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

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 𝐶([0,𝑇),𝐻^𝑠(𝑅))∩𝐶¹([0,𝑇),𝐻^𝑠−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 𝐻^𝑠(𝑅) with 1≤𝑠≤3/2 is developed.

1. Introduction

Camassa and Holm [1] used the Hamiltonian method to derive a completely integrable wave equation𝑢_𝑡−𝑢_𝑥𝑥𝑡+2𝑘𝑢_𝑥+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,(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 𝑘=0, (1.1) has traveling wave solutions of the form 𝑐e^{−|𝑥−𝑐𝑡|}, called peakons, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations (see [9–11]). For 𝑘>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 𝐻^𝑠(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 equation𝑢_𝑡−𝑢_𝑡𝑥𝑥+2𝑘𝑢_𝑥+𝑎𝑢^𝑚𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥+𝛽𝜕_𝑥𝑢^𝑥^𝑁,(1.2) where 𝑚≥1 and 𝑁≥1 are natural numbers, and 𝑎, 𝑘, and 𝛽 are arbitrary constants. Obviously, (1.2) reduces to (1.1) if we set 𝑎=3, 𝑚=1, and 𝛽=0. Actually, Wu and Yin [34] consider a nonlinearly dissipative Camassa-Holm equation which includes a nonlinearly dissipative term 𝐿(𝑢), where 𝐿 is a differential operator or a quasidifferential operator. Therefore, we can regard the term 𝛽𝜕_𝑥[(𝑢_𝑥)^𝑁] as a nonlinearly dissipative term for the dissipative Camassa-Holm equation (1.2).

Due to the term 𝛽𝜕_𝑥[(𝑢_𝑥)^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 𝑢₀(𝑥)∈𝐻^𝑠(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 𝜕_𝑥[(𝑢_𝑥)^𝑁]. Lai and Wu [33] only studied a generalized Camassa-Holm equation in the case where 𝛽≥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 𝐶([0,𝑇),𝐻^𝑠(𝑅))⋂𝐶¹([0,𝑇),𝐻^𝑠−1(𝑅)) with 𝑠>3/2. Secondly, it is shown that the existence of weak solutions in lower order Sobolev space 𝐻^𝑠(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 𝜙(𝑡,𝑥) with compact support in [0,+∞)×𝑅 is denoted by 𝐶^∞0. 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.

In order to study the existence of solutions for (1.2), we consider its Cauchy problem in the form𝑢_𝑡−𝑢_𝑡𝑥𝑥=−2𝑘𝑢_𝑥−𝑎𝑚+1𝑢^𝑚+1_𝑥=−𝑘𝑢_𝑥+2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥+𝛽𝜕_𝑥𝑢^𝑥^𝑁−𝑎𝑚+1𝑢^𝑚+1_𝑥+12𝜕^3𝑥𝑢²−1𝑢(0,𝑥)=𝑢₀2𝜕^𝑥𝑢^2𝑥+𝛽𝜕_𝑥𝑢^𝑥^𝑁,(𝑥),(2.2) which is equivalent to𝑢_𝑡+𝑢𝑢_𝑥=Λ⁻²−𝑘𝑢−𝑎𝑚+1𝑢^𝑚+1_𝑥+Λ⁻²𝑢𝑢_𝑥−12Λ⁻²𝑢(0,𝑥)=𝑢₀𝜕_𝑥𝑢^2𝑥+𝛽Λ⁻²𝜕_𝑥𝑢^𝑥^𝑁,(𝑥).(2.3)

Now, we state our main results.

Theorem 2.1. Let 𝑢₀(𝑥)∈𝐻^𝑠(𝑅) with 𝑠>3/2. Then problem (2.2) or problem (2.3) has a unique solution 𝑢(𝑡,𝑥)∈𝐶([0,𝑇);𝐻^𝑠(𝑅))⋂𝐶¹([0,𝑇);𝐻^𝑠−1(𝑅)) where 𝑇>0 depends on ‖𝑢₀‖_{𝐻𝑠(𝑅)}.

Theorem 2.2. Suppose that 𝑢₀(𝑥)∈𝐻^𝑠 with 1≤𝑠≤3/2 and ‖𝑢_0𝑥‖_𝐿∞<∞. Then there exists a 𝑇>0 such that (1.2) subject to initial value 𝑢₀(𝑥) has a weak solution 𝑢(𝑡,𝑥)∈𝐿²([0,𝑇],𝐻^𝑠) in the sense of distribution and 𝑢_𝑥∈𝐿^∞([0,𝑇]×𝑅).

3. Local Well-Posedness

We consider the abstract quasilinear evolution equation𝑑𝑣𝑑𝑡+𝐴(𝑣)𝑣=𝑓(𝑣),𝑡≥0,𝑣(0)=𝑣⁰.(3.1) Let 𝑋 and 𝑌 be Hilbert spaces such that 𝑌 is continuously and densely embedded in 𝑋, and let 𝑄∶𝑌→𝑋 be a topological isomorphism. Let 𝐿(𝑌,𝑋) be the space of all bounded linear operators from 𝑌 to 𝑋. If 𝑋=𝑌, we denote this space by 𝐿(𝑋). We state the following conditions in which 𝜌₁,𝜌₂,𝜌₃, and 𝜌₄ are constants depending on max{‖𝑦‖_𝑌,‖𝑧‖_𝑌}.(i)𝐴(𝑦)∈𝐿(𝑌,𝑋) for 𝑦∈𝑋 with ‖(𝐴(𝑦)−𝐴(𝑧))𝑤‖_𝑋≤𝜌₁‖𝑦−𝑧‖_𝑋‖𝑤‖_𝑌,𝑦,𝑧,𝑤∈𝑌,(3.2) and 𝐴(𝑦)∈𝐺(𝑋,1,𝛽) (i.e., 𝐴(𝑦) is quasi-𝑚-accretive), uniformly on bounded sets in 𝑌.(ii)𝑄𝐴(𝑦)𝑄⁻¹=𝐴(𝑦)+𝐵(𝑦), where 𝐵(𝑦)∈𝐿(𝑋) is bounded, uniformly on bounded sets in 𝑌. Moreover, ‖(𝐵(𝑦)−𝐵(𝑧))𝑤‖_𝑋≤𝜌₂‖𝑦−𝑧‖_𝑌‖𝑤‖_𝑋,𝑦,𝑧∈𝑌,𝑤∈𝑋.(3.3)(iii)𝑓∶𝑌→𝑌 extends to a map from 𝑋 into 𝑋 is bounded on bounded sets in 𝑌, and satisfies‖‖𝑓𝑓((𝑦𝑦))−−𝑓𝑓((𝑧𝑧))‖‖_𝑌𝑋≤𝜌₃‖𝑦−𝑧‖_𝑌,𝑦,𝑧∈𝑌,≤𝜌₄‖𝑦−𝑧‖_𝑋,𝑦,𝑧∈𝑌.(3.4)

Kato Theorem (see [35])
Assume that (i), (ii), and (iii) hold. If 𝑣₀∈𝑌, there is a maximal 𝑇>0 depending only on ‖𝑣₀‖_𝑌, and a unique solution 𝑣 to problem (3.1) such that 𝑣=𝑣⋅,𝑣₀∈𝐶([0,𝑇);𝑌)𝐶¹([0,𝑇);𝑋).(3.5) Moreover, the map 𝑣₀→𝑣(⋅,𝑣₀) is a continuous map from 𝑌 to the space 𝐶([0,𝑇);𝑌)𝐶¹([0,𝑇);𝑋).(3.6)

For problem (2.3), we set 𝐴(𝑢)=𝑢𝜕_𝑥, 𝑌=𝐻^𝑠(𝑅), 𝑋=𝐻^𝑠−1(𝑅), Λ=(1−𝜕^2𝑥)^1/2, 𝑓(𝑢)=Λ⁻²−𝑘𝑢−𝑎𝑚+1𝑢^𝑚+1_𝑥+Λ⁻²𝑢𝑢_𝑥−12Λ⁻²𝜕_𝑥𝑢^2𝑥+𝛽Λ⁻²𝜕_𝑥𝑢^𝑥^𝑁,(3.7) and 𝑄=Λ. In order to prove Theorem 2.1, we only need to check that 𝐴(𝑢) and 𝑓(𝑢) satisfy assumptions (i)–(iii).

Lemma 3.1. The operator 𝐴(𝑢)=𝑢𝜕_𝑥 with 𝑢∈𝐻^𝑠(𝑅), 𝑠>3/2 belongs to 𝐺(𝐻^𝑠−1,1,𝛽).

Lemma 3.2. Let 𝐴(𝑢)=𝑢𝜕_𝑥 with 𝑢∈𝐻^𝑠 and 𝑠>3/2. Then 𝐴(𝑢)∈𝐿(𝐻^𝑠,𝐻^𝑠−1) for all 𝑢∈𝐻^𝑠. Moreover, ‖(𝐴(𝑢)−𝐴(𝑧))𝑤‖_𝐻^𝑠−1≤𝜌₁‖𝑢−𝑧‖_𝐻^𝑠−1‖𝑤‖_𝐻^𝑠,𝑢,𝑧,𝑤∈𝐻^𝑠(𝑅).(3.8)

Lemma 3.3. For 𝑠>3/2, 𝑢,𝑧∈𝐻^𝑠 and 𝑤∈𝐻^𝑠−1, it holds that 𝐵(𝑢)=[Λ,𝑢𝜕_𝑥]Λ⁻¹∈𝐿(𝐻^𝑠−1) for 𝑢∈𝐻^𝑠 and ‖(𝐵(𝑢)−𝐵(𝑧))𝑤‖_𝐻^𝑠−1≤𝜌₂‖𝑢−𝑧‖_𝐻^𝑠‖𝑤‖_𝐻^𝑠−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 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.(3.10)

Lemma 3.5. Let 𝑢,𝑧∈𝐻^𝑠 with 𝑠>3/2, then 𝑓(𝑢) is bounded on bounded sets in 𝐻^𝑠 and satisfies ‖𝑓‖𝑓(𝑢()𝑢−)−𝑓𝑓(𝑧()𝑧‖)‖_𝐻^𝑠_𝐻𝑠−1≤𝜌₃‖𝑢−𝑧‖_𝐻^𝑠,(3.11)≤𝜌₄‖𝑢−𝑧‖_𝐻^𝑠−1.(3.12)

Proof. Using the algebra property of the space 𝐻^𝑠0 with 𝑠₀>1/2, we have ‖𝑓(𝑢)−𝑓(𝑧)‖_𝐻^𝑠≤𝑐‖‖‖Λ⁻²−𝑘𝑢−𝑚𝑎+1𝑢^𝑚+1_𝑥−−𝑘𝑧−𝑎𝑚+1𝑧^𝑚+1_𝑥‖‖‖_𝐻𝑠+‖‖Λ⁻²𝑢𝑢_𝑥−𝑧𝑧_𝑥‖‖_𝐻^𝑠+‖‖Λ⁻²𝜕_𝑥𝑢^2𝑥−𝑧^2𝑥‖‖_𝐻^𝑠+‖‖Λ⁻²𝜕_𝑥𝑢^𝑥^𝑁−Λ⁻²𝜕_𝑥𝑧^𝑥^𝑁‖‖_𝐻𝑠≤𝑐‖𝑢−𝑧‖_𝐻𝑠−1+‖‖𝑢^𝑚+1−𝑧^𝑚+1‖‖_𝐻𝑠−1+‖‖𝑢𝑢_𝑥−𝑧𝑧_𝑥‖‖_𝐻𝑠−1+‖‖𝑢^2𝑥−𝑧^2𝑥‖‖_𝐻𝑠−1≤𝑐‖+𝑢‖‖−(𝑢𝑧_𝑥‖)^𝑁_𝐻𝑠−(𝑧_𝑥)^𝑁‖‖_𝐻𝑠−11+^𝑚_𝑗=0‖𝑢‖^{𝐻𝑚−𝑗}_𝑠‖𝑧‖^𝑗𝐻_𝑠+‖𝑢‖_𝐻^𝑠+‖𝑧‖_𝐻^𝑠+^𝑁−1_𝑗=0‖‖𝑢_𝑥‖‖^{𝐻𝑁−𝑗}_𝑠−1‖‖𝑧_𝑥‖‖^𝑗𝐻_𝑠−1≤𝜌₃‖𝑢−𝑧‖_𝐻^𝑠,(3.13) from which we obtain (3.11).
Applying Lemma 3.4, 𝑢𝑢_𝑥=(1/2)(𝑢²)_𝑥, 𝑠>3/2,‖𝑢‖_𝐿^∞≤𝑐‖𝑢‖_𝐻^𝑠−1 and ‖𝑢_𝑥‖_𝐿^∞≤𝑐‖𝑢‖_𝐻^𝑠, we get ‖𝑓(𝑢)−𝑓(𝑧)‖_𝐻^𝑠−1≤𝑐⎡⎢⎢⎣‖𝑢−𝑧‖_𝐻𝑠−2+‖‖𝑢^𝑚+1−𝑧^𝑚+1‖‖_𝐻𝑠−2+‖‖𝑢²−𝑧²‖‖_𝐻𝑠−2+‖‖(𝑢_𝑥−𝑧_𝑥)(𝑢_𝑥+𝑧_𝑥)‖‖_𝐻𝑠−2+‖‖‖‖(𝑢_𝑥−𝑧_𝑥)^𝑁−1_𝑗=0𝑢^{𝑥𝑁−1−𝑗}𝑧^𝑗𝑥≤𝑐‖𝑢−𝑧‖_𝐻𝑠−1‖‖‖‖_𝐻𝑠−2⎤⎥⎥⎦1+^𝑚_𝑗=0‖𝑢‖^{𝐻𝑚−𝑗}_𝑠−1‖𝑧‖^𝑗𝐻_𝑠−1+‖𝑢‖_𝐻^𝑠−1+‖𝑧‖_𝐻^𝑠−1+‖𝑢‖_𝐻^𝑠+‖𝑧‖_𝐻^𝑠+^𝑁−1_𝑗=0‖‖𝑢_𝑥‖‖^{𝐻𝑁−𝑗}_𝑠−1‖‖𝑧_𝑥‖‖^𝑗𝐻_𝑠−1≤𝜌₄‖𝑢−𝑧‖_𝐻^𝑠−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 𝑢(𝑡,𝑥)∈𝐶([0,𝑇);𝐻^𝑠(𝑅))𝐶¹[0,𝑇);𝐻^𝑠−1(𝑅).(3.15)

4. Existence of Weak Solutions

For 𝑠≥2, using the first equation of system (2.2) derives𝑑𝑑𝑡_𝑅𝑢²+𝑢^2𝑥+2𝛽^𝑡0𝑢^𝑥𝑁+1𝑑𝜏𝑑𝑥=0,(4.1) from which we have the conservation law_𝑅𝑢²+𝑢^2𝑥+2𝛽^𝑡0𝑢^𝑥𝑁+1𝑑𝜏𝑑𝑥=_𝑅𝑢²⁰+𝑢^20𝑥𝑑𝑥.(4.2)

Lemma 4.1 (Kato and Ponce [36]). If 𝑟>0, then 𝐻^𝑟⋂𝐿^∞ is an algebra. Moreover, ‖𝑢𝑣‖_𝑟≤𝑐‖𝑢‖_𝐿^∞‖𝑣‖_𝑟+‖𝑢‖_𝑟‖𝑣‖_𝐿^∞,(4.3) where 𝑐 is a constant depending only on 𝑟.

Lemma 4.2 (Kato and Ponce [36]). Let 𝑟>0. If 𝑢∈𝐻^𝑟⋂𝑊^1,∞ and 𝑣∈𝐻^𝑟−1⋂𝐿^∞, then ‖‖Λ^𝑟,𝑢𝑣‖‖_𝐿2≤𝑐‖‖𝜕_𝑥𝑢‖‖_𝐿∞‖‖Λ^𝑟−1𝑣‖‖_𝐿2+‖Λ^𝑟𝑢‖_𝐿²‖𝑣‖_𝐿^∞.(4.4)

Lemma 4.3. Let 𝑠≥2 and the function 𝑢(𝑡,𝑥) is a solution of problem (2.2) and the initial data 𝑢₀(𝑥)∈𝐻^𝑠(𝑅). Then the following inequality holds ‖𝑢‖_𝐿^∞≤‖𝑢‖_𝐻¹≤‖‖𝑢₀‖‖_𝐻1𝑒^{∫|𝛽|𝑡0‖𝑢𝑥‖𝐿∞𝑁−1𝑑𝜏}.(4.5)
For 𝑞∈(0,𝑠−1], there is a constant 𝑐, which only depends on 𝑚, 𝑁, 𝑘, 𝑎, and 𝛽, such that _𝑅Λ^𝑞+1𝑢²𝑑𝑥≤+_𝑅𝑐_𝑡0Λ^𝑞+1𝑢₀²𝑑𝑥+𝑐^𝑡0‖‖𝑢_𝑥‖‖_𝐿∞‖𝑢‖^2𝐻_𝑞+11+‖𝑢‖^{𝐿𝑚−1∞}𝑑𝜏‖𝑢‖^2𝐻_𝑞+1‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞𝑑𝜏.(4.6)
For 𝑞∈[0,𝑠−1], there is a constant 𝑐, which only depends on 𝑚,𝑁,𝑘,𝑎, and 𝛽, such that ‖‖𝑢_𝑡‖‖_𝐻𝑞≤𝑐‖𝑢‖_𝐻^𝑞+11+1+‖𝑢‖^{𝐿𝑚−1∞}‖𝑢‖_𝐻¹+‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞.(4.7)

Proof. Using ‖𝑢‖^2𝐻₁=∫_𝑅(𝑢²+𝑢^2𝑥)𝑑𝑥 and (4.2) derives (4.5).
Using 𝜕^2𝑥=−Λ²+1 and the Parseval equality gives rise to _𝑅Λ^𝑞𝑢Λ^𝑞𝜕^3𝑥𝑢²𝑑𝑥=−2_𝑅Λ^𝑞+1𝑢Λ^𝑞+1𝑢𝑢_𝑥𝑑𝑥+2_𝑅(Λ^𝑞𝑢)Λ^𝑞𝑢𝑢_𝑥𝑑𝑥.(4.8)
For 𝑞∈(0,𝑠−1], applying (Λ^𝑞𝑢)Λ^𝑞 to both sides of the first equation of system (2.3) and integrating with respect to 𝑥 by parts, we have the identity 12𝑑𝑑𝑡_𝑅(Λ^𝑞𝑢)²+Λ^𝑞𝑢_𝑥²𝑑𝑥=−−𝑎_𝑅(Λ^𝑞𝑢)Λ^𝑞𝑢^𝑚𝑢_𝑥𝑑𝑥_𝑅+Λ^𝑞+1𝑢Λ^𝑞+1𝑢𝑢_𝑥𝑑𝑥+12_𝑅Λ^𝑞𝑢_𝑥Λ^𝑞𝑢^2𝑥𝑑𝑥_𝑅(Λ^𝑞𝑢)Λ^𝑞𝑢𝑢_𝑥𝑑𝑥−𝛽_𝑅Λ^𝑞𝑢_𝑥Λ^𝑞𝑢^𝑥^𝑁𝑑𝑥.(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 _𝑅(Λ^𝑞𝑢)Λ^𝑞𝑢^𝑚𝑢_𝑥𝑑𝑥=≤𝑐_𝑅‖𝑢(‖Λ^𝑞𝑢_𝐻𝑞)Λ^𝑞𝑢^𝑚𝑢_𝑥−𝑢^𝑚Λ^𝑞𝑢_𝑥𝑑𝑥+_𝑅(Λ^𝑞𝑢)𝑢^𝑚Λ^𝑞𝑢_𝑥𝑑𝑥+1𝑚‖𝑢‖^{𝐿𝑚−1∞}‖‖𝑢_𝑥‖‖_𝐿∞‖𝑢‖_𝐻^𝑞+‖‖𝑢_𝑥‖‖_𝐿∞‖𝑢‖^𝐿𝑚−1_∞‖𝑢‖_𝐻^𝑞2‖𝑢‖^{𝐿𝑚−1∞}‖‖𝑢_𝑥‖‖_𝐿∞‖Λ^𝑞𝑢‖^2𝐿₂≤𝑐‖𝑢‖^2𝐻_𝑞‖𝑢‖^𝐿𝑚−1_∞‖‖𝑢_𝑥‖‖_𝐿∞.(4.10) Using the above estimate to the second term yields _𝑅Λ^𝑞+1𝑢Λ^𝑞+1𝑢𝑢_𝑥𝑑𝑥≤𝑐‖𝑢‖^2𝐻𝑞+1‖‖𝑢_𝑥‖‖_𝐿∞.(4.11) For the third term, using the Cauchy-Schwartz inequality and Lemma 4.1, we obtain _𝑅Λ^𝑞𝑢_𝑥Λ^𝑞𝑢^2𝑥𝑑𝑥≤‖‖Λ^𝑞𝑢_𝑥‖‖_𝐿2‖‖Λ^𝑞𝑢^2𝑥‖‖_𝐿²≤≤𝑐𝑐‖‖𝑢𝑢‖‖_𝐻^𝑞+1_2𝐻𝑞+1‖‖𝑢_𝑥‖‖_𝐿∞‖‖𝑢_𝑥‖‖_𝐻𝑞+‖‖𝑢_𝑥‖‖_𝐿∞‖‖𝑢_𝑥‖‖_𝐻𝑞‖‖𝑢_𝑥‖‖_𝐿∞.(4.12)
For the last term in (4.9), using Lemma 4.1 repeatedly results in ||||_𝑅Λ^𝑞𝑢_𝑥Λ^𝑞𝑢_𝑥^𝑁𝑑𝑥||||≤‖‖𝑢_𝑥‖‖_𝐻𝑞‖‖𝑢^𝑁𝑥‖‖_𝐻𝑞≤𝑐‖𝑢‖^2𝐻_𝑞+1‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞.(4.13) It follows from (4.9) to (4.13) that there exists a constant 𝑐 depending only on 𝑚,𝑁 and the coefficients of (1.2) such that 12𝑑𝑑𝑡_𝑅(Λ^𝑞𝑢)²+Λ^𝑞𝑢_𝑥²𝑑𝑥≤𝑐‖‖𝑢^𝑥‖‖_𝐿∞‖𝑢‖^2𝐻_𝑞+11+‖𝑢‖^{𝐿𝑚−1∞}+𝑐‖𝑢‖^2𝐻𝑞+1‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞.(4.14) Integrating both sides of the above inequality with respect to 𝑡 results in inequality (4.6).
To estimate the norm of 𝑢_𝑡, we apply the operator (1−𝜕^2𝑥)⁻¹ to both sides of the first equation of system (2.3) to obtain the equation 𝑢_𝑡=1−𝜕^2𝑥⁻¹−2𝑘𝑢^𝑥+𝜕_𝑥−𝑎𝑚+1𝑢^𝑚+1+12𝜕^2𝑥𝑢²−12𝑢^2𝑥+𝛽𝜕^𝑥𝑢^𝑥^𝑁.(4.15) Applying (Λ^𝑞𝑢_𝑡)Λ^𝑞 to both sides of (4.15) for 𝑞∈(0,𝑠−1] gives rise to _𝑅Λ^𝑞𝑢_𝑡²𝑑𝑥=_𝑅Λ^𝑞𝑢_𝑡Λ^𝑞−2𝜕^𝑥−2𝑘𝑢−𝑎𝑚+1𝑢^𝑚+1+12𝜕^2𝑥𝑢²−12𝑢^2𝑥+𝛽𝜕^𝑥𝑢^𝑥^𝑁𝑑𝜏.(4.16) For the right-hand side of (4.16), we have _𝑅Λ^𝑞_𝑅𝑢_𝑡Λ^𝑞−2−2𝑘𝑢_𝑥𝑑𝑥≤𝑐‖‖𝑢_𝑡‖‖_𝐻𝑞‖𝑢‖_𝐻^𝑞,≤Λ𝑐‖‖^𝑞𝑢𝑢^𝑡_𝑡1−𝜕^2𝑥⁻¹Λ^𝑞𝜕_𝑥−𝑚𝑎+1𝑢^𝑚+1−12𝑢^2𝑥𝑑𝑥‖‖_𝐻𝑞_𝑅1+𝜉²^𝑞−1×_𝑅−𝑚𝑎+1𝑢^𝑚(𝜉−𝜂)𝑢̂(𝜂)−21𝑢^𝑥(𝜉−𝜂)𝑢_𝑥(𝜂)𝑑𝜂²^1/2≤𝑐‖‖𝑢_𝑡‖‖_𝐻𝑞‖𝑢‖_𝐻¹‖𝑢‖_𝐻^𝑞+11+‖𝑢‖^{𝐿𝑚−1∞}.(4.17) Since Λ^𝑞𝑢_𝑡1−𝜕^2𝑥⁻¹Λ^𝑞𝜕^2𝑥𝑢𝑢_𝑥𝑑𝑥=−Λ^𝑞𝑢_𝑡Λ^𝑞𝑢𝑢_𝑥𝑑𝑥+Λ^𝑞𝑢_𝑡1−𝜕^2𝑥⁻¹Λ^𝑞𝑢𝑢_𝑥𝑑𝑥,(4.18) using Lemma 4.1, ‖𝑢𝑢_𝑥‖_𝐻^𝑞≤𝑐‖(𝑢²)_𝑥‖_𝐻^𝑞≤𝑐‖𝑢‖_𝐿^∞‖𝑢‖_𝐻^𝑞+1 and ‖𝑢‖_𝐿^∞≤‖𝑢‖_𝐻¹, we have Λ^𝑞𝑢_𝑡Λ^𝑞𝑢𝑢_𝑥𝑑𝑥≤𝑐‖‖𝑢_𝑡‖‖_𝐻𝑞‖‖𝑢𝑢_𝑥‖‖_𝐻𝑞Λ_𝑞≤𝑐‖‖𝑢_𝑡‖‖_𝐻𝑞‖𝑢‖_𝐻¹‖𝑢‖_𝐻^𝑞+1,𝑢_𝑡1−𝜕^2𝑥⁻¹Λ^𝑞𝑢𝑢_𝑥𝑑𝑥≤𝑐‖‖𝑢_𝑡‖‖_𝐻𝑞‖𝑢‖_𝐻¹‖𝑢‖_𝐻^𝑞+1.(4.19) Using the Cauchy-Schwartz inequality and Lemma 4.1 yields ||||_𝑅Λ^𝑞𝑢_𝑡1−𝜕^2𝑥⁻¹Λ^𝑞𝜕_𝑥𝑢^𝑁𝑥𝑑𝑥||||≤𝑐‖‖𝑢_𝑡‖‖_𝐻𝑞‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞‖𝑢‖_𝐻^𝑞+1.(4.20) Applying (4.17)–(4.20) into (4.16) yields the inequality ‖‖𝑢_𝑡‖‖_𝐻𝑞≤𝑐‖𝑢‖_𝐻^𝑞+11+1+‖𝑢‖^{𝐿𝑚−1∞}‖𝑢‖_𝐻¹+‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞.(4.21) This completes the proof of Lemma 4.3.

Defining𝜙(𝑥)=𝑒^{1/(𝑥2−1)}0,,||𝑥𝑥||<≥11,,(4.22) and 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(𝑅) and 𝑠>0.

It follows from Theorem 2.1 that for each 𝜀 the Cauchy problem𝑢_𝑡−𝑢_𝑡𝑥𝑥=𝜕_𝑥−2𝑘𝑢−𝑎𝑚𝑢(+0,1𝑥𝑢)=^𝑚+1𝑢_𝜀0+21𝜕^3𝑥𝑢²−12𝜕^𝑥𝑢^2𝑥+𝛽𝜕_𝑥𝑢^𝑥^𝑁,(𝑥),𝑥∈𝑅,(4.23) has a unique solution 𝑢_𝜀(𝑡,𝑥)∈𝐶^∞([0,𝑇);𝐻^∞).

Lemma 4.4. Under the assumptions of problem (4.23), the following estimates hold for any 𝜀 with 0<𝜀<1/4 and 𝑠>0‖‖‖‖𝑢𝑢_{𝜀0𝑥𝜀0}‖‖_𝐿∞≤𝑐₁‖‖𝑢_0𝑥‖‖_𝐿∞,‖‖𝑢‖‖_𝐻𝑞≤𝑐₁,if𝑞≤𝑠,_𝜀0‖‖𝑢‖‖_𝐻𝑞≤𝑐₁𝜀^{(𝑠−𝑞)/4},if𝑞>𝑠,_𝜀0−𝑢‖‖₀𝑢‖‖_𝐻𝑞≤𝑐₁𝜀^{(𝑠−𝑞)/4},if𝑞≤𝑠,_𝜀0−𝑢₀‖‖_𝐻𝑠=𝑜(1),(4.24) where 𝑐₁ is a constant independent of 𝜀.

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

Lemma 4.5. If 𝑢₀(𝑥)∈𝐻^𝑠(𝑅) with 𝑠∈[1,3/2] such that ‖𝑢_0𝑥‖_𝐿^∞<∞. Let 𝑢_𝜀0 be defined as in system (4.23). Then there exist two positive constants 𝑇 and 𝑐, which are independent of 𝜀, such that the solution 𝑢_𝜀 of problem (4.23) satisfies ‖𝑢_𝜀𝑥‖_𝐿^∞≤𝑐 for any 𝑡∈[0,𝑇).

Proof. Using notation 𝑢=𝑢_𝜀 and differentiating both sides of the first equation of problem (4.23) or (4.15) with respect to 𝑥 give rise to 𝑢_𝑡𝑥+12𝜕^2𝑥𝑢²−12𝑢^2𝑥=2𝑘𝑢+𝑎𝑚+1𝑢^𝑚+1−12𝑢²−𝛽𝑢^𝑁𝑥−Λ⁻²2𝑘𝑢+𝑎𝑚+1𝑢^𝑚+1−12𝑢²+12𝑢^2𝑥−𝛽𝑢^𝑁𝑥.(4.25) Letting 𝑝>0 be an integer and multiplying the above equation by (𝑢_𝑥)^2𝑝+1 and then integrating the resulting equation with respect to 𝑥 yield the equality 12𝑝+2𝑑𝑑𝑡_𝑅𝑢_𝑥^2𝑝+2𝑑𝑥+𝑝2𝑝+2_𝑅𝑢_𝑥^2𝑝+3=𝑑𝑥_𝑅−𝑢_𝑥^2𝑝+12𝑘𝑢+𝑚𝑎+1𝑢^𝑚+1−12𝑢²−𝛽𝑢^𝑁𝑥𝑑𝑥_𝑅𝑢_𝑥^2𝑝+1Λ⁻²2𝑘𝑢+𝑚𝑎+1𝑢^𝑚+1−𝑢2²+21𝑢^2𝑥−𝛽𝑢^𝑁𝑥𝑑𝑥.(4.26) Applying the Hölder's inequality yields 12𝑝+2𝑑𝑑𝑡_𝑅𝑢_𝑥^2𝑝+2𝑑𝑥≤||2𝑘||_𝑅|𝑢|^2𝑝+2𝑑𝑥^1/(2𝑝+2)+𝑎𝑚+1_𝑅||𝑢^𝑚+1||^2𝑝+2𝑑𝑥^1/(2𝑝+2)+12_𝑅||𝑢²||^2𝑝+2𝑑𝑥^1/(2𝑝+2)+𝛽_𝑅||𝑢^𝑁𝑥||^2𝑝+2𝑑𝑥^1/(2𝑝+2)+_𝑅||𝐺||^2𝑝+2𝑑𝑥^1/(2𝑝+2)_𝑅||𝑢_𝑥||^2𝑝+2𝑑𝑥^{(2𝑝+1)/(2𝑝+2)}+𝑝2𝑝+2‖‖𝑢^𝑥‖‖_𝐿∞_𝑅||𝑢_𝑥||^2𝑝+2𝑑𝑥,(4.27) or 𝑑𝑑𝑡_𝑅𝑢_𝑥^2𝑝+2𝑑𝑥^1/(2𝑝+2)≤||2𝑘||_𝑅|𝑢|^2𝑝+2𝑑𝑥^1/(2𝑝+2)+𝑎𝑚+1_𝑅||𝑢^𝑚+1||^2𝑝+2𝑑𝑥^1/(2𝑝+2)+12_𝑅||𝑢²||^2𝑝+2𝑑𝑥^1/(2𝑝+2)+𝛽_𝑅||𝑢^𝑁𝑥||^2𝑝+2𝑑𝑥^1/(2𝑝+2)+_𝑅||𝐺||^2𝑝+2𝑑𝑥^1/(2𝑝+2)+𝑝2𝑝+2‖‖𝑢^𝑥‖‖_𝐿∞_𝑅||𝑢_𝑥||^2𝑝+2𝑑𝑥^1/(2𝑝+2),(4.28) where 𝐺=Λ⁻²2𝑘𝑢+𝑚𝑎+1𝑢^𝑚+1−𝑢²2+21𝑢^2𝑥−𝛽𝑢^𝑁𝑥.(4.29) Since ‖𝑓‖_𝐿^𝑝→‖𝑓‖_𝐿^∞ as 𝑝→∞ for any 𝑓∈𝐿^∞⋂𝐿², integrating both sides of the inequality (4.28) with respect to 𝑡 and taking the limit as 𝑝→∞ result in the estimate ‖‖𝑢_𝑥‖‖_𝐿∞≤‖‖𝑢_0𝑥‖‖_𝐿∞+^𝑡0𝑐‖𝑢‖_𝐿^∞+‖‖𝑢²‖‖_𝐿∞+‖‖𝑢^𝑚+1‖‖_𝐿∞+𝛽‖‖𝑢_𝑥‖‖^𝑁𝐿_∞+‖𝐺‖_𝐿^∞+12‖‖𝑢^𝑥‖‖^2𝐿_∞𝑑𝜏.(4.30) Using the algebra property of 𝐻^𝑠0(𝑅) with 𝑠₀>1/2 yields (‖𝑢_𝜀‖_𝐻^(1/2)+ means that there exists a sufficiently small 𝛿>0 such that ‖𝑢_𝜀‖_(1/2)+=‖𝑢_𝜀‖_𝐻^1/2+𝛿)‖𝐺‖_𝐿^∞≤𝑐‖𝐺‖_𝐻^(1/2)+≤𝑐‖‖‖Λ⁻²2𝑘𝑢+𝑚𝑎+1𝑢^𝑚+1−𝑢²2+21𝑢^2𝑥−𝛽𝑢^𝑁𝑥‖‖‖_𝐻(1/2)+≤≤𝑐𝑐‖‖𝑢𝑢‖‖^𝐻1_𝐻1+‖𝑢‖^2𝐻₁+‖𝑢‖^𝐻𝑚+1₁+‖‖Λ⁻²(𝑢^2𝑥)‖‖_𝐻(1/2)++‖‖Λ⁻²(𝑢^𝑁𝑥)‖‖_𝐻(1/2)+≤𝑐‖𝑢‖_𝐻1+‖𝑢‖^2𝐻₁+‖𝑢‖^𝐻𝑚+1₁+‖‖𝑢^2𝑥‖‖_𝐻0+‖‖𝑢^𝑁𝑥‖‖_𝐻0≤𝑐𝑒+_{𝑐∫𝑡0‖𝑢𝑥‖𝐿∞𝑁−1𝑑𝜏}‖𝑢‖^2𝐻₁+‖𝑢‖^𝐻𝑚+1₁+‖‖𝑢_𝑥‖‖_𝐿∞‖𝑢‖_𝐻¹+‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞‖𝑢‖_𝐻¹1+‖‖𝑢^𝑥‖‖_𝐿∞+‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞,(4.31) in which Lemma 4.3 is used. Therefore, we get ^𝑡0‖𝐺‖_𝐿^∞𝑑𝜏≤𝑐^𝑡0𝑒^{𝑐∫𝜏0‖𝑢𝑥‖𝐿∞𝑁−1𝑑𝜉}1+‖‖𝑢^𝑥‖‖_𝐿∞+‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞𝑑𝜏.(4.32)
From (4.30) and (4.32), one has ‖‖𝑢_𝑥‖‖_𝐿∞≤‖‖𝑢_0𝑥‖‖_𝐿∞+𝑐^𝑡0‖‖𝑢^𝑥‖‖^2𝐿_∞+‖‖𝑢_𝑥‖‖^𝑁𝐿_∞+𝑒^{𝑐∫𝑡0‖𝑢𝑥‖𝐿∞𝑁−1𝑑𝜏}+𝑒^{𝑐∫𝜏0‖𝑢𝑥‖𝐿∞𝑁−1𝑑𝜉}1+‖‖𝑢^𝑥‖‖_𝐿∞+‖‖𝑢_𝑥‖‖^𝐿𝑁−1_∞𝑑𝜏.(4.33)
From Lemma 4.4, it follows from the contraction mapping principle that there is a 𝑇>0 such that the equation ‖𝑊‖_𝐿^∞=‖‖𝑢_0𝑥‖‖_𝐿∞+𝑐^𝑡0‖𝑊‖^2𝐿∞+‖𝑊‖^𝑁𝐿_∞+𝑒^{𝑐∫𝑡0‖𝑊‖𝐿∞𝑁−1𝑑𝜏}+𝑒^{𝑐∫𝜏0‖𝑊‖𝐿∞𝑁−1𝑑𝜉}1+‖𝑊‖_𝐿^∞+‖𝑊‖^𝐿𝑁−1_∞𝑑𝜏(4.34) has a unique solution 𝑊∈𝐶[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 𝜀 such that ‖𝑢_𝑥‖_𝐿^∞≤𝑊(𝑡) for arbitrary 𝑡∈[0,𝑇], which leads to the conclusion of Lemma 4.5.
Using Lemmas 4.3 and 4.5, notation 𝑢_𝜀=𝑢 and Gronwall's inequality results in the inequalities ‖‖‖‖𝑢𝑢_𝜀𝜀𝑡‖‖_𝐻𝑞≤𝐶_𝑇𝑒^𝐶𝑇,‖‖_𝐻𝑟≤𝐶_𝑇𝑒^𝐶𝑇,(4.35) where 𝑞∈(0,𝑠], 𝑟∈(0,𝑠−1] and 𝐶_𝑇 depends on 𝑇. It follows from Aubin's compactness theorem that there is a subsequence of {𝑢_𝜀}, denoted by {𝑢_𝜀𝑛}, such that {𝑢_𝜀𝑛} and their temporal derivatives {𝑢_𝜀𝑛𝑡} are weakly convergent to a function 𝑢(𝑡,𝑥) and its derivative 𝑢_𝑡 in 𝐿²([0,𝑇],𝐻^𝑠) and 𝐿²([0,𝑇],𝐻^𝑠−1), respectively. Moreover, for any real number 𝑅₁>0, {𝑢_𝜀𝑛} is convergent to the function 𝑢 strongly in the space 𝐿²([0,𝑇],𝐻^𝑞(−𝑅₁,𝑅₁)) for 𝑞∈[0,𝑠) and {𝑢_𝜀𝑛𝑡} converges to 𝑢_𝑡 strongly in the space 𝐿²([0,𝑇],𝐻^𝑟(−𝑅₁,𝑅₁)) for 𝑟∈[0,𝑠−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 {𝑢_𝜀𝑛𝑥}(𝜀_𝑛→0) is bounded in the space 𝐿^∞. Thus, the sequences {𝑢_𝜀𝑛} and {𝑢_𝜀𝑛𝑥} are weakly convergent to 𝑢 and 𝑢_𝑥 in 𝐿²[0,𝑇],𝐻^𝑟(−𝑅,𝑅) for any 𝑟∈[0,𝑠−1), respectively. Therefore, 𝑢 satisfies the equation −^𝑇0_𝑅𝑢𝑔_𝑡−𝑔_𝑥𝑥𝑡𝑑𝑥𝑑𝑡=^𝑇0_𝑅2𝑘𝑢+𝑚𝑎+1𝑢^𝑚+1+12𝑢^2𝑥𝑔_𝑥−12𝑢²𝑔_𝑥𝑥𝑥−𝛽𝑢_𝑥^𝑁𝑔_𝑥𝑑𝑥𝑑𝑡,(4.36) with 𝑢(0,𝑥)=𝑢₀(𝑥) and 𝑔∈𝐶^∞0. Since 𝑋=𝐿¹([0,𝑇]×𝑅) is a separable Banach space and {𝑢_𝜀𝑛𝑥} is a bounded sequence in the dual space 𝑋^∗=𝐿^∞([0,𝑇]×𝑅) of 𝑋, there exists a subsequence of {𝑢_𝜀𝑛𝑥}, still denoted by {𝑢_𝜀𝑛𝑥}, weakly star convergent to a function 𝑣 in 𝐿^∞([0,𝑇]×𝑅). It derives from the {𝑢_𝜀𝑛𝑥} weakly convergent to 𝑢_𝑥 in 𝐿²([0,𝑇]×𝑅) that 𝑢_𝑥=𝑣 almost everywhere. Thus, we obtain 𝑢_𝑥∈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).