The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq1_HTML.gif

  • Shaoyong Lai1Email author,

    Affiliated with

    • Nan Li1 and

      Affiliated with

      • Yonghong Wu2

        Affiliated with

        Boundary Value Problems20132013:26

        DOI: 10.1186/1687-2770-2013-26

        Received: 15 November 2012

        Accepted: 27 January 2013

        Published: 12 February 2013

        Abstract

        The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space C ( [ 0 , ) × R ) L ( [ 0 , ) ; H 1 ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq2_HTML.gif under the assumption that the initial value u 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq3_HTML.gif only belongs to the space H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq1_HTML.gif. The limit of viscous approximations, a one-sided super bound estimate and a space-time higher-norm estimate for the equation are established to prove the existence of the global weak solution.

        MSC:35G25, 35L05.

        Keywords

        global weak solution Camassa-Holm type equation existence

        1 Introduction

        In this work, we investigate the Cauchy problem for the nonlinear model
        u t u t x x + x f ( u ) = 2 u x u x x + u u x x x λ u 2 N + 1 + β u 2 m u x x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ1_HTML.gif
        (1)

        where λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq4_HTML.gif, β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq5_HTML.gif, f ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq6_HTML.gif is a polynomial with order n, N and m are nonnegative integers. When f ( u ) = 2 k u + 3 2 u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq7_HTML.gif, λ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq8_HTML.gif, β = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq9_HTML.gif, Eq. (1) is the standard Camassa-Holm equation [13]. In fact, the nonlinear term λ u 2 N + 1 + β u 2 m u x x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq10_HTML.gif can be regarded as a weakly dissipative term for the Camassa-Holm model (see [4, 5]). Here we coin (1) a weakly dissipative Camassa-Holm equation.

        To link with previous works, we review several works on global weak solutions for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions of the standard Camassa-Holm equation have been proved by Constantin and Escher [6], Constantin and Molinet [7] and Danchin [8, 9] under the sign condition imposing on the initial value. Xin and Zhang [10] established the global existence of a weak solution for the Camassa-Holm equation in the energy space H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq1_HTML.gif without imposing the sign conditions on the initial value, and the uniqueness of the weak solution was obtained under certain conditions on the solution [11]. Under the sign condition for the initial value, Yin and Lai [12] proved the existence and uniqueness results of a global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. Lai and Wu [13] obtained the existence of a local weak solution for Eq. (1) in the lower-order Sobolev space H s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq11_HTML.gif with 1 s 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq12_HTML.gif. For other meaningful methods to handle the problems relating to dynamic properties of the Camassa-Holm equation and other partial differential equations, the reader is referred to [1419]. Coclite et al. [20] used the analysis presented in [10, 11] and investigated global weak solutions for a generalized hyperelastic-rod wave equation (or a generalized Camassa-Holm equation), namely, λ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq13_HTML.gif, β = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq9_HTML.gif in Eq. (1). The existence of a strongly continuous semigroup of global weak solutions for the generalized hyperelastic-rod equation with any initial value in the space H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq1_HTML.gif was established in [20]. Up to now, the existence result of the global weak solution for the weakly dissipative Camassa-Holm equation (1) has not been found in the literature. This constitutes the motivation of this work.

        The objective of this work is to study the existence of global weak solutions for the Eq. (1) in the space C ( [ 0 , ) × R ) L ( [ 0 , ) ; H 1 ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq14_HTML.gif under the assumption u 0 ( x ) H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq15_HTML.gif. The key elements in our analysis include some new a priori one-sided upper bound and space-time higher-norm estimates on the first-order derivatives of the solution. Also, the limit of viscous approximations for the equation is used to establish the existence of the global weak solution. Here we should mention that the approaches used in this work come from Xin and Zhang [10] and Coclite et al. [20].

        The rest of this paper is as follows. The main result is given in Section 2. In Section 3, we present a viscous problem of Eq. (1) and give a corresponding well-posedness result. An upper bound, a higher integrability estimate and some basic compactness properties for the viscous approximations are also established in Section 3. Strong compactness of the derivative of the viscous approximations is obtained in Section 4, where the main result for the existence of Eq. (1) is proved.

        2 Main result

        Consider the Cauchy problem for Eq. (1)
        { u t u t x x + x f ( u ) = 2 u x u x x + u u x x x λ u 2 N + 1 + β u 2 m u x x , u ( 0 , x ) = u 0 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ2_HTML.gif
        (2)
        which is equivalent to
        { u t + u u x + P x = 0 , P x = Λ 2 x [ f ( u ) + 1 2 ( u x 2 u 2 ) β u 2 m u x ] + Λ 2 [ λ u 2 N + 1 + 2 m β u 2 m 1 u x 2 ] , u ( 0 , x ) = u 0 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ3_HTML.gif
        (3)
        where the operator Λ 2 = 1 2 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq16_HTML.gif. For a fixed 1 p 0 < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq17_HTML.gif, one has
        Λ 2 g ( x ) = 1 2 R e | x y | g ( y ) d y for  g ( x ) L p 0 ( R ) , 1 < p 0 < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equa_HTML.gif
        In fact, as proved in [13], problem (2) satisfies the following conservation law:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ4_HTML.gif
        (4)

        Now we introduce the definition of a weak solution to Cauchy problem (2) or (3).

        Definition 1 A continuous function u : [ 0 , ) × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq18_HTML.gif is said to be a global weak solution to Cauchy problem (3) if
        1. (i)

          u C ( [ 0 , ) × R ) L ( [ 0 , ) ; H 1 ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq19_HTML.gif;

           
        2. (ii)

          u ( t , ) H 1 ( R ) u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq20_HTML.gif;

           
        3. (iii)

          u = u ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq21_HTML.gif satisfies (3) in the sense of distributions and takes on the initial value pointwise.

           

        The main result of this paper is stated as follows.

        Theorem 1 Assume u 0 ( x ) H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq15_HTML.gif. Then Cauchy problem (2) or (3) has a global weak solution u ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq22_HTML.gif in the sense of Definition 1. Furthermore, the weak solution satisfies the following properties.
        1. (a)
          There exists a positive constant c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq23_HTML.gif depending on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq24_HTML.gif and the coefficients of Eq. (1) such that the following one-sided L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq25_HTML.gif norm estimate on the first-order spatial derivative holds:
          u ( t , x ) x 4 t + c 0 , for ( t , x ) [ 0 , ) × R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ5_HTML.gif
          (5)
           
        2. (b)
          Let 0 < γ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq26_HTML.gif, T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq27_HTML.gif and a , b R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq28_HTML.gif, a < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq29_HTML.gif. Then there exists a positive constant c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq30_HTML.gif depending only on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq31_HTML.gif, γ, T, a, b and the coefficients of Eq. (1) such that the following space higher integrability estimate holds:
          0 t a b | u ( t , x ) x | 2 + γ d x c 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ6_HTML.gif
          (6)
           

        3 Viscous approximations

        Defining
        ϕ ( x ) = { e 1 x 2 1 , | x | < 1 , 0 , | x | 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ7_HTML.gif
        (7)
        and setting the mollifier ϕ ε ( x ) = ε 1 4 ϕ ( ε 1 4 x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq32_HTML.gif with 0 < ε < 1 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq33_HTML.gif and u ε , 0 = ϕ ε u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq34_HTML.gif, we know that u ε , 0 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq35_HTML.gif for any u 0 H s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq36_HTML.gif, s > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq37_HTML.gif (see Lai and Wu [13]). In fact, choosing the mollifier properly, we have
        u ε , 0 H 1 ( R ) u 0 H 1 ( R ) and u ε , 0 u 0 in  H 1 ( R ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ8_HTML.gif
        (8)
        The existence of a weak solution to Cauchy problem (3) will be established by proving the compactness of a sequence of smooth functions { u ε } ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq38_HTML.gif solving the following viscous problem:
        { u ε t + u ε u ε x + P ε x = ε 2 u ε x 2 , P ε x = Λ 2 x [ f ( u ε ) 1 2 u ε 2 + 1 2 ( u ε x ) 2 β u ε 2 m u ε x ] P ε x = + λ Λ 2 ( u ε ) 2 N + 1 + 2 m β Λ 2 [ u ε 2 m 1 ( u ε x ) 2 ] , u ε ( 0 , x ) = u ε , 0 ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ9_HTML.gif
        (9)

        Now we start our analysis by establishing the following well-posedness result for problem (9).

        Lemma 3.1 Provided that u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq39_HTML.gif, for any σ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq40_HTML.gif, there exists a unique solution u ε C ( [ 0 , ) ; H σ ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq41_HTML.gif to Cauchy problem (9). Moreover, for any t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq42_HTML.gif, it holds that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ10_HTML.gif
        (10)
        or
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ11_HTML.gif
        (11)

        Proof For any σ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq43_HTML.gif and u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq39_HTML.gif, we have u ε , 0 C ( [ 0 , ) ; H σ ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq44_HTML.gif. From Theorem 2.3 in [21], we conclude that problem (9) has a unique solution u ε C ( [ 0 , ) ; H σ ( R ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq45_HTML.gif for an arbitrary σ > 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq46_HTML.gif.

        We know that the first equation in system (9) is equivalent to the form
        u ε t 3 u ε t x 2 + f ( u ε ) x = 2 u ε x 2 u ε x 2 + u ε 3 u ε x 3 λ u ε 2 N + 1 + β u ε 2 m 2 u ε x 2 + ε ( 2 u ε x 2 4 u ε x 4 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ12_HTML.gif
        (12)
        from which we derive that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ13_HTML.gif
        (13)

        which completes the proof. □

        From Lemma 3.1 and (8), we have
        u ε L ( R ) u ε H 1 ( R ) u ε , 0 H 1 ( R ) u 0 H 1 ( R ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ14_HTML.gif
        (14)
        Differentiating the first equation of problem (9) with respect to x and writing u ε x = q ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq47_HTML.gif, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ15_HTML.gif
        (15)
        Lemma 3.2 Let 0 < γ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq26_HTML.gif, T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq27_HTML.gif and a , b R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq28_HTML.gif, a < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq29_HTML.gif. Then there exists a positive constant c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq30_HTML.gif depending only on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq24_HTML.gif, γ, T, a, b and the coefficients of Eq. (1), but independent of ε, such that the space higher integrability estimate holds
        0 T a b | u ε ( t , x ) x | 2 + γ d x c 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ16_HTML.gif
        (16)

        where u ε = u ε ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq48_HTML.gif is the unique solution of problem (9).

        The proof is similar to that of Proposition 3.2 presented in Xin and Zhang [10] (also see Coclite et al. [20]). Here we omit it.

        Lemma 3.3 There exists a positive constant C depending only on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq49_HTML.gif and the coefficients of Eq. (1) such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ17_HTML.gif
        (17)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ18_HTML.gif
        (18)

        where u ε = u ε ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq48_HTML.gif is the unique solution of system (9).

        Due to strong similarities with the proof of Lemma 5.1 presented in Coclite et al. [20], we do not prove Lemma 3.3 here.

        Lemma 3.4 Assume that u ε = u ε ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq48_HTML.gif is the unique solution of (9). For an arbitrary T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq27_HTML.gif, there exists a positive constant C depending only on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq31_HTML.gif and the coefficients of Eq. (1) such that the following one-sided L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq25_HTML.gif norm estimate on the first-order spatial derivative holds:
        u ε ( t , x ) x 4 t + C for ( t , x ) [ 0 , ) × R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ19_HTML.gif
        (19)
        Proof From (15) and Lemma 3.3, we know that there exists a positive constant C depending only on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq31_HTML.gif and the coefficients of Eq. (1) such that Q ε ( t , x ) L ( R ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq50_HTML.gif. Therefore,
        q ε t + u ε q ε x + 1 2 q ε 2 + β ( u ε ) 2 m q ε = Q ε ( t , x ) C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ20_HTML.gif
        (20)
        Let f = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq51_HTML.gif be the solution of
        d f d t + 1 2 f 2 + β ( u ) 2 m f = C , t > 0 , f ( 0 ) = u ε , 0 x L ( R ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ21_HTML.gif
        (21)
        where u ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq52_HTML.gif is the value of u ε ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq53_HTML.gif when sup x R q ε ( t , x ) = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq54_HTML.gif. From the comparison principle for parabolic equations, we get
        q ε ( t , x ) f ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ22_HTML.gif
        (22)
        Using (14) and β ( u ) 2 m f ρ 2 f 2 + 1 4 ρ 2 β 2 ( u ) 4 m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq55_HTML.gif, we derive that
        d f d t = C 1 2 f 2 β ( u ) 2 m f C 1 2 f 2 + ρ 2 f 2 + 1 4 ρ 2 β 2 ( u ) 4 m C 1 4 f 2 + C 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ23_HTML.gif
        (23)
        where 1 4 ρ 2 β 2 ( u ) 4 m C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq56_HTML.gif and ρ = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq57_HTML.gif. Setting M 0 = C + C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq58_HTML.gif, we obtain
        d f d t + 1 4 f 2 M 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ24_HTML.gif
        (24)

        Letting F ( t ) = 4 t + 2 M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq59_HTML.gif, we have d F ( t ) d t + 1 4 F 2 ( t ) M 0 = 4 M 0 t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq60_HTML.gif. From the comparison principle for ordinary differential equations, we get f ( t ) F ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq61_HTML.gif for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq42_HTML.gif. Therefore, by this and (22), the estimate (19) is proved. □

        Lemma 3.5 For u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq62_HTML.gif, there exists a sequence { ε j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq63_HTML.gif tending to zero and a function u L ( [ 0 , ) ; H 1 ( R ) ) H 1 ( [ 0 , T ] × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq64_HTML.gif such that, for each T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq65_HTML.gif, it holds that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ25_HTML.gif
        (25)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ26_HTML.gif
        (26)

        where u ε = u ε ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq48_HTML.gif is the unique solution of (9).

        Lemma 3.6 There exists a sequence { ε j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq63_HTML.gif tending to zero and a function Q L ( [ 0 , ) × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq66_HTML.gif such that for each 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq67_HTML.gif,
        Q ε j Q strongly in L loc p ( [ 0 , ) × R ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ27_HTML.gif
        (27)

        The proofs of Lemmas 3.5 and 3.6 are similar to those of Lemmas 5.2 and 5.3 in [20]. Here we omit their proofs.

        Throughout this paper, we use overbars to denote weak limits (the space in which these weak limits are taken is L loc r ( [ 0 , ) × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq68_HTML.gif with 1 < r < 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq69_HTML.gif).

        Lemma 3.7 There exists a sequence { ε j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq63_HTML.gif tending to zero and two functions q L loc p ( [ 0 , ) × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq70_HTML.gif, q 2 ¯ L loc r ( [ 0 , ) × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq71_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ28_HTML.gif
        (28)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ29_HTML.gif
        (29)
        for each 1 < p < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq72_HTML.gif and 1 < r < 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq69_HTML.gif. Moreover,
        q 2 ( t , x ) q 2 ¯ ( t , x ) for almost every ( t , x ) [ 0 , ) × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ30_HTML.gif
        (30)
        and
        u x = q in the sense of distributions on [ 0 , ) × R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ31_HTML.gif
        (31)

        Proof (28) and (29) are a direct consequence of Lemmas 3.1 and 3.2. Inequality (30) is valid because of the weak convergence in (29). Finally, (31) is a consequence of the definition of  q ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq73_HTML.gif, Lemma 3.5 and (28). □

        In the following, for notational convenience, we replace the sequence { u ε j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq74_HTML.gif, { q ε j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq75_HTML.gif and { Q ε j } j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq76_HTML.gif by { u ε } ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq77_HTML.gif, { q ε } ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq78_HTML.gif and { Q ε } ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq79_HTML.gif, respectively.

        Using (28), we conclude that for any convex function η C 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq80_HTML.gif with η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq81_HTML.gif being bounded and Lipschitz continuous on R and for any 1 < p < 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq72_HTML.gif, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ32_HTML.gif
        (32)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ33_HTML.gif
        (33)
        Multiplying Eq. (15) by η ( q ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq82_HTML.gif yields
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ34_HTML.gif
        (34)
        Lemma 3.8 For any convex η C 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq80_HTML.gif with η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq81_HTML.gif being bounded and Lipschitz continuous on R, it holds that
        η ( q ) t ¯ + x ( u η ( q ) ¯ ) q η ( q ) ¯ 1 2 η ( q ) q 2 ¯ β u 2 m q η ( q ) ¯ + Q ( t , x ) η ( q ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ35_HTML.gif
        (35)

        in the sense of distributions on [ 0 , ) × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq83_HTML.gif. Here q η ( q ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq84_HTML.gif and η ( q ) q 2 ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq85_HTML.gif denote the weak limits of q ε η ( q ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq86_HTML.gif and q ε 2 η ( q ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq87_HTML.gif in L loc r ( [ 0 , ) × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq68_HTML.gif, 1 < r < 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq88_HTML.gif, respectively.

        Proof In (34), by the convexity of η, (14), Lemmas 3.5, 3.6 and 3.7, taking limit for ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq89_HTML.gif gives rise to the desired result. □

        Remark 3.9 From (28) and (29), we know that
        q = q + + q = q + ¯ + q ¯ , q 2 = ( q + ) 2 + ( q ) 2 , q 2 ¯ = ( q + ) 2 ¯ + ( q ) 2 ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ36_HTML.gif
        (36)
        almost everywhere in [ 0 , ) × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq83_HTML.gif, where ξ + : = ξ χ [ 0 , + ) ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq90_HTML.gif, ξ : = ξ χ ( , 0 ] ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq91_HTML.gif for ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq92_HTML.gif. From Lemma 3.4 and (28), we have
        q ε ( t , x ) , q ( t , x ) 4 t + C for  t > 0 , x R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ37_HTML.gif
        (37)

        where C is a constant depending only on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq31_HTML.gif and the coefficients of Eq. (1).

        Lemma 3.10 In the sense of distributions on [ 0 , ) × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq83_HTML.gif, it holds that
        q t + x ( u q ) = 1 2 q 2 ¯ β u 2 m q + Q ( t , x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ38_HTML.gif
        (38)

        Proof Using (15), Lemmas 3.5 and 3.6, (28), (29) and (31), the conclusion (38) holds by taking limit for ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq93_HTML.gif in (15). □

        The next lemma contains a generalized formulation of (38).

        Lemma 3.11 For any η C 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq80_HTML.gif with η L ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq94_HTML.gif, it holds that
        η ( q ) t + x ( u η ( q ) ) = q η ( q ) + ( 1 2 q 2 ¯ q 2 ) η ( q ) β u 2 m q η ( q ) + Q ( t , x ) η ( q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ39_HTML.gif
        (39)

        in the sense of distributions on [ 0 , ) × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq83_HTML.gif.

        Proof Let { ω δ } δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq95_HTML.gif be a family of mollifiers defined on R. Denote q δ ( t , x ) : = ( q ( t , ) ω δ ) ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq96_HTML.gif, where the ⋆ is the convolution with respect to x variable. Multiplying (38) by η ( q δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq97_HTML.gif yields
        η ( q δ ) t = η ( q δ ) q δ t = η ( q δ ) [ 1 2 q 2 ¯ ω δ β u 2 m q δ + Q ( t , x ) ω δ q 2 ω δ u q x ω δ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ40_HTML.gif
        (40)
        and
        x ( u η ( q δ ) ) = q η ( q δ ) + u η ( q δ ) ( q δ x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ41_HTML.gif
        (41)

        Using the boundedness of η, η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq81_HTML.gif and letting δ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq98_HTML.gif in the above two equations, we obtain (39). □

        4 Strong convergence of q ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq73_HTML.gif

        Now, we will prove the strong convergence result, i.e.,
        x u ε x u as  ε 0  in  L loc 2 ( [ 0 , ) × R ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ42_HTML.gif
        (42)

        which is one of key statements to derive that u ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq22_HTML.gif is a global weak solution required in Theorem 1.

        Lemma 4.1 Assume u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq39_HTML.gif. It holds that
        lim t 0 R q 2 ( t , x ) d x = lim t 0 R q 2 ¯ ( t , x ) d x = R ( u 0 x ) 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ43_HTML.gif
        (43)
        Lemma 4.2 If u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq39_HTML.gif, for each M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq99_HTML.gif, it holds that
        lim t 0 R ( η M ± ( q ) ¯ ( t , x ) η M ± ( q ( t , x ) ) ) d x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ44_HTML.gif
        (44)
        where
        η M ( ξ ) : = { 1 2 ξ 2 if | ξ | M , M | ξ | 1 2 M 2 if | ξ | > M , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ45_HTML.gif
        (45)

        and η M + ( ξ ) : = η M ( ξ ) χ [ 0 , + ) ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq100_HTML.gif, η M ( ξ ) : = η M ( ξ ) χ ( , 0 ] ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq101_HTML.gif, ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq92_HTML.gif.

        Lemma 4.3 Let M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq99_HTML.gif. Then for each ξ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq92_HTML.gif,
        { η M ( ξ ) = 1 2 ξ 2 1 2 ( M | ξ | ) 2 χ ( , M ) ( M , ) ( ξ ) , η M ( ξ ) ξ = ξ + ( M | ξ | ) sign ( ξ ) χ ( , M ) ( M , ) ( ξ ) , η M + ( ξ ) = 1 2 ( ξ + ) 2 1 2 ( M ξ ) 2 χ ( M , ) ( ξ ) , ( η M + ) ( ξ ) = ξ + + ( M ξ ) χ ( M , ) ( ξ ) , η M ( ξ ) = 1 2 ( ξ ) 2 1 2 ( M + ξ ) 2 χ ( , M ) ( ξ ) , ( η M ) ( ξ ) = ξ ( M + ξ ) χ ( , M ) ( ξ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ46_HTML.gif
        (46)

        The proofs of Lemmas 4.1, 4.2 and 4.3 can be found in [10] or [20].

        Lemma 4.4 Assume u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq39_HTML.gif. Then for almost all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq42_HTML.gif,
        1 2 R ( ( q + ) 2 ¯ q + 2 ) ( t , x ) d x 0 t R Q ( s , x ) [ q + ¯ ( s , x ) q + ( s , x ) ] d s d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ47_HTML.gif
        (47)
        Lemma 4.5 For any t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq42_HTML.gif, M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq99_HTML.gif and u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq39_HTML.gif, it holds that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ48_HTML.gif
        (48)

        We do not provide the proofs of Lemmas 4.4 and 4.5 since they are similar to those of Lemmas 6.4 and 6.5 in Coclite et al. [20].

        Lemma 4.6 Assume u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq39_HTML.gif. Then it has
        q 2 ¯ = q 2 almost everywhere in [ 0 , ) × ( , ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ49_HTML.gif
        (49)

        Proof

        Applying Lemmas 4.4 and 4.5 gives rise to
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ50_HTML.gif
        (50)
        From Lemma 3.6, we know that there exists a constant L > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq102_HTML.gif, depending only on u 0 H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq49_HTML.gif, such that
        Q ( t , x ) L ( [ 0 , ) × R ) L . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ51_HTML.gif
        (51)
        By Remark 3.9 and Lemma 4.3, one has
        q + + ( η M ) ( q ) = q ( M + q ) χ ( , M ) , q + ¯ + ( η M ) ( q ) ¯ = q ( M + q ) χ ( , M ) ( q ) ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ52_HTML.gif
        (52)
        Thus, by the convexity of the map ξ ξ + + ( η M ) ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq103_HTML.gif, we get
        0 [ q + ¯ q + ] + [ ( η M ) ( q ) ¯ ( η M ) ( q ) ] = ( M + q ) χ ( , M ) ( M + q ) χ ( , M ) ( q ) ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ53_HTML.gif
        (53)
        Using (51) derives
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ54_HTML.gif
        (54)
        Since ξ ( M + ξ ) χ ( , M ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq104_HTML.gif is concave, choosing M large enough, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ55_HTML.gif
        (55)
        Then, from (50) and (55), we have
        0 R ( 1 2 [ ( q + ) 2 ¯ ( q + ) 2 ] + [ η M ¯ η M ] ) ( t , x ) d x c M 0 t R ( 1 2 [ ( q + ) 2 ¯ ( q + ) 2 ] + [ η M ¯ η M ] ) ( t , x ) d s d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ56_HTML.gif
        (56)
        By using the Gronwall inequality, for each t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq42_HTML.gif, we have
        0 R ( 1 2 [ ( q + ) 2 ¯ ( q + ) 2 ] + [ η M ¯ η M ] ) ( t , x ) d x = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equb_HTML.gif
        By the Fatou lemma, Remark 3.9 and (30), letting M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_IEq105_HTML.gif, we obtain
        0 R ( q 2 ¯ q 2 ) ( t , x ) d x = 0 for t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ57_HTML.gif
        (57)

        which completes the proof. □

        Proof of the main result Using (8), (10) and Lemma 3.5, we know that the conditions (i) and (ii) in Definition 1 are satisfied. We have to verify (iii). Due to Lemma 4.2 and Lemma 4.6, we have
        q ε q in  L loc 2 ( [ 0 , ) × R ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-26/MediaObjects/13661_2012_Article_274_Equ58_HTML.gif
        (58)

        From Lemma 3.5, (27) and (58), we know that u is a distributional solution to problem (3). In addition, inequalities (5) and (6) are deduced from Lemmas 3.2 and 3.4. The proof of Theorem 1 is completed. □

        Declarations

        Acknowledgements

        Thanks are given to referees whose comments and suggestions were very helpful for revising 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
        (2)
        Department of Mathematics and Statistics, Curtin University

        References

        1. Camassa R, Holm D: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993, 71(11):1661-1664. 10.1103/PhysRevLett.71.1661MathSciNetView Article
        2. Constantin A, Lannes D: The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 2009, 193(2):165-186.MathSciNetView Article
        3. Johnson RS: The Camassa-Holm Korteweg-de Vries and related models for water waves. J. Fluid Mech. 2002, 455(1):63-82.MathSciNet
        4. Wu SY, Yin ZY: Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation. J. Differ. Equ. 2009, 246: 4309-4321. 10.1016/j.jde.2008.12.008MathSciNetView Article
        5. Wu SY, Yin ZY: Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation. J. Math. Phys. 2006, 47(1):1-12.MathSciNetView Article
        6. Constantin A, Escher J: Global weak solutions for a shallow water equation. Indiana Univ. Math. J. 1998, 47(11):1527-1545.MathSciNet
        7. Constantin A, Molinet L: Global weak solutions for a shallow water equation. Commun. Math. Phys. 2000, 211(1):45-61. 10.1007/s002200050801MathSciNetView Article
        8. Danchin R: A few remarks on the Camassa-Holm equation. Differ. Integral Equ. 2001, 14: 953-988.MathSciNet
        9. Danchin R: A note on well-posedness for Camassa-Holm equation. J. Differ. Equ. 2003, 192: 429-444. 10.1016/S0022-0396(03)00096-2MathSciNetView Article
        10. Xin Z, Zhang P: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 2000, 53(11):1411-1433. 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5MathSciNetView Article
        11. Xin Z, Zhang P: On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Commun. Partial Differ. Equ. 2002, 27: 1815-1844. 10.1081/PDE-120016129MathSciNetView Article
        12. Yin Z, Lai SY: Global existence of weak solutions for a shallow water equation. Comput. Math. Appl. 2010, 60: 2645-2652. 10.1016/j.camwa.2010.08.094MathSciNetView Article
        13. Lai SY, Wu YH: The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. J. Differ. Equ. 2010, 248: 2038-2063. 10.1016/j.jde.2010.01.008MathSciNetView Article
        14. Lai SY, Wu HY: Local well-posedness and weak solutions for a weakly dissipative Camassa-Holm equation. Sci. China Ser. A 2010, 40(9):901-920. (in Chinese)MathSciNet
        15. Zhang S, Yin ZY: Global weak solutions to DGH. Nonlinear Anal. 2010, 72: 1690-1700. 10.1016/j.na.2009.09.008MathSciNetView Article
        16. Zhou Y: Blow-up of solutions to the DGH equation. J. Funct. Anal. 2007, 250(2):227-248.MathSciNetView Article
        17. Tian LX, Zhu MY: Blow-up and local weak solution for a modified two-component Camassa-Holm equations. Bound. Value Probl. 2012., 2012: Article ID 52
        18. Faramarz T, Mohammad S: Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source. Bound. Value Probl. 2012., 2012: Article ID 50
        19. Zhang Y, Liu DM, Mu CL, Zheng P: Blow-up for an evolution p-Laplace system with nonlocal sources and inner absorptions. Bound. Value Probl. 2011., 2011: Article ID 29
        20. Coclite GM, Holden H, Karlsen KH: Global weak solutions to a generalized hyperelastic-rod wave equation. SIAM J. Math. Anal. 2005, 37: 1044-1069. 10.1137/040616711MathSciNetView Article
        21. Coclite GM, Holden H, Karlsen KH: Well-posedness for a parabolic-elliptic system. Discrete Contin. Dyn. Syst. 2005, 13(6):659-682.MathSciNet

        Copyright

        © Lai et al.; licensee Springer 2013

        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.