Open Access Research

Local well-posedness and persistence properties for a model containing both Camassa-Holm and Novikov equation

Shouming Zhou1*, Chun Wu1 and Baoshuai Zhang2

Author Affiliations

1 College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, P.R. China

2 School of Economics Management, Chongqing Normal University, Chongqing, 401331, P.R. China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:9  doi:10.1186/1687-2770-2014-9


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/9


Received:25 June 2013
Accepted:10 December 2013
Published:8 January 2014

© 2014 Zhou et al.; licensee Springer.

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.

Abstract

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M3">View MathML</a>. This equation is a generalization of the famous equation of Camassa-Holm and the Novikov equation. The local well-posedness of strong solutions for this equation in Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M5">View MathML</a> is obtained, and persistence properties of the strong solutions are studied. Furthermore, under appropriate hypotheses, the existence of its weak solutions in low order Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M7">View MathML</a> is established.

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

1 Introduction

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M8">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M3">View MathML</a>.

Obviously, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M13">View MathML</a>, equation (1.1) becomes the Camassa-Holm equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M14">View MathML</a>

(1.2)

where the variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M15">View MathML</a> represents the fluid velocity at time t and in the spatial direction x, and k is a nonnegative parameter related to the critical shallow water speed [1]. The Camassa-Holm equation (1.2) is also a model for the propagation of axially symmetric waves in hyperelastic rods (cf.[2]). It is well known that equation (1.2) has also a bi-Hamiltonian structure [3,4] and is completely integrable (see [5,6] and the in-depth discussion in [7,8]). In [9], Qiao has shown that the Camassa-Holm spectral problem yields two different integrable hierarchies of nonlinear evolution equations, one is of negative order CH hierachy while the other one is of positive order CH hierarchy. Its solitary waves are smooth if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M16">View MathML</a> and peaked in the limiting case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M17">View MathML</a> (cf.[1]). The orbital stability of the peaked solitons is proved in [10], and the stability of the smooth solitons is considered in [11]. It is worth pointing out that solutions of this type are not mere abstractions: the peakons replicate a feature that is characteristic for the waves of great height - waves of largest amplitude that are exact solutions of the governing equations for irrotational water waves (cf.[12-14]). The explicit interaction of the peaked solitons is given in [15] and all possible explicit single soliton solutions are shown in [16]. The Cauchy problem for the Camassa-Holm equation (1.2) has been studied extensively. It has been shown that this problem is locally well-posed for initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M19">View MathML</a>[17-19]. Moreover, it has global strong solutions and also admits finite time blow-up solutions [17,18,20,21]. On the other hand, it also has global weak solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M20">View MathML</a>[22-25]. The advantage of the Camassa-Holm equation in comparison with the KdV equation (1.2) lies in the fact that the Camassa-Holm equation has peaked solitons and models the peculiar wave breaking phenomena [1,21].

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M23">View MathML</a>, equation (1.1) becomes a generalized Camassa-Holm equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M24">View MathML</a>

(1.3)

Wazwaz [26,27] studied the solitary wave solutions for the generalized Camassa-Holm equation (1.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M13">View MathML</a>, and the peakon wave solutions for this equation were studied in [28-30], and the periodic blow-up solutions and limit forms for (1.3) were obtained in [31]. In [30,32], the authors have given the traveling waves solution, peaked solitary wave solutions for (1.3).

On the other hand, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M29">View MathML</a> in (1.1) we found the Novikov equation [33]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M30">View MathML</a>

(1.4)

The Novikov equation (1.4) possesses a matrix Lax pair, many conserved densities, a bi-Hamiltonian structure as well as peakon solutions [34]. These apparently exotic waves replicate a feature that is characteristic of the waves of great height-waves of largest amplitude that are exact solutions of the governing equations for water waves, as far as the details are concerned [13,35,36]. The Novikov equation possesses the explicit formulas for multipeakon solutions [37]. It has been shown that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces and in Sobolev spaces and possesses the persistence properties [38,39]. In [40,41], the authors showed that the data-to-solution map for equation (1.4) is not uniformly continuous on bounded subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M31">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32">View MathML</a>. Analogous to the Camassa-Holm equation, the Novikov equation shows the blow-up phenomenon [42] and has global weak solutions [43]. Recently, Zhao and Zhou [44] discussed the symbolic analysis and exact traveling wave solutions of a modified Novikov equation, which is new in that it has a nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M33">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M34">View MathML</a>.

Other integrable CH-type equations with cubic nonlinearity have been discovered:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M35">View MathML</a>

(1.5)

where γ is a constant. equation (1.5) was independently proposed by Fokas [45], by Fuchssteiner [46], and Olver and Rosenau [47] as a new generalization of integrable system by using the general method of tri-Hamiltonian duality to the bi-Hamiltonian representation of the modified Korteweg-de Vries equation. Later, it was obtained by Qiao [48,49] from the two-dimensional Euler equations, where the variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M37">View MathML</a> represent, respectively, the velocity of the fluid and its potential density. Ivanov and Lyons [50] obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of woks by Qiao [48,49]. It was shown that equation (1.5) admits the Lax-pair and the Cauchy problem (1.5) may be solved by the inverse scattering transform method. The formation of singularities and the existence of peaked traveling-wave solutions for equation (1.5) was investigated in [51]. The well-posedness, blow-up mechanism, and persistence properties are given in [52]. It was also found that equation (1.5) is related to the short-pulse equation derived by Schäfer and Wayne [53].

Applying the method of pseudoparabolic regularization, Lai and Wu [54] investigated the local well-posedness and existence of weak solutions for the following generalized Camassa-Holm equation with dissipative term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M38">View MathML</a>

(1.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M39">View MathML</a>, and a, k, β are constants. Hakkaev and Kirchev [55] studied the local well-posedness and orbital stability of solitary wave solution for equation (1.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M42">View MathML</a>.

Motivated by the results mentioned above, the goal of this paper is to establish the well-posedness of strong solutions and weak solutions for problem (1.1). First, we use Kato’s theorem to obtain the existence and uniqueness of strong solutions for equation (1.1).

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32">View MathML</a>. Then there exists a maximal<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M45">View MathML</a>, and a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M46">View MathML</a>to the problem (1.1) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M47">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M48">View MathML</a>

is continuous.

In [38,56,57], the spatial decay rates for the strong solution to the Camassa-Holm Novikov equation were established provided that the corresponding initial datum decays at infinity. This kind of property is so-called the persistence property. Similarly, for equation (1.1), we also have the following persistence properties for the strong solution.

Theorem 1.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M49">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M50">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M51">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M52">View MathML</a> (respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M53">View MathML</a>), then the corresponding strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M54">View MathML</a>to equation (1.1) satisfies for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M56">View MathML</a>

uniformly in the time interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57">View MathML</a>.

Theorem 1.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M59">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M60">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M50">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M62">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M63">View MathML</a> (respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M65">View MathML</a>), then the corresponding strong solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M54">View MathML</a>to equation (1.1) satisfies for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M68">View MathML</a>

uniformly in the time interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57">View MathML</a>.

Remark 1.1 The notations mean that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M70">View MathML</a>

Finally, we have the following theorem for the existence of a weak solution for equation (1.1).

Theorem 1.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M71">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M7">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M73">View MathML</a>. Then there exists a life span<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55">View MathML</a>such that problem (1.1) has a weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M75">View MathML</a>in the sense of a distribution and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M76">View MathML</a>.

The plan of this paper is as follows. In the next section, the local well-posedness and persistence properties of strong solutions for the problem (1.1) are established, and Theorems 1.1-1.3 are proved. The existence of weak solutions for the problem (1.1) is proved in Section 3, and this proves Theorem 1.4.

2 Well-posedness and persistence properties of strong solutions

Notation The space of all infinitely differentiable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M77">View MathML</a> with compact support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M78">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M79">View MathML</a>. Let p be any constant with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M80">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M81">View MathML</a> to be the space of all measurable functions f such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M82">View MathML</a>. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M83">View MathML</a> with the standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M84">View MathML</a>. For any real number s, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M85">View MathML</a> denote the Sobolev space with the norm defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M86">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M87">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M88">View MathML</a> denote the class of continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M91">View MathML</a>.

Proof of Theorem 1.1 To prove well-posedness we apply Kato’s semigroup approach [58]. For this, we rewrite the Cauchy problem of equation (1.1) as follows for the transport equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M92">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M93">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M95">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M99">View MathML</a>. Following closely the considerations made in [17,54,59], we obtain the statement of Theorem 1.1. □

Proof of Theorem 1.2 We introduce the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M100">View MathML</a>. The first step we will give estimates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M101">View MathML</a>. Integrating the both sides with respect to x variable by multiplying the first equation of (2.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M102">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M103">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M104">View MathML</a>

(2.2)

Note that the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M105">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M106">View MathML</a>

are true. Moreover, using Hölder’s inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M107">View MathML</a>

From equation (2.2) we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M108">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M109">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M110">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M111">View MathML</a>. From the above inequality we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M112">View MathML</a>

where we use

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M113">View MathML</a>

Because of Gronwall’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M114">View MathML</a>

Next, we will give estimates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M115">View MathML</a>. Differentiating (2.1) with respect to the x-variable produces the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M116">View MathML</a>

(2.3)

Multiplying this equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M117">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M103">View MathML</a>, integrating the result in the x-variable, and using integration by parts:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M119">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M120">View MathML</a>

where we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M121">View MathML</a>. Then passing to the limit in this inequality and using Gronwall’s inequality one can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M122">View MathML</a>

We shall now repeat the arguments using the weight

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M123">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M124">View MathML</a>. Observe that for all N we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M125">View MathML</a>

(2.4)

Using the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M126">View MathML</a>, from (2.1) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M127">View MathML</a>

and from (2.3), we also obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M128">View MathML</a>

We need to eliminate the second derivatives in the second term in the above equality. Thus, combining integration by parts and equation (2.4) we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M129">View MathML</a>

Hence, as in the weightless case, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M130">View MathML</a>

A simple calculation shows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M131">View MathML</a>, depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M52">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M133">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M134">View MathML</a>

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M135">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M136">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M137">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M138">View MathML</a>

Thus, it follows that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M131">View MathML</a> which depends only on M, m, n, k, a, and T, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M140">View MathML</a>

Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M141">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M143">View MathML</a>

Finally, taking the limit as N goes to infinity we find that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M145">View MathML</a>

which completes the proof of Theorem 1.2. □

Next, we give a simple proof for Theorem 1.3.

Proof of Theorem 1.3 We should use Theorem 1.3 to prove this theorem.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M146">View MathML</a>, integrating equation (2.1) over the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M147">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M148">View MathML</a>

(2.5)

From Theorem 1.2, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M149">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M150">View MathML</a>

We shall show that the last term in equation (2.5) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M151">View MathML</a>; thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M152">View MathML</a>

From the given condition and Theorem 1.2. we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M153">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M154">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M155">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M156">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M157">View MathML</a>

From equation (2.5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M158">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M159">View MathML</a>, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M160">View MathML</a>

By the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M146">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M162">View MathML</a>

uniformly in the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M57">View MathML</a>. This completes the proof of Theorem 1.3. □

3 Existence of solution of the regularized equation

In order to prove Theorem 1.4, we consider the regularized problem for equation (1.1) in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M164">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M167">View MathML</a> and a, k are constants. One can easily check that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M168">View MathML</a>, equation (3.1) is equivalent to the IVP (1.1).

Before giving the proof of Theorem 1.4, we give several lemmas.

Lemma 3.1 (See [54])

Letpandqbe real numbers such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M169">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M170">View MathML</a>

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M71">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32">View MathML</a>. Then the Cauchy problem (3.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M173">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55">View MathML</a>depends on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M175">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M176">View MathML</a>, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M173">View MathML</a>exists for all time. In particular, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M178">View MathML</a>, the corresponding solution is a classical globally defined solution of problem (3.1).

Proof First, we note that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165">View MathML</a> and any s, the integral operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M180">View MathML</a>

defines a bounded linear operator on the indicated Sobolev spaces.

To prove the existence of a solution to the problem (3.1), we apply the operator to both sides of equation (3.1) and then integrate the resulting equations with regard to t. This leads to the following equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M182">View MathML</a>

(3.2)

Suppose that is the operator in the right-hand side of equation (3.2). For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M185">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M186">View MathML</a> is an algebra for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M187">View MathML</a>, we have the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M188">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M32">View MathML</a>, by Lemma 3.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M190">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M191">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M193">View MathML</a> only depend on n. Suppose that both u and v are in the closed ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M194">View MathML</a> of radius R about the zero function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M88">View MathML</a>; by the above inequalities, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M196">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M197">View MathML</a> and C only depend on a, k, m, n. Choosing T sufficiently small such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M198">View MathML</a>, we know that is a contraction. Applying the above inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M200">View MathML</a>

Taking T sufficiently small so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M201">View MathML</a>, we deduce that maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M194">View MathML</a> to itself. It follows from the contraction-mapping principle that the mapping has a unique fixed point u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M194">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M176">View MathML</a>, multiplying the first equation of the system (3.1) by 2u, integrating with respect to x, one derives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M207">View MathML</a>

from which we have the conservation law

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M208">View MathML</a>

(3.3)

The global existence result follows from the integral from equation (3.2) and equation (3.3). □

Now we study the norms of solutions of equation (3.1) using energy estimates. First, recall the following two lemmas.

Lemma 3.3 (See [58])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M209">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M210">View MathML</a>is an algebra, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M211">View MathML</a>

herecis a constant depending only onr.

Lemma 3.4 (See [58])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M209">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M213">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M214">View MathML</a>denotes the commutator of the linear operatorsAandB, andcis a constant depending only onr.

Theorem 3.1Suppose that, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M178">View MathML</a>, the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M216">View MathML</a>are a solution of equation (3.1) corresponding to the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18">View MathML</a>. Then the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M218">View MathML</a>

(3.4)

For any real number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M219">View MathML</a>, there exists a constantcdepending only onqsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M220">View MathML</a>

(3.5)

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M221">View MathML</a>, there is a constantcindependent ofϵsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M222">View MathML</a>

(3.6)

Proof Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M223">View MathML</a> and (3.3) derives (3.4).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M224">View MathML</a> and the Parseval equality gives rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M225">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M219">View MathML</a>, applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M227">View MathML</a> to both sides of the first equation of (3.1), respectively, and integrating with regard to x again,using integration by parts, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M228">View MathML</a>

(3.7)

We will estimate the terms on the right-hand side of (3.7) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M229">View MathML</a>

(3.8)

Using the above estimate to the second term on the right-hand side of equation (3.7) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M230">View MathML</a>

(3.9)

For the fourth term on the right-hand side of equation (3.7), using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M231">View MathML</a>

(3.10)

For the last term on the right-hand side of equation (3.7), using Lemma 3.3 repeatedly results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M232">View MathML</a>

(3.11)

It follows from equations (3.7)-(3.11) that there exists a constant c depending only on a, m, n, s such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M233">View MathML</a>

Integrating both sides of the above inequality with respect to t results in inequality (3.5).

To estimate the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M234">View MathML</a>, we apply the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M235">View MathML</a> to both sides of the first equation of the system (3.1) to obtain the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M236">View MathML</a>

(3.12)

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M237">View MathML</a> to both sides of equation (3.12) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M219">View MathML</a> gives rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M239">View MathML</a>

(3.13)

For the right-hand of equation (3.13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M240">View MathML</a>

(3.14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M241">View MathML</a>

(3.15)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M242">View MathML</a>

(3.16)

Using Lemma 3.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M244">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M245">View MathML</a>

(3.17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M246">View MathML</a>

(3.18)

By the Cauchy-Schwartz inequality and Lemma 3.3, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M247">View MathML</a>

(3.19)

Substituting equations (3.14)-(3.19) into equation (3.13) yields the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M248">View MathML</a>

(3.20)

with a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M249">View MathML</a>. This completes the proof of Theorem 3.1. □

For a real number s with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M250">View MathML</a>, suppose that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M251">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M253">View MathML</a> be the convolution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M254">View MathML</a> of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M256">View MathML</a> be such that the Fourier transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M257">View MathML</a> of ϕ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M260">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M261">View MathML</a>. Thus we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M262">View MathML</a>. It follows from Theorem 3.1 that for each ϵ satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165">View MathML</a>, the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M264">View MathML</a>

(3.21)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M265">View MathML</a>, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M266">View MathML</a> may depend on ϵ.

For an arbitrary positive Sobolev exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M250">View MathML</a>, we give the following lemma.

Lemma 3.5For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M18">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M250">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M270">View MathML</a>, the following estimates hold for anyϵwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M165">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M272">View MathML</a>

(3.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M273">View MathML</a>

(3.23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M274">View MathML</a>

(3.24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M275">View MathML</a>

(3.25)

wherecis a constant independent ofϵ.

Proof This proof is similar to that of Lemma 5 in [60] and Lemma 4.5 in [61], we omit it here. □

Remark 3.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M276">View MathML</a>, using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M278">View MathML</a>, equations (3.4), (3.22), and (3.23), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M279">View MathML</a>

(3.26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M280">View MathML</a> is independent of ϵ.

Theorem 3.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M71">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M282">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M73">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M253">View MathML</a>be defined as in the system (3.21). Then there exist two constants, cand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55">View MathML</a>, which are independent ofϵ, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M286">View MathML</a>of problem (3.21) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M287">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M288">View MathML</a>.

Proof Using the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M289">View MathML</a> and differentiating equation (3.21) or equation (3.12) with respect to x give rise to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M290">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M291">View MathML</a> be an integer and multiplying the above equality by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M292">View MathML</a>, then integrating the resulting equation with respect to x, and using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M293">View MathML</a>

we find the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M294">View MathML</a>

(3.27)

Applying Hölder’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M295">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M296">View MathML</a>. Furthermore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M297">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M109">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M110">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M111">View MathML</a>, integrating the above inequality with respect to t and taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M110">View MathML</a> result in the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M302">View MathML</a>

(3.28)

Using the algebraic property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M304">View MathML</a> and the inequality (3.26) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M305">View MathML</a>

(3.29)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M306">View MathML</a>

where c is a constant independent of ϵ. Using (3.6), (3.29), and the above inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M307">View MathML</a>

(3.30)

where c is independent of ϵ. Furthermore, for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M308">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M309">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M310">View MathML</a>. By (3.6) and (3.26), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M311">View MathML</a>

(3.31)

Making use of the Gronwall inequality with equation (3.5), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M289">View MathML</a>, and equation (3.26), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M314">View MathML</a>

(3.32)

From equations (3.22)-(3.23) and (3.31)-(3.32), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M315">View MathML</a>

(3.33)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M316">View MathML</a>, applying equations (3.28), (3.30), and (3.33), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M317">View MathML</a>

It follows from the contraction-mapping principle that there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55">View MathML</a> such that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M319">View MathML</a>

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M320">View MathML</a>. From the above inequality, we know that the variable T only depends on c and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M321">View MathML</a>. Using the theorem present on p.51 in [28] or Theorem II in Section 1.1 in [62] one derives that there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M249">View MathML</a> independent of ϵ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M324">View MathML</a> for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M142">View MathML</a>, which leads to the conclusion of Theorem 3.2. □

Using equations (3.5)-(3.6) in Theorem 3.1 and Theorem 3.2, with the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M326">View MathML</a> and with Gronwall’s inequality, results in the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M327">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M328">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M330">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M288">View MathML</a>. It follows from Aubin’s compactness theorem that there is a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M332">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M333">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M333">View MathML</a> and their temporal derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M335">View MathML</a> are weakly convergent to a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M46">View MathML</a> and its derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M234">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M338">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M339">View MathML</a>, respectively. Moreover, for any real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M333">View MathML</a> is convergent to the function u strongly in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M342">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M329">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M335">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M234">View MathML</a> strongly in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M346">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M330">View MathML</a>. Thus, we can prove the existence of a weak solution to equation (1.1).

Proof of Theorem 1.4 From Theorem 3.2, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M348">View MathML</a> is bounded in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M349">View MathML</a>. Thus, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351">View MathML</a> are weakly convergent to u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M352">View MathML</a> in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M353">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M330">View MathML</a>, separately. Hence, u satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M355">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M356">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M357">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M358">View MathML</a> is a separable Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351">View MathML</a> is a bounded sequence in the dual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M360">View MathML</a> of X, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351">View MathML</a>, weakly star convergent to a function v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M363">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M351">View MathML</a> weakly converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M352">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M366">View MathML</a>, as a result <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M367">View MathML</a> almost everywhere. Thus, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M368">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.

Acknowledgements

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

References

  1. Camassa, R, Holm, D, Hyman, J: A new integrable shallow water equation. Adv. Appl. Mech.. 31, 1–33 (1994)

  2. Constantin, A, Strauss, WA: Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A. 270, 140–148 (2000). Publisher Full Text OpenURL

  3. Fokas, AS, Fuchssteiner, B: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica, D. 4, 47–66 (1981). Publisher Full Text OpenURL

  4. Lenells, J: Conservation laws of the Camassa-Holm equation. J. Phys. A. 38, 869–880 (2005). Publisher Full Text OpenURL

  5. Camassa, R, Holm, D: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett.. 71, 1661–1664 (1993). PubMed Abstract | Publisher Full Text OpenURL

  6. Constantin, A: On the scattering problem for the Camassa-Holm equation. Proc. R. Soc. Lond.. 457, 953–970 (2001). Publisher Full Text OpenURL

  7. Boutet, A, Monvel, D, Shepelsky, D: Riemann-Hilbert approach for the Camassa-Holm equation on the line. C. R. Math. Acad. Sci. Paris. 343, 627–632 (2006). Publisher Full Text OpenURL

  8. Constantin, A, Gerdjikov, V, Ivanov, R: Inverse scattering transform for the Camassa-Holm equation. Inverse Probl.. 22, 2197–2207 (2006). Publisher Full Text OpenURL

  9. Qiao, ZJ: The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold. Commun. Math. Phys.. 239, 309–341 (2003). Publisher Full Text OpenURL

  10. Constantin, A, Strauss, WA: Stability of peakons. Commun. Pure Appl. Math.. 53, 603–610 (2000). Publisher Full Text OpenURL

  11. Constantin, A, Strauss, WA: Stability of the Camassa-Holm solitons. J. Nonlinear Sci.. 12, 415–422 (2002). Publisher Full Text OpenURL

  12. Constantin, A: The trajectories of particles in Stokes waves. Invent. Math.. 166, 523–535 (2006). Publisher Full Text OpenURL

  13. Constantin, A, Escher, J: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math.. 173, 559–568 (2011). Publisher Full Text OpenURL

  14. Toland, JF: Stokes waves. Topol. Methods Nonlinear Anal.. 7, 1–48 (1996)

  15. Beals, R, Sattinger, D, Szmigielski, J: Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math.. 140, 190–206 (1998). Publisher Full Text OpenURL

  16. Qiao, ZJ, Zhang, GP: On peaked and smooth solitons for the Camassa-Holm equation. Europhys. Lett.. 73, 657–663 (2006). Publisher Full Text OpenURL

  17. Constantin, A, Escher, J: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci.. 26, 303–328 (1998)

  18. Li, YA, Olver, PJ: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ.. 162, 27–63 (2000). Publisher Full Text OpenURL

  19. Rodriguez-Blanco, G: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal.. 46, 309–327 (2001). Publisher Full Text OpenURL

  20. Constantin, A: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier. 50, 321–362 (2000). Publisher Full Text OpenURL

  21. Constantin, A, Escher, J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math.. 181, 229–243 (1998). Publisher Full Text OpenURL

  22. Bressan, A, Constantin, A: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal.. 183, 215–239 (2007). Publisher Full Text OpenURL

  23. Constantin, A, Escher, J: Global weak solutions for a shallow water equation. Indiana Univ. Math. J.. 47, 1527–1545 (1998)

  24. Constantin, A, Molinet, L: Global weak solutions for a shallow water equation. Commun. Math. Phys.. 211, 45–61 (2000). Publisher Full Text OpenURL

  25. Xin, ZP, Zhang, P: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math.. 53, 1411–1433 (2000). Publisher Full Text OpenURL

  26. Wazwaz, A: Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations. Phys. Lett. A. 352, 500–504 (2006). Publisher Full Text OpenURL

  27. Wazwaz, A: New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations. Appl. Math. Comput.. 186, 130–141 (2007). Publisher Full Text OpenURL

  28. Liu, ZR, Ouyang, ZY: A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations. Phys. Lett. A. 366, 377–381 (2007). Publisher Full Text OpenURL

  29. Liu, ZR, Qian, ZF: Peakons and their bifurcation in a generalized Camassa-Holm equation. Int. J. Bifurc. Chaos. 11, 781–792 (2001). Publisher Full Text OpenURL

  30. Tian, LX, Song, XY: New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos Solitons Fractals. 21, 621–637 (2004)

  31. Liu, ZR, Guo, BL: Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation. Prog. Nat. Sci.. 18, 259–266 (2008). Publisher Full Text OpenURL

  32. Shen, JW, Xu, W: Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation. Chaos Solitons Fractals. 26, 1149–1162 (2005). Publisher Full Text OpenURL

  33. Novikov, YS: Generalizations of the Camassa-Holm equation. J. Phys. A. 42, Article ID 342002 (2009)

  34. Hone, ANW, Wang, JP: Integrable peakon equations with cubic nonlinearity. J. Phys. A, Math. Theor.. 41, Article ID 372002 (2008)

  35. Constantin, A: The trajectories of particles in Stokes waves. Invent. Math.. 166, 523–535 (2006). Publisher Full Text OpenURL

  36. Constantin, A, Escher, J: Particle trajectories in solitary water waves. Bull. Am. Math. Soc.. 44, 423–431 (2007). Publisher Full Text OpenURL

  37. Hone, ANW, Lundmark, H, Szmigielski, J: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm equation. Dyn. Partial Differ. Equ.. 6, 253–289 (2009). Publisher Full Text OpenURL

  38. Ni, LD, Zhou, Y: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ.. 250, 3002–3201 (2011). Publisher Full Text OpenURL

  39. Yan, W, Li, Y, Zhang, Y: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ.. 253, 298–318 (2012). Publisher Full Text OpenURL

  40. Grayshan, K: Peakon solutions of the Novikov equation and properties of the data-to-solution map. J. Math. Anal. Appl.. 397, 515–521 (2013). Publisher Full Text OpenURL

  41. Himonas, A, Holliman, C: The Cauchy problem for the Novikov equation. Nonlinearity. 25, 449–479 (2012). Publisher Full Text OpenURL

  42. Jiang, ZH, Ni, LD: Blow-up phenomena for the integrable Novikov equation. J. Math. Anal. Appl.. 385, 551–558 (2012). Publisher Full Text OpenURL

  43. Wu, SY, Yin, ZY: Global weak solutions for the Novikov equation. J. Phys. A, Math. Theor.. 44, Article ID 055202 (2011)

  44. Zhao, L, Zhou, S: Symbolic analysis and exact travelling wave solutions to a new modified Novikov equation. Appl. Math. Comput.. 217, 590–598 (2010). Publisher Full Text OpenURL

  45. Fokas, AS: The Korteweg-de Vries equation and beyond. Acta Appl. Math.. 39, 295–305 (1995). Publisher Full Text OpenURL

  46. Fuchssteiner, B: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation. Physica, D. 95, 229–243 (1996). Publisher Full Text OpenURL

  47. Olver, PJ, Rosenau, P: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E. 53, 1900–1906 (1996). Publisher Full Text OpenURL

  48. Qiao, ZJ: A new integrable equation with cuspons and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M369">View MathML</a>-shape-peaks solitons. J. Math. Phys.. 47, Article ID 112701 (2006)

  49. Qiao, ZJ: New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/9/mathml/M371">View MathML</a>-shape peak solitons. J. Math. Phys.. 48, Article ID 082701 (2007)

  50. Ivanov, RI, Lyons, T: Dark solitons of the Qiao’s hierarchy. J. Math. Phys.. 53, Article ID 123701 (2012)

  51. Gui, GL, Liu, Y, Olver, PL, Qu, CZ: Wave-breaking and peakons for a modified Camassa-Holm equation. Commun. Math. Phys.. 319, 731–759 (2012)

  52. Fu, Y, Gui, GL, Liu, Y, Qu, CZ: On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity. J. Differ. Equ.. 255, 1905–1938 (2013). Publisher Full Text OpenURL

  53. Schäfer-Wayne, T: Propagation of ultra-short optical pulses in cubic nonlinear media. Physica, D. 196, 90–105 (2004). Publisher Full Text OpenURL

  54. Lai, SY, Wu, YH: The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. J. Differ. Equ.. 248, 2038–2063 (2010). Publisher Full Text OpenURL

  55. Hakkaev, S, Kirchev, K: Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation. Commun. Partial Differ. Equ.. 30, 761–781 (2005). Publisher Full Text OpenURL

  56. Himonas, AA, Misiolek, G, Ponce, G, Zhou, Y: Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Commun. Math. Phys.. 271, 511–522 (2007). Publisher Full Text OpenURL

  57. Ni, LD, Zhou, Y: A new asymptotic behavior of solutions to the Camassa-Holm equation. Proc. Am. Math. Soc.. 140, 607–614 (2012). Publisher Full Text OpenURL

  58. Kato, T: Quasi-linear equations of evolution with applications to partial differential equations. Spectral Theory and Differential Equations, pp. 25–70. Springer, Berlin (1975)

  59. Mu, CL, Zhou, SM, Zeng, R: Well-posedness and blow-up phenomena for a higher order shallow water equation. J. Differ. Equ.. 251, 3488–3499 (2011). Publisher Full Text OpenURL

  60. Bona, J, Smith, R: The initial value problem for the Korteweg-de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A. 278, 555–601 (1975). Publisher Full Text OpenURL

  61. Lai, SY, Wu, YH: A model containing both the Camassa-Holm and Degasperis-Procesi equations. J. Math. Anal. Appl.. 374, 458–469 (2011). Publisher Full Text OpenURL

  62. Walter, W: Differential and Integral Inequalities, Springer, New York (1970)