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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 285040, 15 pages
http://dx.doi.org/10.1155/2011/285040
Research Article

The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

Received 17 January 2011; Revised 3 August 2011; Accepted 8 August 2011

Academic Editor: Yuri V.Β Rogovchenko

Copyright Β© 2011 Shaoyong Lai. 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

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in space 𝐢([0,𝑇),𝐻𝑠(𝑅))∩𝐢1([0,𝑇),π»π‘ βˆ’1(𝑅)) with 𝑠>3/2. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space 𝐻𝑠(𝑅) with 1≀𝑠≀3/2 is developed.

1. Introduction

Camassa and Holm [1] used the Hamiltonian method to derive a completely integrable wave equationπ‘’π‘‘βˆ’π‘’π‘₯π‘₯𝑑+2π‘˜π‘’π‘₯+3𝑒𝑒π‘₯=2𝑒π‘₯𝑒π‘₯π‘₯+𝑒𝑒π‘₯π‘₯π‘₯,(1.1) by retaining two terms that are usually neglected in the small amplitude, shallow water limit. Its alternative derivation as a model for water waves can be found in Constantin and Lannes [2] and Johnson [3]. Equation (1.1) also models wave current interaction [4], while Dai [5] derived it as a model in elasticity (see Constantin and Strauss [6]). Moreover, it was pointed out in Lakshmanan [7] that the Camassa-Holm equation (1.1) could be relevant to the modeling of tsunami waves (see Constantin and Johnson [8]).

In fact, a huge amount of work has been carried out to investigate the dynamic properties of (1.1). For π‘˜=0, (1.1) has traveling wave solutions of the form 𝑐eβˆ’|π‘₯βˆ’π‘π‘‘|, called peakons, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations (see [9–11]). For π‘˜>0, its solitary waves are stable solitons [6, 11]. It was shown in [12–14] that the inverse spectral or scattering approach was a powerful tool to handle Camassa-Holm equation. Equation (1.1) is a completely integrable infinite-dimensional Hamiltonian system (in the sense that for a large class of initial data, the flow is equivalent to a linear flow at constant speed [15]). It should be emphasized that (1.1) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group (see [16, 17]), and this geometric illustration leads to a proof that the Least Action Principle holds. It is worthwhile to mention that Xin and Zhang [18] proved that the global existence of the weak solution in the energy space 𝐻1(𝑅) without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under some conditions on the solution [19]. Coclite et al. [20] extended the analysis presented in [18, 19] and obtained many useful dynamic properties to other equations (also see [21–24]). Li and Olver [25] established the local well-posedness in the Sobolev space 𝐻𝑠(𝑅) with 𝑠>3/2 for (1.1) and gave conditions on the initial data that lead to finite time blowup of certain solutions. It was shown in Constantin and Escher [26] that the blowup occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. After wave breaking, the solution can be continued uniquely either as a global conservative weak solution [21] or a global dissipative solution [22]. For peakons, these possibilities are explicitly illustrated in the paper [27]. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water models, the reader is referred to [10, 28–32] and the references therein.

In this paper, motivated by the work in [25, 33], we study the following generalized Camassa-Holm equationπ‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯+2π‘˜π‘’π‘₯+π‘Žπ‘’π‘šπ‘’π‘₯=2𝑒π‘₯𝑒π‘₯π‘₯+𝑒𝑒π‘₯π‘₯π‘₯+π›½πœ•π‘₯𝑒π‘₯𝑁,(1.2) where π‘šβ‰₯1 and 𝑁β‰₯1 are natural numbers, and π‘Ž, π‘˜, and 𝛽 are arbitrary constants. Obviously, (1.2) reduces to (1.1) if we set π‘Ž=3, π‘š=1, and 𝛽=0. Actually, Wu and Yin [34] consider a nonlinearly dissipative Camassa-Holm equation which includes a nonlinearly dissipative term 𝐿(𝑒), where 𝐿 is a differential operator or a quasidifferential operator. Therefore, we can regard the term π›½πœ•π‘₯[(𝑒π‘₯)𝑁] as a nonlinearly dissipative term for the dissipative Camassa-Holm equation (1.2).

Due to the term π›½πœ•π‘₯[(𝑒π‘₯)𝑁] in (1.2), the conservation laws in previous works [10, 25] for (1.1) lose their powers to obtain some bounded estimates of the solution for (1.2). A new conservation law different from those presented in [10, 25] will be established to prove the local existence and uniqueness of the solution to (2.3) subject to initial value 𝑒0(π‘₯)βˆˆπ»π‘ (𝑅) with 𝑠>3/2. We should address that all the generalized versions of the Camassa-Holm equation in previous works (see [17, 25, 34]) do not involve the nonlinear term πœ•π‘₯[(𝑒π‘₯)𝑁]. Lai and Wu [33] only studied a generalized Camassa-Holm equation in the case where 𝛽β‰₯0 and 𝑁 is an odd number. Namely, (1.2) with 𝛽<0 and arbitrary positive integer 𝑁 was not investigated in [33].

The main tasks of this paper are two-fold. Firstly, by using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for (1.2) with any 𝛽 and arbitrary positive integer 𝑁 in space 𝐢([0,𝑇),𝐻𝑠⋂𝐢(𝑅))1([0,𝑇),π»π‘ βˆ’1(𝑅)) with 𝑠>3/2. Secondly, it is shown that the existence of weak solutions in lower order Sobolev space 𝐻𝑠(𝑅) with 1≀𝑠≀3/2. The ideas of proving the second result come from those presented in Li and Olver [25].

2. Main Results

Firstly, we give some notation.

The space of all infinitely differentiable functions πœ™(𝑑,π‘₯) with compact support in [0,+∞)×𝑅 is denoted by 𝐢∞0. 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 byβ€–β„Žβ€–π»π‘ =ξ‚΅ξ€œπ‘…ξ‚€||πœ‰||1+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.

In order to study the existence of solutions for (1.2), we consider its Cauchy problem in the formπ‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯=βˆ’2π‘˜π‘’π‘₯βˆ’π‘Žξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯+2𝑒π‘₯𝑒π‘₯π‘₯+𝑒𝑒π‘₯π‘₯π‘₯+π›½πœ•π‘₯𝑒π‘₯𝑁=βˆ’π‘˜π‘’π‘₯βˆ’π‘Žξ€·π‘’π‘š+1π‘š+1ξ€Έπ‘₯+12πœ•3π‘₯𝑒2βˆ’12πœ•π‘₯𝑒2π‘₯ξ€Έ+π›½πœ•π‘₯𝑒π‘₯𝑁,𝑒(0,π‘₯)=𝑒0(π‘₯),(2.2) which is equivalent to𝑒𝑑+𝑒𝑒π‘₯=Ξ›βˆ’2ξ‚ƒπ‘Žβˆ’π‘˜π‘’βˆ’ξ€·π‘’π‘š+1π‘š+1ξ€Έξ‚„π‘₯+Ξ›βˆ’2𝑒𝑒π‘₯ξ€Έβˆ’12Ξ›βˆ’2πœ•π‘₯𝑒2π‘₯ξ€Έ+π›½Ξ›βˆ’2πœ•π‘₯𝑒π‘₯𝑁,𝑒(0,π‘₯)=𝑒0(π‘₯).(2.3)

Now, we state our main results.

Theorem 2.1. Let 𝑒0(π‘₯)βˆˆπ»π‘ (𝑅) with 𝑠>3/2. Then problem (2.2) or problem (2.3) has a unique solution 𝑒(𝑑,π‘₯)∈𝐢([0,𝑇);𝐻𝑠⋂𝐢(𝑅))1([0,𝑇);π»π‘ βˆ’1(𝑅)) where 𝑇>0 depends on ‖𝑒0‖𝐻𝑠(𝑅).

Theorem 2.2. Suppose that 𝑒0(π‘₯)βˆˆπ»π‘  with 1≀𝑠≀3/2 and ‖𝑒0π‘₯β€–πΏβˆž<∞. Then there exists a 𝑇>0 such that (1.2) subject to initial value 𝑒0(π‘₯) has a weak solution 𝑒(𝑑,π‘₯)∈𝐿2([0,𝑇],𝐻𝑠) in the sense of distribution and 𝑒π‘₯∈𝐿∞([0,𝑇]×𝑅).

3. Local Well-Posedness

We consider the abstract quasilinear evolution equation𝑑𝑣𝑑𝑑+𝐴(𝑣)𝑣=𝑓(𝑣),𝑑β‰₯0,𝑣(0)=𝑣0.(3.1) Let 𝑋 and π‘Œ be Hilbert spaces such that π‘Œ is continuously and densely embedded in 𝑋, and let π‘„βˆΆπ‘Œβ†’π‘‹ be a topological isomorphism. Let 𝐿(π‘Œ,𝑋) be the space of all bounded linear operators from π‘Œ to 𝑋. If 𝑋=π‘Œ, we denote this space by 𝐿(𝑋). We state the following conditions in which 𝜌1,𝜌2,𝜌3, and 𝜌4 are constants depending on max{β€–π‘¦β€–π‘Œ,β€–π‘§β€–π‘Œ}.(i)𝐴(𝑦)∈𝐿(π‘Œ,𝑋) for π‘¦βˆˆπ‘‹ with β€–(𝐴(𝑦)βˆ’π΄(𝑧))π‘€β€–π‘‹β‰€πœŒ1β€–π‘¦βˆ’π‘§β€–π‘‹β€–π‘€β€–π‘Œ,𝑦,𝑧,π‘€βˆˆπ‘Œ,(3.2) and 𝐴(𝑦)∈𝐺(𝑋,1,𝛽) (i.e., 𝐴(𝑦) is quasi-π‘š-accretive), uniformly on bounded sets in π‘Œ.(ii)𝑄𝐴(𝑦)π‘„βˆ’1=𝐴(𝑦)+𝐡(𝑦), where 𝐡(𝑦)∈𝐿(𝑋) is bounded, uniformly on bounded sets in π‘Œ. Moreover, β€–(𝐡(𝑦)βˆ’π΅(𝑧))π‘€β€–π‘‹β‰€πœŒ2β€–π‘¦βˆ’π‘§β€–π‘Œβ€–π‘€β€–π‘‹,𝑦,π‘§βˆˆπ‘Œ,π‘€βˆˆπ‘‹.(3.3)(iii)π‘“βˆΆπ‘Œβ†’π‘Œ extends to a map from 𝑋 into 𝑋 is bounded on bounded sets in π‘Œ, and satisfies‖𝑓(𝑦)βˆ’π‘“(𝑧)β€–π‘Œβ‰€πœŒ3β€–π‘¦βˆ’π‘§β€–π‘Œβ€–,𝑦,π‘§βˆˆπ‘Œ,‖𝑓(𝑦)βˆ’π‘“(𝑧)π‘‹β‰€πœŒ4β€–π‘¦βˆ’π‘§β€–π‘‹,𝑦,π‘§βˆˆπ‘Œ.(3.4)

Kato Theorem (see [35])
Assume that (i), (ii), and (iii) hold. If 𝑣0βˆˆπ‘Œ, there is a maximal 𝑇>0 depending only on ‖𝑣0β€–π‘Œ, and a unique solution 𝑣 to problem (3.1) such that 𝑣=𝑣⋅,𝑣0ξ€Έ[ξ™πΆβˆˆπΆ(0,𝑇);π‘Œ)1([0,𝑇);𝑋).(3.5) Moreover, the map 𝑣0→𝑣(β‹…,𝑣0) is a continuous map from π‘Œ to the space [𝐢𝐢(0,𝑇);π‘Œ)1([0,𝑇);𝑋).(3.6)

For problem (2.3), we set 𝐴(𝑒)=π‘’πœ•π‘₯, π‘Œ=𝐻𝑠(𝑅), 𝑋=π»π‘ βˆ’1(𝑅), Ξ›=(1βˆ’πœ•2π‘₯)1/2, 𝑓(𝑒)=Ξ›βˆ’2ξ‚ƒπ‘Žβˆ’π‘˜π‘’βˆ’ξ€·π‘’π‘š+1π‘š+1ξ€Έξ‚„π‘₯+Ξ›βˆ’2𝑒𝑒π‘₯ξ€Έβˆ’12Ξ›βˆ’2πœ•π‘₯𝑒2π‘₯ξ€Έ+π›½Ξ›βˆ’2πœ•π‘₯𝑒π‘₯𝑁,(3.7) and 𝑄=Ξ›. In order to prove Theorem 2.1, we only need to check that 𝐴(𝑒) and 𝑓(𝑒) satisfy assumptions (i)–(iii).

Lemma 3.1. The operator 𝐴(𝑒)=π‘’πœ•π‘₯ with π‘’βˆˆπ»π‘ (𝑅), 𝑠>3/2 belongs to 𝐺(π»π‘ βˆ’1,1,𝛽).

Lemma 3.2. Let 𝐴(𝑒)=π‘’πœ•π‘₯ with π‘’βˆˆπ»π‘  and 𝑠>3/2. Then 𝐴(𝑒)∈𝐿(𝐻𝑠,π»π‘ βˆ’1) for all π‘’βˆˆπ»π‘ . Moreover, β€–(𝐴(𝑒)βˆ’π΄(𝑧))π‘€β€–π»π‘ βˆ’1β‰€πœŒ1β€–π‘’βˆ’π‘§β€–π»π‘ βˆ’1‖𝑀‖𝐻𝑠,𝑒,𝑧,π‘€βˆˆπ»π‘ (𝑅).(3.8)

Lemma 3.3. For 𝑠>3/2, 𝑒,π‘§βˆˆπ»π‘  and π‘€βˆˆπ»π‘ βˆ’1, it holds that 𝐡(𝑒)=[Ξ›,π‘’πœ•π‘₯]Ξ›βˆ’1∈𝐿(π»π‘ βˆ’1) for π‘’βˆˆπ»π‘  and β€–(𝐡(𝑒)βˆ’π΅(𝑧))π‘€β€–π»π‘ βˆ’1β‰€πœŒ2β€–π‘’βˆ’π‘§β€–π»π‘ β€–π‘€β€–π»π‘ βˆ’1.(3.9)

Proofs of the above Lemmas 3.1–3.3 can be found in [29] or [31].

Lemma 3.4 (see [35]). 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,ifπ‘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,ifπ‘Ÿ<2.(3.10)

Lemma 3.5. Let 𝑒,π‘§βˆˆπ»π‘  with 𝑠>3/2, then 𝑓(𝑒) is bounded on bounded sets in 𝐻𝑠 and satisfies ‖𝑓(𝑒)βˆ’π‘“(𝑧)β€–π»π‘ β‰€πœŒ3β€–π‘’βˆ’π‘§β€–π»π‘ ,(3.11)‖𝑓(𝑒)βˆ’π‘“(𝑧)β€–π»π‘ βˆ’1β‰€πœŒ4β€–π‘’βˆ’π‘§β€–π»π‘ βˆ’1.(3.12)

Proof. Using the algebra property of the space 𝐻𝑠0 with 𝑠0>1/2, we have ‖𝑓(𝑒)βˆ’π‘“(𝑧)β€–π»π‘ ξ‚Έβ€–β€–β€–Ξ›β‰€π‘βˆ’2ξ‚΅ξ‚ƒπ‘Žβˆ’π‘˜π‘’βˆ’ξ€·π‘’π‘š+1π‘š+1ξ€Έξ‚„π‘₯βˆ’ξ‚ƒπ‘Žβˆ’π‘˜π‘§βˆ’ξ€·π‘§π‘š+1π‘š+1ξ€Έξ‚„π‘₯‖‖‖𝐻𝑠+β€–β€–Ξ›βˆ’2𝑒𝑒π‘₯βˆ’π‘§π‘§π‘₯‖‖𝐻𝑠+β€–β€–Ξ›βˆ’2πœ•π‘₯𝑒2π‘₯βˆ’π‘§2π‘₯‖‖𝐻𝑠+β€–β€–Ξ›βˆ’2πœ•π‘₯𝑒π‘₯ξ€Έπ‘ξ‚„βˆ’Ξ›βˆ’2πœ•π‘₯𝑧π‘₯ξ€Έπ‘ξ‚„β€–β€–π»π‘ ξ‚Ήξ€Ίβ‰€π‘β€–π‘’βˆ’π‘§β€–π»π‘ βˆ’1+β€–β€–π‘’π‘š+1βˆ’π‘§π‘š+1β€–β€–π»π‘ βˆ’1+‖‖𝑒𝑒π‘₯βˆ’π‘§π‘§π‘₯β€–β€–π»π‘ βˆ’1+‖‖𝑒2π‘₯βˆ’π‘§2π‘₯β€–β€–π»π‘ βˆ’1+β€–β€–(𝑒π‘₯)π‘βˆ’(𝑧π‘₯)π‘β€–β€–π»π‘ βˆ’1ξ€»β‰€π‘β€–π‘’βˆ’π‘§β€–π»π‘ ξƒ¬1+π‘šξ“π‘—=0β€–π‘’β€–π»π‘šβˆ’π‘—π‘ β€–π‘§β€–π‘—π»π‘ +‖𝑒‖𝐻𝑠+‖𝑧‖𝐻𝑠+π‘βˆ’1𝑗=0‖‖𝑒π‘₯β€–β€–π»π‘βˆ’π‘—π‘ βˆ’1‖‖𝑧π‘₯β€–β€–π‘—π»π‘ βˆ’1ξƒ­β‰€πœŒ3β€–π‘’βˆ’π‘§β€–π»π‘ ,(3.13) from which we obtain (3.11).
Applying Lemma 3.4, 𝑒𝑒π‘₯=(1/2)(𝑒2)π‘₯, 𝑠>3/2,β€–π‘’β€–πΏβˆžβ‰€π‘β€–π‘’β€–π»π‘ βˆ’1 and ‖𝑒π‘₯β€–πΏβˆžβ‰€π‘β€–π‘’β€–π»π‘ , we get ‖𝑓(𝑒)βˆ’π‘“(𝑧)β€–π»π‘ βˆ’1βŽ‘βŽ’βŽ’βŽ£β‰€π‘β€–π‘’βˆ’π‘§β€–π»π‘ βˆ’2+β€–β€–π‘’π‘š+1βˆ’π‘§π‘š+1β€–β€–π»π‘ βˆ’2+‖‖𝑒2βˆ’π‘§2β€–β€–π»π‘ βˆ’2+β€–β€–(𝑒π‘₯βˆ’π‘§π‘₯)(𝑒π‘₯+𝑧π‘₯)β€–β€–π»π‘ βˆ’2+β€–β€–β€–β€–(𝑒π‘₯βˆ’π‘§π‘₯)π‘βˆ’1𝑗=0𝑒π‘₯π‘βˆ’1βˆ’π‘—π‘§π‘—π‘₯β€–β€–β€–β€–π»π‘ βˆ’2⎀βŽ₯βŽ₯βŽ¦β‰€π‘β€–π‘’βˆ’π‘§β€–π»π‘ βˆ’11+π‘šξ“π‘—=0β€–π‘’β€–π»π‘šβˆ’π‘—π‘ βˆ’1β€–π‘§β€–π‘—π»π‘ βˆ’1+β€–π‘’β€–π»π‘ βˆ’1+β€–π‘§β€–π»π‘ βˆ’1+‖𝑒‖𝐻𝑠+‖𝑧‖𝐻𝑠+π‘βˆ’1𝑗=0‖‖𝑒π‘₯β€–β€–π»π‘βˆ’π‘—π‘ βˆ’1‖‖𝑧π‘₯β€–β€–π‘—π»π‘ βˆ’1ξƒ­β‰€πœŒ4β€–π‘’βˆ’π‘§β€–π»π‘ βˆ’1,(3.14) which completes the proof of (3.12).

Proof of Theorem 2.1. Using the Kato Theorem, Lemmas 3.1–3.3, and 3.5, we know that system (2.2) or problem (2.3) has a unique solution [𝑒(𝑑,π‘₯)∈𝐢(0,𝑇);𝐻𝑠𝐢(𝑅))1ξ€·[0,𝑇);π»π‘ βˆ’1ξ€Έ(𝑅).(3.15)

4. Existence of Weak Solutions

For 𝑠β‰₯2, using the first equation of system (2.2) derivesπ‘‘ξ€œπ‘‘π‘‘π‘…ξ‚΅π‘’2+𝑒2π‘₯ξ€œ+2𝛽𝑑0𝑒π‘₯𝑁+1ξ‚Άπ‘‘πœπ‘‘π‘₯=0,(4.1) from which we have the conservation lawξ€œπ‘…ξ‚΅π‘’2+𝑒2π‘₯ξ€œ+2𝛽𝑑0𝑒π‘₯𝑁+1ξ‚Άξ€œπ‘‘πœπ‘‘π‘₯=𝑅𝑒20+𝑒20π‘₯𝑑π‘₯.(4.2)

Lemma 4.1 (Kato and Ponce [36]). If π‘Ÿ>0, then π»π‘Ÿβ‹‚πΏβˆž is an algebra. Moreover, β€–π‘’π‘£β€–π‘Ÿξ€·β‰€π‘β€–π‘’β€–πΏβˆžβ€–π‘£β€–π‘Ÿ+β€–π‘’β€–π‘Ÿβ€–π‘£β€–πΏβˆžξ€Έ,(4.3) where 𝑐 is a constant depending only on π‘Ÿ.

Lemma 4.2 (Kato and Ponce [36]). Let π‘Ÿ>0. If π‘’βˆˆπ»π‘Ÿβ‹‚π‘Š1,∞ and π‘£βˆˆπ»π‘Ÿβˆ’1β‹‚πΏβˆž, then β€–β€–ξ€ΊΞ›π‘Ÿξ€»π‘£β€–β€–,𝑒𝐿2ξ€·β€–β€–πœ•β‰€π‘π‘₯π‘’β€–β€–πΏβˆžβ€–β€–Ξ›π‘Ÿβˆ’1𝑣‖‖𝐿2+β€–Ξ›π‘Ÿπ‘’β€–πΏ2β€–π‘£β€–πΏβˆžξ€Έ.(4.4)

Lemma 4.3. Let 𝑠β‰₯2 and the function 𝑒(𝑑,π‘₯) is a solution of problem (2.2) and the initial data 𝑒0(π‘₯)βˆˆπ»π‘ (𝑅). Then the following inequality holds β€–π‘’β€–πΏβˆžβ‰€β€–π‘’β€–π»1≀‖‖𝑒0‖‖𝐻1π‘’βˆ«|𝛽|𝑑0‖𝑒π‘₯β€–πΏβˆžπ‘βˆ’1π‘‘πœ.(4.5)
For π‘žβˆˆ(0,π‘ βˆ’1], there is a constant 𝑐, which only depends on π‘š, 𝑁, π‘˜, π‘Ž, and 𝛽, such that ξ€œπ‘…ξ€·Ξ›π‘ž+1𝑒2ξ€œπ‘‘π‘₯β‰€π‘…ξ€·Ξ›π‘ž+1𝑒0ξ€Έ2ξ€œπ‘‘π‘₯+𝑐𝑑0‖‖𝑒π‘₯β€–β€–πΏβˆžβ€–π‘’β€–2π»π‘ž+1ξ€·1+β€–π‘’β€–πΏπ‘šβˆ’1βˆžξ€Έξ€œπ‘‘πœ+𝑐𝑑0‖𝑒‖2π»π‘ž+1‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1βˆžπ‘‘πœ.(4.6)
For π‘žβˆˆ[0,π‘ βˆ’1], there is a constant 𝑐, which only depends on π‘š,𝑁,π‘˜,π‘Ž, and 𝛽, such that β€–β€–π‘’π‘‘β€–β€–π»π‘žβ‰€π‘β€–π‘’β€–π»π‘ž+1ξ‚€ξ€·1+1+β€–π‘’β€–πΏπ‘šβˆ’1βˆžξ€Έβ€–π‘’β€–π»1+‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1βˆžξ‚.(4.7)

Proof. Using ‖𝑒‖2𝐻1=βˆ«π‘…(𝑒2+𝑒2π‘₯)𝑑π‘₯ and (4.2) derives (4.5).
Using πœ•2π‘₯=βˆ’Ξ›2+1 and the Parseval equality gives rise to ξ€œπ‘…Ξ›π‘žπ‘’Ξ›π‘žπœ•3π‘₯𝑒2ξ€Έξ€œπ‘‘π‘₯=βˆ’2π‘…ξ€·Ξ›π‘ž+1π‘’ξ€ΈΞ›π‘ž+1𝑒𝑒π‘₯ξ€Έξ€œπ‘‘π‘₯+2𝑅(Ξ›π‘žπ‘’)Ξ›π‘žξ€·π‘’π‘’π‘₯𝑑π‘₯.(4.8)
For π‘žβˆˆ(0,π‘ βˆ’1], applying (Ξ›π‘žπ‘’)Ξ›π‘ž to both sides of the first equation of system (2.3) and integrating with respect to π‘₯ by parts, we have the identity 12π‘‘ξ€œπ‘‘π‘‘π‘…ξ‚€(Ξ›π‘žπ‘’)2+ξ€·Ξ›π‘žπ‘’π‘₯ξ€Έ2ξ‚ξ€œπ‘‘π‘₯=βˆ’π‘Žπ‘…(Ξ›π‘žπ‘’)Ξ›π‘žξ€·π‘’π‘šπ‘’π‘₯ξ€Έβˆ’ξ€œπ‘‘π‘₯π‘…ξ€·Ξ›π‘ž+1π‘’ξ€ΈΞ›π‘ž+1𝑒𝑒π‘₯ξ€Έ1𝑑π‘₯+2ξ€œπ‘…ξ€·Ξ›π‘žπ‘’π‘₯ξ€ΈΞ›π‘žξ€·π‘’2π‘₯ξ€Έ+ξ€œπ‘‘π‘₯𝑅(Ξ›π‘žπ‘’)Ξ›π‘žξ€·π‘’π‘’π‘₯ξ€Έξ€œπ‘‘π‘₯βˆ’π›½π‘…Ξ›π‘žπ‘’π‘₯Ξ›π‘žξ‚ƒξ€·π‘’π‘₯𝑁𝑑π‘₯.(4.9) We will estimate the terms on the right-hand side of (4.9) separately. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2, we have ξ€œπ‘…(Ξ›π‘žπ‘’)Ξ›π‘žξ€·π‘’π‘šπ‘’π‘₯ξ€Έξ€œπ‘‘π‘₯=𝑅(Ξ›π‘žξ€ΊΞ›π‘’)π‘žξ€·π‘’π‘šπ‘’π‘₯ξ€Έβˆ’π‘’π‘šΞ›π‘žπ‘’π‘₯ξ€»ξ€œπ‘‘π‘₯+𝑅(Ξ›π‘žπ‘’)π‘’π‘šΞ›π‘žπ‘’π‘₯𝑑π‘₯β‰€π‘β€–π‘’β€–π»π‘žξ€·π‘šβ€–π‘’β€–πΏπ‘šβˆ’1βˆžβ€–β€–π‘’π‘₯β€–β€–πΏβˆžβ€–π‘’β€–π»π‘ž+‖‖𝑒π‘₯β€–β€–πΏβˆžβ€–π‘’β€–πΏπ‘šβˆ’1βˆžβ€–π‘’β€–π»π‘žξ€Έ+12β€–π‘’β€–πΏπ‘šβˆ’1βˆžβ€–β€–π‘’π‘₯β€–β€–πΏβˆžβ€–Ξ›π‘žπ‘’β€–2𝐿2≀𝑐‖𝑒‖2π»π‘žβ€–π‘’β€–πΏπ‘šβˆ’1βˆžβ€–β€–π‘’π‘₯β€–β€–πΏβˆž.(4.10) Using the above estimate to the second term yields ξ€œπ‘…ξ€·Ξ›π‘ž+1π‘’ξ€ΈΞ›π‘ž+1𝑒𝑒π‘₯𝑑π‘₯≀𝑐‖𝑒‖2π»π‘ž+1‖‖𝑒π‘₯β€–β€–πΏβˆž.(4.11) For the third term, using the Cauchy-Schwartz inequality and Lemma 4.1, we obtain ξ€œπ‘…ξ€·Ξ›π‘žπ‘’π‘₯ξ€ΈΞ›π‘žξ€·π‘’2π‘₯‖‖Λ𝑑π‘₯β‰€π‘žπ‘’π‘₯‖‖𝐿2β€–β€–Ξ›π‘žξ€·π‘’2π‘₯‖‖𝐿2β‰€π‘β€–π‘’β€–π»π‘ž+1‖‖𝑒π‘₯β€–β€–πΏβˆžβ€–β€–π‘’π‘₯β€–β€–π»π‘ž+‖‖𝑒π‘₯β€–β€–πΏβˆžβ€–β€–π‘’π‘₯β€–β€–π»π‘žξ€Έβ‰€π‘β€–π‘’β€–2π»π‘ž+1‖‖𝑒π‘₯β€–β€–πΏβˆž.(4.12)
For the last term in (4.9), using Lemma 4.1 repeatedly results in ||||ξ€œπ‘…ξ€·Ξ›π‘žπ‘’π‘₯ξ€ΈΞ›π‘žξ€·π‘’π‘₯𝑁||||≀‖‖𝑒𝑑π‘₯π‘₯β€–β€–π»π‘žβ€–β€–π‘’π‘π‘₯β€–β€–π»π‘žβ‰€π‘β€–π‘’β€–2π»π‘ž+1‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1∞.(4.13) It follows from (4.9) to (4.13) that there exists a constant 𝑐 depending only on π‘š,𝑁 and the coefficients of (1.2) such that 12π‘‘ξ€œπ‘‘π‘‘π‘…ξ‚ƒ(Ξ›π‘žπ‘’)2+ξ€·Ξ›π‘žπ‘’π‘₯ξ€Έ2‖‖𝑒𝑑π‘₯≀𝑐π‘₯β€–β€–πΏβˆžβ€–π‘’β€–2π»π‘ž+1ξ€·1+β€–π‘’β€–πΏπ‘šβˆ’1βˆžξ€Έ+𝑐‖𝑒‖2π»π‘ž+1‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1∞.(4.14) Integrating both sides of the above inequality with respect to 𝑑 results in inequality (4.6).
To estimate the norm of 𝑒𝑑, we apply the operator (1βˆ’πœ•2π‘₯)βˆ’1 to both sides of the first equation of system (2.3) to obtain the equation 𝑒𝑑=ξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1ξ‚ƒβˆ’2π‘˜π‘’π‘₯+πœ•π‘₯ξ‚€βˆ’π‘Žπ‘’π‘š+1π‘š+1+12πœ•2π‘₯𝑒2ξ€Έβˆ’12𝑒2π‘₯+π›½πœ•π‘₯𝑒π‘₯𝑁.ξ‚„ξ‚„(4.15) Applying (Ξ›π‘žπ‘’π‘‘)Ξ›π‘ž to both sides of (4.15) for π‘žβˆˆ(0,π‘ βˆ’1] gives rise to ξ€œπ‘…ξ€·Ξ›π‘žπ‘’π‘‘ξ€Έ2ξ€œπ‘‘π‘₯=π‘…ξ€·Ξ›π‘žπ‘’π‘‘ξ€ΈΞ›π‘žβˆ’2ξ‚ƒπœ•π‘₯ξ‚€π‘Žβˆ’2π‘˜π‘’βˆ’π‘’π‘š+1π‘š+1+12πœ•2π‘₯𝑒2ξ€Έβˆ’12𝑒2π‘₯+π›½πœ•π‘₯𝑒π‘₯ξ€Έπ‘ξ‚„ξ‚„π‘‘πœ.(4.16) For the right-hand side of (4.16), we have ξ€œπ‘…ξ€·Ξ›π‘žπ‘’π‘‘ξ€ΈΞ›π‘žβˆ’2ξ€·βˆ’2π‘˜π‘’π‘₯‖‖𝑒𝑑π‘₯β‰€π‘π‘‘β€–β€–π»π‘žβ€–π‘’β€–π»π‘ž,ξ€œπ‘…ξ€·Ξ›π‘žπ‘’π‘‘ξ€Έξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1Ξ›π‘žπœ•π‘₯ξ‚€βˆ’π‘Žπ‘’π‘š+1π‘š+1βˆ’12𝑒2π‘₯‖‖𝑒𝑑π‘₯β‰€π‘π‘‘β€–β€–π»π‘žξƒ©ξ€œπ‘…ξ€·1+πœ‰2ξ€Έπ‘žβˆ’1Γ—ξ‚Έξ€œπ‘…ξ‚ƒβˆ’π‘Žξ‚Šπ‘’π‘š+1π‘š(1πœ‰βˆ’πœ‚)̂𝑒(πœ‚)βˆ’2𝑒π‘₯(πœ‰βˆ’πœ‚)𝑒π‘₯(ξ‚„ξ‚Ήπœ‚)π‘‘πœ‚2ξƒͺ1/2β€–β€–π‘’β‰€π‘π‘‘β€–β€–π»π‘žβ€–π‘’β€–π»1β€–π‘’β€–π»π‘ž+1ξ€·1+β€–π‘’β€–πΏπ‘šβˆ’1βˆžξ€Έ.(4.17) Since ξ€œξ€·Ξ›π‘žπ‘’π‘‘ξ€Έξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1Ξ›π‘žπœ•2π‘₯𝑒𝑒π‘₯ξ€Έξ€œξ€·Ξ›π‘‘π‘₯=βˆ’π‘žπ‘’π‘‘ξ€ΈΞ›π‘žξ€·π‘’π‘’π‘₯ξ€Έξ€œξ€·Ξ›π‘‘π‘₯+π‘žπ‘’π‘‘ξ€Έξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1Ξ›π‘žξ€·π‘’π‘’π‘₯𝑑π‘₯,(4.18) using Lemma 4.1, ‖𝑒𝑒π‘₯β€–π»π‘žβ‰€π‘β€–(𝑒2)π‘₯β€–π»π‘žβ‰€π‘β€–π‘’β€–πΏβˆžβ€–π‘’β€–π»π‘ž+1 and β€–π‘’β€–πΏβˆžβ‰€β€–π‘’β€–π»1, we have ξ€œξ€·Ξ›π‘žπ‘’π‘‘ξ€ΈΞ›π‘žξ€·π‘’π‘’π‘₯‖‖𝑒𝑑π‘₯β‰€π‘π‘‘β€–β€–π»π‘žβ€–β€–π‘’π‘’π‘₯β€–β€–π»π‘žβ€–β€–π‘’β‰€π‘π‘‘β€–β€–π»π‘žβ€–π‘’β€–π»1β€–π‘’β€–π»π‘ž+1,ξ€œξ€·Ξ›π‘žπ‘’π‘‘ξ€Έξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1Ξ›π‘žξ€·π‘’π‘’π‘₯‖‖𝑒𝑑π‘₯β‰€π‘π‘‘β€–β€–π»π‘žβ€–π‘’β€–π»1β€–π‘’β€–π»π‘ž+1.(4.19) Using the Cauchy-Schwartz inequality and Lemma 4.1 yields ||||ξ€œπ‘…ξ€·Ξ›π‘žπ‘’π‘‘ξ€Έξ€·1βˆ’πœ•2π‘₯ξ€Έβˆ’1Ξ›π‘žπœ•π‘₯𝑒𝑁π‘₯ξ€Έ||||‖‖𝑒𝑑π‘₯β‰€π‘π‘‘β€–β€–π»π‘žβ€–β€–π‘’π‘₯β€–β€–πΏπ‘βˆ’1βˆžβ€–π‘’β€–π»π‘ž+1.(4.20) Applying (4.17)–(4.20) into (4.16) yields the inequality β€–β€–π‘’π‘‘β€–β€–π»π‘žβ‰€π‘β€–π‘’β€–π»π‘ž+1ξ‚€ξ€·1+1+β€–π‘’β€–πΏπ‘šβˆ’1βˆžξ€Έβ€–π‘’β€–π»1+‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1βˆžξ‚.(4.21) This completes the proof of Lemma 4.3.

Definingξ‚»π‘’πœ™(π‘₯)=1/(π‘₯2βˆ’1),|π‘₯|<1,0,|π‘₯|β‰₯1,(4.22) and 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‘ (𝑅) and 𝑠>0.

It follows from Theorem 2.1 that for each πœ€ the Cauchy problemπ‘’π‘‘βˆ’π‘’π‘‘π‘₯π‘₯=πœ•π‘₯ξ‚€π‘Žβˆ’2π‘˜π‘’βˆ’π‘’π‘š+1π‘š+1+12πœ•3π‘₯𝑒2ξ€Έβˆ’12πœ•π‘₯𝑒2π‘₯ξ€Έ+π›½πœ•π‘₯𝑒π‘₯𝑁,𝑒(0,π‘₯)=π‘’πœ€0(π‘₯),π‘₯βˆˆπ‘…,(4.23) has a unique solution π‘’πœ€(𝑑,π‘₯)∈𝐢∞([0,𝑇);𝐻∞).

Lemma 4.4. Under the assumptions of problem (4.23), the following estimates hold for any πœ€ with 0<πœ€<1/4 and 𝑠>0β€–β€–π‘’πœ€0π‘₯β€–β€–πΏβˆžβ‰€π‘1‖‖𝑒0π‘₯β€–β€–πΏβˆž,β€–β€–π‘’πœ€0β€–β€–π»π‘žβ‰€π‘1‖‖𝑒,ifπ‘žβ‰€π‘ ,πœ€0β€–β€–π»π‘žβ‰€π‘1πœ€(π‘ βˆ’π‘ž)/4‖‖𝑒,ifπ‘ž>𝑠,πœ€0βˆ’π‘’0β€–β€–π»π‘žβ‰€π‘1πœ€(π‘ βˆ’π‘ž)/4‖‖𝑒,ifπ‘žβ‰€π‘ ,πœ€0βˆ’π‘’0‖‖𝐻𝑠=π‘œ(1),(4.24) where 𝑐1 is a constant independent of πœ€.

The proof of this Lemma can be found in Lai and Wu [33].

Lemma 4.5. If 𝑒0(π‘₯)βˆˆπ»π‘ (𝑅) with π‘ βˆˆ[1,3/2] such that ‖𝑒0π‘₯β€–πΏβˆž<∞. Let π‘’πœ€0 be defined as in system (4.23). Then there exist two positive constants 𝑇 and 𝑐, which are independent of πœ€, such that the solution π‘’πœ€ of problem (4.23) satisfies β€–π‘’πœ€π‘₯β€–πΏβˆžβ‰€π‘ for any π‘‘βˆˆ[0,𝑇).

Proof. Using notation 𝑒=π‘’πœ€ and differentiating both sides of the first equation of problem (4.23) or (4.15) with respect to π‘₯ give rise to 𝑒𝑑π‘₯+12πœ•2π‘₯𝑒2βˆ’12𝑒2π‘₯π‘Ž=2π‘˜π‘’+π‘’π‘š+1π‘š+1βˆ’12𝑒2βˆ’π›½π‘’π‘π‘₯βˆ’Ξ›βˆ’2ξ‚ƒπ‘Ž2π‘˜π‘’+π‘’π‘š+1π‘š+1βˆ’12𝑒2+12𝑒2π‘₯βˆ’π›½π‘’π‘π‘₯ξ‚„.(4.25) Letting 𝑝>0 be an integer and multiplying the above equation by (𝑒π‘₯)2𝑝+1 and then integrating the resulting equation with respect to π‘₯ yield the equality 1𝑑2𝑝+2ξ€œπ‘‘π‘‘π‘…ξ€·π‘’π‘₯ξ€Έ2𝑝+2𝑝𝑑π‘₯+ξ€œ2𝑝+2𝑅𝑒π‘₯ξ€Έ2𝑝+3=ξ€œπ‘‘π‘₯𝑅𝑒π‘₯ξ€Έ2𝑝+1ξ‚€π‘Ž2π‘˜π‘’+π‘’π‘š+1π‘š+1βˆ’12𝑒2βˆ’π›½π‘’π‘π‘₯ξ‚βˆ’ξ€œπ‘‘π‘₯𝑅𝑒π‘₯ξ€Έ2𝑝+1Ξ›βˆ’2ξ‚Έπ‘Ž2π‘˜π‘’+π‘’π‘š+1π‘š+1βˆ’π‘’22+12𝑒2π‘₯βˆ’π›½π‘’π‘π‘₯𝑑π‘₯.(4.26) Applying the HΓΆlder's inequality yields 1𝑑2𝑝+2ξ€œπ‘‘π‘‘π‘…ξ€·π‘’π‘₯ξ€Έ2𝑝+2ξƒ―||||ξ‚΅ξ€œπ‘‘π‘₯≀2π‘˜π‘…|𝑒|2𝑝+2𝑑π‘₯1/(2𝑝+2)+π‘Žξ‚΅ξ€œπ‘š+1𝑅||π‘’π‘š+1||2𝑝+2𝑑π‘₯1/(2𝑝+2)+12ξ‚΅ξ€œπ‘…||𝑒2||2𝑝+2𝑑π‘₯1/(2𝑝+2)ξ‚΅ξ€œ+𝛽𝑅||𝑒𝑁π‘₯||2𝑝+2𝑑π‘₯1/(2𝑝+2)+ξ‚΅ξ€œπ‘…||𝐺||2𝑝+2𝑑π‘₯1/(2𝑝+2)ξƒ°ξ‚΅ξ€œπ‘…||𝑒π‘₯||2𝑝+2𝑑π‘₯(2𝑝+1)/(2𝑝+2)+𝑝‖‖𝑒2𝑝+2π‘₯β€–β€–πΏβˆžξ€œπ‘…||𝑒π‘₯||2𝑝+2𝑑π‘₯,(4.27) or π‘‘ξ‚΅ξ€œπ‘‘π‘‘π‘…ξ€·π‘’π‘₯ξ€Έ2𝑝+2𝑑π‘₯1/(2𝑝+2)≀||||ξ‚΅ξ€œ2π‘˜π‘…|𝑒|2𝑝+2𝑑π‘₯1/(2𝑝+2)+π‘Žξ‚΅ξ€œπ‘š+1𝑅||π‘’π‘š+1||2𝑝+2𝑑π‘₯1/(2𝑝+2)+12ξ‚΅ξ€œπ‘…||𝑒2||2𝑝+2𝑑π‘₯1/(2𝑝+2)ξ‚΅ξ€œ+𝛽𝑅||𝑒𝑁π‘₯||2𝑝+2𝑑π‘₯1/(2𝑝+2)+ξ‚΅ξ€œπ‘…||𝐺||2𝑝+2𝑑π‘₯1/(2𝑝+2)+𝑝‖‖𝑒2𝑝+2π‘₯β€–β€–πΏβˆžξ‚΅ξ€œπ‘…||𝑒π‘₯||2𝑝+2𝑑π‘₯1/(2𝑝+2),(4.28) where 𝐺=Ξ›βˆ’2ξ‚Έπ‘Ž2π‘˜π‘’+π‘’π‘š+1π‘š+1βˆ’π‘’22+12𝑒2π‘₯βˆ’π›½π‘’π‘π‘₯ξ‚Ή.(4.29) Since β€–π‘“β€–πΏπ‘β†’β€–π‘“β€–πΏβˆž as π‘β†’βˆž for any π‘“βˆˆπΏβˆžβ‹‚πΏ2, integrating both sides of the inequality (4.28) with respect to 𝑑 and taking the limit as π‘β†’βˆž result in the estimate ‖‖𝑒π‘₯β€–β€–πΏβˆžβ‰€β€–β€–π‘’0π‘₯β€–β€–πΏβˆž+ξ€œπ‘‘0π‘ξ‚ƒξ€·β€–π‘’β€–πΏβˆž+‖‖𝑒2β€–β€–πΏβˆž+β€–β€–π‘’π‘š+1β€–β€–πΏβˆžβ€–β€–π‘’+𝛽π‘₯β€–β€–π‘πΏβˆž+β€–πΊβ€–πΏβˆžξ€Έ+12‖‖𝑒π‘₯β€–β€–2πΏβˆžξ‚„π‘‘πœ.(4.30) Using the algebra property of 𝐻𝑠0(𝑅) with 𝑠0>1/2 yields (β€–π‘’πœ€β€–π»(1/2)+ means that there exists a sufficiently small 𝛿>0 such that β€–π‘’πœ€β€–(1/2)+=β€–π‘’πœ€β€–π»1/2+𝛿)β€–πΊβ€–πΏβˆžβ‰€π‘β€–πΊβ€–π»(1/2)+β€–β€–β€–Ξ›β‰€π‘βˆ’2ξ‚Έπ‘Ž2π‘˜π‘’+π‘’π‘š+1π‘š+1βˆ’π‘’22+12𝑒2π‘₯βˆ’π›½π‘’π‘π‘₯‖‖‖𝐻(1/2)+≀𝑐‖𝑒‖𝐻1+‖𝑒‖2𝐻1+β€–π‘’β€–π»π‘š+11+β€–β€–Ξ›βˆ’2(𝑒2π‘₯)‖‖𝐻(1/2)++β€–β€–Ξ›βˆ’2(𝑒𝑁π‘₯)‖‖𝐻(1/2)+≀𝑐‖𝑒‖𝐻1+‖𝑒‖2𝐻1+β€–π‘’β€–π»π‘š+11+‖‖𝑒2π‘₯‖‖𝐻0+‖‖𝑒𝑁π‘₯‖‖𝐻0≀𝑐‖𝑒‖𝐻1+‖𝑒‖2𝐻1+β€–π‘’β€–π»π‘š+11+‖‖𝑒π‘₯β€–β€–πΏβˆžβ€–π‘’β€–π»1+‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1βˆžβ€–π‘’β€–π»1ξ‚β‰€π‘π‘’π‘βˆ«π‘‘0‖𝑒π‘₯β€–πΏβˆžπ‘βˆ’1π‘‘πœξ‚€β€–β€–π‘’1+π‘₯β€–β€–πΏβˆž+‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1βˆžξ‚,(4.31) in which Lemma 4.3 is used. Therefore, we get ξ€œπ‘‘0β€–πΊβ€–πΏβˆžξ€œπ‘‘πœβ‰€π‘π‘‘0π‘’π‘βˆ«πœ0‖𝑒π‘₯β€–πΏβˆžπ‘βˆ’1π‘‘πœ‰ξ‚€β€–β€–π‘’1+π‘₯β€–β€–πΏβˆž+‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1βˆžξ‚π‘‘πœ.(4.32)
From (4.30) and (4.32), one has ‖‖𝑒π‘₯β€–β€–πΏβˆžβ‰€β€–β€–π‘’0π‘₯β€–β€–πΏβˆžξ€œ+𝑐𝑑0‖‖𝑒π‘₯β€–β€–2𝐿∞+‖‖𝑒π‘₯β€–β€–π‘πΏβˆž+π‘’π‘βˆ«π‘‘0‖𝑒π‘₯β€–πΏβˆžπ‘βˆ’1π‘‘πœ+π‘’π‘βˆ«πœ0‖𝑒π‘₯β€–πΏβˆžπ‘βˆ’1π‘‘πœ‰ξ‚€β€–β€–π‘’1+π‘₯β€–β€–πΏβˆž+‖‖𝑒π‘₯β€–β€–πΏπ‘βˆ’1βˆžξ‚ξ‚„π‘‘πœ.(4.33)
From Lemma 4.4, it follows from the contraction mapping principle that there is a 𝑇>0 such that the equation β€–π‘Šβ€–πΏβˆž=‖‖𝑒0π‘₯β€–β€–πΏβˆžξ€œ+𝑐𝑑0ξ‚ƒβ€–π‘Šβ€–2𝐿∞+β€–π‘Šβ€–π‘πΏβˆž+π‘’π‘βˆ«π‘‘0β€–π‘Šβ€–πΏβˆžπ‘βˆ’1π‘‘πœ+π‘’π‘βˆ«πœ0β€–π‘Šβ€–πΏβˆžπ‘βˆ’1π‘‘πœ‰ξ€·1+β€–π‘Šβ€–πΏβˆž+β€–π‘Šβ€–πΏπ‘βˆ’1βˆžξ€Έξ‚„π‘‘πœ(4.34) has a unique solution π‘ŠβˆˆπΆ[0,𝑇]. Using the Theorem presented at page 51 in [25] or Theorem 2 in Section 1.1 presented in [37] yields that there are constants 𝑇>0 and 𝑐>0 independent of πœ€ such that ‖𝑒π‘₯β€–πΏβˆžβ‰€π‘Š(𝑑) for arbitrary π‘‘βˆˆ[0,𝑇], which leads to the conclusion of Lemma 4.5.
Using Lemmas 4.3 and 4.5, notation π‘’πœ€=𝑒 and Gronwall's inequality results in the inequalities β€–β€–π‘’πœ€β€–β€–π»π‘žβ‰€πΆπ‘‡π‘’πΆπ‘‡,β€–β€–π‘’πœ€π‘‘β€–β€–π»π‘Ÿβ‰€πΆπ‘‡π‘’πΆπ‘‡,(4.35) where π‘žβˆˆ(0,𝑠], π‘Ÿβˆˆ(0,π‘ βˆ’1] and 𝐢𝑇 depends on 𝑇. It follows from Aubin's compactness theorem that there is a subsequence of {π‘’πœ€}, denoted by {π‘’πœ€π‘›}, such that {π‘’πœ€π‘›} and their temporal derivatives {π‘’πœ€π‘›π‘‘} are weakly convergent to a function 𝑒(𝑑,π‘₯) and its derivative 𝑒𝑑 in 𝐿2([0,𝑇],𝐻𝑠) and 𝐿2([0,𝑇],π»π‘ βˆ’1), respectively. Moreover, for any real number 𝑅1>0, {π‘’πœ€π‘›} is convergent to the function 𝑒 strongly in the space 𝐿2([0,𝑇],π»π‘ž(βˆ’π‘…1,𝑅1)) for π‘žβˆˆ[0,𝑠) and {π‘’πœ€π‘›π‘‘} converges to 𝑒𝑑 strongly in the space 𝐿2([0,𝑇],π»π‘Ÿ(βˆ’π‘…1,𝑅1)) for π‘Ÿβˆˆ[0,π‘ βˆ’1]. Thus, we can prove the existence of a weak solution to (2.2).

Proof of Theorem 2.2. From Lemma 4.5, we know that {π‘’πœ€π‘›π‘₯}(πœ€π‘›β†’0) is bounded in the space 𝐿∞. Thus, the sequences {π‘’πœ€π‘›} and {π‘’πœ€π‘›π‘₯} are weakly convergent to 𝑒 and 𝑒π‘₯ in 𝐿2[0,𝑇],π»π‘Ÿ(βˆ’π‘…,𝑅) for any π‘Ÿβˆˆ[0,π‘ βˆ’1), respectively. Therefore, 𝑒 satisfies the equation βˆ’ξ€œπ‘‡0ξ€œπ‘…π‘’ξ€·π‘”π‘‘βˆ’π‘”π‘₯π‘₯π‘‘ξ€Έξ€œπ‘‘π‘₯𝑑𝑑=𝑇0ξ€œπ‘…π‘Žξ‚ƒξ‚€2π‘˜π‘’+π‘’π‘š+1π‘š+1+12𝑒2π‘₯𝑔π‘₯βˆ’12𝑒2𝑔π‘₯π‘₯π‘₯ξ€·π‘’βˆ’π›½π‘₯𝑁𝑔π‘₯𝑑π‘₯𝑑𝑑,(4.36) with 𝑒(0,π‘₯)=𝑒0(π‘₯) and π‘”βˆˆπΆβˆž0. Since 𝑋=𝐿1([0,𝑇]×𝑅) is a separable Banach space and {π‘’πœ€π‘›π‘₯} is a bounded sequence in the dual space π‘‹βˆ—=𝐿∞([0,𝑇]×𝑅) of 𝑋, there exists a subsequence of {π‘’πœ€π‘›π‘₯}, still denoted by {π‘’πœ€π‘›π‘₯}, weakly star convergent to a function 𝑣 in 𝐿∞([0,𝑇]×𝑅). It derives from the {π‘’πœ€π‘›π‘₯} weakly convergent to 𝑒π‘₯ in 𝐿2([0,𝑇]×𝑅) that 𝑒π‘₯=𝑣 almost everywhere. Thus, we obtain 𝑒π‘₯∈𝐿∞([0,𝑇]×𝑅).

Acknowledgments

The author is very grateful to the reviewers for their helpful and valuable comments, which have led to a meaningful improvement of the paper. This work is supported by the Key Project of Chinese Ministry of Education (109140).

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