The [InlineEquation not available: see fulltext.] stability of strong solutions to a generalized BBM equation

Abstract

A nonlinear generalized Benjamin-Bona-Mahony equation is investigated. Using the estimates of strong solutions derived from the equation itself, we establish the [InlineEquation not available: see fulltext.] stability of the solutions under the assumption that the initial value [InlineEquation not available: see fulltext.] lies in the space [InlineEquation not available: see fulltext.].

MSC: 35G25, 35L05.

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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: mailto:[email protected]

Web End [email protected] 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 the L1(R) stability of the solutions under the assumption that the initial value u0(x) lies in the

space H1(R).

MSC: 35G25; 35L05

Keywords: generalized BBM equation; strong solutions; L1 stability

1 Introduction

Benjamin, Bona and Mahony [] established the BBM model

ut + aux buxxt + k

u

x = , ()

where a, b and k are constants. Equation () is often used as an alternative to the KdV equation which describes unidirectional propagation of weakly long dispersive waves []. As a model that characterizes long waves in nonlinear dispersive media, the BBM equation, like the KdV equation, 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 uids (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 []. The Lq (q ) asymptotic 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 for KP-BBM and ZK-BBM equations.

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

ut + aux buxxt + k

um

x = , ()

where a = , b > and k = are constants, and m is an integer.

2014 Lai and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons

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The objective of this work is to investigate Eq. (). Using the methods of the Kruzkovs device of doubling the variables presented in Kruzkovs paper [], we obtain the L stability of strong solutions. Namely, for any solutions u(t, x) and u(t, x) satisfying Eq. (), we will derive that

[vextenddouble][vextenddouble]u

(t, x) u(t, 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) and u(, x) H

[vextenddouble][vextenddouble]

L(R) cect

[vextenddouble][vextenddouble]u

(, x) u(, x)

[vextenddouble][vextenddouble]

(R). From our knowledge, we state that the L 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 establishes the proofs of the main result.

2 Several lemmas

Let T = [, T]R for an arbitrary T > . We denote the space of all innitely dierentiable

functions f (t, x) with compact support in [, T] R by C(T). We dene () to be a

function which is innitely dierentiable on (, +) such that () , () = for || and

[integraltext]

() d = . For any number > , we let () = (

) . Then we have

that () is a function in C(, ) and

() , () = if || ,|()| c ,

[integraltext]

() d = .

()

Assume that the function v(x) is locally integrable in (, ). We dene the approxima

tion of function v(x) as

v(x) =

[integraldisplay]

v(y) dy, > .

We call x a Lebesgue point of function v(x) if

lim

|xx|

x y

[vextendsingle][vextendsingle]v(x)

v(x)

[vextendsingle][vextendsingle]

dx = .

At any Lebesgue point x of the function v(x), we have lim v(x) = v(x). Since the set

of points which are not Lebesgue points of v(x) has measure zero, we get v(x) v(x) as

almost everywhere.

We introduce notations connected with the concept of a characteristic cone. For any M > , we dene N > supt[,) u L(R) < . Let

[Omegainv] denote the cone {(t, x) : |x| M

Nt, < t < T = min(T, MN)}. We let S represent the cross section of the cone

[Omegainv] by the

plane t = , [, T]. Let Hr = {x : |x| r}, where r > .

Lemma . ([]) Let the function v(t, x) be bounded and measurable in cylinder = [, T] Hr. If for any (, min[r, T]) and any number (, ), then the function

V =

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]|

t

|,

t+

T,|

xy

|,|

x+y

|r

[vextendsingle][vextendsingle]v(t,

x) v(, y)

[vextendsingle][vextendsingle]

dx dt dy d

satises lim V = .

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

R

u + bux[parenrightbig]dx =

R

u(, x) + bux(, x)

[parenrightbig]

dx, ()

from which we have

u L(R) c u H

(R), ()

where c only depends on b.We write the equivalent form of Eq. () in the form

ut +

au + kum

x = , ()

where the operator g =

b

b |xy|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

[vextenddouble][vextenddouble]K

u(t, x)

[integraltext]

[vextenddouble][vextenddouble]

L(R) < C,

[vextenddouble][vextenddouble]P

u(t, x)

[vextenddouble][vextenddouble]

L(R) < C,

where the constant C is independent of time t.

Proof We have

Ku(t, x) = b [integraldisplay]

e

b |xy|

au(t, y) + kum(t, y)

[bracketrightbig]

dy ()

and

[vextendsingle][vextendsingle]P

u(t, x)

[vextendsingle][vextendsingle]

=

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]x

b [integraldisplay]

e

[bracketrightbig]

dy

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

b |xy|

au(t, y) + kum(t, y)

=

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

be

b x

[integraldisplay]

e

x au(t, y) + kum(t, y)

[bracketrightbig]

dy

+

be

b x

[integraldisplay]

x e

au(t, y) + kum(t, y)

[bracketrightbig]

dy

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

b

[integraldisplay]

e

b |xy|

[vextendsingle][vextendsingle]au(t,

y) + kum(t, y)

[vextendsingle][vextendsingle]

dy. ()

b |xy| dy = b and ()-(), we obtain the proof of Lemma ..

Lemma . Let u be the strong solution of Eq. (), f (t, x) C(T). Then

[integraldisplay][integraldisplay]T [braceleftbig]|u

k|ft sign(u k)Pu(t, x)f

Using ()-(), the integral

[integraltext]

[bracerightbig]

dx dt = , ()

where k is an arbitrary constant.

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Proof Let (u) be an arbitrary twice smooth function on the line < u < . We mul

tiply Eq. () by the function (u)f (t, x), where f (t, x) C(T). Integrating over T and

transferring the derivatives with respect to t and x to the test function f , we obtain

[integraldisplay][integraldisplay]T [braceleftbig] (u)f

t (u)Pu(t, x)f

[bracerightbig]

dx dt = . ()

Let (u) be an approximation of the function |u k| and set (u) = (u). Letting

, 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(T),

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay]

sign(u u)

Pu(t, x) Pu(t, x)

f dx

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

c

[integraldisplay]

|

u u| dx, ()

where c depends on u H

(R) and u H

(R) and f .

Proof Using (), we have

[vextendsingle][vextendsingle]P

u (t, x) Pu(t, x)

[vextendsingle][vextendsingle]

=

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

be

b x

[integraldisplay]

e

x au(t, y) + kum(t, y) au(t, y) kum(t, y)

[bracketrightbig]

dy

+

be

au(t, y) + kum(t, y) au(t, y) kum(t, y)

[bracketrightbig]

dy

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

b x

[integraldisplay]

x e

c

[integraldisplay]

e

[vextendsingle][vextendsingle]

dy

b |xy|

[vextendsingle][vextendsingle]au

(t, y) + kum(t, y) au(t, y) kum(t, y)

c

[integraldisplay]

e

[vextendsingle][vextendsingle]

dy, ()

b |xy|

[vextendsingle][vextendsingle]u

(t, y) u(t, y)

(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

[vextenddouble][vextenddouble]u

(t, ) u(t, )

in which we have used u L u H

(R) and u L u H

[vextenddouble][vextenddouble]

L(R) cect u u L

(R), ()

where c depends on u H

(R) and u H

(R).

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Proof For an arbitrary T > , set T = [, T] R. Let f (t, x) C(T). We assume that

f (t, x) = outside the cylinder

= (t, x)

[bracerightbig]= [, T ] Hr, < min(T, r). ()

We dene

g = f

t + ,x + y

t

x y

= f ( )(), ()

where ( ) = (t+ , x+y) and () = (t , xy). The function () is dened in (). Note that

gt + g = ft( )(), gx + gy = fx( )(). ()

Taking u = u(t, x) and k = u(, y) and assuming f (t, x) = outside the cylinder , from

Lemma ., we have

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T [braceleftbig][vextendsingle][vextendsingle]u

(t, x) u(, y)

[vextendsingle][vextendsingle]g

t

+ sign

u(t, x) u(, y)

Pu(t, x)g[bracerightbig]dx dt dy d = . ()

Similarly, it holds

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T [braceleftbig][vextendsingle][vextendsingle]u

(, y) u(t, x)

[vextendsingle][vextendsingle]g

+ sign

u(, y) u(t, x)

Pu(, y)g[bracerightbig]dx dt dy d = , ()

from which we obtain

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T [vextendsingle][vextendsingle]u

(t, x) u(, y)

[vextendsingle][vextendsingle](g

t + g )

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

+ [integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

sign

u(t, x) u(, y)

[parenrightbig][parenleftbig]

Pu(t, x) Pu(, y)

g dx dt dy d

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= [integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

I dx dt dy d +

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

I dx dt dy d

[vextendsingle][vextendsingle][vextendsingle][vextendsingle].

()

We will show that

[integraldisplay][integraldisplay]T [vextendsingle][vextendsingle]u

(t, x) u(t, x)

[vextendsingle][vextendsingle]f

t

+

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay][integraldisplay]T

u(t, x) u(t, x)

[parenrightbig][bracketleftbig]

Pu(t, x) Pu(t, x)

f dx dt

[vextendsingle][vextendsingle][vextendsingle][vextendsingle].

sign

()

We note that the rst term in the integrand of () can be represented in the form

Y = Y

t, x, , y, u(t, x), u(, y) (). ()

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By the choice of g, we have Y = outside the region

(t, x; , y)

[bracerightbig]=

t + T , |t | , |x + y| r , |x y|

[bracerightbigg]

()

and

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

Y dx dt dy d

= [integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T [bracketleftbig]Y[parenleftbig]t,

x, , y, u(t, x), u(, y)

[parenrightbig]

Y

t, x, t, x, u(t, x), u(t, x)

+ [integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

Y

[parenrightbig][bracketrightbig]

() dx dt dy d

t, x, t, x, u(t, x), u(t, x)

() dx dt dy d = J() + J. ()

Considering the estimate |()|

c

and the expression of function Y, we have

[vextendsingle][vextendsingle]J

()

[vextendsingle][vextendsingle]

c

[bracketleftbigg]

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]|

t

|,

t+

T,|

xy

|,|

x+y

|r

[vextendsingle][vextendsingle]u

(t, x)

, ()

where the constant c does not depend on . Using Lemma ., we obtain J() as

. The integral J does not depend on . In fact, substituting t = , t = , x = ,

xy

= and noting that

[integraldisplay]

u(, y)

[vextendsingle][vextendsingle]

dx dt dy d

[integraldisplay]

(, ) d d = , ()

we have

J =

[integraldisplay][integraldisplay]T

Y

, , , , u(, ), u(, )

[parenrightbig][braceleftbigg][integraldisplay]

[integraldisplay]

(, ) d d

d d

=

[integraldisplay][integraldisplay]T

Y

t, x, t, x, u(t, x), u(t, x)

[parenrightbig]dx dt. ()

Hence

lim

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

Y dx dt dy d =

[integraldisplay][integraldisplay]T

Y

t, x, t, x, u(t, x), u(t, x)

[parenrightbig]dx dt. ()

Since

I = sign

u(t, x) u(, y)

[parenrightbig][parenleftbig]Pu (t, x) Pu(, y)

f () ()

and

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

I dx dt dy d =

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T [bracketleftbig]I

(t, x, , y) I(t, x, t, x)

[bracketrightbig]

dx dt dy d

+ [integraldisplay][integraldisplay][integraldisplay][integraldisplay]T T

I(t, x, t, x) dx dt dy d = K() + K, ()

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we obtain

[vextendsingle][vextendsingle]K

()

[vextendsingle][vextendsingle]

c

+

[integraldisplay][integraldisplay][integraldisplay][integraldisplay]|

t

|,

t+

T,|

xy

|,|

x+y

|r

[vextendsingle][vextendsingle]P

u (t, x)

. ()

By Lemmas . and ., we have K() as . Using (), we have

K =

[integraldisplay][integraldisplay]T

I

Pu(, y)

[vextendsingle][vextendsingle]

dx dt dy d

, , , , u(, ), u(, )

[parenrightbig][braceleftbigg][integraldisplay]

[integraldisplay]

(, ) d d

d d

=

[integraldisplay][integraldisplay]T

I

t, x, t, x, u(t, x), u(t, x)

[parenrightbig]

dx dt

=

[integraldisplay][integraldisplay]T

sign

u(t, x) u(t, x)

[parenrightbig][parenleftbig]Pu (t, x) Pu(t, x)

f (t, x) dx dt. ()

From () and (), we prove that inequality () holds.

Let

w(t) =

[integraldisplay]

[vextendsingle][vextendsingle]u

(t, x) u(t, x)

[vextendsingle][vextendsingle]

dx. ()

We dene the following increasing function

() =

[integraldisplay]

() d

() = ()

[parenrightbig]

()

and choose two numbers and (, T), < . In (), we choose

f =

(t ) (t )

(t, x), < min(, T ), ()

where

(t, x) = h(t, x) = h

|x| + Nt M + h

, h > . ()

When h is suciently small, we note that function (t, x) = outside the cone [Omegainv] and f (t, x) = outside the set

. For (t, x) [Omegainv], we have the relation

t + N|x| = ,

which derives

t . ()

Applying (), ()-() and the increasing properties of , we have the inequality

[integraldisplay][integraldisplay]T [braceleftbig][bracketleftbig]

(t ) (t )

h

[vextendsingle]u

(t, x) u(t, x)

[vextendsingle][bracerightbig]

dx dt

+

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay][integraldisplay]T [bracketleftbig]

(t ) (t )

[bracketrightbig][bracketleftbig]P

u (t, x) Pu(t, x)

B(t, x)h(t, x) dx dt

[vextendsingle][vextendsingle][vextendsingle][vextendsingle],

()

where B(t, x) = sign[u(t, x) u(t, x)].

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From (), we obtain

[integraldisplay][integraldisplay]T [braceleftbig][bracketleftbig]

(t ) (t )

h

[vextendsingle][vextendsingle]u

(t, x) u(t, x)

[vextendsingle][vextendsingle][bracerightbig]

dx dt

+

[integraldisplay]

(t ) (t )

[parenrightbig][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay]

Pu(t, x) Pu(t, x)

B(t, x)h(t, x) dx

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

dt. ()

T

Using Lemma ., we have

[integraldisplay][integraldisplay]T [braceleftbig][bracketleftbig]

(t ) (t )

h

[vextendsingle][vextendsingle]u

(t, x) u(t, x)

[vextendsingle][vextendsingle][bracerightbig]

dx dt

T

+ c

[integraldisplay]

(t ) (t )

[parenrightbig] [integraldisplay]

|

u u| dx dt, ()

where c is dened in Lemma ..Letting h in () and letting M , we have

[integraldisplay]

T

[braceleftbigg][bracketleftbig]

(t ) (t )

[bracketrightbig] [integraldisplay]

[vextendsingle][vextendsingle]u

(t, x) u(t, x)

[vextendsingle][vextendsingle]

dx

dt

T

+ c

[integraldisplay]

(t ) (t )

[parenrightbig][parenleftbigg][integraldisplay]

|

u u| dx

dt. ()

By the properties of the function () for min(, T ), we have

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay]

T

T

(t )w(t) dt w()

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[integraldisplay]

(t )

w(t) w()

[bracketrightbig] dt

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

c

[integraldisplay]

+

[vextendsingle][vextendsingle]w(t)

w()

[vextendsingle][vextendsingle]

dt as , ()

where c is independent of .

Set

L() =

[integraldisplay]

T

T

(t )w(t) dt =

[integraldisplay]

[integraldisplay]

t

() dw(t) dt. ()

Using the similar proof of (), we get

L () =

[integraldisplay]

T

(t )w(t) dt w() as , ()

from which we obtain

L() L()

[integraldisplay]

w(

) d as . ()

Similarly, we have

L() L()

[integraldisplay]

w(

) d as . ()

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Then we get

L() L()

[integraldisplay]

w(

) d as . ()

Letting , and t, from (), () and (), for any t [, T], we have

[integraldisplay]

[vextendsingle][vextendsingle]u

(t, x) u(t, x)

[vextendsingle][vextendsingle]

dx

[integraldisplay]

[vextendsingle][vextendsingle]u

(, x) u(, x)

[vextendsingle][vextendsingle]

dx

t

+ c

[integraldisplay]

dx dt, ()

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

[integraldisplay]

[vextendsingle][vextendsingle]u

(t, x) u(t, x)

[vextendsingle][vextendsingle]

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 nal version of the paper. All authors read and approved the nal 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: 2 January 2014

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doi:10.1186/1029-242X-2014-3Cite this article as: Lai and Wang: The L1 stability of strong solutions to a generalized BBM equation. Journal of Inequalities and Applications 2014 2014:3.