On the Study of Local Solutions for a Generalized Camassa-Holm Equation

Abstract

The pseudoparabolic regularization technique is employed to study the local well-posedness of strong solutions for a nonlinear dispersive model, which includes the famous Camassa-Holm equation. The local well-posedness is established in the Sobolev space 𝐻^𝑠(𝑅) with 𝑠>3/2 via a limiting procedure.

1. Introduction

In recent years, extensive research has been carried out worldwide to study highly nonlinear equations including the Camassa-Holm (CH) equation and its various generalizations [1–6]. It is shown in [7–9] that the inverse spectral or scattering approach is a powerful technique to handle the Camassa-Holm equation and analyze its dynamics. It is pointed out in [10–12] that the CH equation gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group, and this geometric illustration leads to a proof that the Least Action Principle holds. Li and Olver [13] established the local well-posedness to the CH model in the Sobolev space 𝐻^𝑠(𝑅) with 𝑠>3/2 and gave conditions on the initial data that lead to finite time blow-up of certain solutions. Constantin and Escher [14] proved that the blow-up occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. Hakkaev and Kirchev [15] investigated a generalized form of the Camassa-Holm equation with high order nonlinear terms and obtained the orbit stability of the traveling wave solutions under certain assumptions. Lai and Wu [16] discussed a generalized Camassa-Holm model and acquired its local existence and uniqueness. Recently, Li et al. [17] investigated the generalized Camassa-Holm equation 𝑢_𝑡−𝑢_𝑡𝑥𝑥+𝑘𝑢^𝑚𝑢_𝑥+(𝑚+3)𝑢^𝑚+1𝑢_𝑥=(𝑚+2)𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝑢^𝑚+1𝑢_𝑥𝑥𝑥,(1.1) where 𝑚≥0 is a natural number and 𝑘≥0. The authors in [17] assume that the initial value satisfies the sign condition and establish the global existence of solutions for (1.1).

In this paper, we will study the following generalization of (1.1): 𝑢_𝑡−𝑢_𝑡𝑥𝑥+𝑘𝑢^𝑚𝑢_𝑥+(𝑚+3)𝑢^𝑚+1𝑢_𝑥=(𝑚+2)𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝑢^𝑚+1𝑢_𝑥𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥,(1.2) where 𝑚≥0 is a natural number, 𝑘≥0, and 𝜆 is a constant.

The objective of this paper is to study the local well-posedness of (1.2). Its local well-posedness of strong solutions in the Sobolev space 𝐻^𝑠(𝑅) with 𝑠>3/2 is investigated by using the pseudoparabolic regularization method. Comparing with the work by Li et al. [17], (1.2) considered in this paper possesses a conservation law different to that in [17] (see Lemma 3.2 in Section 3). Also (1.2) contains a dissipative term 𝜆(𝑢−𝑢_𝑥𝑥), which causes difficulty to establish its local and global existence in the Sobolev space. It should be mentioned that the existence and uniqueness of local strong solutions for the generalized nonlinear Camassa-Holm models like (1.2) have never been investigated in the literatures.

The organization of this work is as follows. The main result is given in Section 2. Section 3 establishes several lemmas, and the last section gives the proof of the main result.

2. Main Result

Firstly, we introduce several notations.

𝐿^𝑝=𝐿^𝑝(𝑅)(1≤𝑝<+∞) is the space of all measurable functions ℎ such that ‖ℎ‖^𝑝𝐿_𝑝=∫_𝑅|ℎ(𝑡,𝑥)|^𝑝𝑑𝑥<∞. We define 𝐿^∞=𝐿^∞(𝑅) with the standard norm ‖ℎ‖_𝐿^∞=inf_𝑚(𝑒)=0sup_{𝑥∈𝑅⧵𝑒}|ℎ(𝑡,𝑥)|. For any real number 𝑠, 𝐻^𝑠=𝐻^𝑠(𝑅) denotes the Sobolev space with the norm defined by ‖ℎ‖_𝐻^𝑠=_𝑅1+|𝜉|²^𝑠|||ℎ(𝑡,𝜉)|||²𝑑𝜉^1/2<∞,(2.1) where ℎ(𝑡,𝜉)=∫_𝑅𝑒^{−𝑖𝑥𝜉}ℎ(𝑡,𝑥)𝑑𝑥.

For 𝑇>0 and nonnegative number 𝑠, 𝐶([0,𝑇);𝐻^𝑠(𝑅)) denotes the Frechet space of all continuous 𝐻^𝑠-valued functions on [0,𝑇). We set Λ=(1−𝜕^2𝑥)^1/2. For simplicity, throughout this paper, we let 𝑐 denote any positive constant that is independent of parameter 𝜀.

We consider the Cauchy problem of (1.2) 𝑢_𝑡−𝑢_𝑡𝑥𝑥=−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2𝑢^𝑚𝑢𝑢^2𝑥(0+,𝑥𝑢)^𝑚=𝑢_𝑥𝑢𝑢_𝑥𝑥0+𝜆𝑢−𝑢_𝑥𝑥,𝑘≥0,𝑚≥0,(𝑥).(2.2)

Now, we give our main results for problem of (2.2).

Theorem 2.1. Suppose that the initial function 𝑢₀(𝑥) belongs to the Sobolev space 𝐻^𝑠(𝑅) with 𝑠>3/2 and 𝜆 is a constant. Then, there is a 𝑇>0, which depends on ‖𝑢₀‖_𝐻𝑠, such that problem (2.2) has a unique solution 𝑢(𝑡,𝑥) satisfying 𝑢(𝑡,𝑥)∈𝐶([0,𝑇);𝐻^𝑠(𝑅))𝐶¹[0,𝑇);𝐻^𝑠−1(𝑅).(2.3)

3. Local Well-Posedness

In order to prove Theorem 2.1, we consider the associated regularized problem 𝑢_𝑡−𝑢_𝑡𝑥𝑥+𝜀𝑢_{𝑡𝑥𝑥𝑥𝑥}=−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2𝑢(0,𝑥)𝑢=^𝑚𝑢𝑢^2𝑥₀+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥,(𝑥),(3.1) where the parameter 𝜀 satisfies 0<𝜀<1/4.

Lemma 3.1. For 𝑠≥1 and 𝑓(𝑥)∈𝐻^𝑠(𝑅) and letting 𝑘₁>0 be an integer such that 𝑘₁≤𝑠−1, 𝑓,𝑓′,…,𝑓^𝑘1 are uniformly continuous bounded functions that converge to 0 at 𝑥=±∞.

The proof of Lemma 3.1 was stated on page 559 by Bona and Smith [18].

Lemma 3.2. If 𝑢(𝑡,𝑥)∈𝐻^𝑠(𝑠>7/2) is a solution to problem (3.1), it holds that _𝑅𝑢²+𝑢^2𝑥+𝜀𝑢^2𝑥𝑥𝑑𝑥=_𝑅𝑢²⁰+𝑢^20𝑥+𝜀𝑢^20𝑥𝑥𝑑𝑥+2𝜆^𝑡0_𝑅𝑢²+𝑢^2𝑥𝑑𝑥.(3.2)

Proof. Using Lemma 3.1, we have 𝑢(𝑡,±∞)=𝑢_𝑥(𝑡,±∞)=𝑢_𝑥𝑥(𝑡,±∞)=𝑢_𝑥𝑥𝑥(𝑡,±∞)=0. The integration by parts results in _𝑅𝑢^𝑚+2𝑢_𝑥𝑥𝑥𝑑𝑥==−_𝑅(𝑚𝑢+^𝑚+22𝑑)𝑢_𝑥𝑥𝑅=𝑢^𝑚+2𝑢_𝑥𝑥∣^+∞−∞−(𝑚+2)_𝑅𝑢^𝑚+1𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥𝑢^𝑚+1𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥.(3.3)
Direct calculation and integration by parts give rise to 12𝑑𝑑𝑡_𝑅𝑢²=+𝑢^2𝑥+𝜀𝑢^2𝑥𝑥𝑑𝑥_𝑅=𝑢𝑢_𝑡+𝑢_𝑥𝑢_𝑡𝑥+𝜀𝑢_𝑥𝑥𝑢_𝑡𝑥𝑥𝑑𝑥_𝑅=𝑢𝑢_𝑡−𝑢_𝑡𝑥𝑥+𝜀𝑢_{𝑡𝑥𝑥𝑥𝑥}𝑑𝑥_𝑅𝑢−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2=𝑢^𝑚𝑢^2𝑥+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥𝑑𝑥_𝑅=𝑢(𝑚+2)𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝑢^𝑚+1𝑢_𝑥𝑥𝑥𝑑𝑥+𝜆_𝑅𝑢²−𝑢𝑢_𝑥𝑥𝑑𝑥_𝑅=𝜆_𝑅(𝑚+2)𝑢^𝑚+1𝑢_𝑥𝑢_𝑥𝑥+𝑢^𝑚+2𝑢_𝑥𝑥𝑥𝑑𝑥+𝜆_𝑅𝑢²+𝑢^2𝑥𝑑𝑥𝑢²+𝑢^2𝑥𝑑𝑥,(3.4) in which we have used (3.3). From (3.4), we obtain the conservation law (3.2).

Lemma 3.3. Let 𝑠≥7/2. The function 𝑢(𝑡,𝑥) is a solution of problem (3.1) and the initial value 𝑢₀(𝑥)∈𝐻^𝑠. Then, the following inequality holds: ‖‖𝑢𝑢‖‖^2𝐻_12𝐻1≤_𝑅𝑢²⁰+𝑢^20𝑥+𝜀𝑢^20𝑥𝑥𝑑𝑥,𝑖𝑓𝜆≤0,≤𝑒^2𝜆𝑡_𝑅𝑢²⁰+𝑢^20𝑥+𝜀𝑢^20𝑥𝑥𝑑𝑥,𝑖𝑓𝜆>0.(3.5) For 𝑞∈(0,𝑠−1], there is a constant 𝑐 independent of 𝜀 such that _𝑅Λ^𝑞+1𝑢²𝑑𝑥≤+_𝑅𝑐_𝑡0Λ^𝑞+1𝑢₀²+𝜀Λ^𝑞𝑢_0𝑥𝑥²𝑑𝑥‖𝑢‖^2𝐻_𝑞+1||𝜆||+‖𝑢‖^{𝐿𝑚−1∞}+‖𝑢‖^𝑚𝐿_∞‖‖𝑢_𝑥‖‖_𝐿∞+‖𝑢‖^𝐿𝑚−1_∞‖‖𝑢_𝑥‖‖^2𝐿_∞𝑑𝜏.(3.6) For 𝑞∈[0,𝑠−1], there is a constant 𝑐 independent of 𝜀 such that (1−2𝜀)‖‖𝑢_𝑡‖‖_𝐻𝑞≤𝑐‖𝑢‖_𝐻^𝑞+1||𝜆||+‖𝑢‖^{𝐿𝑚−1∞}+‖𝑢‖^𝑚𝐿_∞‖𝑢‖_𝐻¹+‖𝑢‖^𝑚𝐿_∞‖‖𝑢_𝑥‖‖_𝐿∞+‖𝑢‖^𝐿𝑚−1_∞‖‖𝑢_𝑥‖‖^2𝐿_∞.(3.7)

The proof of this lemma is similar to that of Lemma  3.5 in [17]. Here we omit it.

Lemma 3.4. Let 𝑟 and 𝑞 be real numbers such that −𝑟<𝑞≤𝑟. Then, ‖𝑢𝑣‖_𝐻^𝑞≤𝑐‖𝑢‖_𝐻^𝑟‖𝑣‖_𝐻^𝑞,𝑖𝑓𝑟>1‖𝑢𝑣‖_{𝐻𝑟+𝑞−1/2}2,≤𝑐‖𝑢‖_𝐻^𝑟‖𝑣‖_𝐻^𝑞,𝑖𝑓𝑟<12.(3.8)

This lemma can be found in [19] or [20].

Lemma 3.5. Let 𝑢₀(𝑥)∈𝐻^𝑠(𝑅) with 𝑠>3/2. Then, the Cauchy problem (3.1) has a unique solution 𝑢(𝑡,𝑥)∈𝐶([0,𝑇];𝐻^𝑠(𝑅)), where 𝑇>0 depends on ‖𝑢₀‖_{𝐻𝑠(𝑅)}. If 𝑠≥7/2, the solution 𝑢∈𝐶([0,+∞);𝐻^𝑠) exists for all time.

Proof. Letting 𝐷=(1−𝜕^2𝑥+𝜀𝜕^4𝑥)⁻¹, we know that 𝐷∶𝐻^𝑠→𝐻^𝑠+4 is a bounded linear operator. Applying the operator 𝐷 on both sides of the first equation of system (3.1) and then integrating the resultant equation with respect to 𝑡 over the interval (0,𝑡), we get 𝑢(𝑡,𝑥)=𝑢₀(𝑥)+^𝑡0𝐷−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2𝑢^𝑚𝑢^2𝑥+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥𝑑𝑡.(3.9) Suppose that both 𝑢 and 𝑣 are in the closed ball 𝐵_𝑀0(0) of radius 𝑀₀ about the zero function in 𝐶([0,𝑇];𝐻^𝑠(𝑅)) and 𝐴 is the operator in the right-hand side of (3.9). For any fixed 𝑡∈[0,𝑇], we obtain ‖‖‖^𝑡0𝐷−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2−𝑢^𝑚𝑢^2𝑥+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥𝑑𝑡_𝑡0𝐷−𝑘𝑚+1𝑣^𝑚+1_𝑥−𝑚+3𝑚+2𝑣^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑣^𝑚+2𝑣^𝑚𝑣^2𝑥+𝑣^𝑚𝑣_𝑥𝑣_𝑥𝑥+𝜆𝑣−𝑣_𝑥𝑥𝑑𝑡‖‖‖_𝐻𝑠≤𝑇𝐶₁sup_{0≤𝑡≤𝑇}‖𝑢−𝑣‖_𝐻^𝑠+sup_{0≤𝑡≤𝑇}‖‖𝑢^𝑚+1−𝑣^𝑚+1‖‖_𝐻𝑠+sup_{0≤𝑡≤𝑇}‖‖𝑢^𝑚+2−𝑣^𝑚+2‖‖_𝐻𝑠+sup_{0≤𝑡≤𝑇}‖‖𝐷𝜕_𝑥𝑢^𝑚𝑢^2𝑥−𝑣^𝑚𝑣^2𝑥‖‖_𝐻^𝑠+sup_{0≤𝑡≤𝑇}‖‖𝐷𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥−𝑣^𝑚𝑣_𝑥𝑣_𝑥𝑥‖‖_𝐻^𝑠,(3.10) where 𝐶₁ may depend on 𝜀. The algebraic property of 𝐻^𝑠0(𝑅) with 𝑠₀>1/2 derives ‖‖𝑢^𝑚+2−𝑣^𝑚+2‖‖_𝐻𝑠=‖‖(𝑢−𝑣)𝑢^𝑚+1+𝑢^𝑚𝑣+⋯+𝑢𝑣^𝑚+𝑣^𝑚+1‖‖_𝐻^𝑠≤‖(𝑢−𝑣)‖_𝐻^𝑠𝑚+1_𝑗=0≤𝑀_0𝑚+1‖𝑢‖^{𝐻𝑚+1−𝑗}_𝑠‖𝑣‖^𝑗𝐻_𝑠,‖‖𝑢_𝑚+1‖𝑢−𝑣‖_𝐻^𝑠,‖‖𝐷𝜕_𝑥−𝑣^𝑚+1‖‖_𝐻𝑠≤𝑀^𝑚0‖𝑢−𝑣‖_𝐻^𝑠,𝑢^𝑚𝑢^2𝑥−𝑣^𝑚𝑣^2𝑥‖‖_𝐻^𝑠≤‖‖𝐷𝜕_𝑥𝑢^𝑚𝑢^2𝑥−𝑣^2𝑥‖‖_𝐻^𝑠+‖‖𝐷𝜕_𝑥𝑣^2𝑥(𝑢^𝑚−𝑣^𝑚)‖‖_𝐻𝑠≤≤𝐶𝐶𝑀‖‖𝑢^𝑚_0𝑚+1𝑢^2𝑥−𝑣^2𝑥‖‖_𝐻^𝑠−1+‖‖𝑣^2𝑥(𝑢^𝑚−𝑣^𝑚)‖‖_𝐻𝑠−1‖𝑢−𝑣‖_𝐻^𝑠.(3.11) Using the first inequality of Lemma 3.4 gives rise to ‖‖𝐷𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥−𝑣^𝑚𝑣_𝑥𝑣_𝑥𝑥‖‖_𝐻^𝑠=‖‖‖12𝐷𝑢^𝑚𝑢^2𝑥_𝑥−𝑣^𝑚𝑣^2𝑥_𝑥‖‖‖_𝐻𝑠≤12‖‖𝐷𝑢^𝑚𝑢^2𝑥≤𝐶‖‖𝑢_𝑚−𝑣^2𝑥_𝑥‖‖_𝐻^𝑠+‖‖𝐷𝑣^2𝑥_𝑥(𝑢^𝑚−𝑣^𝑚)‖‖_𝐻𝑠≤𝐶‖𝑢_𝑚𝑢^2𝑥−𝑣^2𝑥_𝑥‖‖_𝐻𝑠−2+‖‖𝑣^2𝑥_𝑥(𝑢^𝑚−𝑣^𝑚)‖‖_𝐻𝑠−2≤𝐶𝑀_0𝑚+1‖_𝐻^𝑠‖‖𝑢^2𝑥−𝑣^2𝑥‖‖_𝐻𝑠−1+‖‖𝑣^2𝑥‖‖_𝐻𝑠−1‖𝑢^𝑚−𝑣^𝑚‖_𝐻^𝑠‖𝑢−𝑣‖_𝐻^𝑠,(3.12) where 𝐶 may depend on 𝜀. From (3.11)-(3.12), we obtain ‖𝐴𝑢−𝐴𝑣‖_𝐻^𝑠≤𝜃‖𝑢−𝑣‖_𝐻^𝑠,(3.13) where 𝜃=𝑇𝐶₂(𝑀^𝑚0+𝑀^0𝑚+1) and 𝐶₂ is independent of 0<𝑡<𝑇. Choosing 𝑇 sufficiently small such that 𝜃<1, we know that 𝐴 is a contraction. Similarly, it follows from (3.10) that ‖𝐴𝑢‖_𝐻^𝑠≤‖‖𝑢₀‖‖_𝐻𝑠+𝜃‖𝑢‖_𝐻^𝑠.(3.14) Choosing 𝑇 sufficiently small such that 𝜃𝑀₀+‖𝑢₀‖_𝐻𝑠<𝑀₀, we deduce that 𝐴 maps 𝐵_𝑀0(0) to itself. It follows from the contraction-mapping principle that the mapping 𝐴 has a unique fixed point 𝑢 in 𝐵_𝑀0(0). It completes the proof.

From the above and Lemma 3.2, we have _𝑅𝑢²+𝑢^2𝑥+𝜀𝑢^2𝑥𝑥𝑑𝑥≤𝑒^2|𝜆|𝑡_𝑅𝑢²⁰+𝑢^20𝑥+𝜀𝑢^20𝑥𝑥𝑑𝑥.(3.15) Therefore, ‖‖𝑢_𝑥‖‖_𝐿∞≤𝐶_𝜀𝑒^2|𝜆|𝑡_𝑅𝑢²⁰+𝑢^20𝑥+𝜀𝑢^20𝑥𝑥𝑑𝑥,(3.16) which together with Lemma 3.3 completes the proof of the global existence.

Setting 𝜙_𝜀(𝑥)=𝜀^−1/4𝜙(𝜀^−1/4𝑥) with 0<𝜀<1/4 and 𝑢_𝜀0=𝜙_𝜀⋆𝑢₀, we know that 𝑢_𝜀0∈𝐶^∞ for any 𝑢₀∈𝐻^𝑠,𝑠>0. From Lemma 3.5, it derives that the Cauchy problem 𝑢_𝑡−𝑢_𝑡𝑥𝑥+𝜀𝑢_{𝑡𝑥𝑥𝑥𝑥}=−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2𝑢(0,𝑥)=𝑢𝑢^𝑚_𝜀0𝑢^2𝑥+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥,(𝑥),𝑥∈𝑅,(3.17) has a unique solution 𝑢_𝜀(𝑡,𝑥)∈𝐶^∞([0,∞);𝐻^∞).

Furthermore, we have the following.

Lemma 3.6. For 𝑠>0,𝑢₀∈𝐻^𝑠, it holds that ‖‖‖‖𝑢𝑢_𝜀0𝑥‖‖_𝐿∞≤𝑐‖‖𝑢_0𝑥‖‖_𝐿∞,(3.18)_𝜀0‖‖𝑢‖‖_𝐻𝑞≤𝑐,𝑖𝑓𝑞≤𝑠,(3.19)_𝜀0‖‖𝑢‖‖_𝐻𝑞≤𝑐𝜀^{(𝑠−𝑞)/4},𝑖𝑓𝑞>𝑠,(3.20)_𝜀0−𝑢‖‖₀𝑢‖‖_𝐻𝑞≤𝑐𝜀^{(𝑠−𝑞)/4},𝑖𝑓𝑞≤𝑠,(3.21)_𝜀0−𝑢₀‖‖_𝐻𝑠=𝑜(1),(3.22) where 𝑐 is a constant independent of 𝜀.

The proof of Lemma 3.6 can be found in [16].

Remark 3.7. For 𝑠≥1, using ‖𝑢_𝜀‖_𝐿∞≤𝑐‖𝑢_𝜀‖_𝐻1/2+≤𝑐‖𝑢_𝜀‖_𝐻1, ‖𝑢_𝜀‖^2𝐻₁≤𝑐∫_𝑅(𝑢^2𝜀+𝑢^2𝜀𝑥)𝑑𝑥, (3.5), (3.19), and (3.20), we know that, ‖‖𝑢_𝜀‖‖^2𝐿_∞≤𝑐‖‖𝑢_𝜀‖‖^2𝐻₁≤𝑐𝑒^2|𝜆|𝑡_𝑅𝑢^2𝜀0≤𝑐𝑒_2|𝜆|𝑡+𝑢^2𝜀0𝑥+𝜀𝑢^{2𝜀0𝑥𝑥}𝑑𝑥≤𝑐𝑒_|2𝜆|𝑡‖‖𝑢^𝜀0‖‖^2𝐻₁+𝜀‖‖𝑢_𝜀0‖‖^2𝐻₂≤𝑐₀𝑐+𝑐𝜀×𝜀^(𝑠−2)/2𝑒^2|𝜆|𝑡,(3.23) where 𝑐₀ is independent of 𝜀 and 𝑡.

Lemma 3.8. Suppose 𝑢₀(𝑥)∈𝐻^𝑠(𝑅) with 𝑠≥1 such that ‖𝑢_0𝑥‖_𝐿∞<∞. Let 𝑢_𝜀0 be defined as in system (3.17). Then, there exist two positive constants 𝑇 and 𝑐, which are independent of 𝜀, such that the solution 𝑢_𝜀 of problem (3.17) satisfies ‖𝑢_𝜀𝑥‖_𝐿∞≤𝑐 for any 𝑡∈[0,𝑇).

Here we omit the proof of Lemma 3.8 since it is similar to Lemma  3.9 presented in [17].

Lemma 3.9 (see Li and Olver [13]). If 𝑢 and 𝑓 are functions in 𝐻^𝑞+1∩{‖𝑢_𝑥‖_𝐿∞<∞}, then ||||_𝑅Λ^𝑞𝑢Λ^𝑞(𝑢𝑓)_𝑥𝑑𝑥||||≤⎧⎪⎪⎨⎪⎪⎩𝑐^𝑞‖𝑓‖_𝐻^𝑞+1‖𝑢‖^2𝐻_𝑞,𝑞∈1𝑐2,1_𝑞,‖𝑓‖_𝐻^𝑞+1‖𝑢‖_𝐻^𝑞‖𝑢‖_𝐿∞𝑞+∈‖‖𝑓_𝑥(0‖‖,∞^𝐿_∞‖)𝑢.‖^2𝐻_𝑞+‖𝑓‖_𝐻^𝑞‖𝑢‖_𝐻^𝑞‖‖𝑢_𝑥‖‖_𝐿∞,(3.24)

Lemma 3.10 (see Lai and Wu [16]). For 𝑢,𝑣∈𝐻^𝑠(𝑅) with 𝑠>3/2, 𝑤=𝑢−𝑣,𝑞>1/2, and a natural number 𝑛, it holds that ||||_𝑅Λ^𝑠𝑤Λ^𝑠𝑢^𝑛+1−𝑣^𝑛+1_𝑥𝑑𝑥||||≤𝑐‖𝑤‖_𝐻^𝑠‖𝑤‖_𝐻^𝑞‖𝑣‖_𝐻^𝑠+1+‖𝑤‖^2𝐻_𝑠.(3.25)

Lemma 3.11 (see Lai and Wu [16]). If 1/2<𝑞<min{1,𝑠−1} and 𝑠>3/2, then for any functions 𝑤,𝑓 defined on 𝑅, it holds that ||||_𝑅||||Λ^𝑞𝑤Λ^𝑞−2(𝑤𝑓)_𝑥𝑑𝑥||||≤𝑐‖𝑤‖^2𝐻_𝑞‖𝑓‖_𝐻^𝑞,(3.26)_𝑅Λ^𝑞𝑤Λ^𝑞−2𝑤_𝑥𝑓_𝑥_𝑥𝑑𝑥||||≤𝑐‖𝑤‖^2𝐻_𝑞‖𝑓‖_𝐻^𝑠.(3.27)

Lemma 3.12. For problem (3.17), 𝑠>3/2, and 𝑢₀∈𝐻^𝑠(𝑅), there exist two positive constants 𝑐 and 𝑀, which are independent of 𝜀, such that the following inequalities hold for any sufficiently small 𝜀 and 𝑡∈[0,𝑇): ‖‖𝑢‖‖_𝜀𝑢_𝜀‖‖_𝐻𝑠≤𝑀𝑒^𝑐𝑡,‖‖𝑢‖‖_{𝐻1𝑠+𝑘}≤𝜀^−𝑘1/4𝑀𝑒^𝑐𝑡,𝑘₁>0,_𝜀𝑡‖‖_{𝐻1𝑠+𝑘}≤𝜀^{−(𝑘1+1)/4}𝑀𝑒^𝑐𝑡,𝑘₁>−1.(3.28)

Slightly modifying the methods presented in [16] can complete the proof of Lemma 3.12.

Our next step is to demonstrate that 𝑢_𝜀 is a Cauchy sequence. Let 𝑢_𝜀 and 𝑢_𝛿 be solutions of problem (3.17), corresponding to the parameters 𝜀 and 𝛿, respectively, with 0<𝜀<𝛿<1/4, and let 𝑤=𝑢_𝜀−𝑢_𝛿. Then, 𝑤 satisfies the problem (1−𝜀)=𝑤_𝑡1−−𝜀𝜕𝑤_{𝑥𝑥𝑡2𝑥}+(𝛿−𝜀)𝑢_𝛿𝑡+𝑢_{𝛿𝑥𝑥𝑡}−⁻¹𝜕−_𝑥𝜀𝑤^𝑡+(𝛿−𝜀)𝑢_𝛿𝑡−𝑚𝑘+1𝜕^𝑥𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1−𝜕_𝑥𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2+𝑢_𝑚𝜀𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤+𝜕_𝑥𝑢^𝜀𝑚+1−𝑢^𝜀𝑚+1𝜕_𝑥𝑢_𝛿𝑢_𝜀𝑥𝑢_𝜀𝑥𝑥−𝑢^𝑚𝛿𝑢_𝛿𝑥𝑢_𝛿𝑥𝑥−1𝑚+2𝜕^𝑥𝑢^𝜀𝑚+2𝑤(𝑥,0)=𝑤₀−𝑢^𝛿𝑚+2+𝜆𝑤,(3.29)(𝑥)=𝑢_𝜀0(𝑥)−𝑢_𝛿0(𝑥).(3.30)

Lemma 3.13. For 𝑠>3/2, 𝑢₀∈𝐻^𝑠(𝑅), there exists 𝑇>0 such that the solution 𝑢_𝜀 of (3.17) is a Cauchy sequence in 𝐶([0,𝑇];𝐻^𝑠(𝑅))⋂𝐶¹([0,𝑇];𝐻^𝑠−1(𝑅)).

Proof. For 𝑞 with 1/2<𝑞<min{1,𝑠−1}, multiplying both sides of (3.29) by Λ^𝑞𝑤Λ^𝑞 and then integrating with respect to 𝑥 give rise to 12𝑑𝑑𝑡_𝑅(1−𝜀)(Λ^𝑞=(𝜀−𝛿)_𝑅𝑤)²+𝜀Λ^𝑞𝑤_𝑥²𝑑𝑥+(𝛿−𝜀)_𝑅(Λ^𝑞𝑤)Λ^𝑞𝑢_𝛿𝑡+Λ^𝑞𝑢_{𝛿𝑥𝑥𝑡}𝑑𝑥−𝜀_𝑅Λ^𝑞𝑤Λ^𝑞−2𝑤_𝑡𝑑𝑥Λ^𝑞𝑤Λ^𝑞−2𝑢_𝛿𝑡𝑑𝑥−1𝑚+2_𝑅−𝑘(Λ^𝑞𝑤)Λ^𝑞𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥𝑚+1_𝑅−Λ^𝑞𝑤Λ^𝑞−2𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1_𝑥𝑑𝑥−_𝑅Λ^𝑞𝑤Λ^𝑞−2𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥_𝑅+Λ^𝑞𝑤Λ^𝑞−2𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤_𝑥𝑑𝑥−_𝑅Λ^𝑞𝑤Λ^𝑞−2𝜕_𝑥𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1𝜕_𝑥𝑢_𝛿_𝑥𝑑𝑥_𝑅Λ^𝑞𝑤Λ^𝑞−2𝑢^𝑚𝜀𝑢_𝜀𝑥𝑢_𝜀𝑥𝑥−𝑢^𝑚𝛿𝑢_𝛿𝑥𝑢_𝛿𝑥𝑥𝑑𝑥+𝜆_𝑅Λ^𝑞𝑤Λ^𝑞𝑤𝑑𝑥.(3.31) It follows from the Schwarz inequality that 𝑑𝑑𝑡(1−𝜀)(Λ^𝑞≤𝑐‖Λ_𝑞𝑤)²+𝜀Λ^𝑞𝑤_𝑥²𝑑𝑥+||𝜆||𝑤‖_𝐿²_𝑅(𝛿−𝜀)‖‖Λ^𝑞𝑢_𝛿𝑡‖‖_𝐿2+‖‖Λ^𝑞𝑢_{𝛿𝑥𝑥𝑡}‖‖_𝐿2+𝜀‖‖Λ^𝑞−2𝑤_𝑡‖‖_𝐿2+(𝛿−𝜀)‖‖Λ^𝑞−2𝑢_𝛿𝑡‖‖_𝐿2||||Λ(Λ^𝑞𝑤)²𝑑𝑥+||||_𝑅Λ^𝑞𝑤Λ^𝑞𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥_𝑞||||+||||𝑤Λ^𝑞−2𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1_𝑥𝑑𝑥||||+||||Λ^𝑞𝑤Λ^𝑞−2𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥_𝑅||||+||||Λ^𝑞𝑤Λ^𝑞−2𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤_𝑥𝑑𝑥||||+||||_𝑅Λ^𝑞𝑤Λ^𝑞−2𝜕_𝑥𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1𝜕_𝑥𝑢_𝛿_𝑥𝑑𝑥_𝑅||||Λ^𝑞𝑤Λ^𝑞−2𝑢^𝑚𝜀𝑢_𝜀𝑥𝑢_𝜀𝑥𝑥−𝑢^𝑚𝛿𝑢_𝛿𝑥𝑢_𝛿𝑥𝑥𝑑𝑥||||.(3.32) Using the first inequality in Lemma 3.9, we have ||||_𝑅Λ^𝑞𝑤Λ^𝑞𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥||||=≤𝑐||||‖_𝑅𝑤Λ‖^𝑞𝑤_2𝐻𝑞Λ^𝑞𝑤𝑔_𝑚+1_𝑥𝑑𝑥||||‖‖𝑔_𝑚+1‖‖_𝐻𝑞+1,(3.33) where 𝑔_𝑚+1=∑^𝑚+1𝑗=0𝑢^{𝜀𝑚+1−𝑗}𝑢^𝑗𝛿. For the last three terms in (3.32), using Lemmas 3.4 and 3.12, 1/2<𝑞<min{1,𝑠−1}, 𝑠>3/2, the algebra property of 𝐻^𝑠0 with 𝑠₀>1/2, and (3.23), we have ||||_𝑅Λ^𝑞𝑤||||Λ^𝑞−2𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤_𝑥𝑑𝑥||||≤𝑐‖𝑤‖^2𝐻_𝑞‖‖𝑢_𝜀‖‖^𝐻𝑚+1_𝑠,(3.34)_𝑅Λ^𝑞≤𝑤Λ𝑐‖𝑤^𝑞−2‖𝜕_𝑥𝐻𝑞𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1𝜕_𝑥𝑢_𝛿_𝑥𝑑𝑥||||‖‖𝑢_𝛿‖‖_𝐻𝑠‖‖𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1‖‖_𝐻𝑞||||≤𝑐‖𝑤‖^2𝐻_𝑞‖‖𝑢_𝛿‖‖_𝐻𝑠,(3.35)_𝑅Λ^𝑞≤𝑤Λ𝑐‖𝑤^𝑞−2‖𝑢^𝑚𝜀_𝐻𝑞𝑢_𝜀𝑥𝑢_𝜀𝑥𝑥−𝑢^𝑚𝛿𝑢_𝛿𝑥𝑢_𝛿𝑥𝑥𝑑𝑥||||‖‖𝑢^𝑚𝜀−𝑢^𝑚𝛿𝑢^2𝜀𝑥_𝑥+𝑢^𝑚𝛿𝑢^2𝜀𝑥−𝑢^2𝛿𝑥_𝑥‖‖_𝐻𝑞−2≤≤𝑐𝑐‖‖𝑤𝑤‖‖_𝐻^𝑞_𝐻𝑞‖‖𝑢^𝑚𝜀−𝑢^𝑚𝛿𝑢^2𝜀𝑥_𝑥‖‖_𝐻𝑞−1+‖‖𝑢^𝑚𝛿𝑢^2𝜀𝑥−𝑢^2𝛿𝑥_𝑥‖‖_𝐻𝑞−2‖‖𝑢^𝑚𝜀−𝑢^𝑚𝛿‖‖_𝐻𝑞‖‖𝑢^2𝜀𝑥_𝑥‖‖_𝐻𝑞−1+‖‖𝑢^𝑚𝛿‖‖_𝐻𝑠‖‖𝑢^2𝜀𝑥−𝑢^2𝛿𝑥_𝑥‖‖_𝐻𝑞−2≤𝑐‖𝑤‖_𝐻𝑞≤𝑐‖𝑤‖_2𝐻𝑞‖𝑤‖_𝐻^𝑞‖‖𝑔_𝑚−1‖‖_𝐻𝑞‖𝑢‖^2𝐻_𝑠+‖‖𝑢^𝑚𝛿‖‖_𝐻𝑠‖‖𝑢_𝜀𝑥+𝑢_𝛿𝑥‖‖_𝐻𝑞‖𝑤‖_𝐻^𝑞‖‖𝑔_𝑚−1‖‖_𝐻𝑞‖𝑢‖^2𝐻_𝑠+‖‖𝑢^𝑚𝛿‖‖_𝐻𝑠‖‖𝑢_𝜀𝑥+𝑢_𝛿𝑥‖‖_𝐻𝑞.(3.36) Using (3.26), we derive that the inequality ||||_𝑅Λ^𝑞𝑤Λ^𝑞−2𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥||||≤=𝑐||||‖‖_𝑅𝑔Λ^𝑞_𝑚+1𝑤Λ^𝑞−2𝑤𝑔_𝑚+1_𝑥𝑑𝑥||||‖‖_𝐻𝑞‖𝑤‖^2𝐻_𝑞(3.37) holds for some constant 𝑐, where 𝑔_𝑚+1=∑^𝑚+1𝑗=0𝑢^{𝜀𝑚+1−𝑗}𝑢^𝑗𝛿. Using the algebra property of 𝐻^𝑞 with 𝑞>1/2, 𝑞+1<𝑠 and Lemma 3.11, we have ‖𝑔_𝑚‖_𝐻𝑞+1≤𝑐 for 𝑡∈(0,𝑇]. Then, it follows from (3.28) and (3.33)–(3.37) that there is a constant 𝑐 depending on 𝑇 such that the estimate 𝑑𝑑𝑡_𝑅(1−𝜀)(Λ^𝑞𝑤)²+𝜀Λ^𝑞𝑤_𝑥²𝑑𝑥≤𝑐𝛿^𝛾‖𝑤‖_𝐻^𝑞+‖𝑤‖^2𝐻_𝑞(3.38) holds for any 𝑡∈[0,𝑇), where 𝛾=1 if 𝑠≥3+𝑞 and 𝛾=(1+𝑠−𝑞)/4 if 𝑠<3+𝑞. Integrating (3.38) with respect to 𝑡, one obtains the estimate 12‖𝑤‖^2𝐻𝑞=12_𝑅≤(Λ^𝑞𝑤)²𝑑𝑥_𝑅≤(1−𝜀)(Λ^𝑞𝑤)²+𝜀(Λ^𝑞𝑤)²𝑑𝑥_𝑅Λ^𝑞𝑤₀²+𝜀Λ^𝑞𝑤_0𝑥²𝑑𝑥+𝑐^𝑡0𝛿^𝛾‖𝑤‖_𝐻^𝑞+‖𝑤‖^2𝐻_𝑞𝑑𝜏.(3.39) Applying the Gronwall inequality and using (3.20) and (3.22) yield ‖𝑢‖_𝐻^𝑞≤𝑐𝛿^{(𝑠−𝑞)/4}𝑒^𝑐𝑡+𝛿^𝛾𝑒^𝑐𝑡−1(3.40) for any 𝑡∈[0,𝑇).
Multiplying both sides of (3.29) by Λ^𝑠𝑤Λ^𝑠 and integrating the resultant equation with respect to 𝑥, one obtains 12𝑑𝑑𝑡_𝑅(1−𝜀)(Λ^𝑠=(𝜀−𝛿)_𝑅𝑤)²+𝜀Λ^𝑠𝑤_𝑥²𝑑𝑥+(𝛿−𝜀)_𝑅(Λ^𝑠𝑤)Λ^𝑠𝑢_𝛿𝑡+Λ^𝑠𝑢_{𝛿𝑥𝑥𝑡}𝑑𝑥−𝜀_𝑅Λ^𝑠𝑤Λ^𝑠−2𝑤_𝑡𝑑𝑥Λ^𝑠𝑤Λ^𝑠−2𝑢_𝛿𝑡𝑑𝑥−𝑘𝑚+1_𝑅−1(Λ^𝑠𝑤)Λ^𝑠𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1_𝑥𝑑𝑥𝑚+2_𝑅−(Λ^𝑠𝑤)Λ^𝑠𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥−_𝑅Λ^𝑠𝑤Λ^𝑠−2𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥_𝑅+Λ^𝑠𝑤Λ^𝑠−2𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤_𝑥𝑑𝑥−_𝑅Λ^𝑠𝑤Λ^𝑠−2𝜕_𝑥𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1𝜕_𝑥𝑢_𝛿_𝑥𝑑𝑥_𝑅Λ^𝑠𝑤Λ^𝑠−2𝑢^𝑚𝜀𝑢_𝜀𝑥𝑢_𝜀𝑥𝑥−𝑢^𝑚𝛿𝑢_𝛿𝑥𝑢_𝛿𝑥𝑥𝑑𝑥+𝜆_𝑅(Λ^𝑠𝑤)²𝑑𝑥.(3.41) From Lemma 3.12, we have ||||_𝑅Λ^𝑠𝑤Λ^𝑠−2𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥||||≤𝑐₃‖‖𝑔_𝑚+1‖‖_𝐻𝑠‖𝑤‖^2𝐻_𝑠.(3.42) From Lemma 3.10, it holds that ||||_𝑅Λ^𝑠𝑤Λ^𝑠𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥||||≤𝑐‖𝑤‖_𝐻^𝑠‖𝑤‖_𝐻^𝑞‖‖𝑢_𝛿‖‖_𝐻𝑠+1+‖𝑤‖^2𝐻_𝑠.(3.43) Using the Cauchy-Schwartz inequality and the algebra property of 𝐻^𝑠0 with 𝑠₀>1/2, for 𝑠>3/2, we have ||||_𝑅Λ=^𝑠𝑤||||Λ^𝑠−2𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤_𝑥𝑑𝑥_𝑅||||≤𝑐‖ΛΛ^𝑞_𝑠𝑤Λ^𝑠−2𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤_𝑥𝑑𝑥||||𝑤‖_𝐿²‖‖Λ^𝑠−2𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤_𝑥‖‖_𝐿2≤𝑐‖𝑤‖_𝐻^𝑞‖‖𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤‖‖_𝐻𝑠−1||||≤𝑐‖‖𝑢_{𝜀𝐻𝑚+1𝑠}‖‖‖𝑤‖^2𝐻_𝑠,(3.44)_𝑅Λ≤^𝑠𝑤𝑐‖Λ𝑤‖^𝑠−2_𝐻𝑠𝜕_𝑥𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1𝜕_𝑥𝑢_𝛿_𝑥𝑑𝑥||||‖‖Λ^𝑠−2𝜕_𝑥𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1𝜕_𝑥𝑢_𝛿_𝑥‖‖_𝐿2||||≤𝑐‖‖𝑢_𝛿‖‖_𝐻𝑠‖‖𝑔_𝑚‖‖_𝐻𝑠‖𝑤‖^2𝐻_𝑠,(3.45)_𝑅Λ≤^𝑠𝑤𝑐‖Λ𝑤‖^𝑠−2_𝐻𝑠𝑢^𝑚𝜀𝑢_𝜀𝑥𝑢_𝜀𝑥𝑥−𝑢^𝑚𝛿𝑢_𝛿𝑥𝑢_𝛿𝑥𝑥𝑑𝑥||||‖‖𝑢^𝑚𝜀−𝑢^𝑚𝛿𝑢^2𝜀𝑥_𝑥‖‖_𝐻𝑠−2+‖‖𝑢^𝑚𝛿𝑢^2𝜀𝑥−𝑢^2𝛿𝑥_𝑥‖‖_𝐻𝑠−2≤𝑐‖𝑤‖_𝐻𝑠‖‖𝑢^𝑚𝜀−𝑢^𝑚𝛿‖‖_𝐻𝑠‖‖𝑢^2𝜀𝑥_𝑥‖‖_𝐻𝑠−2+‖‖𝑢^𝑚𝛿‖‖_𝐻𝑠‖‖𝑢^2𝜀𝑥−𝑢^2𝛿𝑥_𝑥‖‖_𝐻𝑠−2≤𝑐‖𝑤‖_2𝐻𝑠,(3.46) in which we have used Lemma 3.4 and the bounded property of ‖𝑢_𝜀‖_𝐻^𝑠 and ‖𝑢_𝛿‖_𝐻^𝑠 (see Remark 3.7). It follows from (3.41)–(3.46) and the inequalities (3.28) and (3.40) that there exists a constant 𝑐 depending on 𝑚 such that 𝑑𝑑𝑡_𝑅(1−𝜀)(Λ^𝑠≤2𝛿‖‖𝑢_𝛿𝑡𝑤)²+𝜀Λ^𝑠𝑤_𝑥²𝑑𝑥‖‖_𝐻𝑠+‖‖𝑢_{𝛿𝑥𝑥𝑡}‖‖_𝐻𝑠+‖‖Λ^𝑠−2𝑤_𝑡‖‖_𝐿2+‖‖Λ^𝑠−2𝑢_𝛿𝑡‖‖‖𝑤‖_𝐻𝑠≤+𝑐𝑐𝛿‖_𝛾1𝑤‖^2𝐻_𝑠+‖𝑤‖_𝐻^𝑞‖𝑤‖_𝐻^𝑠‖‖𝑢_𝛿‖‖_𝐻𝑠+1‖𝑤‖_𝐻^𝑠+‖𝑤‖^2𝐻_𝑠,(3.47) where 𝛾₁=min(1/4,(𝑠−𝑞−1)/4)>0. Integrating (3.47) with respect to 𝑡 leads to the estimate 12‖𝑤‖^2𝐻𝑠≤≤_𝑅(1−𝜀)(Λ^𝑠𝑤)²+𝜀Λ^𝑠𝑤_𝑥²𝑑𝑥_𝑅Λ^𝑠𝑤₀²+𝜀Λ^𝑠𝑤_0𝑥²𝑑𝑥+𝑐^𝑡0𝛿^𝛾1‖𝑤‖_𝐻^𝑠+‖𝑤‖^2𝐻_𝑠𝑑𝜏.(3.48) It follows from the Gronwall inequality and (3.48) that ‖𝑤‖_𝐻^𝑠≤≤𝑐₁2_𝑅Λ^𝑠𝑤₀²+𝜀Λ^𝑠𝑤_0𝑥²𝑑𝑥^1/2𝑒^𝑐𝑡+𝛿^𝛾1𝑒^𝑐𝑡−1‖‖𝑤₀‖‖_𝐻𝑠+𝛿^3/4𝑒^𝑐𝑡+𝛿^𝛾1𝑒^𝑐𝑡−1,(3.49) where 𝑐₁ is independent of 𝜀 and 𝛿.
Then, (3.22) and the above inequality show that ‖𝑤‖_𝐻^𝑠⟶0as𝜀⟶0,𝛿⟶0.(3.50) Next, we consider the convergence of the sequence {𝑢_𝜀𝑡}. Multiplying both sides of (3.29) by Λ^𝑠−1𝑤_𝑡Λ^𝑠−1 and integrating the resultant equation with respect to 𝑥, we obtain (1−𝜀)‖‖𝑤_𝑡‖‖^2𝐻_𝑠−1+1𝑚+2_𝑅Λ^𝑠−1+𝑤_𝑡Λ^𝑠−1𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2_𝑥𝑑𝑥_𝑅=−𝜀Λ^𝑠−1𝑤_𝑡Λ^𝑠−1𝑤_𝑥𝑥𝑡+(𝛿−𝜀)Λ^𝑠−1𝑤_𝑡Λ^𝑠−1𝑢_𝛿𝑡+𝑢_{𝛿𝑥𝑥𝑡}𝑑𝑥_𝑅−𝜕Λ^𝑠−1_𝑥𝑤_𝑡Λ^𝑠−3−𝜀𝑤^𝑡+(𝛿−𝜀)𝑢_𝛿𝑡−𝑚𝑘+1𝜕^𝑥𝑢^𝜀𝑚+1−𝑢^𝛿𝑚+1−𝜕_𝑥𝑢^𝜀𝑚+2−𝑢^𝛿𝑚+2+𝑢_𝑚𝜀𝜕_𝑥𝑢^𝜀𝑚+1𝜕_𝑥𝑤+𝜕_𝑥𝑢^𝜀𝑚+1−𝑢^𝜀𝑚+1𝜕_𝑥𝑢_𝛿𝑢_𝜀𝑥𝑢_𝜀𝑥𝑥−𝑢^𝑚𝛿𝑢_𝛿𝑥𝑢_𝛿𝑥𝑥𝑑𝑥+𝜆_𝑅Λ^𝑠−1𝑤_𝑡Λ^𝑠−1𝑤𝑑𝑥.(3.51)
It follows from inequalities (3.28) and the Schwartz inequality that there is a constant 𝑐 depending on 𝑇 and 𝑚 such that (1−𝜀)‖‖𝑤_𝑡‖‖^2𝐻_𝑠−1≤𝑐𝛿^1/2+‖𝑤‖_𝐻^𝑠+‖𝑤‖_𝑠−1‖‖𝑤_𝑡‖‖_𝐻𝑠−1+𝜀‖‖𝑤_𝑡‖‖^2𝐻_𝑠−1(3.52) Hence 12‖‖𝑤^𝑡‖‖^2𝐻_𝑠−1≤(1−2𝜀)‖‖𝑤_𝑡‖‖^2𝐻_𝑠−1≤𝑐𝛿^1/2+‖𝑤‖_𝐻^𝑠+‖𝑤‖_𝐻^𝑠−1‖‖𝑤_𝑡‖‖_𝐻𝑠−1,(3.53) which results in 12‖‖𝑤^𝑡‖‖_𝐻𝑠−1≤𝑐𝛿^1/2+‖𝑤‖_𝐻^𝑠+‖𝑤‖_𝐻^𝑠−1.(3.54)
It follows from (3.40) and (3.50) that 𝑤_𝑡→0 as 𝜀, 𝛿→0 in the 𝐻^𝑠−1 norm. This implies that 𝑢_𝜀 is a Cauchy sequence in the spaces 𝐶([0,𝑇);𝐻^𝑠(𝑅)) and 𝐶([0,𝑇);𝐻^𝑠−1(𝑅)), respectively. The proof is completed.

4. Proof of the Main Result

We consider the problem (1−𝜀)𝑢_𝑡−𝜀𝑢_𝑡𝑥𝑥=1−𝜕^2𝑥⁻¹−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2𝑢(0,𝑥)=𝑢𝑢^𝑚_𝜀0𝑢^2𝑥+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝜆𝑢,(𝑥).(4.1) Letting 𝑢(𝑡,𝑥) be the limit of the sequence 𝑢_𝜀 and taking the limit in problem (4.1) as 𝜀→0, from Lemma 3.13, we know that 𝑢 is a solution of the problem 𝑢_𝑡=1−𝜕^2𝑥⁻¹−𝑘𝑚+1𝑢^𝑚+1_𝑥−𝑚+3𝑚+2𝑢^𝑚+2_𝑥+1−(𝑚+1)𝜕_𝑥𝑚+2𝜕^3𝑥𝑢^𝑚+2𝑢(0,𝑢𝑥^𝑚)𝑢=^2𝑥𝑢+₀𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥+𝜆𝑢,(𝑥),(4.2) and hence 𝑢 is a solution of problem (4.2) in the sense of distribution. In particular, if 𝑠≥4, 𝑢 is also a classical solution. Let 𝑢 and 𝑣 be two solutions of (4.2) corresponding to the same initial value 𝑢₀ such that 𝑢, 𝑣∈𝐶([0,𝑇);𝐻^𝑠(𝑅)). Then, 𝑤=𝑢−𝑣 satisfies the Cauchy problem 𝑤_𝑡=1−𝜕^2𝑥⁻¹𝜕^𝑥−𝑘𝑚+1𝑤𝑔^𝑚−𝑚+3𝑚+2𝑤𝑔^𝑚+1+1−𝜕_𝑥𝑚+2𝜕^2𝑥𝑤𝑔_𝑚+1𝑢^𝑚+1𝜕_𝑥𝑤−𝜕𝑤^𝑥(0𝑢,𝑥)^𝑚+1=−0𝑣.^𝑚+1𝜕_𝑥𝑣+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥−𝑣^𝑚𝑣_𝑥𝑣_𝑥𝑥+𝜆𝑤,(4.3) For any 1/2<𝑞<min{1,𝑠−1}, applying the operator Λ^𝑞𝑤Λ^𝑞 to both sides of (4.3) and integrating the resultant equation with respect to 𝑥, we obtain the equality 12𝑑𝑑𝑡‖𝑤‖^2𝐻𝑞=_𝑅(Λ^𝑞𝑤)Λ^𝑞−2𝜕^𝑥−𝑘𝑚+1𝑤𝑔^𝑚−𝑚+3𝑚+2𝑤𝑔^𝑚+1+1−𝜕_𝑥𝑚+2𝜕^2𝑥𝑤𝑔_𝑚+1𝑢^𝑚+1𝜕_𝑥𝑤−𝜕_𝑥𝑢^𝑚+1−𝑣^𝑚+1𝜕_𝑥𝑣+𝑢^𝑚𝑢_𝑥𝑢_𝑥𝑥−𝑣^𝑚𝑣_𝑥𝑣_𝑥𝑥𝑑𝑥+||𝜆||‖𝑤‖^2𝐻𝑞.(4.4) By the similar estimates presented in Lemma 3.13, we have 𝑑𝑑𝑡‖𝑤‖^2𝐻𝑞≤𝑐̃‖𝑤‖^2𝐻_𝑞.(4.5) Using the Gronwall inequality leads to the conclusion that ‖𝑤‖_𝐻^𝑞≤0×𝑒𝑐̃^𝑡=0(4.6) for 𝑡∈[0,𝑇). This completes the proof.

Acknowledgments

Thanks are given to the referees whose suggestions were very helpful in improving the paper. This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).