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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 164876, 17 pages
http://dx.doi.org/10.1155/2012/164876
Research Article

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

School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 23 May 2012; Revised 30 June 2012; Accepted 18 July 2012

Academic Editor: Yong HongΒ Wu

Copyright Β© 2012 Meng Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝐻𝑠(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 𝐻𝑠(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 π‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯+π‘˜π‘’π‘šπ‘’π‘₯+(π‘š+3)π‘’π‘š+1𝑒π‘₯=(π‘š+2)π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯+π‘’π‘š+1𝑒π‘₯π‘₯π‘₯,(1.1) where π‘šβ‰₯0 is a natural number and π‘˜β‰₯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): π‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯+π‘˜π‘’π‘šπ‘’π‘₯+(π‘š+3)π‘’π‘š+1𝑒π‘₯=(π‘š+2)π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯+π‘’π‘š+1𝑒π‘₯π‘₯π‘₯ξ€·+πœ†π‘’βˆ’π‘’π‘₯π‘₯ξ€Έ,(1.2) where π‘šβ‰₯0 is a natural number, π‘˜β‰₯0, and πœ† 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 𝐻𝑠(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 πœ†(π‘’βˆ’π‘’π‘₯π‘₯), 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.

𝐿𝑝=𝐿𝑝(𝑅)(1≀𝑝<+∞) is the space of all measurable functions β„Ž such that β€–β„Žβ€–π‘πΏπ‘=βˆ«π‘…|β„Ž(𝑑,π‘₯)|𝑝𝑑π‘₯<∞. We define 𝐿∞=𝐿∞(𝑅) with the standard norm β€–β„Žβ€–πΏβˆž=infπ‘š(𝑒)=0supπ‘₯βˆˆπ‘…β§΅π‘’|β„Ž(𝑑,π‘₯)|. For any real number 𝑠, 𝐻𝑠=𝐻𝑠(𝑅) denotes the Sobolev space with the norm defined by β€–β„Žβ€–π»π‘ =ξ‚΅ξ€œπ‘…ξ€·1+|πœ‰|2𝑠||||||β„Ž(𝑑,πœ‰)2ξ‚Άπ‘‘πœ‰1/2<∞,(2.1) where ξβˆ«β„Ž(𝑑,πœ‰)=π‘…π‘’βˆ’π‘–π‘₯πœ‰β„Ž(𝑑,π‘₯)𝑑π‘₯.

For 𝑇>0 and nonnegative number 𝑠, 𝐢([0,𝑇);𝐻𝑠(𝑅)) denotes the Frechet space of all continuous 𝐻𝑠-valued functions on [0,𝑇). We set Ξ›=(1βˆ’πœ•2π‘₯)1/2. For simplicity, throughout this paper, we let 𝑐 denote any positive constant that is independent of parameter πœ€.

We consider the Cauchy problem of (1.2) π‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯π‘˜=βˆ’ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ€·+πœ†π‘’βˆ’π‘’π‘₯π‘₯𝑒,π‘˜β‰₯0,π‘šβ‰₯0,(0,π‘₯)=𝑒0(π‘₯).(2.2)

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

Theorem 2.1. Suppose that the initial function 𝑒0(π‘₯) belongs to the Sobolev space 𝐻𝑠(𝑅) with 𝑠>3/2 and πœ† is a constant. Then, there is a 𝑇>0, which depends on ‖𝑒0‖𝐻𝑠, such that problem (2.2) has a unique solution 𝑒(𝑑,π‘₯) satisfying [𝑒(𝑑,π‘₯)∈𝐢(0,𝑇);𝐻𝑠𝐢(𝑅))1ξ€·[0,𝑇);π»π‘ βˆ’1ξ€Έ(𝑅).(2.3)

3. Local Well-Posedness

In order to prove Theorem 2.1, we consider the associated regularized problem π‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯+πœ€π‘’π‘‘π‘₯π‘₯π‘₯π‘₯π‘˜=βˆ’ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ€·+πœ†π‘’βˆ’π‘’π‘₯π‘₯ξ€Έ,𝑒(0,π‘₯)=𝑒0(π‘₯),(3.1) where the parameter πœ€ satisfies 0<πœ€<1/4.

Lemma 3.1. For 𝑠β‰₯1 and 𝑓(π‘₯)βˆˆπ»π‘ (𝑅) and letting π‘˜1>0 be an integer such that π‘˜1β‰€π‘ βˆ’1, 𝑓,𝑓′,…,π‘“π‘˜1 are uniformly continuous bounded functions that converge to 0 at π‘₯=±∞.

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

Lemma 3.2. If 𝑒(𝑑,π‘₯)βˆˆπ»π‘ (𝑠>7/2) is a solution to problem (3.1), it holds that ξ€œπ‘…ξ€·π‘’2+𝑒2π‘₯+πœ€π‘’2π‘₯π‘₯ξ€Έξ€œπ‘‘π‘₯=𝑅𝑒20+𝑒20π‘₯+πœ€π‘’20π‘₯π‘₯ξ€Έξ€œπ‘‘π‘₯+2πœ†π‘‘0ξ€œπ‘…ξ€·π‘’2+𝑒2π‘₯𝑑π‘₯.(3.2)

Proof. Using Lemma 3.1, we have 𝑒(𝑑,±∞)=𝑒π‘₯(𝑑,±∞)=𝑒π‘₯π‘₯(𝑑,±∞)=𝑒π‘₯π‘₯π‘₯(𝑑,±∞)=0. The integration by parts results in ξ€œπ‘…π‘’π‘š+2𝑒π‘₯π‘₯π‘₯ξ€œπ‘‘π‘₯=π‘…π‘’π‘š+2𝑑𝑒π‘₯π‘₯=π‘’π‘š+2𝑒π‘₯π‘₯∣+βˆžβˆ’βˆžξ€œβˆ’(π‘š+2)π‘…π‘’π‘š+1𝑒π‘₯𝑒π‘₯π‘₯ξ€œπ‘‘π‘₯=βˆ’(π‘š+2)π‘…π‘’π‘š+1𝑒π‘₯𝑒π‘₯π‘₯𝑑π‘₯.(3.3)
Direct calculation and integration by parts give rise to 12π‘‘ξ€œπ‘‘π‘‘π‘…ξ€·π‘’2+𝑒2π‘₯+πœ€π‘’2π‘₯π‘₯ξ€Έ=ξ€œπ‘‘π‘₯𝑅𝑒𝑒𝑑+𝑒π‘₯𝑒𝑑π‘₯+πœ€π‘’π‘₯π‘₯𝑒𝑑π‘₯π‘₯ξ€Έ=ξ€œπ‘‘π‘₯π‘…π‘’ξ€·π‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯+πœ€π‘’π‘‘π‘₯π‘₯π‘₯π‘₯ξ€Έ=ξ€œπ‘‘π‘₯π‘…π‘’ξ‚ƒβˆ’π‘˜ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ€·+πœ†π‘’βˆ’π‘’π‘₯π‘₯ξ€Έξ‚„=ξ€œπ‘‘π‘₯𝑅𝑒(π‘š+2)π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯+π‘’π‘š+1𝑒π‘₯π‘₯π‘₯ξ€»ξ€œπ‘‘π‘₯+πœ†π‘…ξ€·π‘’2βˆ’π‘’π‘’π‘₯π‘₯ξ€Έ=ξ€œπ‘‘π‘₯𝑅(π‘š+2)π‘’π‘š+1𝑒π‘₯𝑒π‘₯π‘₯+π‘’π‘š+2𝑒π‘₯π‘₯π‘₯ξ€»ξ€œπ‘‘π‘₯+πœ†π‘…ξ€·π‘’2+𝑒2π‘₯ξ€Έξ€œπ‘‘π‘₯=πœ†π‘…ξ€·π‘’2+𝑒2π‘₯𝑑π‘₯,(3.4) in which we have used (3.3). From (3.4), we obtain the conservation law (3.2).

Lemma 3.3. Let 𝑠β‰₯7/2. The function 𝑒(𝑑,π‘₯) is a solution of problem (3.1) and the initial value 𝑒0(π‘₯)βˆˆπ»π‘ . Then, the following inequality holds: ‖𝑒‖2𝐻1β‰€ξ€œπ‘…ξ€·π‘’20+𝑒20π‘₯+πœ€π‘’20π‘₯π‘₯𝑑π‘₯,π‘–π‘“πœ†β‰€0,‖𝑒‖2𝐻1≀𝑒2πœ†π‘‘ξ€œπ‘…ξ€·π‘’20+𝑒20π‘₯+πœ€π‘’20π‘₯π‘₯𝑑π‘₯,π‘–π‘“πœ†>0.(3.5) For π‘žβˆˆ(0,π‘ βˆ’1], there is a constant 𝑐 independent of πœ€ such that ξ€œπ‘…ξ€·Ξ›π‘ž+1𝑒2ξ€œπ‘‘π‘₯β‰€π‘…ξ‚ƒξ€·Ξ›π‘ž+1𝑒0ξ€Έ2ξ€·Ξ›+πœ€π‘žπ‘’0π‘₯π‘₯ξ€Έ2ξ‚„ξ€œπ‘‘π‘₯+𝑐𝑑0‖𝑒‖2π»π‘ž+1ξ‚€||πœ†||+ξ€·β€–π‘’β€–πΏπ‘šβˆ’1∞+β€–π‘’β€–π‘šπΏβˆžξ€Έβ€–β€–π‘’π‘₯β€–β€–πΏβˆž+β€–π‘’β€–πΏπ‘šβˆ’1βˆžβ€–β€–π‘’π‘₯β€–β€–2πΏβˆžξ‚π‘‘πœ.(3.6) For π‘žβˆˆ[0,π‘ βˆ’1], there is a constant 𝑐 independent of πœ€ such that ‖‖𝑒(1βˆ’2πœ€)π‘‘β€–β€–π»π‘žβ‰€π‘β€–π‘’β€–π»π‘ž+1ξ€·||πœ†||+ξ€·β€–π‘’β€–πΏπ‘šβˆ’1∞+β€–π‘’β€–π‘šπΏβˆžξ€Έβ€–π‘’β€–π»1+β€–π‘’β€–π‘šπΏβˆžβ€–β€–π‘’π‘₯β€–β€–πΏβˆž+β€–π‘’β€–πΏπ‘šβˆ’1βˆžβ€–β€–π‘’π‘₯β€–β€–2πΏβˆžξ‚.(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 π‘Ÿ and π‘ž be real numbers such that βˆ’π‘Ÿ<π‘žβ‰€π‘Ÿ. Then, β€–π‘’π‘£β€–π»π‘žβ‰€π‘β€–π‘’β€–π»π‘Ÿβ€–π‘£β€–π»π‘ž1,π‘–π‘“π‘Ÿ>2,β€–π‘’π‘£β€–π»π‘Ÿ+π‘žβˆ’1/2β‰€π‘β€–π‘’β€–π»π‘Ÿβ€–π‘£β€–π»π‘ž1,π‘–π‘“π‘Ÿ<2.(3.8)

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

Lemma 3.5. Let 𝑒0(π‘₯)βˆˆπ»π‘ (𝑅) with 𝑠>3/2. Then, the Cauchy problem (3.1) has a unique solution 𝑒(𝑑,π‘₯)∈𝐢([0,𝑇];𝐻𝑠(𝑅)), where 𝑇>0 depends on ‖𝑒0‖𝐻𝑠(𝑅). If 𝑠β‰₯7/2, the solution π‘’βˆˆπΆ([0,+∞);𝐻𝑠) exists for all time.

Proof. Letting 𝐷=(1βˆ’πœ•2π‘₯+πœ€πœ•4π‘₯)βˆ’1, we know that π·βˆΆπ»π‘ β†’π»π‘ +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 𝑑 over the interval (0,𝑑), we get 𝑒(𝑑,π‘₯)=𝑒0(ξ€œπ‘₯)+𝑑0π·ξ‚ƒβˆ’π‘˜ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ€·+πœ†π‘’βˆ’π‘’π‘₯π‘₯𝑑𝑑.(3.9) Suppose that both 𝑒 and 𝑣 are in the closed ball 𝐡𝑀0(0) of radius 𝑀0 about the zero function in 𝐢([0,𝑇];𝐻𝑠(𝑅)) and 𝐴 is the operator in the right-hand side of (3.9). For any fixed π‘‘βˆˆ[0,𝑇], we obtain β€–β€–β€–ξ€œπ‘‘0π·ξ‚ƒβˆ’π‘˜ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ€·+πœ†π‘’βˆ’π‘’π‘₯π‘₯ξ€Έξ‚„βˆ’ξ€œπ‘‘π‘‘π‘‘0π·ξ‚ƒβˆ’π‘˜ξ€·π‘£π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘£π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘£π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘£π‘šπ‘£2π‘₯ξ€Έ+π‘£π‘šπ‘£π‘₯𝑣π‘₯π‘₯ξ€·+πœ†π‘£βˆ’π‘£π‘₯π‘₯‖‖‖𝑑𝑑𝐻𝑠≀𝑇𝐢1ξ‚΅sup0β‰€π‘‘β‰€π‘‡β€–π‘’βˆ’π‘£β€–π»π‘ +sup0β‰€π‘‘β‰€π‘‡β€–β€–π‘’π‘š+1βˆ’π‘£π‘š+1‖‖𝐻𝑠+sup0β‰€π‘‘β‰€π‘‡β€–β€–π‘’π‘š+2βˆ’π‘£π‘š+2‖‖𝐻𝑠+sup0β‰€π‘‘β‰€π‘‡β€–β€–π·πœ•π‘₯ξ€Ίπ‘’π‘šπ‘’2π‘₯βˆ’π‘£π‘šπ‘£2π‘₯‖‖𝐻𝑠+sup0β‰€π‘‘β‰€π‘‡β€–β€–π·ξ€Ίπ‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯βˆ’π‘£π‘šπ‘£π‘₯𝑣π‘₯π‘₯‖‖𝐻𝑠,(3.10) where 𝐢1 may depend on πœ€. The algebraic property of 𝐻𝑠0(𝑅) with 𝑠0>1/2 derives β€–β€–π‘’π‘š+2βˆ’π‘£π‘š+2‖‖𝐻𝑠=‖‖𝑒(π‘’βˆ’π‘£)π‘š+1+π‘’π‘šπ‘£+β‹―+π‘’π‘£π‘š+π‘£π‘š+1‖‖𝐻𝑠≀‖(π‘’βˆ’π‘£)β€–π»π‘ π‘š+1𝑗=0β€–π‘’β€–π»π‘š+1βˆ’π‘—π‘ β€–π‘£β€–π‘—π»π‘ ,≀𝑀0π‘š+1β€–π‘’βˆ’π‘£β€–π»π‘ ,β€–β€–π‘’π‘š+1βˆ’π‘£π‘š+1β€–β€–π»π‘ β‰€π‘€π‘š0β€–π‘’βˆ’π‘£β€–π»π‘ ,β€–β€–π·πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯βˆ’π‘£π‘šπ‘£2π‘₯ξ€Έβ€–β€–π»π‘ β‰€β€–β€–π·πœ•π‘₯ξ€Ίπ‘’π‘šξ€·π‘’2π‘₯βˆ’π‘£2π‘₯‖‖𝐻𝑠+β€–β€–π·πœ•π‘₯𝑣2π‘₯(π‘’π‘šβˆ’π‘£π‘š)ξ€»β€–β€–π»π‘ ξ€·β€–β€–π‘’β‰€πΆπ‘šξ€·π‘’2π‘₯βˆ’π‘£2π‘₯ξ€Έβ€–β€–π»π‘ βˆ’1+‖‖𝑣2π‘₯(π‘’π‘šβˆ’π‘£π‘š)β€–β€–π»π‘ βˆ’1≀𝐢𝑀0π‘š+1β€–π‘’βˆ’π‘£β€–π»π‘ .(3.11) Using the first inequality of Lemma 3.4 gives rise to β€–β€–π·ξ€Ίπ‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯βˆ’π‘£π‘šπ‘£π‘₯𝑣π‘₯π‘₯‖‖𝐻𝑠=β€–β€–β€–12π·ξ€Ίπ‘’π‘šξ€·π‘’2π‘₯ξ€Έπ‘₯βˆ’π‘£π‘šξ€·π‘£2π‘₯ξ€Έπ‘₯‖‖‖𝐻𝑠≀12ξ‚€β€–β€–π·ξ€Ίπ‘’π‘šξ€·π‘’2π‘₯βˆ’π‘£2π‘₯ξ€Έπ‘₯‖‖𝐻𝑠+‖‖𝐷𝑣2π‘₯ξ€Έπ‘₯(π‘’π‘šβˆ’π‘£π‘š)ξ€»β€–β€–π»π‘ ξ‚ξ‚€β€–β€–π‘’β‰€πΆπ‘šξ€·π‘’2π‘₯βˆ’π‘£2π‘₯ξ€Έπ‘₯β€–β€–π»π‘ βˆ’2+‖‖𝑣2π‘₯ξ€Έπ‘₯(π‘’π‘šβˆ’π‘£π‘š)β€–β€–π»π‘ βˆ’2ξ‚ξ€·β‰€πΆβ€–π‘’π‘šβ€–π»π‘ β€–β€–π‘’2π‘₯βˆ’π‘£2π‘₯β€–β€–π»π‘ βˆ’1+‖‖𝑣2π‘₯β€–β€–π»π‘ βˆ’1β€–π‘’π‘šβˆ’π‘£π‘šβ€–π»π‘ ξ€Έβ‰€πΆπ‘€0π‘š+1β€–π‘’βˆ’π‘£β€–π»π‘ ,(3.12) where 𝐢 may depend on πœ€. From (3.11)-(3.12), we obtain β€–π΄π‘’βˆ’π΄π‘£β€–π»π‘ β‰€πœƒβ€–π‘’βˆ’π‘£β€–π»π‘ ,(3.13) where πœƒ=𝑇𝐢2(π‘€π‘š0+𝑀0π‘š+1) and 𝐢2 is independent of 0<𝑑<𝑇. Choosing 𝑇 sufficiently small such that πœƒ<1, we know that 𝐴 is a contraction. Similarly, it follows from (3.10) that ‖𝐴𝑒‖𝐻𝑠≀‖‖𝑒0‖‖𝐻𝑠+πœƒβ€–π‘’β€–π»π‘ .(3.14) Choosing 𝑇 sufficiently small such that πœƒπ‘€0+‖𝑒0‖𝐻𝑠<𝑀0, we deduce that 𝐴 maps 𝐡𝑀0(0) to itself. It follows from the contraction-mapping principle that the mapping 𝐴 has a unique fixed point 𝑒 in 𝐡𝑀0(0). It completes the proof.

From the above and Lemma 3.2, we have ξ€œπ‘…ξ€·π‘’2+𝑒2π‘₯+πœ€π‘’2π‘₯π‘₯𝑑π‘₯≀𝑒2|πœ†|π‘‘ξ€œπ‘…ξ€·π‘’20+𝑒20π‘₯+πœ€π‘’20π‘₯π‘₯𝑑π‘₯.(3.15) Therefore, ‖‖𝑒π‘₯β€–β€–πΏβˆžβ‰€πΆπœ€π‘’2|πœ†|π‘‘ξ€œπ‘…ξ€·π‘’20+𝑒20π‘₯+πœ€π‘’20π‘₯π‘₯𝑑π‘₯,(3.16) which together with Lemma 3.3 completes the proof of the global existence.

Setting πœ™πœ€(π‘₯)=πœ€βˆ’1/4πœ™(πœ€βˆ’1/4π‘₯) with 0<πœ€<1/4 and π‘’πœ€0=πœ™πœ€β‹†π‘’0, we know that π‘’πœ€0∈𝐢∞ for any 𝑒0βˆˆπ»π‘ ,𝑠>0. From Lemma 3.5, it derives that the Cauchy problem π‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯+πœ€π‘’π‘‘π‘₯π‘₯π‘₯π‘₯π‘˜=βˆ’ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ€·+πœ†π‘’βˆ’π‘’π‘₯π‘₯ξ€Έ,𝑒(0,π‘₯)=π‘’πœ€0(π‘₯),π‘₯βˆˆπ‘…,(3.17) has a unique solution π‘’πœ€(𝑑,π‘₯)∈𝐢∞([0,∞);𝐻∞).

Furthermore, we have the following.

Lemma 3.6. For 𝑠>0,𝑒0βˆˆπ»π‘ , it holds that β€–β€–π‘’πœ€0π‘₯β€–β€–πΏβˆžβ€–β€–π‘’β‰€π‘0π‘₯β€–β€–πΏβˆž,‖‖𝑒(3.18)πœ€0β€–β€–π»π‘žβ€–β€–π‘’β‰€π‘,π‘–π‘“π‘žβ‰€π‘ ,(3.19)πœ€0β€–β€–π»π‘žβ‰€π‘πœ€(π‘ βˆ’π‘ž)/4‖‖𝑒,π‘–π‘“π‘ž>𝑠,(3.20)πœ€0βˆ’π‘’0β€–β€–π»π‘žβ‰€π‘πœ€(π‘ βˆ’π‘ž)/4‖‖𝑒,π‘–π‘“π‘žβ‰€π‘ ,(3.21)πœ€0βˆ’π‘’0‖‖𝐻𝑠=π‘œ(1),(3.22) where 𝑐 is a constant independent of πœ€.

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

Remark 3.7. For 𝑠β‰₯1, using β€–π‘’πœ€β€–πΏβˆžβ‰€π‘β€–π‘’πœ€β€–π»1/2+β‰€π‘β€–π‘’πœ€β€–π»1, β€–π‘’πœ€β€–2𝐻1βˆ«β‰€π‘π‘…(𝑒2πœ€+𝑒2πœ€π‘₯)𝑑π‘₯, (3.5), (3.19), and (3.20), we know that, β€–β€–π‘’πœ€β€–β€–2πΏβˆžβ€–β€–π‘’β‰€π‘πœ€β€–β€–2𝐻1≀𝑐𝑒2|πœ†|π‘‘ξ€œπ‘…ξ€·π‘’2πœ€0+𝑒2πœ€0π‘₯+πœ€π‘’2πœ€0π‘₯π‘₯𝑑π‘₯≀𝑐𝑒2|πœ†|π‘‘ξ‚€β€–β€–π‘’πœ€0β€–β€–2𝐻1‖‖𝑒+πœ€πœ€0β€–β€–2𝐻2≀𝑐𝑒|2πœ†|𝑑𝑐+π‘πœ€Γ—πœ€(π‘ βˆ’2)/2≀𝑐0𝑒2|πœ†|𝑑,(3.23) where 𝑐0 is independent of πœ€ and 𝑑.

Lemma 3.8. Suppose 𝑒0(π‘₯)βˆˆπ»π‘ (𝑅) with 𝑠β‰₯1 such that ‖𝑒0π‘₯β€–πΏβˆž<∞. Let π‘’πœ€0 be defined as in system (3.17). Then, there exist two positive constants 𝑇 and 𝑐, which are independent of πœ€, such that the solution π‘’πœ€ of problem (3.17) satisfies β€–π‘’πœ€π‘₯β€–πΏβˆžβ‰€π‘ for any π‘‘βˆˆ[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 𝑒 and 𝑓 are functions in π»π‘ž+1∩{‖𝑒π‘₯β€–πΏβˆž<∞}, then ||||ξ€œπ‘…Ξ›π‘žπ‘’Ξ›π‘ž(𝑒𝑓)π‘₯||||β‰€βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘π‘‘π‘₯π‘žβ€–π‘“β€–π»π‘ž+1‖𝑒‖2π»π‘žξ‚€1,π‘žβˆˆ2ξ‚„,𝑐,1π‘žξ€·β€–π‘“β€–π»π‘ž+1β€–π‘’β€–π»π‘žβ€–π‘’β€–πΏβˆž+‖‖𝑓π‘₯β€–β€–πΏβˆžβ€–π‘’β€–2π»π‘ž+β€–π‘“β€–π»π‘žβ€–π‘’β€–π»π‘žβ€–β€–π‘’π‘₯β€–β€–πΏβˆžξ‚,π‘žβˆˆ(0,∞).(3.24)

Lemma 3.10 (see Lai and Wu [16]). For 𝑒,π‘£βˆˆπ»π‘ (𝑅) with 𝑠>3/2, 𝑀=π‘’βˆ’π‘£,π‘ž>1/2, and a natural number 𝑛, it holds that ||||ξ€œπ‘…Ξ›π‘ π‘€Ξ›π‘ ξ€·π‘’π‘›+1βˆ’π‘£π‘›+1ξ€Έπ‘₯||||𝑑π‘₯β‰€π‘β€–π‘€β€–π»π‘ β€–π‘€β€–π»π‘žβ€–π‘£β€–π»π‘ +1+‖𝑀‖2𝐻𝑠.(3.25)

Lemma 3.11 (see Lai and Wu [16]). If 1/2<π‘ž<min{1,π‘ βˆ’1} and 𝑠>3/2, then for any functions 𝑀,𝑓 defined on 𝑅, it holds that ||||ξ€œπ‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2(𝑀𝑓)π‘₯||||𝑑π‘₯≀𝑐‖𝑀‖2π»π‘žβ€–π‘“β€–π»π‘ž||||ξ€œ,(3.26)π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2𝑀π‘₯𝑓π‘₯ξ€Έπ‘₯||||𝑑π‘₯≀𝑐‖𝑀‖2π»π‘žβ€–π‘“β€–π»π‘ .(3.27)

Lemma 3.12. For problem (3.17), 𝑠>3/2, and 𝑒0βˆˆπ»π‘ (𝑅), there exist two positive constants 𝑐 and 𝑀, which are independent of πœ€, such that the following inequalities hold for any sufficiently small πœ€ and π‘‘βˆˆ[0,𝑇): β€–β€–π‘’πœ€β€–β€–π»π‘ β‰€π‘€π‘’π‘π‘‘,β€–β€–π‘’πœ€β€–β€–π»1𝑠+π‘˜β‰€πœ€βˆ’π‘˜1/4𝑀𝑒𝑐𝑑,π‘˜1‖‖𝑒>0,πœ€π‘‘β€–β€–π»1𝑠+π‘˜β‰€πœ€βˆ’(π‘˜1+1)/4𝑀𝑒𝑐𝑑,π‘˜1>βˆ’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 π‘’πœ€ is a Cauchy sequence. Let π‘’πœ€ and 𝑒𝛿 be solutions of problem (3.17), corresponding to the parameters πœ€ and 𝛿, respectively, with 0<πœ€<𝛿<1/4, and let 𝑀=π‘’πœ€βˆ’π‘’π›Ώ. Then, 𝑀 satisfies the problem (1βˆ’πœ€)π‘€π‘‘βˆ’πœ€π‘€π‘₯π‘₯𝑑+𝑒(π›Ώβˆ’πœ€)𝛿𝑑+𝑒𝛿π‘₯π‘₯𝑑=ξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1ξ‚ƒβˆ’πœ€π‘€π‘‘+(π›Ώβˆ’πœ€)π‘’π›Ώπ‘‘βˆ’π‘˜πœ•π‘š+1π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έβˆ’πœ•π‘₯ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έβˆ’πœ•π‘₯ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀+πœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑒𝛿+ξ€Ίπ‘’π‘šπœ€π‘’πœ€π‘₯π‘’πœ€π‘₯π‘₯βˆ’π‘’π‘šπ›Ώπ‘’π›Ώπ‘₯𝑒𝛿π‘₯π‘₯ξ€»ξ‚„βˆ’1πœ•π‘š+2π‘₯ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2𝑀+πœ†π‘€,(3.29)(π‘₯,0)=𝑀0(π‘₯)=π‘’πœ€0(π‘₯)βˆ’π‘’π›Ώ0(π‘₯).(3.30)

Lemma 3.13. For 𝑠>3/2, 𝑒0βˆˆπ»π‘ (𝑅), there exists 𝑇>0 such that the solution π‘’πœ€ of (3.17) is a Cauchy sequence in 𝐢([0,𝑇];𝐻𝑠⋂𝐢(𝑅))1([0,𝑇];π»π‘ βˆ’1(𝑅)).

Proof. For π‘ž with 1/2<π‘ž<min{1,π‘ βˆ’1}, multiplying both sides of (3.29) by Ξ›π‘žπ‘€Ξ›π‘ž and then integrating with respect to π‘₯ give rise to 12π‘‘ξ€œπ‘‘π‘‘π‘…ξ‚ƒ(1βˆ’πœ€)(Ξ›π‘žπ‘€)2ξ€·Ξ›+πœ€π‘žπ‘€π‘₯ξ€Έ2ξ‚„ξ€œπ‘‘π‘₯=(πœ€βˆ’π›Ώ)𝑅(Ξ›π‘žΞ›π‘€)ξ€Ίξ€·π‘žπ‘’π›Ώπ‘‘ξ€Έ+ξ€·Ξ›π‘žπ‘’π›Ώπ‘₯π‘₯π‘‘ξ€œξ€Έξ€»π‘‘π‘₯βˆ’πœ€π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2π‘€π‘‘ξ€œπ‘‘π‘₯+(π›Ώβˆ’πœ€)π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2𝑒𝛿𝑑1𝑑π‘₯βˆ’ξ€œπ‘š+2𝑅(Ξ›π‘žπ‘€)Ξ›π‘žξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯βˆ’π‘˜π‘‘π‘₯ξ€œπ‘š+1π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπ‘₯ξ€œπ‘‘π‘₯βˆ’π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯βˆ’ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀π‘₯ξ€œπ‘‘π‘₯βˆ’π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπœ•π‘₯𝑒𝛿π‘₯+ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€Ίπ‘’π‘šπœ€π‘’πœ€π‘₯π‘’πœ€π‘₯π‘₯βˆ’π‘’π‘šπ›Ώπ‘’π›Ώπ‘₯𝑒𝛿π‘₯π‘₯ξ€»ξ€œπ‘‘π‘₯+πœ†π‘…Ξ›π‘žπ‘€Ξ›π‘žπ‘€π‘‘π‘₯.(3.31) It follows from the Schwarz inequality that π‘‘ξ€œξ‚ƒ(𝑑𝑑1βˆ’πœ€)(Ξ›π‘žπ‘€)2ξ€·Ξ›+πœ€π‘žπ‘€π‘₯ξ€Έ2𝑑π‘₯β‰€π‘β€–Ξ›π‘žπ‘€β€–πΏ2ξ€Ίξ€·β€–β€–Ξ›(π›Ώβˆ’πœ€)π‘žπ‘’π›Ώπ‘‘β€–β€–πΏ2+β€–β€–Ξ›π‘žπ‘’π›Ώπ‘₯π‘₯𝑑‖‖𝐿2ξ€Έβ€–β€–Ξ›+πœ€π‘žβˆ’2𝑀𝑑‖‖𝐿2β€–β€–Ξ›+(π›Ώβˆ’πœ€)π‘žβˆ’2𝑒𝛿𝑑‖‖𝐿2ξ€»+||πœ†||ξ€œπ‘…(Ξ›π‘žπ‘€)2||||ξ€œπ‘‘π‘₯+π‘…Ξ›π‘žπ‘€Ξ›π‘žξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯||||||||ξ€œΞ›π‘‘π‘₯π‘žπ‘€Ξ›π‘žβˆ’2ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπ‘₯||||+||||ξ€œΞ›π‘‘π‘₯π‘žπ‘€Ξ›π‘žβˆ’2ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯||||+||||ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀π‘₯||||+||||ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπœ•π‘₯𝑒𝛿π‘₯||||+||||ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€Ίπ‘’π‘šπœ€π‘’πœ€π‘₯π‘’πœ€π‘₯π‘₯βˆ’π‘’π‘šπ›Ώπ‘’π›Ώπ‘₯𝑒𝛿π‘₯π‘₯ξ€»||||ξƒ°.𝑑π‘₯(3.32) Using the first inequality in Lemma 3.9, we have ||||ξ€œπ‘…Ξ›π‘žπ‘€Ξ›π‘žξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯||||=||||ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘žξ€·π‘€π‘”π‘š+1ξ€Έπ‘₯||||𝑑π‘₯≀𝑐‖𝑀‖2π»π‘žβ€–β€–π‘”π‘š+1β€–β€–π»π‘ž+1,(3.33) where π‘”π‘š+1=βˆ‘π‘š+1𝑗=0π‘’πœ€π‘š+1βˆ’π‘—π‘’π‘—π›Ώ. For the last three terms in (3.32), using Lemmas 3.4 and 3.12, 1/2<π‘ž<min{1,π‘ βˆ’1}, 𝑠>3/2, the algebra property of 𝐻𝑠0 with 𝑠0>1/2, and (3.23), we have ||||ξ€œπ‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€·πœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀π‘₯||||𝑑π‘₯≀𝑐‖𝑀‖2π»π‘žβ€–β€–π‘’πœ€β€–β€–π»π‘š+1𝑠||||ξ€œ,(3.34)π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€·πœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπœ•π‘₯𝑒𝛿π‘₯||||𝑑π‘₯β‰€π‘β€–π‘€β€–π»π‘žβ€–β€–π‘’π›Ώβ€–β€–π»π‘ β€–β€–π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1β€–β€–π»π‘žβ‰€π‘β€–π‘€β€–2π»π‘žβ€–β€–π‘’π›Ώβ€–β€–π»π‘ ,||||ξ€œ(3.35)π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€Ίπ‘’π‘šπœ€π‘’πœ€π‘₯π‘’πœ€π‘₯π‘₯βˆ’π‘’π‘šπ›Ώπ‘’π›Ώπ‘₯𝑒𝛿π‘₯π‘₯ξ€»||||𝑑π‘₯β‰€π‘β€–π‘€β€–π»π‘žβ€–β€–ξ€·π‘’π‘šπœ€βˆ’π‘’π‘šπ›Ώπ‘’ξ€Έξ€·2πœ€π‘₯ξ€Έπ‘₯+π‘’π‘šπ›Ώξ€Ίπ‘’2πœ€π‘₯βˆ’π‘’2𝛿π‘₯ξ€»π‘₯β€–β€–π»π‘žβˆ’2β‰€π‘β€–π‘€β€–π»π‘žξ‚€β€–β€–ξ€·π‘’π‘šπœ€βˆ’π‘’π‘šπ›Ώπ‘’ξ€Έξ€·2πœ€π‘₯ξ€Έπ‘₯β€–β€–π»π‘žβˆ’1+β€–β€–π‘’π‘šπ›Ώξ€Ίπ‘’2πœ€π‘₯βˆ’π‘’2𝛿π‘₯ξ€»π‘₯β€–β€–π»π‘žβˆ’2ξ‚β‰€π‘β€–π‘€β€–π»π‘žξ‚€β€–β€–π‘’π‘šπœ€βˆ’π‘’π‘šπ›Ώβ€–β€–π»π‘žβ€–β€–ξ€·π‘’2πœ€π‘₯ξ€Έπ‘₯β€–β€–π»π‘žβˆ’1+β€–β€–π‘’π‘šπ›Ώβ€–β€–π»π‘ β€–β€–ξ€Ίπ‘’2πœ€π‘₯βˆ’π‘’2𝛿π‘₯ξ€»π‘₯β€–β€–π»π‘žβˆ’2ξ‚β‰€π‘β€–π‘€β€–π»π‘žξ€·β€–π‘€β€–π»π‘žβ€–β€–π‘”π‘šβˆ’1β€–β€–π»π‘žβ€–π‘’β€–2𝐻𝑠+β€–β€–π‘’π‘šπ›Ώβ€–β€–π»π‘ β€–β€–π‘’πœ€π‘₯+𝑒𝛿π‘₯β€–β€–π»π‘žβ€–π‘€β€–π»π‘žξ€Έβ‰€π‘β€–π‘€β€–2π»π‘žξ€·β€–β€–π‘”π‘šβˆ’1β€–β€–π»π‘žβ€–π‘’β€–2𝐻𝑠+β€–β€–π‘’π‘šπ›Ώβ€–β€–π»π‘ β€–β€–π‘’πœ€π‘₯+𝑒𝛿π‘₯β€–β€–π»π‘žξ€Έ.(3.36) Using (3.26), we derive that the inequality ||||ξ€œπ‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯||||=||||ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘žβˆ’2ξ€·π‘€π‘”π‘š+1ξ€Έπ‘₯||||‖‖𝑔𝑑π‘₯β‰€π‘π‘š+1β€–β€–π»π‘žβ€–π‘€β€–2π»π‘ž(3.37) holds for some constant 𝑐, where π‘”π‘š+1=βˆ‘π‘š+1𝑗=0π‘’πœ€π‘š+1βˆ’π‘—π‘’π‘—π›Ώ. Using the algebra property of π»π‘ž with π‘ž>1/2, π‘ž+1<𝑠 and Lemma 3.11, we have β€–π‘”π‘šβ€–π»π‘ž+1≀𝑐 for ξ‚π‘‘βˆˆ(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 π‘‘ξ€œπ‘‘π‘‘π‘…ξ‚ƒ(1βˆ’πœ€)(Ξ›π‘žπ‘€)2ξ€·Ξ›+πœ€π‘žπ‘€π‘₯ξ€Έ2𝛿𝑑π‘₯β‰€π‘π›Ύβ€–π‘€β€–π»π‘ž+‖𝑀‖2π»π‘žξ€Έ(3.38) holds for any ξ‚π‘‘βˆˆ[0,𝑇), where 𝛾=1 if 𝑠β‰₯3+π‘ž and 𝛾=(1+π‘ βˆ’π‘ž)/4 if 𝑠<3+π‘ž. Integrating (3.38) with respect to 𝑑, one obtains the estimate 12‖𝑀‖2π»π‘ž=12ξ€œπ‘…(Ξ›π‘žπ‘€)2β‰€ξ€œπ‘‘π‘₯𝑅(1βˆ’πœ€)(Ξ›π‘žπ‘€)2+πœ€(Ξ›π‘žπ‘€)2ξ€»β‰€ξ€œπ‘‘π‘₯π‘…ξ‚ƒξ€·Ξ›π‘žπ‘€0ξ€Έ2ξ€·Ξ›+πœ€π‘žπ‘€0π‘₯ξ€Έ2ξ‚„ξ€œπ‘‘π‘₯+𝑐𝑑0ξ€·π›Ώπ›Ύβ€–π‘€β€–π»π‘ž+‖𝑀‖2π»π‘žξ€Έπ‘‘πœ.(3.39) Applying the Gronwall inequality and using (3.20) and (3.22) yield β€–π‘’β€–π»π‘žβ‰€π‘π›Ώ(π‘ βˆ’π‘ž)/4𝑒𝑐𝑑+π›Ώπ›Ύξ€·π‘’π‘π‘‘ξ€Έβˆ’1(3.40) for any ξ‚π‘‘βˆˆ[0,𝑇).
Multiplying both sides of (3.29) by Λ𝑠𝑀Λ𝑠 and integrating the resultant equation with respect to π‘₯, one obtains 12π‘‘ξ€œπ‘‘π‘‘π‘…ξ‚ƒ(1βˆ’πœ€)(Λ𝑠𝑀)2ξ€·Ξ›+πœ€π‘ π‘€π‘₯ξ€Έ2ξ‚„ξ€œπ‘‘π‘₯=(πœ€βˆ’π›Ώ)𝑅(Λ𝑠Λ𝑀)𝑠𝑒𝛿𝑑+Λ𝑠𝑒𝛿π‘₯π‘₯π‘‘ξ€œξ€Έξ€»π‘‘π‘₯βˆ’πœ€π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2π‘€π‘‘ξ€œπ‘‘π‘₯+(π›Ώβˆ’πœ€)π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2π‘’π›Ώπ‘‘π‘˜π‘‘π‘₯βˆ’ξ€œπ‘š+1𝑅(Λ𝑠𝑀)Ξ›π‘ ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπ‘₯βˆ’1𝑑π‘₯ξ€œπ‘š+2𝑅(Λ𝑠𝑀)Ξ›π‘ ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯ξ€œπ‘‘π‘₯βˆ’π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯βˆ’ξ€œπ‘‘π‘₯π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀π‘₯ξ€œπ‘‘π‘₯βˆ’π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπœ•π‘₯𝑒𝛿π‘₯+ξ€œπ‘‘π‘₯π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€Ίπ‘’π‘šπœ€π‘’πœ€π‘₯π‘’πœ€π‘₯π‘₯βˆ’π‘’π‘šπ›Ώπ‘’π›Ώπ‘₯𝑒𝛿π‘₯π‘₯ξ€»ξ€œπ‘‘π‘₯+πœ†π‘…(Λ𝑠𝑀)2𝑑π‘₯.(3.41) From Lemma 3.12, we have ||||ξ€œπ‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯||||𝑑π‘₯≀𝑐3β€–β€–π‘”π‘š+1‖‖𝐻𝑠‖𝑀‖2𝐻𝑠.(3.42) From Lemma 3.10, it holds that ||||ξ€œπ‘…Ξ›π‘ π‘€Ξ›π‘ ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯||||𝑑π‘₯β‰€π‘β€–π‘€β€–π»π‘ β€–π‘€β€–π»π‘žβ€–β€–π‘’π›Ώβ€–β€–π»π‘ +1+‖𝑀‖2𝐻𝑠.(3.43) Using the Cauchy-Schwartz inequality and the algebra property of 𝐻𝑠0 with 𝑠0>1/2, for 𝑠>3/2, we have ||||ξ€œπ‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀π‘₯||||=||||ξ€œπ‘‘π‘₯π‘…Ξ›π‘žπ‘€Ξ›π‘ βˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀π‘₯||||𝑑π‘₯≀𝑐‖Λ𝑠𝑀‖𝐿2β€–β€–Ξ›π‘ βˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀π‘₯‖‖𝐿2β‰€π‘β€–π‘€β€–π»π‘žβ€–β€–πœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯π‘€β€–β€–π»π‘ βˆ’1β€–β€–π‘’β‰€π‘πœ€π»π‘š+1𝑠‖‖‖𝑀‖2𝐻𝑠,||||ξ€œ(3.44)π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπœ•π‘₯𝑒𝛿π‘₯||||𝑑π‘₯β‰€π‘β€–π‘€β€–π»π‘ β€–β€–Ξ›π‘ βˆ’2ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έπœ•π‘₯𝑒𝛿π‘₯‖‖𝐿2β€–β€–π‘’β‰€π‘π›Ώβ€–β€–π»π‘ β€–β€–π‘”π‘šβ€–β€–π»π‘ β€–π‘€β€–2𝐻𝑠,||||ξ€œ(3.45)π‘…Ξ›π‘ π‘€Ξ›π‘ βˆ’2ξ€Ίπ‘’π‘šπœ€π‘’πœ€π‘₯π‘’πœ€π‘₯π‘₯βˆ’π‘’π‘šπ›Ώπ‘’π›Ώπ‘₯𝑒𝛿π‘₯π‘₯ξ€»||||𝑑π‘₯β‰€π‘β€–π‘€β€–π»π‘ ξ‚€β€–β€–ξ€·π‘’π‘šπœ€βˆ’π‘’π‘šπ›Ώπ‘’ξ€Έξ€·2πœ€π‘₯ξ€Έπ‘₯β€–β€–π»π‘ βˆ’2+β€–β€–π‘’π‘šπ›Ώξ€Ίπ‘’2πœ€π‘₯βˆ’π‘’2𝛿π‘₯ξ€»π‘₯β€–β€–π»π‘ βˆ’2ξ‚β‰€π‘β€–π‘€β€–π»π‘ ξ‚€β€–β€–π‘’π‘šπœ€βˆ’π‘’π‘šπ›Ώβ€–β€–π»π‘ β€–β€–ξ€·π‘’2πœ€π‘₯ξ€Έπ‘₯β€–β€–π»π‘ βˆ’2+β€–β€–π‘’π‘šπ›Ώβ€–β€–π»π‘ β€–β€–ξ€Ίπ‘’2πœ€π‘₯βˆ’π‘’2𝛿π‘₯ξ€»π‘₯β€–β€–π»π‘ βˆ’2≀𝑐‖𝑀‖2𝐻𝑠,(3.46) in which we have used Lemma 3.4 and the bounded property of β€–π‘’πœ€β€–π»π‘  and ‖𝑒𝛿‖𝐻𝑠 (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 π‘š such that π‘‘ξ€œπ‘‘π‘‘π‘…ξ‚ƒ(1βˆ’πœ€)(Λ𝑠𝑀)2ξ€·Ξ›+πœ€π‘ π‘€π‘₯ξ€Έ2‖‖𝑒𝑑π‘₯≀2𝛿𝛿𝑑‖‖𝐻𝑠+‖‖𝑒𝛿π‘₯π‘₯𝑑‖‖𝐻𝑠+β€–β€–Ξ›π‘ βˆ’2𝑀𝑑‖‖𝐿2+β€–β€–Ξ›π‘ βˆ’2𝑒𝛿𝑑‖‖‖𝑀‖𝐻𝑠+𝑐‖𝑀‖2𝐻𝑠+β€–π‘€β€–π»π‘žβ€–π‘€β€–π»π‘ β€–β€–π‘’π›Ώβ€–β€–π»π‘ +1𝛿≀𝑐𝛾1‖𝑀‖𝐻𝑠+‖𝑀‖2𝐻𝑠,(3.47) where 𝛾1=min(1/4,(π‘ βˆ’π‘žβˆ’1)/4)>0. Integrating (3.47) with respect to 𝑑 leads to the estimate 12‖𝑀‖2π»π‘ β‰€ξ€œπ‘…ξ‚ƒ(1βˆ’πœ€)(Λ𝑠𝑀)2ξ€·Ξ›+πœ€π‘ π‘€π‘₯ξ€Έ2ξ‚„β‰€ξ€œπ‘‘π‘₯𝑅Λ𝑠𝑀0ξ€Έ2ξ€·Ξ›+πœ€π‘ π‘€0π‘₯ξ€Έ2ξ‚„ξ€œπ‘‘π‘₯+𝑐𝑑0𝛿𝛾1‖𝑀‖𝐻𝑠+‖𝑀‖2π»π‘ ξ€Έπ‘‘πœ.(3.48) It follows from the Gronwall inequality and (3.48) that ‖𝑀‖𝐻𝑠≀2ξ€œπ‘…ξ‚ƒξ€·Ξ›π‘ π‘€0ξ€Έ2ξ€·Ξ›+πœ€π‘ π‘€0π‘₯ξ€Έ2𝑑π‘₯1/2𝑒𝑐𝑑+𝛿𝛾1ξ€·π‘’π‘π‘‘ξ€Έβˆ’1≀𝑐1‖‖𝑀0‖‖𝐻𝑠+𝛿3/4𝑒𝑐𝑑+𝛿𝛾1𝑒𝑐𝑑,βˆ’1(3.49) where 𝑐1 is independent of πœ€ and 𝛿.
Then, (3.22) and the above inequality show that β€–π‘€β€–π»π‘ βŸΆ0asπœ€βŸΆ0,π›ΏβŸΆ0.(3.50) Next, we consider the convergence of the sequence {π‘’πœ€π‘‘}. Multiplying both sides of (3.29) by Ξ›π‘ βˆ’1π‘€π‘‘Ξ›π‘ βˆ’1 and integrating the resultant equation with respect to π‘₯, we obtain (‖‖𝑀1βˆ’πœ€)𝑑‖‖2π»π‘ βˆ’1+1ξ€œπ‘š+2π‘…ξ€·Ξ›π‘ βˆ’1π‘€π‘‘ξ€ΈΞ›π‘ βˆ’1ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έπ‘₯+ξ€œπ‘‘π‘₯π‘…ξ€Ίξ€·Ξ›βˆ’πœ€π‘ βˆ’1π‘€π‘‘Ξ›ξ€Έξ€·π‘ βˆ’1𝑀π‘₯π‘₯𝑑Λ+(π›Ώβˆ’πœ€)π‘ βˆ’1π‘€π‘‘ξ€ΈΞ›π‘ βˆ’1𝑒𝛿𝑑+𝑒𝛿π‘₯π‘₯𝑑=ξ€œξ€Έξ€»π‘‘π‘₯π‘…ξ€·Ξ›π‘ βˆ’1π‘€π‘‘ξ€ΈΞ›π‘ βˆ’3ξ‚ƒβˆ’πœ€π‘€π‘‘+(π›Ώβˆ’πœ€)π‘’π›Ώπ‘‘βˆ’π‘˜πœ•π‘š+1π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’π›Ώπ‘š+1ξ€Έβˆ’πœ•π‘₯ξ€·π‘’πœ€π‘š+2βˆ’π‘’π›Ώπ‘š+2ξ€Έβˆ’πœ•π‘₯ξ€Ίπœ•π‘₯ξ€·π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑀+πœ•π‘₯ξ€·π‘’πœ€π‘š+1βˆ’π‘’πœ€π‘š+1ξ€Έπœ•π‘₯𝑒𝛿+ξ€Ίπ‘’π‘šπœ€π‘’πœ€π‘₯π‘’πœ€π‘₯π‘₯βˆ’π‘’π‘šπ›Ώπ‘’π›Ώπ‘₯𝑒𝛿π‘₯π‘₯ξ€»ξ‚„ξ€œπ‘‘π‘₯+πœ†π‘…Ξ›π‘ βˆ’1π‘€π‘‘Ξ›π‘ βˆ’1𝑀𝑑π‘₯.(3.51)
It follows from inequalities (3.28) and the Schwartz inequality that there is a constant 𝑐 depending on 𝑇 and π‘š such that ‖‖𝑀(1βˆ’πœ€)𝑑‖‖2π»π‘ βˆ’1𝛿≀𝑐1/2+‖𝑀‖𝐻𝑠+β€–π‘€β€–π‘ βˆ’1ξ€Έβ€–β€–π‘€π‘‘β€–β€–π»π‘ βˆ’1‖‖𝑀+πœ€π‘‘β€–β€–2π»π‘ βˆ’1(3.52) Hence 12‖‖𝑀𝑑‖‖2π»π‘ βˆ’1‖‖𝑀≀(1βˆ’2πœ€)𝑑‖‖2π»π‘ βˆ’1𝛿≀𝑐1/2+‖𝑀‖𝐻𝑠+β€–π‘€β€–π»π‘ βˆ’1ξ€Έβ€–β€–π‘€π‘‘β€–β€–π»π‘ βˆ’1,(3.53) which results in 12β€–β€–π‘€π‘‘β€–β€–π»π‘ βˆ’1𝛿≀𝑐1/2+‖𝑀‖𝐻𝑠+β€–π‘€β€–π»π‘ βˆ’1ξ€Έ.(3.54)
It follows from (3.40) and (3.50) that 𝑀𝑑→0 as πœ€, 𝛿→0 in the π»π‘ βˆ’1 norm. This implies that π‘’πœ€ is a Cauchy sequence in the spaces 𝐢([0,𝑇);𝐻𝑠(𝑅)) and 𝐢([0,𝑇);π»π‘ βˆ’1(𝑅)), respectively. The proof is completed.

4. Proof of the Main Result

We consider the problem (1βˆ’πœ€)π‘’π‘‘βˆ’πœ€π‘’π‘‘π‘₯π‘₯=ξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1ξ‚ƒβˆ’π‘˜ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ‚„+πœ†π‘’,𝑒(0,π‘₯)=π‘’πœ€0(π‘₯).(4.1) Letting 𝑒(𝑑,π‘₯) be the limit of the sequence π‘’πœ€ and taking the limit in problem (4.1) as πœ€β†’0, from Lemma 3.13, we know that 𝑒 is a solution of the problem 𝑒𝑑=ξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1ξ‚ƒβˆ’π‘˜ξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯βˆ’π‘š+3ξ€·π‘’π‘š+2π‘š+2ξ€Έπ‘₯+1πœ•π‘š+23π‘₯ξ€·π‘’π‘š+2ξ€Έβˆ’(π‘š+1)πœ•π‘₯ξ€·π‘’π‘šπ‘’2π‘₯ξ€Έ+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯ξ‚„+πœ†π‘’,𝑒(0,π‘₯)=𝑒0(π‘₯),(4.2) and hence 𝑒 is a solution of problem (4.2) in the sense of distribution. In particular, if 𝑠β‰₯4, 𝑒 is also a classical solution. Let 𝑒 and 𝑣 be two solutions of (4.2) corresponding to the same initial value 𝑒0 such that 𝑒, π‘£βˆˆπΆ([0,𝑇);𝐻𝑠(𝑅)). Then, 𝑀=π‘’βˆ’π‘£ satisfies the Cauchy problem 𝑀𝑑=ξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1ξ‚†πœ•π‘₯ξ‚ƒβˆ’π‘˜π‘š+1π‘€π‘”π‘šβˆ’π‘š+3π‘š+2π‘€π‘”π‘š+1+1πœ•π‘š+22π‘₯ξ€·π‘€π‘”π‘š+1ξ€Έβˆ’πœ•π‘₯ξ€·π‘’π‘š+1ξ€Έπœ•π‘₯π‘€βˆ’πœ•π‘₯ξ€·π‘’π‘š+1βˆ’π‘£π‘š+1ξ€Έπœ•π‘₯𝑣+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯βˆ’π‘£π‘šπ‘£π‘₯𝑣π‘₯π‘₯+πœ†π‘€,𝑀(0,π‘₯)=0.(4.3) For any 1/2<π‘ž<min{1,π‘ βˆ’1}, applying the operator Ξ›π‘žπ‘€Ξ›π‘ž to both sides of (4.3) and integrating the resultant equation with respect to π‘₯, we obtain the equality 12𝑑𝑑𝑑‖𝑀‖2π»π‘ž=ξ€œπ‘…(Ξ›π‘žπ‘€)Ξ›π‘žβˆ’2ξ‚†πœ•π‘₯ξ‚ƒβˆ’π‘˜π‘š+1π‘€π‘”π‘šβˆ’π‘š+3π‘š+2π‘€π‘”π‘š+1+1πœ•π‘š+22π‘₯ξ€·π‘€π‘”π‘š+1ξ€Έβˆ’πœ•π‘₯ξ€·π‘’π‘š+1ξ€Έπœ•π‘₯π‘€βˆ’πœ•π‘₯ξ€·π‘’π‘š+1βˆ’π‘£π‘š+1ξ€Έπœ•π‘₯𝑣+π‘’π‘šπ‘’π‘₯𝑒π‘₯π‘₯βˆ’π‘£π‘šπ‘£π‘₯𝑣π‘₯π‘₯||πœ†||𝑑π‘₯+‖𝑀‖2π»π‘ž.(4.4) By the similar estimates presented in Lemma 3.13, we have 𝑑𝑑𝑑‖𝑀‖2π»π‘žβ‰€Μƒπ‘β€–π‘€β€–2π»π‘ž.(4.5) Using the Gronwall inequality leads to the conclusion that β€–π‘€β€–π»π‘žβ‰€0×𝑒̃𝑐𝑑=0(4.6) for ξ‚π‘‘βˆˆ[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).

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