Open Access

The local strong and weak solutions to a generalized Novikov equation

Boundary Value Problems20132013:134

DOI: 10.1186/1687-2770-2013-134

Received: 8 December 2012

Accepted: 3 May 2013

Published: 20 May 2013

Abstract

A nonlinear partial differential equation, which includes the Novikov equation as a special case, is investigated. The well-posedness of local strong solutions for the equation in the Sobolev space H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq1_HTML.gif with s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif is established. Although the H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq3_HTML.gif-norm of the solutions to the nonlinear model does not remain constant, the existence of its local weak solutions in the lower order Sobolev space H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq1_HTML.gif with 1 s 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq4_HTML.gif is established under the assumptions u 0 H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq5_HTML.gif and u 0 x L < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq6_HTML.gif.

MSC:35Q35, 35Q51.

Keywords

local strong solution local weak solution generalized Novikov equation

1 Introduction

Novikov [1] derived the integrable equation with cubic nonlinearities
u t u t x x + 4 u 2 u x = 3 u u x u x x + u 2 u x x x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ1_HTML.gif
(1)

which has been investigated by many scholars. Grayshan [2] studied both the periodic and the non-periodic Cauchy problem for Eq. (1) and discussed continuity results for the data-to-solution map in the Sobolev spaces. A Galerkin-type approximation method was used in Himonas and Holliman’s paper [3] to establish the well-posedness of Eq. (1) in the Sobolev space H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq1_HTML.gif with s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif on both the line and the circle. Hone et al. [4] applied the scattering theory to find non-smooth explicit soliton solutions with multiple peaks for Eq. (1). This multiple peak property is common with the Camassa-Holm and Degasperis-Procesi equations (see [57]). A matrix Lax pair for Eq. (1) was acquired in [8, 9] and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Sufficient conditions on the initial data to guarantee the formation of singularities in finite time for Eq. (1) were given in Jiang and Li [10]. Mi and Mu [11] obtained many dynamic results for a modified Novikov equation with a peak solution. It is shown in Ni and Zhou [12] that the Novikov equation associated with the initial value is locally well-posed in Sobolev space H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq7_HTML.gif with s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif by using the abstract Kato theorem. Two results about the persistence properties of the strong solution for Eq. (1) are established in [12]. Tiglay [13] proved the local well-posedness for the periodic Cauchy problem of the Novikov equation in Sobolev space H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq1_HTML.gif with s > 5 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq8_HTML.gif. The orbit invariants are used to show the existence of a periodic global strong solution if the Sobolev index s 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq9_HTML.gif and a sign condition holds. For analytic initial data, the existence and uniqueness of analytic solutions for Eq. (1) are obtained in [13]. Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, Yan et al. [14] proved that Eq. (1) is locally well-posed in the Besov space under certain assumptions. For other methods to handle the Novikov equation and the related partial differential equations, the reader is referred to [1524] and the references therein.

We note that the coefficients of the terms u 2 u x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq10_HTML.gif, u u x u x x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq11_HTML.gif and u 2 u x x x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq12_HTML.gif in the Novikov equation (1) are 4, 3 and 1, respectively. Namely, 4 = 3 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq13_HTML.gif. This guarantees that the conservation law of Eq. (1) holds
R ( u 2 ( t , x ) + u x 2 ( t , x ) ) d x = R ( u 2 ( 0 , x ) + u x 2 ( 0 , x ) ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equa_HTML.gif

which takes a key role in obtaining various dynamic properties in the previous works.

Motivated by the desire to extend parts of local well-posedness results in [3, 11, 12], we study the following model:
u t u t x x + m u 2 u x = a u u x u x x + b u 2 u x x x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ2_HTML.gif
(2)

where m, a and b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq14_HTML.gif are arbitrary constants. Clearly, letting m = 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq15_HTML.gif, a = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq16_HTML.gif and b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq17_HTML.gif, Eq. (2) becomes the Novikov equation (1).

The objective of this paper is to investigate Eq. (2). Since m, a and b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq14_HTML.gif are arbitrary constants, we do not have the result that the H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq3_HTML.gif norm of the solution of Eq. (2) remains constant. We will apply the Kato theorem for abstract differential equations to prove the existence and uniqueness of local solutions for Eq. (2) subject to the initial value u 0 ( x ) H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq18_HTML.gif ( s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif). In addition, the existence of local weak solutions for Eq. (2) is established in the lower-order Sobolev space H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq1_HTML.gif with 1 s 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq19_HTML.gif under the assumptions u 0 H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq20_HTML.gif and u 0 x L < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq6_HTML.gif.

The rest of this paper is organized as follows. The main results are given in Section 2. The proof of a local well-posedness result is established in Section 3, while the existence of local weak solutions is proved in Section 4.

2 Main results

Firstly, we state some notations.

The space of all infinitely differentiable functions ϕ ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq21_HTML.gif with compact support in [ 0 , + ) × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq22_HTML.gif is denoted by C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq23_HTML.gif. L p = L p ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq24_HTML.gif ( 1 p < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq25_HTML.gif) is the space of all measurable functions h such that h L p p = R | h ( t , x ) | p d x < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq26_HTML.gif. We define L = L ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq27_HTML.gif with the standard norm h L = inf m ( e ) = 0 sup x R e | h ( t , x ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq28_HTML.gif. For any real number s, H s = H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq29_HTML.gif denotes the Sobolev space with the norm defined by
h H s = ( R ( 1 + | ξ | 2 ) s | h ˆ ( t , ξ ) | 2 d ξ ) 1 2 < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equb_HTML.gif

where h ˆ ( t , ξ ) = R e i x ξ h ( t , x ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq30_HTML.gif.

For T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif and nonnegative number s, C ( [ 0 , T ) ; H s ( R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq32_HTML.gif denotes the Frechet space of all continuous H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq7_HTML.gif-valued functions on [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq33_HTML.gif. We set Λ = ( 1 x 2 ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq34_HTML.gif. For simplicity, throughout this article, we let c denote any positive constant which is independent of parameter ε.

Defining
ϕ ( x ) = { e 1 x 2 1 , | x | < 1 , 0 , | x | 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equc_HTML.gif

and setting ϕ ε ( x ) = ε 1 4 ϕ ( ε 1 4 x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq35_HTML.gif with 0 < ε < 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq36_HTML.gif and u ε 0 = ϕ ε u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq37_HTML.gif, we know that u ε 0 C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq38_HTML.gif for any u 0 H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq5_HTML.gif, s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq39_HTML.gif (see [17]).

We consider the Cauchy problem for Eq. (2)
{ u t u t x x + m u 2 u x = a u u x u x x + b u 2 u x x x , u ( 0 , x ) = u 0 ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ3_HTML.gif
(3)
which is equivalent to
{ u t + b u 2 u x = Λ 2 [ ( b m ) u 2 u x + a 6 b 2 ( u u x 2 ) x + 2 b a 2 u x 3 ] , u ( 0 , x ) = u 0 ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ4_HTML.gif
(4)

Now, we give our main results for problem (3).

Theorem 1 Let u 0 ( x ) H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq18_HTML.gif with s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif. Then the Cauchy problem (3) has a unique solution u ( t , x ) C ( [ 0 , T ) ; H s ( R ) ) C 1 ( [ 0 , T ) ; H s 1 ( R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq40_HTML.gif, where T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif depends on u 0 H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq41_HTML.gif.

It follows from Theorem 1 that for each ε satisfying 0 < ε < 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq36_HTML.gif, the Cauchy problem
{ u t u t x x + m u 2 u x = a u u x u x x + b u 2 u x x x , u ( 0 , x ) = u ε 0 ( x ) , x R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ5_HTML.gif
(5)

has a unique solution u ε ( t , x ) C ( [ 0 , T ε ) ; H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq42_HTML.gif, in which T ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq43_HTML.gif may depend on ε. However, we shall show that under certain assumptions, there exist two constants c and T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif, both independent of ε, such that the solution of problem (5) satisfies u ε x L c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq44_HTML.gif for any t [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq45_HTML.gif and there exists a weak solution u ( t , x ) L 2 ( [ 0 , T ] , H s ( R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq46_HTML.gif for problem (3). These results are summarized in the following two theorems.

Theorem 2 If u 0 ( x ) H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq18_HTML.gif with s [ 1 , 3 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq47_HTML.gif such that u 0 x L < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq6_HTML.gif. Let u ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq48_HTML.gif be defined as in system (5). Then there exist two constants c and T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif, which are independent of ε, such that the solution u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq49_HTML.gif of problem (5) satisfies u ε x L c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq44_HTML.gif for any t [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq45_HTML.gif.

Theorem 3 Suppose that u 0 ( x ) H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq50_HTML.gif with 1 s 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq19_HTML.gif and u 0 x L < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq51_HTML.gif. Then there exists a T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif such that problem (3) has a weak solution u ( t , x ) L 2 ( [ 0 , T ] , H s ( R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq46_HTML.gif in the sense of distribution and u x L ( [ 0 , T ] × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq52_HTML.gif.

3 Proof of Theorem 1

Consider the abstract quasi-linear evolution equation
d v d t + A ( v ) v = f ( v ) , t 0 and v ( 0 ) = v 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ6_HTML.gif
(6)
Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let Q : Y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq53_HTML.gif be a topological isomorphism. Let L ( Y , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq54_HTML.gif be the space of all bounded linear operators from Y to X. If X = Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq55_HTML.gif, we denote this space by L ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq56_HTML.gif. We state the following conditions in which ρ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq57_HTML.gif, ρ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq58_HTML.gif, ρ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq59_HTML.gif and ρ 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq60_HTML.gif are constants depending only on max { y Y , z Y } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq61_HTML.gif.
  1. (I)
    A ( y ) L ( Y , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq62_HTML.gif for y X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq63_HTML.gif with
    ( A ( y ) A ( z ) ) w X ρ 1 y z X w Y , y , z , w Y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equd_HTML.gif
     
and A ( y ) G ( X , 1 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq64_HTML.gif (i.e., A ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq65_HTML.gif is quasi-m-accretive), uniformly on bounded sets in Y.
  1. (II)
    Q A ( y ) Q 1 = A ( y ) + B ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq66_HTML.gif, where B ( y ) L ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq67_HTML.gif is bounded, uniformly on bounded sets in Y. Moreover,
    ( B ( y ) B ( z ) ) w X ρ 2 y z Y w X , y , z Y , w X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Eque_HTML.gif
     
  2. (III)
    f : Y Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq68_HTML.gif extends to a map from X into X, is bounded on bounded sets in Y, and satisfies
    f ( y ) f ( z ) Y ρ 3 y z Y , y , z Y , f ( y ) f ( z ) X ρ 4 y z X , y , z X . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equf_HTML.gif
     

Kato theorem (see [25])

Assume that (I), (II) and (III) hold. If v 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq69_HTML.gif, there is a maximal T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif depending only on v 0 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq70_HTML.gif and a unique solution v to problem (6) such that
v = v ( , v 0 ) C ( [ 0 , T ) ; Y ) C 1 ( [ 0 , T ) ; X ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equg_HTML.gif
Moreover, the map v 0 v ( , v 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq71_HTML.gif is a continuous map from Y to the space
C ( [ 0 , T ) ; Y ) C 1 ( [ 0 , T ) ; X ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equh_HTML.gif

We set A ( u ) = b u 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq72_HTML.gif with constant b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq14_HTML.gif, Y = H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq73_HTML.gif, X = H s 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq74_HTML.gif, Λ = ( 1 x 2 ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq75_HTML.gif, f ( u ) = Λ 2 [ ( b m ) u 2 u x + a 6 b 2 ( u u x 2 ) x + 2 b a 2 u x 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq76_HTML.gif and Q = Λ s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq77_HTML.gif. We know that Q is an isomorphism of H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq7_HTML.gif onto H s 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq78_HTML.gif. In order to prove Theorem 1, we only need to check that A ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq79_HTML.gif and f ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq80_HTML.gif satisfy assumptions (I)-(III).

Lemma 3.1 The operator A ( u ) = b u 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq81_HTML.gif with u H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq82_HTML.gif, s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif belongs to G ( H s 1 ( R ) , 1 , β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq83_HTML.gif.

Lemma 3.2 Let A ( u ) = b u 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq81_HTML.gif with u H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq82_HTML.gif and s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif. Then A ( u ) L ( H s ( R ) , H s 1 ( R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq84_HTML.gif for all u H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq82_HTML.gif. Moreover,
( A ( u ) A ( z ) ) w H s 1 ρ 1 u z H s 1 w H s , u , z , w H s ( R ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ7_HTML.gif
(7)
Lemma 3.3 For s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif, u , z H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq85_HTML.gif and w H s 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq86_HTML.gif, it holds that B ( u ) = [ Λ s , b u 2 x ] Λ s L ( H s 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq87_HTML.gif for u H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq88_HTML.gif and
( B ( u ) B ( z ) ) w H s 1 ρ 2 u z H s w H s 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ8_HTML.gif
(8)

The above three lemmas can be found in Ni and Zhou [12].

Lemma 3.4 Let r and q be real numbers such that r < q r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq89_HTML.gif. Then
u v H q c u H r v H q if r > 1 2 , u v H r + q 1 / 2 c u H r v H q if r < 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equi_HTML.gif

This lemma can be found in [25, 26].

Lemma 3.5 Let u , z H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq90_HTML.gif with s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq2_HTML.gif and f ( u ) = Λ 2 [ ( b m ) u 2 u x + a 6 b 2 ( u u x 2 ) x + 2 b a 2 u x 3 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq76_HTML.gif. Then f is bounded on bounded sets in H s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq7_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ10_HTML.gif
(10)
Proof Using the algebra property of the space H s 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq91_HTML.gif with s 0 > 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq92_HTML.gif, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ11_HTML.gif
(11)
which completes the proof of (9). Using s 1 > 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq93_HTML.gif and the first inequality in Lemma 3.4, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ12_HTML.gif
(12)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ13_HTML.gif
(13)
Using (12) and (13) yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ14_HTML.gif
(14)

which completes the proof of inequality (10). □

Proof of Theorem 1 Using the Kato theorem, Lemmas 3.1, 3.2, 3.3 and Lemma 3.5, we know that system (3) or problem (4) has a unique solution
u ( t , x ) C ( [ 0 , T ) ; H s ( R ) ) C 1 ( [ 0 , T ) ; H s 1 ( R ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equj_HTML.gif

 □

4 Proofs of Theorems 2 and 3

Using the first equation of system (3) derives
d d t R ( u 2 + u x 2 ) d x + 2 ( a 3 b ) R u u x 3 d x = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equk_HTML.gif
from which we have the conservation law
R ( u 2 + u x 2 ) d x + 2 ( a 3 b ) 0 t R u u x 3 d x = R ( u 0 2 + u 0 x 2 ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ15_HTML.gif
(15)

Lemma 4.1 (Kato and Ponce [26])

If r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq94_HTML.gif, then H r L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq95_HTML.gif is an algebra. Moreover,
u v r c ( u L v r + u r v L ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equl_HTML.gif

where c is a constant depending only on r.

Lemma 4.2 (Kato and Ponce [26])

Let r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq96_HTML.gif. If u H r W 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq97_HTML.gif and v H r 1 L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq98_HTML.gif, then
[ Λ r , u ] v L 2 c ( x u L Λ r 1 v L 2 + Λ r u L 2 v L ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equm_HTML.gif
Lemma 4.3 Let s > 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq99_HTML.gif and the function u ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq100_HTML.gif is a solution of problem (3) and the initial data u 0 ( x ) H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq18_HTML.gif. Then the following results hold:
u H 1 u 0 H 1 ( R ) e | a 3 b | 2 0 t u x L ( R ) 2 d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ16_HTML.gif
(16)
For q ( 0 , s 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq101_HTML.gif, there is a constant c only depending on m, a and b such that
R ( Λ q + 1 u ) 2 d x R [ ( Λ q + 1 u 0 ) 2 ] d x + c 0 t u H q + 1 2 ( u x L u L + u x L 2 ) d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ17_HTML.gif
(17)
For q [ 0 , s 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq102_HTML.gif, there is a constant c only depending on m, a and b such that
u t H q c u H q + 1 ( u L u H 1 + u L u x L + u x L 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ18_HTML.gif
(18)

Proof Using | 2 u u x | ( u 2 + u x 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq103_HTML.gif, the Gronwall inequality and (15) derives (16).

Using x 2 = Λ 2 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq104_HTML.gif and the Parseval equality gives rise to
R Λ q u Λ q x 3 ( u 3 ) d x = 3 R ( Λ q + 1 u ) Λ q + 1 ( u 2 u x ) d x + 3 R ( Λ q u ) Λ q ( u 2 u x ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equn_HTML.gif
For q ( 0 , s 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq101_HTML.gif, applying ( Λ q u ) Λ q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq105_HTML.gif to both sides of the first equation of system (3) and integrating with respect to x by parts, we have the identity
1 2 d d t R ( ( Λ q u ) 2 + ( Λ q u x ) 2 ) d x = ( m b ) R ( Λ q u ) Λ q ( u 2 u x ) d x b R ( Λ q + 1 u ) Λ q + 1 ( u 2 u x ) d x 2 b R Λ q u Λ q u x 3 d x + ( a 6 b ) R Λ q u Λ q ( u u x u x x ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ19_HTML.gif
(19)
We will estimate the terms on the right-hand side of (19) separately. For the first term, by using the Cauchy-Schwarz inequality and Lemmas 4.1 and 4.2, we have
| R ( Λ q u ) Λ q ( u 2 u x ) d x | = | R ( Λ q u ) [ Λ q ( u 2 u x ) u 2 Λ q u x ] d x + R ( Λ q u ) u 2 Λ q u x d x | c u H q ( 2 u L u x L u H q + u x L u L u H q ) + u L u x L Λ q u L 2 2 c u H q 2 u L u x L . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ20_HTML.gif
(20)
Using the above estimate to the second term yields
| R ( Λ q + 1 u ) Λ q + 1 ( u 2 u x ) d x | c u H q + 1 2 u L u x L . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ21_HTML.gif
(21)
Using the Cauchy-Schwarz inequality and Lemma 4.1, we obtain
| R ( Λ q u x ) Λ q ( u u x 2 ) d x | Λ q u x L 2 Λ q ( u u x 2 ) L 2 c u H q + 1 ( u L u x 2 H q + u H q u x 2 L ) c u H q + 1 2 ( u L u x L + u x L 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ22_HTML.gif
(22)
For the last term in (19), using u ( u x 2 ) x = ( u u x 2 ) x u x u x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq106_HTML.gif results in
| R ( Λ q u ) Λ q ( u u x u x x ) d x | 1 2 | R Λ q u x Λ q ( u u x 2 ) d x | + 1 2 | R Λ q u Λ q [ u x u x 2 ] d x | = K 1 + K 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ23_HTML.gif
(23)
For K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq107_HTML.gif, it follows from (22) that
K 1 c u H q + 1 2 ( u L u x L + u x L 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ24_HTML.gif
(24)
For K 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq108_HTML.gif, applying Lemma 4.1 derives
K 2 c u H q u x u x 2 H q c u H q ( u x L u x 2 H q + u x H q u x 2 L ) c u H q + 1 2 u x L 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ25_HTML.gif
(25)
It follows from (20)-(25) that there exists a constant c such that
1 2 d d t R [ ( Λ q u ) 2 + ( Λ q u x ) 2 ] d x c u H q + 1 2 ( u x L u L + u x L 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ26_HTML.gif
(26)

Integrating both sides of the above inequality with respect to t results in inequality (17).

To estimate the norm of u t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq109_HTML.gif, we apply the operator ( 1 x 2 ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq110_HTML.gif to both sides of the first equation of system (3) to obtain the equation
u t = ( 1 x 2 ) 1 [ m 3 ( u 3 ) x + b 3 x 3 ( u 3 ) 2 b x ( u u x 2 ) + ( a 2 b ) u u x u x x ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ27_HTML.gif
(27)
Applying ( Λ q u t ) Λ q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq111_HTML.gif to both sides of Eq. (27) for q [ 0 , s 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq102_HTML.gif gives rise to
R ( Λ q u t ) 2 d x = R ( Λ q u t ) Λ q 2 [ x ( m 3 u 3 + b 3 x 2 u 3 2 b u u x 2 ) + ( a 2 b ) u u x u x x ] d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ28_HTML.gif
(28)
For the right-hand of Eq. (28), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ29_HTML.gif
(29)
Since
( Λ q u t ) ( 1 x 2 ) 1 Λ q x 2 ( u 2 u x ) d x = ( Λ q u t ) Λ q ( u 2 u x ) d x + ( Λ q u t ) ( 1 x 2 ) 1 Λ q ( u 2 u x ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ30_HTML.gif
(30)
using Lemma 4.1, u 2 u x H q c ( u 3 ) x H q c u L 2 u H q + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq112_HTML.gif and u L u H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq113_HTML.gif, we have
| ( Λ q u t ) Λ q ( u 2 u x ) d x | c u t H q u 2 u x H q c u t H q u L u H 1 u H q + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ31_HTML.gif
(31)
and
| ( Λ q u t ) ( 1 x 2 ) 1 Λ q ( u 2 u x ) d x | c u t H q u L u H 1 u H q + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ32_HTML.gif
(32)
Using the Cauchy-Schwarz inequality and Lemma 4.1 yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ33_HTML.gif
(33)
Applying (29)-(33) into (28) yields the inequality
u t H q c u H q + 1 ( u L u H 1 + u L u x L + u x L 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ34_HTML.gif
(34)

for a constant c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq114_HTML.gif. This completes the proof of Lemma 4.3. □

Lemma 4.4 ([17])

For s > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq39_HTML.gif, u 0 H s ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq20_HTML.gif and u ε 0 = ϕ ε u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq115_HTML.gif, the following estimates hold for any ε with 0 < ε < 1 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq36_HTML.gif
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ35_HTML.gif
(35)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ36_HTML.gif
(36)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ37_HTML.gif
(37)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ38_HTML.gif
(38)

where c is a constant independent of ε.

Proof of Theorem 2 Using notation u = u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq116_HTML.gif and differentiating both sides of the first equation of problem (5) or Eq. (27) with respect to x give rise to
u t x + a 2 b 2 u u x 2 + b u 2 u x x = m b 3 u 3 + Λ 2 [ b m 3 u 3 + a 6 b 2 u u x 2 + 2 b a 2 ( u x 3 ) x ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ39_HTML.gif
(39)
Letting p > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq117_HTML.gif be an integer and multiplying the above equation by ( u x ) 2 p + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq118_HTML.gif and then integrating the resulting equation with respect to x yield the equality
1 2 p + 2 d d t R ( u x ) 2 p + 2 d x + ( a 2 b ) ( p + 1 ) 2 b 2 p + 2 R u ( u x ) 2 p + 3 d x = m b 3 R u 3 ( u x ) 2 p + 1 d x + R ( u x ) 2 p + 1 Λ 2 [ b m 3 u 3 + a 6 b 2 u u x 2 + 2 b a 2 ( u x 3 ) x ] d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ40_HTML.gif
(40)
Applying the Hölder’s inequality yields
1 2 p + 2 d d t R ( u x ) 2 p + 2 d x { | m b | 3 ( R | u 3 | 2 p + 2 d x ) 1 2 p + 2 + ( R | G | 2 p + 2 d x ) 1 2 p + 2 } ( R | u x | 2 p + 2 d x ) 2 p + 1 2 p + 2 + | ( a 2 b ) ( p + 1 ) 2 b 2 p + 2 | u x L u L R | u x | 2 p + 2 d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ41_HTML.gif
(41)
or
d d t ( R ( u x ) 2 p + 2 d x ) 1 2 p + 2 { | m b | 3 ( R | u 3 | 2 p + 2 d x ) 1 2 p + 2 + ( R | G | 2 p + 2 d x ) 1 2 p + 2 } + | ( a 2 b ) ( p + 1 ) 2 b 2 p + 2 | u x L u L ( R | u x | 2 p + 2 d x ) 1 2 p + 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ42_HTML.gif
(42)
where
G = Λ 2 [ b m 3 u 3 + a 6 b 2 u u x 2 + 2 b a 2 ( u x 3 ) x ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equo_HTML.gif
Since f L p f L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq119_HTML.gif as p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq120_HTML.gif for any f L L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq121_HTML.gif, integrating both sides of the inequality (42) with respect to t and taking the limit as p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq120_HTML.gif result in the estimate
u x L u 0 x L + c 0 t [ u L 3 + G L + | a 2 b | 2 u L u x L 2 ] d τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ43_HTML.gif
(43)

where c only depends on m, a, b.

Using the algebraic property of H s 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq91_HTML.gif with s 0 > 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq92_HTML.gif and the inequality (16) yields
u L u H 1 u 0 H 1 e c 0 t u x 2 d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ44_HTML.gif
(44)
and
G L c G H 1 2 + = c Λ 2 [ b m 3 u 3 + a 6 b 2 u u x 2 + 2 b a 2 ( u x 3 ) x ] H 1 2 + c ( u 3 H 0 + u u x 2 H 0 + u x 3 H 0 ) c ( u H 1 3 + u H 1 ( 1 + u x L 2 ) ) c e c 0 t u x 2 d τ ( 1 + u x L 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ45_HTML.gif
(45)
where c is a constant independent of ε. From (45), we have
0 t G L d τ c 0 t e c 0 τ u x 2 d ζ ( 1 + u x L 2 ) d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ46_HTML.gif
(46)
It follows from (43) and (46) that
u x L u 0 x L + c 0 t [ e c 0 τ u x 2 d ζ ( 1 + u x L 2 ) + 1 + u x L 2 ] d τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ47_HTML.gif
(47)
It follows from the contraction mapping principle that there is a T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif such that the equation
W L = u 0 x L + c 0 t [ ( 1 + W L 2 ) exp ( c 0 τ W L 2 d ς ) + 1 + W L 2 ] d τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equp_HTML.gif

has a unique solution W C [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq122_HTML.gif. Using the theorem presented on p.51 in Li and Olver [18] yields that there are constants T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq31_HTML.gif and c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq114_HTML.gif, which are independent of ε, such that u x L W ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq123_HTML.gif for arbitrary t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq124_HTML.gif, which leads to the conclusion of Theorem 2. □

Using Theorem 2, (17), (18) and (44), notation u ε = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq125_HTML.gif and Gronwall’s inequality results in the inequalities
u ε H q c exp [ 0 t ( u ε x L u ε L + u ε x L 2 ) d τ ] c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equq_HTML.gif
and
u ε t H r u ε H r + 1 ( u ε L u ε H 1 + u ε L u ε x L + u ε x L 2 ) c , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equr_HTML.gif

where q ( 0 , s ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq126_HTML.gif, r ( 0 , s 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq127_HTML.gif and t [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq45_HTML.gif. It follows from Aubin’s compactness theorem that there is a subsequence of { u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq128_HTML.gif, denoted by { u ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq129_HTML.gif, such that { u ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq129_HTML.gif and their temporal derivatives { u ε n t } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq130_HTML.gif are weakly convergent to a function u ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq100_HTML.gif and its derivative u t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq109_HTML.gif in L 2 ( [ 0 , T ] , H s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq131_HTML.gif and L 2 ( [ 0 , T ] , H s 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq132_HTML.gif, respectively. Moreover, for any real number R 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq133_HTML.gif, { u ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq129_HTML.gif is convergent to the function u strongly in the space L 2 ( [ 0 , T ] , H q ( R 1 , R 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq134_HTML.gif for q ( 0 , s ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq126_HTML.gif and { u ε n t } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq130_HTML.gif converges to u t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq109_HTML.gif strongly in the space L 2 ( [ 0 , T ] , H r ( R 1 , R 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq135_HTML.gif for r [ 0 , s 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq136_HTML.gif.

Proof of Theorem 3 From Theorem 2, we know that { u ε n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq137_HTML.gif ( ε n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq138_HTML.gif) is bounded in the space L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq139_HTML.gif. Thus, the sequences { u ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq129_HTML.gif, { u ε n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq137_HTML.gif, { u ε n x 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq140_HTML.gif and { u ε n x 3 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq141_HTML.gif are weakly convergent to u, u x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq142_HTML.gif, u x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq143_HTML.gif and u x 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq144_HTML.gif in L 2 ( [ 0 , T ] , H r ( R 1 , R 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq145_HTML.gif for any r [ 0 , s 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq146_HTML.gif, separately. Hence, u satisfies the equation
0 T R u ( g t g x x t ) d x d t = 0 T R ( m 3 u 3 g x b 3 u 3 g x x x + a 2 b 2 u x 3 g a 6 b 2 u u x 2 g x ) d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_Equ48_HTML.gif
(48)

with u ( 0 , x ) = u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq147_HTML.gif and g C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq148_HTML.gif. Since X = L 1 ( [ 0 , T ] × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq149_HTML.gif is a separable Banach space and { u ε n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq137_HTML.gif is a bounded sequence in the dual space X = L ( [ 0 , T ] × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq150_HTML.gif of X, there exists a subsequence of { u ε n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq137_HTML.gif, still denoted by { u ε n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq137_HTML.gif, weakly star convergent to a function v in L ( [ 0 , T ] × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq151_HTML.gif. As { u ε n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq137_HTML.gif weakly converges to u x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq142_HTML.gif in L 2 ( [ 0 , T ] × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq152_HTML.gif, it results that u x = v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq153_HTML.gif almost everywhere. Thus, we obtain u x L ( [ 0 , T ] × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-134/MediaObjects/13661_2012_Article_381_IEq52_HTML.gif. □

Declarations

Acknowledgements

Thanks are given to referees whose comments and suggestions are very helpful to revise our 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).

Authors’ Affiliations

(1)
Department of Mathematics, Southwestern University of Finance and Economics

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© Lai and Wu; licensee Springer. 2013

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