The Uniqueness of Strong Solutions for the Camassa-Holm Equation

Abstract

Assume that there exists a strong solution of the Camassa-Holm equation and the initial value of the solution belongs to the Sobolev space 𝐻¹(𝑅). We provide a new proof of the uniqueness of the strong solution for the equation.

1. Introduction

The integrable Camassa-Holm model [1] 𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥 (1) has been investigated by many scholars. Equation (1) has peaked solitary wave solutions, which takes the form 𝑐𝑒^{−|𝑥−𝑐𝑡|},𝑐∈𝑅. The existence and uniqueness of the global weak solutions for (1) have been given by Constantin and Escher [2] and Constantin and Molinet [3] in which the 𝑚=𝑢₀−𝑢_0𝑥𝑥 is a positive (or negative) Radon measure. The local well-posedness of strong solutions for the Camassa-Holm model and its various generalized forms are provided in [4–8]. For the initial value 𝑢₀ satisfying 𝑢₀−𝑢_0𝑥𝑥≥0 or 𝑢₀−𝑢_0𝑥𝑥≤0, it is shown in [9] that the Camassa-Holm equation has unique global strong solutions in the Sobolev space 𝐻^𝑠(𝑅) with 𝑠>3/2. If the initial data satisfy certain conditions, we know that the local strong solutions blow up in finite time [10, 11]. It means that the slope of the solution becomes unbounded while the solution itself remains bounded. For other techniques to obtain the dynamic properties for the Camassa-Holm equation and other related shallow water equations, the reader is referred to [12–16] and the references therein.

We consider the equivalent form of the Cauchy problem for (1) 𝑢_𝑡+12𝑢²_𝑥+𝜕_𝑥𝑃_𝑢()𝑢()𝑡,𝑥=0,0,𝑥=𝑢₀, (2) where 𝑃_𝑢(𝑡,𝑥)=Λ⁻²(𝑢²+(1/2)𝑢²_𝑥)=(1/2)∫^∞_−∞𝑒^{|𝑥−𝑦|}(𝑢²(𝑡,𝑦)+(1/2)𝑢²_𝑦(𝑡,𝑦))𝑑𝑦. If (1) has a suitable smooth strong solution, we have the conservation law ^∞_−∞𝑢²(𝑡,𝑥)+𝑢²_𝑥()=𝑡,𝑥𝑑𝑥_∞−∞𝑢²(0,𝑥)+𝑢²_𝑥(0,𝑥)𝑑𝑥, (3) which derives ‖𝑢‖_𝐿^∞_(𝑅)≤‖𝑢‖_𝐻¹_(𝑅)=𝑢_0𝐻1(𝑅). (4)

The objective of this work is to give a new proof of the uniqueness for the solutions of the Camassa-Holm equation (1). Firstly, we establish the following inequality: ‖𝑢(𝑡,⋅)−𝑣(𝑡,⋅)‖_𝐿¹_(𝑅)≤𝑐𝑒^𝑐𝑡∞_−∞𝑢₀(𝑥)−𝑣₀(𝑥)𝑑𝑥+^∞_−∞𝑢²_0𝑥−𝑣²_0𝑥𝑑𝑥, (5) where 𝑡∈[0,𝑇₀), functions 𝑢 and 𝑣 are two local or global strong solutions of problem (2) with initial data 𝑢(0,⋅)=𝑢₀∈𝐻¹(𝑅) and 𝑣(0,⋅)=𝑣₀∈𝐻¹(𝑅), respectively. Constant 𝑐 depends on ‖𝑢₀‖_𝐻1(𝑅), ‖𝑣₀‖_𝐻1(𝑅), and the maximum existence time 𝑇₀. Secondly, from (5), we immediately arrive at the goal of the uniqueness. Here we state that the approach to establish (5) is the device of doubling variables which was presented in Kruzkov's paper [17].

This paper is organized as follows. Several lemmas are given in Section 2, while the proofs of the main results are established in Section 3.

2. Notations and Several Lemmas

Set 𝜉_𝑇=[0,𝑇]×𝑅 for an arbitrary 𝑇>0. The space of all infinitely differentiable functions 𝑓(𝑡,𝑥) with compact support in [0,𝑇]×𝑅 is denoted by 𝐶^∞₀(𝜉_𝑇). We define 𝜌(𝜎) as a function which is infinitely differentiable on (−∞,+∞) such that 𝜌(𝜎)≥0, 𝜌(𝜎)=0 for |𝜎|≥1 and ∫^∞_−∞𝜌(𝜎)𝑑𝜎=1. For any number ℎ>0, we let 𝜌_ℎ(𝜎)=𝜌(ℎ⁻¹𝜎)/ℎ. Then we have that 𝜌_ℎ(𝜎) is a function in 𝐶^∞(−∞,∞) and 𝜌_ℎ(𝜎)≥0,𝜌_ℎ(𝜎)|𝜎|𝜌=0if≥ℎ,_ℎ(𝜎≤𝑐)ℎ,^∞_−∞𝜌_ℎ(𝜎)=1. (6) Assume that the function 𝑣(𝑥) is locally integrable in (−∞,∞). We define the approximation of function 𝑣(𝑥) as 𝑣^ℎ(𝑥)=1ℎ^∞_−∞𝜌𝑥−𝑦ℎ𝑣𝑦𝑑𝑦,ℎ>0. (7)

We call 𝑥₀ a Lebesgue point of function 𝑣(𝑥) if lim_ℎ→01ℎ_{|𝑥−𝑥0|≤ℎ}𝑣(𝑥)−𝑣𝑥₀𝑑𝑥=0. (8) At any Lebesgue point 𝑥₀ of the function 𝑣(𝑥), we have lim_ℎ→0𝑣^ℎ(𝑥₀)=𝑣(𝑥₀). Since the set of points which are not Lebesgue points of 𝑣(𝑥) has measured zero, we get 𝑣^ℎ(𝑥)→𝑣(𝑥) as ℎ→0 almost everywhere.

We introduce notation connected with the concept of a characteristic cone. For any 𝑀₀>0, we define 𝑁>sup_{𝑡∈[0,∞)}‖𝑢‖_𝐿^∞_(𝑅)<∞. Let ℧ designate the cone {(𝑡,𝑥)∶|𝑥|≤𝑀₀−𝑁𝑡,0<𝑡<𝑇₀=min(T,𝑀₀𝑁⁻¹)}. We let 𝑆_𝜏 designate the cross section of the cone ℧ by the plane 𝑡=𝜏,𝜏∈[0,𝑇₀].

Let 𝐻_𝑟={𝑥∶|𝑥|≤𝑟}, where 𝑟>0.

Lemma 1 (see [17]). Let the function 𝑣(𝑡,𝑥) be bounded and measurable in cylinder Ω=[0,𝑇]×𝐻_𝑟. If 𝛿∈(0,min[𝑟,𝑇]) and any number ℎ∈(0,𝛿), then the function 𝑉_ℎ=1ℎ^2|()()_|^{𝑡−𝜏/2|≤ℎ,𝛿≤𝑡+𝜏/2≤𝑇−𝛿,}_{𝑥−𝑦/2≤ℎ,𝑥+𝑦/2|≤𝑟−𝛿}𝑣(𝑡,𝑥)−𝑣𝜏,𝑦𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏 (9) satisfies lim_ℎ→0𝑉_ℎ=0.

Lemma 2 (see [17]). If the function |𝜕𝐹(𝑢)/𝜕𝑢| is bounded, then the function 𝐻(𝑢,𝑣)=sign(𝑢−𝑣)(𝐹(𝑢)−𝐹(𝑣)) satisfies the Lipschitz condition in 𝑢 and 𝑣, respectively.

Lemma 3. Let 𝑢₀∈𝐻¹(𝑅). It holds that the function 𝑄_𝑢(𝑡,𝑥)=𝜕_𝑥𝑃_𝑢(𝑡,𝑥)=𝜕_𝑥Λ⁻²(𝑢²+(1/2)𝑢²_𝑥) satisfies 𝑃_𝑢(𝑡,𝑥)_{𝐿∞(𝑅)}𝑃<∞,_𝑢(𝑡,𝑥)_𝐿1(𝑅)𝑃<∞,_𝑢(𝑡,𝑥)_𝐿2(𝑅)𝑄<∞,_𝑢(𝑡,𝑥)_{𝐿∞(𝑅)}𝑄<∞,_𝑢(𝑡,𝑥)_𝐿1(𝑅)𝑄<∞,_𝑢(𝑡,𝑥)_𝐿2(𝑅)<∞. (10)

The proof of Lemma 3 can be found in [13, 15] (see [13, Lemma 5.1]).

Lemma 4. Let 𝑢 be the strong solution of problem (2), 𝑓(𝑡,𝑥)∈𝐶^∞₀(𝜉_𝑇), and 𝑓(0,𝑥)=0. Then |𝑢−𝑘|_𝜉𝑇𝑓_𝑡()1+sign𝑢−𝑘2𝑢²−𝑘²𝑓_𝑥−sign(𝑢−𝑘)𝑄_𝑢(𝑡,𝑥)𝑓𝑑𝑥𝑑𝑡=0, (11) where 𝑘 is an arbitrary constant.

Proof. Let Φ(𝑢) be an arbitrary twice smooth function on the line −∞<𝑢<∞. We multiply (2) by the function Φ^′(𝑢)𝑓(𝑡,𝑥), where 𝑓(𝑡,𝑥)∈𝐶^∞₀(𝜉_𝑇). Integrating over 𝜉_𝑇 and transferring the derivatives with respect to 𝑡 and 𝑥 to the test function 𝑓, for any constant 𝑘, we obtain _𝜉𝑇Φ(𝑢)𝑓_𝑡+^𝑢_𝑘Φ^′(𝑧)𝑧𝑑𝑧𝑓_𝑥−Φ^′(𝑢)𝑄_𝑢(𝑡,𝑥)𝑓𝑑𝑥𝑑𝑡=0, (12) in which we have used ∫^∞_−∞[∫^𝑢_𝑘Φ^′(𝑧)𝑧𝑑𝑧]𝑓_𝑥𝑑𝑥=−∫^∞_−∞[𝑓Φ^′(𝑢)𝑢𝑢_𝑥]𝑑𝑥.
We have ^∞_−∞^𝑢_𝑘Φ^′(𝑧)𝑧𝑑𝑧𝑓_𝑥=𝑑𝑥_∞−∞Φ^′(𝑢)12𝑢²−12𝑘²−12^𝑢_𝑘𝑧²−𝑘²Φ^′′(𝑧)𝑑𝑧𝑓_𝑥𝑑𝑥. (13) Let Φ^ℎ(𝑢) be an approximation of the function |𝑢−𝑘| and set Φ(𝑢)=Φ^ℎ(𝑢). Using the properties of the sign(𝑢−𝑘), (12), and (13) and sending ℎ→0, we have |𝑢−𝑘|_𝜉𝑇𝑓_𝑡()1+sign𝑢−𝑘2𝑢²−𝑘²𝑓_𝑥−sign(𝑢−𝑘)𝑄_𝑢(𝑡,𝑥)𝑓𝑑𝑥𝑑𝑡=0, (14) which completes the proof.

In fact, the proof of (11) can also be found in [17].

Lemma 5. Assume 𝑢 and 𝑣 are two strong solutions of problem (2). It has sign(^∞_−∞𝑢−𝑣)𝑄_𝑢(𝑡,𝑥)−𝑄_𝑣(𝑡,𝑥)𝑓𝑑𝑥≤𝑐^∞_−∞𝑢²₀−𝑣²₀+𝑢²_0𝑥−𝑣²_0𝑥+|𝑢−𝑣|𝑑𝑥. (15)

Proof. We have ^∞_−∞𝑢²_𝑥−𝑣²_𝑥𝑑𝑥=^∞_−∞𝑢²₀+𝑢²_0𝑥−𝑣²₀+𝑣²_0𝑥−𝑑𝑥_∞−∞𝑢²−𝑣²𝑑𝑥,_∞−∞Λ⁻²𝑢²+12𝑢²_𝑥−𝑣²−12𝑣²_𝑥×𝜕_𝑥[𝑢()(𝑓(=1sign𝑡,𝑥−𝑣𝑡,𝑥)]𝑡,𝑥)𝑑𝑥2^∞_−∞𝑒^{−|𝑥−𝑦|}𝑢²+12𝑢²_𝑦−𝑣²−12𝑣²_𝑦𝑑𝑦×𝜕_𝑥[𝑢()(𝑓(≤1sign𝑡,𝑥−𝑣𝑡,𝑥)]𝑡,𝑥)𝑑𝑥2^∞_−∞^∞_−∞𝑢²+12𝑢²_𝑦−𝑣²−12𝑣²_𝑦𝑑𝑦×𝑒^{−|𝑥−𝑦|}𝜕_𝑥[𝑢()(𝑓(≤1sign𝑡,𝑥−𝑣𝑡,𝑥)]𝑡,𝑥)𝑑𝑥2^∞_−∞𝑢²+12𝑢²_𝑦−𝑣²−12𝑣²_𝑦𝑑𝑦×^∞_−∞𝜕_𝑥sign[𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)]𝑓(𝑡,𝑥)𝑑𝑥≤𝑐^∞_−∞𝑢²+12𝑢²_𝑦−𝑣²−12𝑣²_𝑦𝑑𝑦≤𝑐^∞_−∞𝑢²₀−𝑣²₀𝑑𝑥+^∞_−∞𝑢²_0𝑥−𝑣²_0𝑥+𝑑𝑥_∞−∞|𝑢−𝑣|𝑑𝑥, (16) in which we have used the Fubini theorem, ‖𝑢‖_𝐿^∞≤‖𝑢₀‖_𝐻1(𝑅) and ‖𝑣‖_𝐿^∞≤‖𝑣₀‖_𝐻1(𝑅). The proof is completed.

3. Main Results

Theorem 6. Let 𝑢 and 𝑣 be two local or global strong solutions of problem (2) with initial data 𝑢(0,⋅)=𝑢₀∈𝐻¹(𝑅) and 𝑣(0,⋅)=𝑣₀∈𝐻¹(𝑅), respectively. Let 𝑇₀ be the maximum existence time of solutions 𝑢 and 𝑣. For any 𝑡∈[0,𝑇₀), it holds that ‖𝑢(𝑡,⋅)−𝑣(𝑡,⋅)‖_𝐿¹_(𝑅)≤𝑒^𝑐𝑡𝑐1+𝑇₀×^∞_−∞𝑢₀(𝑥)−𝑣₀(𝑥)𝑑𝑥+^∞_−∞𝑢²_0𝑥−𝑣²_0𝑥𝑑𝑥, (17) where 𝑐 depends on ‖𝑢₀‖_𝐻1(𝑅) and ‖𝑣₀‖_𝐻1(𝑅).

From Theorem 6, we immediately obtain the uniqueness result.

Theorem 7. Let 𝑢(𝑡,𝑥) be a strong solution of (1) with 𝑢₀∈𝐻¹(𝑅), and let 𝑇₀ be the maximum existence time of solution 𝑢. Then any strong solution of (1) is unique.

Proof of Theorem 6. For an arbitrary 𝑇>0, set 𝜉_𝑇=[0,𝑇]×𝑅. Let 𝑓(𝑡,𝑥)∈𝐶^∞₀(𝜉_𝑇). We assume that 𝑓(𝑡,𝑥)=0 outside some cylinder ={(𝑡,𝑥)}=[𝛿,𝑇−2𝛿]×𝐻_𝑟−2𝛿,0<2𝛿≤min(𝑇,𝑟). (18) We define 𝑔=𝑓𝑡+𝜏2,𝑥+𝑦2𝜌_ℎ𝑡−𝜏2𝜌_ℎ𝑥−𝑦2=𝑓(⋯)𝜆_ℎ(∗), (19) where (⋯)=((𝑡+𝜏)/2,(𝑥+𝑦)/2) and (∗)=((𝑡−𝜏)/2,(𝑥−𝑦)/2). The function 𝜌(𝜎) is defined in (6). Note that 𝑔_𝑡+𝑔_𝜏=𝑓_𝑡(⋯)𝜆_ℎ(∗),𝑔_𝑥+𝑔_𝑦=𝑓_𝑥(⋯)𝜆_ℎ(∗). (20)
Taking 𝑘=𝑣(𝜏,𝑦) and assuming 𝑓(𝑡,𝑥)=0 outside the cylinder ⨄, from Lemma 4, we have _{𝜉𝑇×𝜉𝑇}𝑢(𝑡,𝑥)−𝑣𝜏,𝑦𝑔_𝑡𝑢()×𝑢+sign𝑡,𝑥−𝑣𝜏,𝑦₂(𝑡,𝑥)2−𝑣²𝜏,𝑦2𝑔_𝑥+sign𝑢(𝑡,𝑥)−𝑣𝜏,𝑦×𝑄_𝑢(𝑡,𝑥)𝑔𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏=0. (21) Similarly, it has _{𝜉𝑇×𝜉𝑇}𝑣𝜏,𝑦−𝑢(𝑡,𝑥)𝑔_𝜏𝑣()×𝑣+sign𝜏,𝑦−𝑢𝑡,𝑥₂𝜏,𝑦2−𝑢²(𝑡,𝑥)2𝑔_𝑦+sign𝑣𝜏,𝑦−𝑢(𝑡,𝑥)×𝑄_𝑣𝜏,𝑦𝑔𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏=0, (22) from which we obtain 0≤_{𝜉𝑇×𝜉𝑇}𝑢(𝑡,𝑥)−𝑣𝜏,𝑦𝑔_𝑡+𝑔_𝜏𝑢()×𝑢+sign𝑡,𝑥−𝑣𝜏,𝑦₂(𝑡,𝑥)2−𝑣²𝜏,𝑦2×𝑔_𝑥+𝑔_𝑦+𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏_{𝜉𝑇×𝜉𝑇}(𝑢()(×𝑄sign𝑡,𝑥−𝑣𝑡,𝑥))_𝑢(𝑡,𝑥)−𝑄_𝑣𝜏,𝑦𝑔𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏=_{𝜉𝑇×𝜉𝑇}𝐼₁+𝐼₂+𝐼₃𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏. (23)
We will show that 0≤_𝜉𝑇|𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)|𝑓_𝑡(𝑢()(×𝑢+sign𝑡,𝑥−𝑣𝑡,𝑥))₂(𝑡,𝑥)2−𝑣²(𝑡,𝑥)2𝑓_𝑥+𝑑𝑥𝑑𝑡_𝜉𝑇(𝑢()(×𝑄sign𝑡,𝑥−𝑣𝑡,𝑥))_𝑢(𝑡,𝑥)−𝑄_𝑣(𝑡,𝑥)𝑓𝑑𝑥𝑑𝑡. (24) We note that the first two terms in the integrand of (23) can be represented in the form 𝑌_ℎ=𝑌𝑡,𝑥,𝜏,𝑦,𝑢(𝑡,𝑥),𝑣𝜏,𝑦𝜆_ℎ(∗). (25) Since ‖𝑢‖_𝐿^∞≤‖𝑢₀‖_𝐻1(𝑅) and ‖𝑣‖_𝐿^∞≤‖𝑣₀‖_𝐻1(𝑅), from Lemma 2, we know 𝑌_ℎ satisfies the Lipschitz condition in 𝑢 and 𝑣, respectively. By the choice of 𝑔, we have 𝑌_ℎ=0 outside the region 𝑡,𝑥;𝜏,𝑦=𝛿≤𝑡+𝜏2||≤𝑇−2𝛿,𝑡−𝜏2≤ℎ,𝑥+𝑦2≤𝑟−2𝛿,𝑥−𝑦2≤ℎ_{𝜉𝑇×𝜉𝑇},𝑌_ℎ=𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏_{𝜉𝑇×𝜉𝑇}𝑌()(()(𝑡,𝑥,𝜏,𝑦,𝑢𝑡,𝑥,𝑣𝜏,𝑦−𝑌𝑡,𝑥,𝑡,𝑥,𝑢𝑡,𝑥,𝑣𝑡,𝑥))×𝜆_ℎ(∗)+𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏_{𝜉𝑇×𝜉𝑇}𝑌(𝑡,𝑥,𝑡,𝑥,𝑢(𝑡,𝑥),𝑣(𝑡,𝑥))×𝜆_ℎ(∗)𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏=𝐽₁(ℎ)+𝐽₂. (26) Considering the estimate |𝜆(∗)|≤𝑐/ℎ² and the expression of function 𝑌_ℎ, we have 𝐽₁(ℎ1)≤𝑐ℎ+ℎ²×^)|₍₎^{|(𝑡−𝜏/2≤ℎ,}_{𝛿≤𝑡+𝜏/2≤𝑇−𝛿,𝑥−𝑦/2≤ℎ,𝑥+𝑦/2≤𝑟−𝛿}𝑣(),𝑡,𝑥−𝑣𝜏,𝑦𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏, (27) where the constant 𝑐 does not depend on ℎ. Using Lemma 1, we obtain 𝐽₁(ℎ)→0 as ℎ→0. The integral 𝐽₂ does not depend on ℎ. In fact, substituting 𝑡=𝛼, (𝑡−𝜏)/2=𝛽, 𝑥=𝜂, (𝑥−𝑦)/2=𝜉 and noting that ^ℎ_−ℎ^∞_−∞𝜆_ℎ𝛽,𝜉𝑑𝜉𝑑𝛽=1, (28) we have 𝐽₂=2²_𝜉𝑇𝑌_ℎ×𝛼,𝜂,𝛼,𝜂,𝑢𝛼,𝜂,𝑣𝛼,𝜂_{ℎ−ℎ∞−∞}𝜆_ℎ𝛽,𝜉𝑑𝜉𝑑𝛽𝑑𝜂𝑑𝛼=4_𝜉𝑇𝑌_ℎ(𝑡,𝑥,𝑡,𝑥,𝑢(𝑡,𝑥),𝑣(𝑡,𝑥))𝑑𝑥𝑑𝑡. (29) Hence lim_{ℎ→0𝜉𝑇×𝜉𝑇}𝑌_ℎ𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏=4_𝜉𝑇𝑌_ℎ(𝑡,𝑥,𝑡,𝑥,𝑢(𝑡,𝑥),𝑣(𝑡,𝑥))𝑑𝑥𝑑𝑡. (30) Since 𝐼₃=sign𝑢(𝑡,𝑥)−𝑣𝜏,𝑦𝑄_𝑢(𝑡,𝑥)−𝑄_𝑣𝜏,𝑦𝑓𝜆_ℎ(∗),_{𝜉𝑇×𝜉𝑇}𝐼₃=𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏_{𝜉𝑇×𝜉𝑇}𝐼₃𝑡,𝑥,𝜏,𝑦−𝐼₃(𝑡,𝑥,𝑡,𝑥)×𝜆_ℎ(∗)+𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏_{𝜉𝑇×𝜉𝑇}𝐼₃(𝑡,𝑥,𝑡,𝑥)𝜆_ℎ(∗)𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏=𝐾₁(ℎ)+𝐾₂, (31) we obtain 𝐾₁(ℎ1)≤𝑐ℎ+ℎ²×^|()_()||^{𝑡−𝜏/2|≤ℎ,}_{𝛿≤𝑡+𝜏/2≤𝑇−𝛿,𝑥−𝑦/2|≤ℎ,𝑥+𝑦/2|≤𝑟−𝛿}𝑄_𝑣(𝑡,𝑥)−𝑄_𝑣𝜏,𝑦𝑑𝑥𝑑𝑡𝑑𝑦𝑑𝜏. (32) By Lemmas 1 and 3, we have 𝐾₁(ℎ)→0 as ℎ→0. Using (28), we have 𝐾₂=2²_𝜉𝑇𝐼₃×𝛼,𝜂,𝛼,𝜂,𝑢𝛼,𝜂,𝑣𝛼,𝜂_{ℎ−ℎ∞−∞}𝜆_ℎ𝛽,𝜉𝑑𝜉𝑑𝛽𝑑𝜂𝑑𝛼=4_𝜉𝑇𝐼₃(𝑡,𝑥,𝑡,𝑥,𝑢(𝑡,𝑥),𝑣(𝑡,𝑥))𝑑𝑥𝑑𝑡=4_𝜉𝑇(𝑢()(×𝑄sign𝑡,𝑥−𝑣𝑡,𝑥))_𝑢(𝑡,𝑥)−𝑄_𝑣(𝑡,𝑥)𝑓(𝑡,𝑥)𝑑𝑥𝑑𝑡. (33) From (30) and (33), we prove that inequality (24) holds.
Let 𝜔(𝑡)=^∞_−∞|𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)|𝑑𝑥. (34) We define 𝜃_ℎ=^𝜎_−∞𝜌_ℎ(𝜎)𝑑𝜎𝜃^′_ℎ(𝜎)=𝜌_ℎ(𝜎)≥0 (35) and choose two numbers 𝜏₁ and 𝜏₂∈(0,𝑇₀), 𝜏₁<𝜏₂. In (24), we choose 𝑓=𝜃_ℎ𝑡−𝜏₁−𝜃_ℎ𝑡−𝜏₂𝜒(),𝜏𝑡,𝑥ℎ<min₁,𝑇₀−𝜏₂, (36) where 𝜒(𝑡,𝑥)=𝜒_𝜀(𝑡,𝑥)=1−𝜃_𝜀|𝑥|+𝑁𝑡−𝑀₀+𝜀,𝜀>0. (37) We note that function 𝜒(𝑡,𝑥)=0 outside the cone ℧ and 𝑓(𝑡,𝑥)=0 outside the set ⨄. For (𝑡,𝑥)∈℧, we have the relations 0=𝜒_𝑡+𝑁𝜒_𝑥≥𝜒_𝑡+𝑁𝜒_𝑥. (38) Applying (24) and (35)–(38), we have the inequality 0≤_𝜉𝑇0𝜌_ℎ𝑡−𝜏₁−𝜌_ℎ𝑡−𝜏₂×𝜒_𝜀|𝑢()(+𝑡,𝑥−𝑣𝑡,𝑥)|𝑑𝑥𝑑𝑡_𝜉𝑇0𝜃_ℎ𝑡−𝜏₁−𝜃_ℎ𝑡−𝜏₂×𝑄_𝑢(𝑡,𝑥)−𝑄_𝑣(𝑡,𝑥)𝐵(𝑡,𝑥)𝜒(𝑡,𝑥)𝑑𝑥𝑑𝑡, (39) where 𝐵(𝑡,𝑥)=sign[𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)].
From (39), we obtain 0≤_𝜉𝑇0𝜌_ℎ𝑡−𝜏₁−𝜌_ℎ𝑡−𝜏₂×𝜒_𝜀|𝑢()(+𝑡,𝑥−𝑣𝑡,𝑥)|𝑑𝑥𝑑𝑡_𝑇00𝜃_ℎ𝑡−𝜏₁−𝜃_ℎ𝑡−𝜏₂×^∞_−∞𝑄_𝑢(𝑡,𝑥)−𝑄_𝑣(𝑡,𝑥)𝐵(𝑡,𝑥)𝜒(𝑡,𝑥)𝑑𝑥𝑑𝑡. (40) Using Lemma 5, we have 0≤_𝜉𝑇0𝜌_ℎ𝑡−𝜏₁−𝜌_ℎ𝑡−𝜏₂×𝜒_𝜀|𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)|𝑑𝑥𝑑𝑡+𝑐^𝑇0₀𝜃_ℎ𝑡−𝜏₁−𝜃_ℎ𝑡−𝜏₂×𝐺+^∞_−∞|𝑢−𝑣|𝑑𝑥𝑑𝑡, (41) where 𝐺=∫^∞_−∞|𝑢²₀−𝑣²₀|𝑑𝑥+∫^∞_−∞|𝑢²_0𝑥−𝑣²_0𝑥|𝑑𝑥 and 𝑐 is defined in Lemma 5.
Letting 𝜀→0 in (41) and sending 𝑀₀→∞, we have 0≤^𝑇0₀𝜌_ℎ𝑡−𝜏₁−𝜌_ℎ𝑡−𝜏₂×^∞_−∞|𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)|𝑑𝑥𝑑𝑡+𝑐^𝑇0₀𝜃_ℎ𝑡−𝜏₁−𝜃_ℎ𝑡−𝜏₂×𝐺+^∞_−∞|𝑢−𝑣|𝑑𝑥𝑑𝑡. (42)
By the properties of the function 𝜌_ℎ(𝜎) for ℎ≤min(𝜏₁,𝑇₀−𝜏₁), we have ^𝑇0₀𝜌_ℎ𝑡−𝜏₁𝜔(𝑡)𝑑𝑡−𝜔𝜏₁=^𝑇0₀𝜌_ℎ𝑡−𝜏₁𝜔(𝑡)−𝜔𝜏₁1𝑑𝑡≤𝑐ℎ^𝜏1_𝜏^+ℎ_1−ℎ𝜔(𝑡)−𝜔𝜏₁𝑑𝑡⟶0asℎ⟶0, (43) where 𝑐 is independent of ℎ.
Set 𝐿𝜏₁=^𝑇0₀𝜃_ℎ𝑡−𝜏₁𝜇(𝑡)𝑑𝑡=^𝑇0₀^𝑡−𝜏1_−∞𝜌_ℎ(𝜎)𝑑𝜎𝜔(𝑡)𝑑𝑡. (44) Using the similar proof of (43), we get 𝐿^′𝜏₁=−^𝑇0₀𝜌_ℎ𝑡−𝜏₁𝜔(𝑡)𝑑𝑡⟶−𝜔𝜏₁asℎ⟶0, (45) from which we obtain 𝐿𝜏₁⟶𝐿(0)−^𝜏1₀𝜔(𝜎)𝑑𝜎asℎ⟶0. (46) Similarly, we have 𝐿𝜏₂⟶𝐿(0)−^𝜏2₀𝜔(𝜎)𝑑𝜎asℎ⟶0. (47) Then, we get 𝐿𝜏₁−𝐿𝜏₂⟶^𝜏2_𝜏1𝜔(𝜎)𝑑𝜎asℎ⟶0. (48) Furthermore, if ℎ→0, we have ^𝑇0₀𝜃_ℎ𝑡−𝜏₁−𝜃_ℎ𝑡−𝜏₂𝐺𝑑𝑡⟶𝐺^𝜏2_𝜏1=𝑑𝑡_∞−∞𝑢²₀−𝑣²₀𝑑𝑥+^∞_−∞𝑢²_0𝑥−𝑣²_0𝑥𝑑𝑥𝜏₂−𝜏₁. (49)
Let 𝜏₁→0 and 𝜏₂→𝑡, and note that 𝑢𝜏₁,𝑥−𝑣𝜏₁,𝑥≤𝑢𝜏₁,𝑥−𝑢₀(𝑥+𝑣𝜏₁),𝑥−𝑣₀(𝑥+𝑢₀)(𝑥)−𝑣₀(𝑥). (50)
Thus, from (42), (43), (48), (49), and (50), for any 𝑡∈[0,𝑇₀], we have ^∞_−∞|𝑢()(≤𝑡,𝑥−𝑣𝑡,𝑥)|𝑑𝑥_∞−∞|𝑢(0,𝑥)−𝑣(0,𝑥)|𝑑𝑥+𝑐𝑇₀^∞_−∞𝑢²₀−𝑣²₀𝑑𝑥+^∞_−∞𝑢²_0𝑥−𝑣²_0𝑥+𝑑𝑥_{𝑡0∞−∞}|𝑢−𝑣|𝑑𝑥, (51) from which we complete the proof of Theorem 6 by using the Gronwall inequality.

Acknowledgments

Thanks are given to referees whose suggestions are very helpful to the paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).