = ,

from which we have the conservation law∫R

(u + ux + εuxx

)dx =

∫R

(u + ux + εuxx

)dx. (.)

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 12 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

The global existence result follows from the integral from equation (.) and equation(.). �

Now we study the norms of solutions of equation (.) using energy estimates. First,recall the following two lemmas.

Lemma . (See []) If r > , then Hr ∩ L∞ is an algebra, and

‖fg‖Hr ≤ c(‖f ‖L∞‖g‖Hr + ‖g‖L∞‖h‖Hr

),

here c is a constant depending only on r.

Lemma . (See []) If r > , then∥∥[�r , f

]g∥∥L ≤ c

(‖∂xf ‖L∞∥∥�r–g

∥∥L +

∥∥�rf∥∥L‖g‖L∞

),

where [A,B] denotes the commutator of the linear operators A and B, and c is a constantdepending only on r.

Theorem . Suppose that, for some s ≥ , the functions u(t,x) are a solution of equation(.) corresponding to the initial data u ∈Hs(R). Then the following inequality holds:

‖u‖H ≤ c∫R

(u + ux + εuxx

)dx = c

∫R

(u + ux + εuxx

)dx. (.)

For any real number q ∈ (, s – ], there exists a constant c depending only on q such that∫R

(�q+u

) dx≤

∫R

[(�q+u

) + ε(�q+uxx

)]dx + c∫ t

‖u‖Hq‖u‖n–L∞ ‖ux‖L∞ dτ

+ c∫ t

‖ux‖L∞

(‖u‖Hq(‖u‖m–

L∞ + ‖u‖nL∞)+ ‖u‖nL∞‖u‖Hq+

)dτ . (.)

For q ∈ [, s – ], there is a constant c independent of ε such that

( – ε)‖ut‖Hq ≤ c‖u‖Hq+( +

(‖u‖m–L∞ + ‖u‖nL∞

)‖u‖H + ‖u‖n–L∞ ‖ux‖L∞). (.)

Proof Using ‖u‖H ≤ c∫R(u + ux)dx and (.) derives (.).

Since ∂x = –� + and the Parseval equality gives rise to∫

R

(�qu

)�q∂

x f dx = –∫R

(�q+u

)�q+f dx +

∫R

(�qu

)�qf dx.

For any q ∈ (, s–], applying (�qu)�q to both sides of the first equation of (.), respec-tively, and integrating with regard to x again,using integration by parts, one obtains

ddt

∫R

((�qu

) + (�qux

) + ε(�quxx

))dx= –a

∫R

(�qu

)�q(umux)dx

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 13 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

–∫R

(�q+u

)�q+(un+ux)dx + n +

∫R

(�qux

)�q(unux)dx

+∫R

(�qu

)�q(un+ux)dx – n

∫R

(�qu

)�q(un–ux)dx. (.)

We will estimate the terms on the right-hand side of (.) separately. For the first term, byusing the Cauchy-Schwartz inequality and Lemmas . and ., we have∫

R

(�qu

)�q(umux)dx = ∫

R

(�qu

)[�q(umux) – um�qux

]dx +

∫R

(�qu

)um�qux dx

≤ c‖u‖Hq(m‖u‖m–

L∞ ‖ux‖L∞‖u‖Hq + ‖u‖m–L∞ ‖ux‖L∞‖u‖Hq

)+m

‖u‖m–L∞ ‖ux‖L∞

∥∥�qu∥∥L

≤ c‖u‖m–L∞ ‖ux‖L∞‖u‖Hq . (.)

Using the above estimate to the second term on the right-hand side of equation (.) yields∫R

(�q+u

)�q+(un+ux)dx = c‖u‖nL∞‖ux‖L∞‖u‖Hq+ . (.)

For the fourth term on the right-hand side of equation (.), using the Cauchy-Schwartzinequality and Lemma ., we obtain∫

R

(�qux

)�q(unux)dx ≤ ∥∥�qux

∥∥L

∥∥�q(unux)∥∥L

≤ c‖u‖Hq+(∥∥unux∥∥L∞‖ux‖Hq + ‖ux‖L∞

∥∥unux∥∥Hq)

≤ c‖u‖nL∞‖ux‖L∞‖u‖Hq+ . (.)

For the last term on the right-hand side of equation (.), using Lemma . repeatedlyresults in∫

R

(�qu

)�q(un–ux)dx ≤ ‖u‖Hq

∥∥un–ux∥∥Hq ≤ ‖u‖Hq‖u‖n–L∞ ‖ux‖L∞ . (.)

It follows from equations (.)-(.) that there exists a constant c depending only on a,m, n, s such that

ddt

∫R

((�qu

) + (�qux

) + ε(�quxx

))dx≤ c‖ux‖L∞

(‖u‖Hq(‖u‖m–

L∞ + ‖u‖nL∞)+ ‖u‖nL∞‖u‖Hq+

)+ ‖u‖Hq‖u‖n–L∞ ‖ux‖L∞ .

Integrating both sides of the above inequality with respect to t results in inequality (.).To estimate the norm of ut , we apply the operator ( – ∂

x )– to both sides of the firstequation of the system (.) to obtain the equation

( – ε)ut – εuxxt =( – ∂

x)–[–εut – ∂x

(ku +

am +

um+)

+

n + ∂x(un+

)–n +

∂x(unux

)–nun–ux

]. (.)

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 14 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

Applying (�qut)�q to both sides of equation (.) for q ∈ (, s – ] gives rise to

( – ε)∫R

(�qut

) dx + ε

∫R

(�quxt

) dx=

∫R

(�qut

)�q–

[–εut +

n +

∂x(un+

)– ∂x

(ku +

am +

um+)–n +

∂x(unux

)–nun–ux

]dx. (.)

For the right-hand of equation (.), we have∫R

(�qut

)�q–(–εut – kux)dx ≤ ε‖ut‖Hq + k‖ut‖Hq‖u‖Hq , (.)

and ∫R

(�qut

)( – ∂

x)–

�q∂x

(–

am +

um+ –n +

unux

)dx

≤ c‖ut‖Hq

{∫R

( + ξ )q–[∫

R

(–

am +

um(ξ – η)u(η)

–n +

unux(ξ – η)ux(η))dη

]}

≤ c‖ut‖Hq‖u‖H‖u‖Hq+(‖u‖m–

L∞ + ‖u‖nL∞). (.)

Since∫R

(�qut

)( – ∂

x)–

�q∂x(un+ux

)dx

= –∫R

(�qut

)�q(un+ux)dx + ∫

R

(�qut

)( – ∂

x)–

�q(un+ux)dx. (.)

Using Lemma ., ‖unux‖Hq ≤ c‖(un+)x‖Hq ≤ c‖u‖nL∞‖u‖Hq+ and ‖u‖L∞ ≤ c‖u‖H , wehave ∫

R

(�qut

)�q(un+ux)dx ≤ c‖ut‖Hq

∥∥un+ux∥∥Hq

≤ c‖ut‖Hq‖u‖nL∞‖u‖Hq+‖u‖H , (.)

and ∫R

(�qut

)( – ∂

x)–

�q(un+ux)dx ≤ c‖ut‖Hq‖u‖nL∞‖u‖Hq+‖u‖H . (.)

By the Cauchy-Schwartz inequality and Lemma ., we get∫R

(�qut

)( – ∂

x)–

�q(un–ux)dx ≤ c‖ut‖Hq‖ux‖L∞‖u‖n–L∞ ‖u‖Hq+ . (.)

Substituting equations (.)-(.) into equation (.) yields the inequality

( – ε)‖ut‖Hq ≤ c‖u‖Hq+( +

(‖u‖m–L∞ + ‖u‖nL∞

)‖u‖H + ‖u‖n–L∞ ‖ux‖L∞)

(.)

with a constant c > . This completes the proof of Theorem .. �

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 15 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

For a real number s with s > , suppose that the function u(x) is in Hs(R), and let uε

be the convolution uε = φεu of the function φε(x) = ε– φ(ε–

x) and u be such that theFourier transform φ of φ satisfies φ ∈ C∞

, φ(ξ ) ≥ and φ(ξ ) = for any ξ ∈ (–, ). Thuswe have uε(x) ∈ C∞. It follows from Theorem . that for each ε satisfying < ε <

, theCauchy problem

⎧⎪⎨⎪⎩ut – uxxt + εuxxxt = ∂x(–ku – a

m+um+) +

n+∂x (un+)

– n+ ∂x(unux) –

nu

n–ux ,u(x, ) = uε(x)

(.)

has a unique solution uε ∈ C∞([,Tε),H∞(R)), in which Tε may depend on ε.For an arbitrary positive Sobolev exponent s > , we give the following lemma.

Lemma . For u ∈ Hs(R) with s > and uε = φε � u, the following estimates hold forany ε with < ε <

:

‖uεx‖L∞ ≤ c‖ux‖L∞ , if q ≤ s, (.)

‖uε‖Hq ≤ cεs–q , if q > s, (.)

‖uε – u‖Hq ≤ cεs–q , if q ≤ s, (.)

‖uε – u‖Hs = o(), (.)

where c is a constant independent of ε.

Proof This proof is similar to that of Lemma in [] and Lemma . in [], we omit ithere. �

Remark . For s ≥ , using ‖uε‖L∞ ≤ c‖uε‖H , ‖uε‖H ≤ c∫R(uε + uεx)dx, equations

(.), (.), and (.), we obtain

‖uε‖L∞ ≤ c‖uε‖H ≤ c∫R

(uε + uεx + uεxx

)dx

≤ c(‖uε‖H + ε‖uε‖H

) ≤ c(c + cε × ε

s–

) ≤ c, (.)

where c is independent of ε.

Theorem . If u(x) ∈Hs(R) with s ∈ [, ] such that ‖ux‖L∞(R) < ∞. Let uε be definedas in the system (.). Then there exist two constants, c and T > , which are independentof ε, such that uε of problem (.) satisfies ‖uεx‖L∞(R) ≤ c for any t ∈ [,T).

Proof Using the notation u = uε and differentiating equation (.) or equation (.) withrespect to x give rise to

( – ε)uxt – εuxxxt –n +

unux +

n + ∂x u

n+

= ku +a

m + um+ –

n +

un+

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 16 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

–( – ∂

x)–[

εuxt + ku +a

m + um+ –

n +

un+

+n +

unux +n∂x

(un–ux

)].

Letting p > be an integer and multiplying the above equality by (ux)p+, then integrat-ing the resulting equation with respect to x, and using

n +

∫R

∂x(un+

)(ux)p+ dx =

∫R

((n + )unux + un+uxx

)(ux)p+ dx

= (n + )∫R

unup+x dx +

p +

∫R

un+∂x(ux)p+ dx

=(n + )(p + )

p +

∫R

unup+x dx,

we find the equality

ddt

– ε

p +

∫R

(ux)p+ dx – ε

∫R

(ux)p+uxxxt dx +p – np +

∫R

unup+x dx

=∫R

(ux)p+(ku +

am +

um+ –

n + un+

)dx –

∫R

(ux)p+( – ∂

x)–

·[εuxt + ku +

am +

um+ –

n + un+ +

n +

unux +n∂x

(un–ux

)]dx. (.)

Applying Hölder’s inequality, we get

– ε

p + ddt

∫R

(ux)p+ dx

≤{ε

(∫R

|uxxxt|p+ dx)

p++ |k|

(∫R

|u|p+ dx)

p++

(∫R

|G|p+ dx)

p+

+a

m +

(∫R

∣∣um+∣∣p+ dx) p+

+

n +

(∫R

∣∣un+∣∣p+ dx) p+

}

·(∫

R

(ux)p+ dx) p+

p++

|p – n|p +

‖ux‖L∞‖u‖nL∞

∫R

|ux|p+ dx,

whereG = (–∂x )–[εuxt +ku+

am+u

m+ – n+u

n+ + n+ unux +

n∂x(u

n–ux)]. Furthermore

– ε

p + ddt

(∫R

(ux)p+ dx)

p+

≤ ε

(∫R

|uxxxt|p+ dx)

p++ |k|

(∫R

|u|p+ dx)

p+

+a

m +

(∫R

∣∣um+∣∣p+ dx) p+

+

n +

(∫R

∣∣un+∣∣p+ dx) p+

+(∫

R

|G|p+ dx)

p++

|p – n|p +

‖ux‖L∞‖u‖nL∞

(∫R

(ux)p+ dx)

p+.

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 17 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

Since ‖f ‖Lp → ‖f ‖L∞ as p→ ∞ for any f ∈ L∞ ∩L, integrating the above inequality withrespect to t and taking the limit as p→ ∞ result in the estimate

( – ε)‖ux‖L∞ ≤ ‖ux‖L∞ +∫ t

[ε‖uxxxt‖L∞ +

‖ux‖L∞‖u‖nL∞‖ux‖L∞

+ c(‖u‖L∞ +

∥∥um+∥∥L∞ +

∥∥un+∥∥L∞ + ‖G‖L∞)]

dτ . (.)

Using the algebraic property of Hs(R) with s > and the inequality (.) leads to

∥∥un+∥∥L∞ ≤ c∥∥un+∥∥

H +

≤ c∥∥un+∥∥H ≤ c‖u‖n+H ≤ c, (.)

and

‖G‖L∞ ≤ c(∥∥�–uxt

∥∥H

+

+∥∥�–(unux)∥∥H

+

+∥∥�–∂x

(un–ux

)∥∥H

+

)+ c

≤ c(‖ut‖L + ∥∥unux∥∥H +

∥∥un–ux∥∥H)+ c

≤ c(‖ut‖L + ∥∥un–ux∥∥L∞‖u‖H +

∥∥un–ux∥∥L∞‖u‖H)+ c

≤ c(‖ut‖L + ‖ux‖L∞ + ‖ux‖L∞

)+ c,

where c is a constant independent of ε. Using (.), (.), and the above inequality, weget

∫ t

‖G‖L∞ dτ ≤ ct + c

∫ t

( + ‖ux‖L∞ + ‖ux‖L∞

)dτ , (.)

where c is independent of ε. Furthermore, for any fixed r ∈ ( , ), there exists a constantcr such that ‖uxxxt‖L∞ ≤ cr‖uxxxt‖Hr ≤ cr‖ut‖Hr+ . By (.) and (.), one has

‖uxxxt‖L∞ ≤ c‖u‖Hr+( + ‖ux‖L∞

). (.)

Making use of the Gronwall inequality with equation (.), with q = s + , u = uε , andequation (.), yields

‖u‖Hr+ ≤(∫

R

(�r+u

) + ε(�r+uxx

) dx)× exp

{c∫ t

(‖ux‖L∞ + ‖ux‖L∞)dτ

}. (.)

From equations (.)-(.) and (.)-(.), we have

‖uxxxt‖L∞ ≤ cεs–r–

( + ‖ux‖L∞

)exp

{c∫ t

(‖ux‖L∞ + ‖ux‖L∞)dτ

}. (.)

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 18 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

For ε < , applying equations (.), (.), and (.), we obtain

‖ux‖L∞ ≤ ‖ux‖L∞ + c∫ t

{ε

s–r

( + ‖ux‖L∞

)exp

(c∫ τ

(‖ux‖L∞ + ‖ux‖L∞)dξ

)+ + ‖ux‖L∞ + ‖ux‖L∞ + ‖ux‖L∞

}dτ .

It follows from the contraction-mapping principle that there is a T > such that the equa-tion

‖W‖L∞ = ‖ux‖L∞ + c∫ t

{ε

s–r

( + ‖W‖L∞

)exp

(c∫ τ

(‖W‖L∞ + ‖W‖L∞)dξ

)+ + ‖W‖L∞ + ‖W‖L∞ + ‖W‖L∞

}dτ

has a unique solution W ∈ C[,T]. From the above inequality, we know that the variableT only depends on c and ‖um ux‖L∞ . Using the theorem present on p. in [] or Theo-rem II in Section . in [] one derives that there are constants T > and c > indepen-dent of ε such that ‖ux‖L∞ ≤ W (t) for arbitrary t ∈ [,T], which leads to the conclusionof Theorem .. �

Using equations (.)-(.) in Theorem . and Theorem ., with the notation uε = uand with Gronwall’s inequality, results in the inequalities

‖uε‖Hq(R) ≤ ‖uε‖Hq+(R) ≤ c exp{c∫ t

( + ‖ux‖L∞(R) + ‖ux‖L∞(R)

)dτ

}≤ c,

and

‖uεt‖Hr (R) ≤ c( + ‖ux‖L∞(R)

) ≤ c,

where q ∈ (, s], r ∈ (, s – ] and t ∈ [,T). It follows from Aubin’s compactness theoremthat there is a subsequence of {uε}, denoted by {uεn}, such that {uεn} and their tempo-ral derivatives {uεnt} are weakly convergent to a function u(x, t) and its derivative ut inL([,T],Hs) and L([,T],Hs–), respectively.Moreover, for any real number R > , {uεn}is convergent to the function u strongly in the space L([,T],Hq(–R,R)) for q ∈ (, s] and{uεnt} converges to ut strongly in the space L([,T],Hr(–R,R)) for r ∈ (, s – ]. Thus,we can prove the existence of a weak solution to equation (.).

Proof of Theorem . From Theorem ., we know that {uεnx}(εn → ) is bounded inthe space L∞. Thus, the sequences uεn , uεnx are weakly convergent to u, ux in the spaceL([,T],Hr(–R,R)) for any r ∈ (, s – ], separately. Hence, u satisfies the equation

–∫ T

∫R

u(gt – gxxt)dxdt =∫ T

∫R

[(ku +

am +

um+ +n +

unux

)gx

–

n + un+gxxx –

nun–uxg

]dxdt

Zhou et al. Boundary Value Problems 2014, 2014:9 Page 19 of 20http://www.boundaryvalueproblems.com/content/2014/1/9

with u(x, ) = u(x) and g ∈ C∞ . Since X = L([,T] × R) is a separable Banach space

and uεnx is a bounded sequence in the dual space X∗ = L∞([,T] × R) of X, there ex-ists a subsequence of uεnx, still denoted by uεnx, weakly star convergent to a function v inL∞([,T]×R). As uεnx weakly converges to ux in L([,T]×R), as a result ux = v almosteverywhere. Thus, we obtain ux ∈ L∞([,T]×R). �

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThis paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authorsread and approved the final manuscript.

Author details1College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, P.R. China. 2School of EconomicsManagement, Chongqing Normal University, Chongqing, 401331, P.R. China.

AcknowledgementsThe authors are very grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatlyimproved the presentation of the paper. This work is supported in part by NSFC grant 11301573 and in part by the fundsof Chongqing Normal University (13XLB006 and 13XWB008) and in part by the Program of Chongqing Innovation TeamProject in University under Grant No. KJTD201308.

Received: 25 June 2013 Accepted: 10 December 2013 Published: 08 Jan 2014

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10.1186/1687-2770-2014-9Cite this article as: Zhou et al.: Local well-posedness and persistence properties for a model containing bothCamassa-Holm and Novikov equation. Boundary Value Problems 2014, 2014:9