The L1 stability of strong solutions to a generalized BBM equation

(1)

R E S E A R C H

Open Access

The

L



stability of strong solutions to a

generalized

BBM

equation

Shaoyong Lai

*

_{and Aiyin Wang}

*_{Correspondence:}

Laishaoy@swufe.edu.cn Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

Abstract

A nonlinear generalized Benjamin-Bona-Mahony equation is investigated. Using the estimates of strong solutions derived from the equation itself, we establish theL1(R) stability of the solutions under the assumption that the initial valueu0(x) lies in the spaceH1_(R).

MSC: 35G25; 35L05

Keywords: generalizedBBMequation; strong solutions;L1stability

1 Introduction

Benjamin, Bona and Mahony [] established theBBMmodel

ut+aux–buxxt+k

u_x= , ()

wherea,b andkare constants. Equation () is often used as an alternative to theKdV

equation which describes unidirectional propagation of weakly long dispersive waves []. As a model that characterizes long waves in nonlinear dispersive media, theBBM equa-tion, like theKdVequation, was formally derived to describe an approximation for surface water waves in a uniform channel. Equation () covers not only the surface waves of long wavelength in liquids, but also hydromagnetic waves in cold plasma, acoustic waves in an-harmonic crystals, and acoustic gravity waves in compressible ﬂuids (see [, ]). Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied in []. Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain are discussed in []. TheLq₍_q_≥_)

asymp-totic property of solutions for the Benjamin-Bona-Mahony-Burgers equations is studied in [] under certain assumptions on the initial data. The tanh technique is employed in [] to get the compact and noncompact solutions forKP-BBMandZK-BBMequations.

Applying the tanh method and the sine-cosine method, Wazwaz [] obtained com-pactons, solitons, solitary patterns and periodic solutions for the following generalized Benjamin-Bona-Mahony equation

ut+aux–buxxt+k

um_x= , ()

wherea= ,b>  andk=  are constants, andm≥ is an integer.

(2)

The objective of this work is to investigate Eq. (). Using the methods of the Kruzkov’s device of doubling the variables presented in Kruzkov’s paper [], we obtain theL_stability

of strong solutions. Namely, for any solutionsu(t,x) andu(t,x) satisfying Eq. (), we will

derive that

u(t,x) –u(t,x)_L₍_R₎≤cectu(,x) –u(,x)_L₍_R₎, t∈[,T], ()

where T is the maximum existence time of solutions u and u and c depends on

u(,x)H₍_R₎ andu_(,x)H₍_R₎. From our knowledge, we state that theL stability of strong solutions for Eq. () has never been acquired in the literature.

This paper is organized as follows. Section  gives several lemmas and Section  estab-lishes the proofs of the main result.

2 Several lemmas

LetηT= [,T]×Rfor an arbitraryT> . We denote the space of all inﬁnitely diﬀerentiable

functionsf(t,x) with compact support in [,T]×RbyC_∞(ηT). We deﬁneγ(σ) to be a

function which is inﬁnitely diﬀerentiable on (–∞, +∞) such thatγ(σ)≥,γ(σ) =  for

|σ| ≥ and_–∞_∞γ(σ)dσ = . For any numberε> , we letγε(σ) =γ(ε –_σ₎

ε . Then we have thatγε(σ) is a function inC∞(–∞,∞) and

γε(σ)≥, γε(σ) =  if|σ| ≥ε,

|γε(σ)| ≤c_ε,

∞

–∞γε(σ)dσ= .

()

Assume that the functionv(x) is locally integrable in (–∞,∞). We deﬁne the approxima-tion of funcapproxima-tionv(x) as

vε₍_x_{) =} 

ε

∞

–∞

γ

x–y

ε v(y)dy, ε> .

We callxa Lebesgue point of functionv(x) if

lim

ε→



ε

|x–x|≤ε

v(x) –v(x)dx= .

At any Lebesgue pointxof the functionv(x), we havelimε→vε(x) =v(x). Since the set

of points which are not Lebesgue points ofv(x) has measure zero, we getvε₍_x₎_→_v₍_x_{) as}

ε→ almost everywhere.

We introduce notations connected with the concept of a characteristic cone. For any

M> , we deﬁneN>sup_t_∈_[,_∞₎uL∞₍_R₎<∞. Letdenote the cone{(t,x) :|x| ≤M–

Nt,  <t<T=min(T,MN–)}. We letSτ represent the cross section of the coneby the planet=τ,τ ∈[,T]. LetHr={x:|x| ≤r}, wherer> .

Lemma . ([]) Let the function v(t,x) be bounded and measurable in cylinder = [,T]×Hr.If for anyδ∈(,min[r,T])and any numberε∈(,δ),then the function

Vε= 

ε

|t–τ

 |≤ε,δ≤t+τ≤T–δ,| x–y

 |≤ε,| x+y

 |≤r–δ

v(t,x) –v(τ,y)dx dt dy dτ

(3)

In fact, for Eq. (), we have the conservation law

R

u+bu_xdx=

R

u(,x) +bu_x(,x)dx, ()

from which we have

uL∞₍_R₎≤cuH₍_R₎, ()

whereconly depends onb.

We write the equivalent form of Eq. () in the form

ut+ –

au+kum_x= , ()

where the operator –_g₌  √b

∞

–∞e

–√

b|x–y|_g₍_y₎_dy_{for any}_g∈_L₍_R_).

Lemma . Let u=u(,x)∈H(R),Ku(t,x) = –(au+kum)and Pu(t,x) =∂xKu(t,x).For

any t∈[,∞),it holds that

Ku(t,x)_L∞₍_R₎<C, Pu(t,x)_L∞₍_R₎<C,

where the constant C is independent of time t.

Proof We have

Ku(t,x) =

 √b

_∞

–∞e

–√

b|x–y|_au₍_t_,_y_{) +}_kum₍_t_,_y₎_dy _()

and

Pu(t,x)=

∂x

 √b

_∞

–∞e

–√

b|x–y|_au₍_t_,_y_{) +}_kum₍_t_,_y₎_dy

=–  be

–√ bx

x

–∞

e  √

by_au₍_t_,_y_{) +}_kum₍_t_,_y₎_dy

+  be

 √ bx

∞

x

e–  √

by_au₍_t_,_y_{) +}_kum₍_t_,_y₎_dy

≤ 

b

∞

–∞

e–  √

b|x–y|_au₍_t_,_y_{) +}_kum₍_t_,_y₎_dy_. _()

Using ()-(), the integral _–∞_∞e–  √

b|x–y|_dy_{= }√_b _{and ()-(), we obtain the proof of}

Lemma ..

Lemma . Let u be the strong solution of Eq. (),f(t,x)∈C_∞(ηT).Then

ηT

|u–k|ft–sign(u–k)Pu(t,x)f

dx dt= , ()

(4)

Proof Let(u) be an arbitrary twice smooth function on the line –∞<u<∞. We mul-tiply Eq. () by the function(u)f(t,x), wheref(t,x)∈C∞ (ηT). Integrating overηTand

transferring the derivatives with respect totandxto the test functionf, we obtain

ηT

(u)ft–(u)Pu(t,x)f

dx dt= . ()

Letε₍_u_{) be an approximation of the function}_|_u_–_k_|_{and set}₍_u_{) =}ε₍_u_{). Letting}

ε→, we complete the proof.

In fact, the proof of () can also be found in [].

Lemma . Assume that u(t,x)and u(t,x)are two strong solutions of Eq. ()associated

with the initial data u=u(,x)and u=u(,x).Then,for any f∈C∞(ηT),

_–∞_∞sign(u–u)

Pu(t,x) –Pu(t,x)

f dx

≤c

_∞

–∞|u–u|dx, ()

where c depends onuH₍_R₎andu_H₍_R₎and f.

Proof Using (), we have

=–  be

–√ bx

x

–∞e

 √

by_au

(t,y) +kum(t,y) –au(t,y) –kum(t,y)

dy

+  be

 √ bx

∞

x

e–  √

by_au

(t,y) +kum(t,y) –au(t,y) –kum(t,y)

dy

≤c

∞

–∞

e–  √

b|x–y|_au

(t,y) +kum(t,y) –au(t,y) –kum(t,y)dy

≤c

_∞

–∞e

–√

b|x–y|_u_₍_t_,_y_{) –}_u_₍_t_,_y₎_dy_, _()

in which we have useduL∞≤ uH₍_R₎ andu_L∞ ≤ u_H₍_R₎. Using the Fubini

theorem completes the proof.

3 Main results

Theorem . Let uand u be two local or global strong solutions of Eq. ()with initial

data u(,x) =u∈H(R)and u(,x) =u∈H(R),respectively.Let Tbe the maximum

existence time of solutions uand u.For any t∈[,T),it holds that

u(t,·) –u(t,·)_L₍_R₎≤cectu–uL₍_R₎, ()

(5)

Proof For an arbitraryT > , setηT= [,T]×R. Letf(t,x)∈C∞(ηT). We assume that

f(t,x) =  outside the cylinder

=(t,x)= [δ,T– δ]×Hr–δ,  < δ≤min(T,r). ()

We deﬁne

g=f

t+τ

 ,

x+y

 γε

t–τ

 γε

x–y

 =f(· · ·)λε(∗), ()

where (· · ·) = (t+τ

 ,

x+y

 ) and (∗) = (

t–τ

 ,

x–y

 ). The functionγε(σ) is deﬁned in (). Note that

gt+gτ =ft(· · ·)λε(∗), gx+gy=fx(· · ·)λε(∗). ()

Takingu=u(t,x) andk=u(τ,y) and assumingf(t,x) =  outside the cylinder

, from Lemma ., we have

ηT×ηT

_u_₍_t_,_x_{) –}_u_₍τ,y)gt

+signu(t,x) –u(τ,y)

Pu(t,x)g

dx dt dy dτ = . ()

Similarly, it holds

ηT×ηT

_u_₍τ,y) –u(t,x)gτ

+signu(τ,y) –u(t,x)

Pu(τ,y)g

dx dt dy dτ = , ()

from which we obtain

≤

ηT×ηT

u(t,x) –u(τ,y)(gt+gτ)

+

ηT×ηT

signu(t,x) –u(τ,y)

Pu(t,x) –Pu(τ,y)

g dx dt dy dτ

=

ηT×ηT

Idx dt dy dτ+

_η

T×ηT

Idx dt dy dτ

. ()

We will show that

≤

ηT

u(t,x) –u(t,x)ft

+

ηT

signu(t,x) –u(t,x)

f dx dt. ()

We note that the ﬁrst term in the integrand of () can be represented in the form

Yε=Y

t,x,τ,y,u(t,x),u(τ,y)

(6)

By the choice ofg, we haveYε=  outside the region

(t,x;τ,y)=

δ≤t+τ

 ≤T– δ,

|t–τ|

 ≤ε,

|x+y|

 ≤r– δ,

|x–y|

 ≤ε

()

and

ηT×ηT

Yεdx dt dy dτ

=

ηT×ηT

Yt,x,τ,y,u(t,x),u(τ,y)

–Yt,x,t,x,u(t,x),u(t,x)

λε(∗)dx dt dy dτ

+

ηT×ηT

Yt,x,t,x,u(t,x),u(t,x)

λε(∗)dx dt dy dτ =J(ε) +J. ()

Considering the estimate|λ(∗)| ≤_εc and the expression of functionYε, we have

_J_₍ε)≤c



ε

|t–τ

 |≤ε,δ≤t+τ≤T–δ,| x–y

 |≤ε,| x+y

 |≤r–δ _u_₍_t_,_x₎

–u(τ,y)dx dt dy dτ

, ()

where the constant cdoes not depend onε. Using Lemma ., we obtainJ(ε)→ as

ε→. The integralJ does not depend onε. In fact, substitutingt=α, t–_τ =β,x=η,

x–y

 =ξ and noting that

ε

–ε

∞

–∞λε(β,ξ)dξdβ= , ()

we have

J= 

ηT

Yε

α,η,α,η,u(α,η),u(α,η)

ε

–ε

∞

–∞

λε(β,ξ)dξdβ

dηdα

= 

ηT

dx dt. ()

Hence

lim

ε→

ηT×ηT

Yεdx dt dy dτ= 

ηT

dx dt. ()

Since

I=sign

u(t,x) –u(τ,y)

Pu(t,x) –Pu(τ,y)

fλε(∗) ()

and

ηT×ηT

Idx dt dy dτ=

ηT×ηT

I(t,x,τ,y) –I(t,x,t,x)

dx dt dy dτ

+

ηT×ηT

(7)

we obtain

K(ε)≤c

ε+ 

ε

|t–τ

 |≤ε,δ≤t+τ≤T–δ,| x–y

 |≤ε,| x+y

 |≤r–δ

Pu(t,x)

–Pu(τ,y)dx dt dy dτ . ()

By Lemmas . and ., we haveK(ε)→ asε→. Using (), we have

K= 

ηT

I

α,η,α,η,u(α,η),u(α,η)

ε

–ε

∞

–∞λε(β,ξ)dξdβ

dηdα

= 

ηT

I

t,x,t,x,u(t,x),u(t,x)

dx dt

= 

ηT

signu(t,x) –u(t,x)

f(t,x)dx dt. ()

From () and (), we prove that inequality () holds. Let

w(t) =

∞

–∞

u(t,x) –u(t,x)dx. ()

We deﬁne the following increasing function

θε(ρ) =

ρ

–∞γε(σ)dσ

θ_ε(ρ) =γε(ρ)≥

()

and choose two numbersτandτ∈(,T),τ<τ. In (), we choose

f =θε(t–τ) –θε(t–τ)

χ(t,x), ε<min(τ,T–τ), ()

where

χ(t,x) =χh(t,x) =  –θh

|x|+Nt–M+h, h> . ()

Whenhis suﬃciently small, we note that function χ(t,x) =  outside the coneand

f(t,x) =  outside the set. For (t,x)∈, we have the relation

χt+N|χx|= ,

which derives

χt≤. ()

Applying (), ()-() and the increasing properties ofθε, we have the inequality

≤

ηT

γε(t–τ) –γε(t–τ)

χhu(t,x) –u(t,x)dx dt

+

ηT_

θε(t–τ) –θε(t–τ)

B(t,x)χh(t,x)dx dt

, ()

(8)

From (), we obtain

≤

ηT_

+ T



_∞

–∞

B(t,x)χh(t,x)dx

dt. ()

Using Lemma ., we have

≤

ηT_

+c T



∞

–∞|u–u|dx dt, ()

wherecis deﬁned in Lemma ..

Lettingh→ in () and lettingM→ ∞, we have

≤ T



∞

–∞

u(t,x) –u(t,x)dx

dt

+c T



∞

–∞

|u–u|dx dt. ()

By the properties of the functionγε(σ) forε≤min(τ,T–τ), we have

T



γε(t–τ)w(t)dt–w(τ)

=

T



γε(t–τ)

w(t) –w(τ)

dt

≤c

ε

τ+ε τ–ε

w(t) –w(τ)dt→ asε→, ()

wherecis independent ofε. Set

L(τ) =

T



θε(t–τ)w(t)dt=

T



t–τ

–∞

γε(σ)dσw(t)dt. ()

Using the similar proof of (), we get

L(τ) = –

T



γε(t–τ)w(t)dt→–w(τ) asε→, ()

from which we obtain

L(τ)→L() –

τ



w(σ)dσ asε→. ()

Similarly, we have

L(τ)→L() –

τ



(9)

Then we get

L(τ) –L(τ)→

τ

τ

w(σ)dσ asε→. ()

Lettingε→,τ→ andτ→t, from (), () and (), for anyt∈[,T], we have

∞

–∞

_u_₍_t_,_x_{) –}_u_₍_t_,_x₎_dx_≤ ∞

–∞

_u__(,_x_{) –}_u__(,_x₎_dx

+c

t



_∞

–∞

u(t,x) –u(t,x)dx dt, ()

from which we complete the proof of Theorem . by using the Gronwall inequality.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The article is a joint work of two authors who contributed equally to the ﬁnal version of the paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors are very grateful to the reviewers for their helpful and valuable comments, which have led to a meaningful improvement of the paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

Received: 25 July 2013 Accepted: 10 December 2013 Published:02 Jan 2014

References

1. Benjamin, TB, Bona, JL, Mahony, JJ: Model equation for long waves in nonlinear dispersive system. Philos. Trans. R. Soc. Lond. Ser. A272, 47-78 (1972)

2. Bona, JL, Smith, R: The initial value problem for the Korteweg-de-Vries equation. Philos. Trans. R. Soc. Lond. Ser. A278, 555-601 (1975)

3. Saut, JC, Tzetkov, N: Global well-posedness for the KP-BBM equations. Appl. Math. Res. Express1, 1-6 (2004) 4. Angulo, J, Scialom, M, Banquet, C: The regularized Benjamin-Ono and BBM equations: well-posedness and nonlinear

stability. J. Diﬀer. Equ.250, 4011-4036 (2011)

5. Rosier, L, Zhang, BY: Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain. J. Diﬀer. Equ.254, 141-178 (2013)

6. Mei, M:LqDecay rates of solutions for Benjamin-Bona-Mahony-Burgers equations. J. Diﬀer. Equ.158, 314-340 (1999) 7. Wazwaz, AM: The extended tanh method for new compact and noncompact solutions for the KP-BBM and ZK-BBM

equations. Chaos Solitons Fractals38, 1505-1516 (2008)

8. Wazwaz, AM: Nonlinear variants of the BBM equation with compact and noncompact physical structures. Chaos Solitons Fractals26, 767-776 (2005)

9. Kruzkov, SN: First order quasi-linear equations in several independent variables. Math. USSR Sb.10, 217-243 (1970) 10.1186/1029-242X-2014-3