The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

Abstract

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that (1−𝜕²_𝑥)𝑢₀∈𝑀⁺(𝑅), 𝑢₀∈𝐻¹(𝑅), and 𝑢₀∈𝐿¹(𝑅), the existence and uniqueness of the global weak solution to the equation are shown to be true.

1. Introduction

The Camassa-Holm equation (C-H equation) 𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥=0,𝑡>0,𝑥∈𝑅, (1.1) as a model for wave motion on shallow water, has a bi-Hamiltonian structure and is completely integrable. After the equation was derived by Camassa and Holm [1], a lot of works was devoted to its investigation of dynamical properties. The local well posedness of solution for initial data 𝑢₀∈𝐻^𝑠(𝑅) with 𝑠>3/2 was given by several authors [2–4]. Under certain assumptions, (1.1) has not only global strong solutions and blow-up solutions [2, 5–7] but also global weak solutions in 𝐻¹(𝑅) (see [8–10]). For other methods to handle the problems related to the Camassa-Holm equation and functional spaces, the reader is referred to [11–14] and the references therein.

To study the effect of the weakly dissipative term on the C-H equation, Guo [15] and Wu and Yin [16] discussed the weakly dissipative C-H equation 𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥=0,𝑡>0,𝑥∈𝑅. (1.2) The global existence of strong solutions and blow-up in finite time were presented in [16] provided that 𝑦₀=(1−𝜕²_𝑥)𝑢₀ changes sign. The sufficient conditions on the infinite propagation speed for (1.2) are offered in [15]. It is found that the local well posedness and the blow-up phenomena of (1.2) are similar to the C-H equation in a finite interval of time. However, there are differences between (1.2) and the C-H equation in their long time behaviors. For example, the global strong solutions of (1.2) decay to zero as time tends to infinite under suitable assumptions, which implies that (1.2) has no traveling wave solutions (see [16]).

Degasperis and Procesi [17] derived the equation (D-P equation) 𝑢_𝑡−𝑢_𝑡𝑥𝑥+4𝑢𝑢_𝑥−3𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥=0,𝑡>0,𝑥∈𝑅, (1.3) as a model for shallow water dynamics. Although the D-P equation (1.3) has a similar form to the C-H equation (1.1), it should be addressed that they are truly different (see [18]). In fact, many researchers have paid their attention to the study of solutions to (1.3). Constantin et al. [19] developed an inverse scattering approach for smooth localized solutions to (1.3). Liu and Yin [20] and Yin [21, 22] investigated the global existence of strong solutions and global weak solutions to (1.3). Henry [23] showed that the smooth solutions to (1.3) have infinite speed of propagation. Coclite and Karlsen [24] obtained global existence results for entropy solutions in 𝐿¹(𝑅)∩𝐵𝑉(𝑅) and in 𝐿²(𝑅)∩𝐿⁴(𝑅).

The weakly dissipative D-P equation 𝑢_𝑡−𝑢_𝑡𝑥𝑥+4𝑢𝑢_𝑥−3𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥=0,𝑡>0,𝑥∈𝑅 (1.4) is investigated by several authors [25–27] to find out the effect of the weakly dissipative term on the D-P equation. The global existence, persistence properties, unique continuation and the infinite propagation speed of the strong solutions to (1.4) are studied in [26]. The blow-up solution modeling wave breaking and the decay of solution were discussed in [27]. The existence and uniqueness of the global weak solution in space 𝑊^1,∞_loc(𝑅₊×𝑅)⋂𝐿^∞_loc(𝑅₊;𝐻¹(𝑅)) were proved (see [25]).

In this paper, we will consider the Cauchy problem for the weakly dissipative shallow water wave equation 𝑢_𝑡−𝑢_𝑡𝑥𝑥+(𝑎+𝑏)𝑢𝑢_𝑥−𝑎𝑢_𝑥𝑢_𝑥𝑥−𝑏𝑢𝑢_𝑥𝑥𝑥+𝜆𝑢−𝑢_𝑥𝑥𝑢()=0,𝑡>0,𝑥∈𝑅,0,𝑥=𝑢₀(𝑥),𝑥∈𝑅, (1.5) where 𝑎>0,𝑏>0, and 𝜆≥0 are arbitrary constants, 𝑢 is the fluid velocity in the 𝑥 direction, 𝜆(𝑢−𝑢_𝑥𝑥) represents the weakly dissipative term. For 𝜆=0, we notice that (1.5) is a special case of the shallow water equation derived by Constantin and Lannes [28].

Since (1.5) is a generalization of the Camassa-Holm equation and the Degasperis-Procesi equation, (1.5) loses some important conservation laws that C-H equation and D-P equation possesses. It needs to be pointed out that Lai and Wu [12] studied global existence and blow-up criteria for (1.5) with 𝜆=0. To the best of our knowledge, the dynamical behaviors related to (1.5) with 𝜆=0, such as the global weak solution in space 𝑊^1,∞_loc(𝑅₊×𝑅)⋂𝐿^∞_loc(𝑅₊;𝐻¹(𝑅)), have not been yet investigated. The objective of this paper is to investigate several dynamical properties of solutions to (1.5). More precisely, we firstly use the Kato theorem [29] to establish the local well-posedness for (1.5) with initial value 𝑢₀∈𝐻^𝑠 with 𝑠>3/2. Then, we present a precise blow-up scenario for (1.5). Provided that 𝑢₀∈𝐻^𝑠(𝑅)⋂𝐿¹(𝑅) and the potential 𝑦₀=(1−𝜕²_𝑥)𝑢₀ does not change sign, the global existence of the strong solution is shown to be true. Finally, under suitable assumptions, the existence and uniqueness of global weak solution in 𝑊^1,∞(𝑅₊×𝑅)⋂𝐿^∞_loc(𝑅₊;𝐻¹(𝑅)) are proved. Our main ideas to prove the existence and uniqueness of the global weak solution come from those presented in Constantin and Molinet [8] and Yin [22].

2. Notations

The space of all infinitely differentiable functions 𝜙(𝑡,𝑥) with compact support in [0,+∞)×𝑅 is denoted by 𝐶^∞₀. Let 1≤𝑝<+∞, and let 𝐿^𝑝=𝐿^𝑝(𝑅) be the space of all measurable functions ℎ(𝑡,𝑥) such that ‖ℎ‖^𝑃_𝐿𝑃=∫_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, let 𝐻^𝑠=𝐻^𝑠(𝑅) denote the Sobolev space with the norm defined by ‖ℎ‖_𝐻𝑠=(∫_𝑅(1+|𝜉|²)^𝑠|ℎ(𝑡,𝜉)|²𝑑𝜉)^1/2<∞, where ℎ(𝑡,𝜉)=∫_𝑅𝑒^{−𝑖𝑥𝜉}ℎ(𝑡,𝑥)𝑑𝑥.

We denote by ∗ the convolution. Let ‖⋅‖_𝑋 denote the norm of Banach space 𝑋 and ⟨⋅,⋅⟩ the 𝐻¹(𝑅), 𝐻⁻¹(𝑅) duality bracket. Let 𝑀(𝑅) be the space of the Radon measures on R with bounded total variation and 𝑀⁺(𝑅) the subset of positive measures. Finally, we write 𝐵𝑉(𝑅) for the space of functions with bounded variation, 𝑉(𝑓) being the total variation of 𝑓∈𝐵𝑉(𝑅).

Note that if 𝐺(𝑥)∶=(1/2)𝑒^−|𝑥|, 𝑥∈𝑅. Then, (1−𝜕²_𝑥)⁻¹𝑓=𝐺∗𝑓 for all 𝑓∈𝐿²(𝑅) and 𝐺∗(𝑢−𝑢_𝑥𝑥)=𝑢. Using this identity, we rewrite problem (1.5) in the form 𝑢_𝑡+𝑏𝑢𝑢_𝑥+𝜕_𝑥𝑎𝐺∗2𝑢²+3𝑏−𝑎2𝑢_𝑥²𝑢()+𝜆𝑢=0,𝑡>0,𝑥∈𝑅,0,𝑥=𝑢₀(𝑥),𝑥∈𝑅, (2.1) which is equivalent to 𝑦_𝑡+𝑏𝑢𝑦_𝑥+𝑎𝑦𝑢_𝑥+𝜆𝑦=0,𝑡>0,𝑥∈𝑅,𝑦=𝑢−𝑢_𝑥𝑥,𝑢(0,𝑥)=𝑢₀(𝑥). (2.2)

3. Preliminaries

Throughout this paper, let {𝜌_𝑛}_𝑛≥1 denote the mollifiers 𝜌_𝑛(𝑥)∶=_𝑅𝜌(𝜉)𝑑𝜉⁻¹𝑛𝜌(𝑛𝑥),𝑥∈𝑅,𝑛≥1, (3.1) where 𝜌∈𝐶^∞_𝑐(𝑅) is defined by 𝜌(𝑥)∶=𝑒^{1/(𝑥2−1)}|𝑥||𝑥|for<1,0for≥1. (3.2) Thus, we get _𝑅𝜌_𝑛(𝑥)𝑑𝑥=1,𝜌_𝑛≥0,𝑥∈𝑅,𝑛≥1. (3.3) Next, we give some useful results.

Lemma 3.1 (see [8]). Let 𝑓∶𝑅→𝑅 be uniformly continuous and bounded. If 𝜇∈𝑀(𝑅), then 𝜌_𝑛∗𝑓𝜇−𝜌_𝑛∗𝑓𝜌_𝑛∗𝜇𝑛⟶⃗∞0in𝐿¹(𝑅). (3.4)

Lemma 3.2 (see [8]). Let 𝑓∶𝑅→𝑅 be uniformly continuous and bounded. If 𝑔∈𝐿^∞(𝑅), then 𝜌_𝑛∗𝑓𝑔−𝜌_𝑛∗𝑓𝜌_𝑛∗𝑔𝑛⟶⃗∞0in𝐿^∞(𝑅). (3.5)

Lemma 3.3 (see [30]). Let 𝑇>0. If 𝑓,𝑔∈𝐿²((0,𝑇);𝐻¹(𝑅)) and 𝑑𝑓/𝑑𝑡,𝑑𝑔/𝑑𝑡∈𝐿²((0,𝑇);𝐻⁻¹(𝑅)), then 𝑓,𝑔 are a.e. equal to a function continuous from [0,𝑇] into 𝐿²(𝑅) and 𝑓(𝑡),𝑔(𝑡)−𝑓(𝑠),𝑔(𝑠)=^𝑡_𝑠𝑑𝑓(𝜏)(𝜏𝑑𝜏,𝑔)𝑑𝜏+^𝑡_𝑠𝑑𝑔(𝜏)(𝜏𝑑𝜏,𝑓)𝑑𝜏 (3.6) for all 𝑠,𝑡∈[0,𝑇].

Lemma 3.4 (see [8]). Assume that 𝑢(𝑡,⋅)∈𝑊^1,1(𝑅) is uniformly bounded in 𝑊^1,1(𝑅) for all 𝑡∈𝑅₊. Then, for a.e. 𝑡∈𝑅₊𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑢𝑑𝑥=_𝑅𝜌_𝑛∗𝑢_𝑡sgn𝜌_𝑛𝑑∗𝑢𝑑𝑥,𝑑𝑡_𝑅𝜌_𝑛∗𝑢_𝑥𝑑𝑥=_𝑅𝜌_𝑛∗𝑢_𝑥𝑡sgn𝜌_𝑛∗𝑢_𝑥𝑑𝑥. (3.7)

4. Global Strong Solution

We firstly present the existence and uniqueness of the local solutions to the problem (2.1).

Theorem 4.1. Let 𝑢₀∈𝐻^𝑠(𝑅),𝑠>3/2. Then, the problem (2.1) has a unique solution 𝑢, such that 𝑢=𝑢(𝑡,𝑥)∈𝐶[0,𝑇);𝐻^𝑠(𝑅)∩𝐶¹[0,𝑇);𝐻^𝑠−1(𝑅), (4.1) where 𝑇>0 depends on ‖𝑢₀‖_{𝐻𝑠(𝑅)}.

Proof. The proof of Theorem 4.1 can be finished by using Kato’s semigroup theory (see [29] or [4]). Here, we omit the detailed proof.

Theorem 4.2. Given 𝑢₀∈𝐻^𝑠,𝑠>3/2, the solution 𝑢=(⋅,𝑢₀) of problem (2.1) blows up in finite time 𝑇<+∞ if and only if lim_𝑡→𝑇infinf_𝑥∈𝑅𝑢_𝑥(𝑡,𝑥)=−∞. (4.2)

Proof. Setting 𝑦(𝑡,𝑥)=𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥), we get 𝑦²_𝐿2=_𝑅𝑢−𝑢_𝑥𝑥²𝑑𝑥=_𝑅𝑢²+2𝑢²_𝑥+𝑢²_𝑥𝑥𝑑𝑥, (4.3) which yields ‖𝑢‖²_𝐻2≤𝑦²_𝐿2≤2‖𝑢‖²_𝐻2. (4.4) Using system (2.2), one has 𝑑𝑑𝑡_𝑅𝑦²(𝑡,𝑥)𝑑𝑥=2_𝑅𝑦𝑦_𝑡𝑑𝑥=−2𝑏_𝑅𝑢𝑦𝑦_𝑥𝑑𝑥−2𝑎_𝑅𝑢_𝑥𝑦²𝑑𝑥−2𝜆_𝑅𝑦²()𝑑𝑥=−2𝑎−𝑏_𝑅𝑢_𝑥𝑦²𝑑𝑥−2𝜆_𝑅𝑦²𝑑𝑥. (4.5) Assume that there is a constant 𝑀>0 such that −𝑢_𝑥(𝑡,𝑥)≤𝑀on[0,𝑇)×𝑅. (4.6) From (4.5), we get 𝑑𝑑𝑡_𝑅𝑦²(𝑡,𝑥)𝑑𝑥≤|2𝑎−𝑏|𝑀_𝑅𝑦²𝑑𝑥−2𝜆_𝑅𝑦²𝑑𝑥. (4.7) Using Gronwall’ inequality, we deduce the ‖𝑢‖_𝐻2 is bounded on [0,𝑇). On the other hand, 𝑢(𝑡,𝑥)=1−𝜕²_𝑥⁻¹𝑦(𝑡,𝑥)=_𝑅𝐺(𝑥−𝑠)𝑦(𝑠)𝑑𝑠. (4.8) Therefore, using (4.4) leads to 𝑢_𝑥𝐿∞≤_𝑅𝐺_𝑥(𝑥−𝑠)𝑦(𝑠)𝑑𝑠≤𝐺_𝑥𝐿2𝑦_𝐿2=12𝑦_𝐿2≤‖𝑢‖_𝐻2. (4.9) It shows that if ‖𝑢‖_𝐻2 is bounded, then ‖𝑢_𝑥‖_𝐿∞ is also bounded. This completes the proof.

We consider the differential equation 𝑞_𝑡[)𝑞()=𝑏𝑢𝑡,𝑞,𝑡∈0,𝑇,𝑥∈𝑅,0,𝑥=𝑥,𝑥∈𝑅, (4.10) where 𝑢 solves (1.5) and 𝑇>0.

Lemma 4.3. Let 𝑢₀∈𝐻^𝑠(𝑅)(𝑠>3); 𝑇 is the maximal existence time of the corresponding solution 𝑢 to (1.5). Then, system (4.10) has a unique solution 𝑞∈𝐶¹([0,𝑇)×𝑅;𝑅). Moreover, the map 𝑞(𝑡,⋅) is an increasing diffeomorphism of 𝑅 with 𝑞_𝑥(𝑡,𝑥)=exp^𝑡₀𝑏𝑢_𝑥()()∈[)𝑦()𝑞𝑠,𝑞𝑠,𝑥𝑑𝑠>0,∀𝑡,𝑥0,𝑇×𝑅,𝑡,𝑞𝑡,𝑥²_𝑥(𝑡,𝑥)=𝑦₀(𝑥)exp^𝑡₀−(𝑎−2𝑏)𝑢_𝑥𝑠,𝑞(𝑠,𝑥)−𝜆𝑑𝑠. (4.11)

Proof. From Theorem 4.1, we have 𝑢∈𝐶¹([0,𝑇);𝐻^𝑠−1(𝑅)) and 𝐻^𝑠−1∈𝐶¹(𝑅). We conclude that both functions 𝑢(𝑡,𝑥) and 𝑢_𝑥(𝑡,𝑥) are bounded, Lipschitz in space, and 𝐶¹ in time. Applying the existence and uniqueness theorem of ordinary differential equations implies that system (4.10) has a unique solution 𝑞∈𝐶¹([0,𝑇)×𝑅,𝑅).
Differentiating (4.10) with respect to 𝑥 leads to𝑑𝑞𝑑𝑡^𝑥=𝑏𝑢_𝑥𝑡,𝑞𝑞_𝑥[)𝑞,𝑡∈0,𝑇,𝑏>0,_𝑥(0,𝑥)=1,𝑥∈𝑅, (4.12) which yields 𝑞_𝑥=exp^𝑡₀𝑏𝑢_𝑥𝑠,𝑞(𝑠,𝑥)𝑑𝑠. (4.13) For every 𝑇^′<𝑇, using the Sobolev embedding theorem gives rise to sup_{()∈𝑠,𝑥0,𝑇′×𝑅}𝑢_𝑥(𝑠,𝑥)<∞. (4.14) It is inferred that there exists a constant 𝐾₀>0 such that 𝑞_𝑥≥𝑒^−𝐾0𝑡 for (𝑡,𝑥)∈[0,𝑇)×𝑅.
By computing directly, we derive 𝑑𝑦()𝑞𝑑𝑡𝑡,𝑞𝑡,𝑥²_𝑥(𝑡,𝑥)=−(𝑎−2𝑏)𝑢_𝑥(𝑡,𝑥)𝑦𝑞²_𝑥−𝜆𝑦𝑞²_𝑥, (4.15)
which results in 𝑦𝑡,𝑞(𝑡,𝑥)𝑞²_𝑥(𝑡,𝑥)=𝑦₀(𝑥)exp^𝑡₀−(𝑎−2𝑏)𝑢_𝑥𝑠,𝑞(𝑠,𝑥)−𝜆𝑑𝑠. (4.16) The proof of Lemma 4.3 is completed.

Theorem 4.4. Let 𝑢₀∈𝐿¹(𝑅)∩𝐻^𝑠(𝑅),𝑠>3/2, and (1−𝜕²_𝑥)𝑢₀≥0 for all 𝑥∈𝑅 (or equivalently (1−𝜕²_𝑥)𝑢₀≤0 for all 𝑥∈𝑅). Then, problem (2.1) has a global strong solution 𝑢=𝑢⋅,𝑢₀∈𝐶[0,∞);𝐻^𝑠(𝑅)∩𝐶¹[0,∞);𝐻^𝑠−1(𝑅). (4.17) Moreover, if 𝑦(𝑡,⋅)=𝑢−𝑢_𝑥𝑥, then one has for all 𝑡∈𝑅₊,(i)𝑦(𝑡,⋅)≥0, 𝑢(𝑡,⋅)≥0, and |𝑢_𝑥(𝑡,⋅)|≤𝑢(𝑡,⋅) on R,(ii)𝑒^−𝜆𝑡‖𝑦₀‖_𝐿1(𝑅)=‖𝑦(𝑡,⋅)‖_𝐿1(𝑅)=‖𝑢(𝑡,⋅)‖_𝐿1(𝑅)=𝑒^−𝜆𝑡‖𝑢₀‖_𝐿1(𝑅) and ‖𝑢_𝑥(𝑡,⋅)‖_{𝐿∞(𝑅)}≤𝑒^−𝜆𝑡‖𝑢₀‖_𝐿1(𝑅),(iii)‖𝑢‖²_𝐻1≤‖𝑢₀‖²_𝐻1exp[(|𝑎−2𝑏|/𝜆)(1−𝑒^−𝜆𝑡)‖𝑢₀‖_𝐿1−2𝜆𝑡].

Proof. Let 𝑢₀∈𝐻^𝑠,𝑠>3/2, and let 𝑇>0 be the maximal existence time of the solution 𝑢 with initial date 𝑢₀ (cf. Theorem 4.1). If 𝑦₀≥0, then Theorem 4.2 ensures that 𝑦≥0 for all 𝑡∈[0,𝑇).
Note that 𝑢₀=𝐺∗𝑦₀ and 𝑦₀=(𝑢₀−𝑢_0,𝑥𝑥)∈𝐿¹(𝑅). By Young’s inequality, we get𝑢_0𝐿1(𝑅)=𝐺∗𝑦₀(𝑡,⋅)_𝐿1(𝑅)≤‖𝐺‖_𝐿1(𝑅)≤‖𝐺‖_𝐿1(𝑅)𝑦₀(𝑡,⋅)_𝐿1(𝑅)≤𝑦_0𝐿1(𝑅). (4.18) Integrating the first equation of problem (2.1) by parts, we get 𝑑𝑑𝑡_𝑅𝑢𝑑𝑥=−𝑏_𝑅𝑢𝑢_𝑥𝑑𝑥−_𝑅𝜕_𝑥𝑎𝐺∗2𝑢²+3𝑏−𝑎2𝑢_𝑥²𝑑𝑥+𝜆_𝑅𝑢𝑑𝑥=0. (4.19) It follows that _𝑅𝑢𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝑢₀𝑑𝑥. (4.20) Since 𝑦=𝑢−𝑢_𝑥𝑥, we have _𝑅𝑦𝑑𝑥=_𝑅𝑢𝑑𝑥−_𝑅𝑢_𝑥𝑥𝑑𝑥=_𝑅𝑢𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝑢₀𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝑢₀𝑑𝑥−_𝑅𝑢_0,𝑥𝑥𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝑦₀𝑑𝑥. (4.21) Given 𝑡∈[0,𝑇), due to 𝑢(𝑡,𝑥)∈𝐻^𝑠,𝑠≥3/2, from Theorem 4.1 and (4.21), we obtain −𝑢_𝑥(𝑡,𝑥)+^𝑥_−∞𝑢𝑑𝑥=^𝑥_−∞𝑢−𝑢_𝑥𝑥𝑑𝑥=^𝑥_−∞≤𝑦𝑑𝑥_∞−∞𝑦𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝑦₀𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝑢₀𝑑𝑥. (4.22) Note that 𝑢=𝐺∗𝑦,  𝑦≥0 on [0,𝑇) and the positivity of 𝐺. Thus, we can infer that 𝑢≥0 on [0,𝑇). From (4.22) we have 𝑢_𝑥(𝑡,𝑥)≥−𝑒^−𝜆𝑡_𝑅𝑢₀𝑑𝑥,∀(𝑡,𝑥)∈[0,𝑇)×𝑅. (4.23) From Theorem 4.2 and (4.23), we find 𝑇=∞. This implies that problem (2.1) has a unique solution 𝑢=𝑢⋅,𝑢₀∈𝐶[0,∞);𝐻^𝑠(𝑅)𝐶¹[0,∞);𝐻^𝑠−1(𝑅3),𝑠≥2. (4.24) Due to 𝑦(𝑡,𝑥)≥0 and 𝑢(𝑡,𝑥)≥0 for all 𝑡≥0, it shows that 𝑢()=𝑒𝑡,𝑥^−𝑥2^𝑥_−∞𝑒^𝜉𝑦(𝜉)𝑒𝑑𝜉+^𝑥2^∞_𝑥𝑒^−𝜉𝑦(𝜉)𝑢𝑑𝜉,_𝑥()𝑒𝑡,𝑥=−^−𝑥2^𝑥_−∞𝑒^𝜉𝑦(𝜉)𝑒𝑑𝜉+^𝑥2^∞_𝑥𝑒^−𝜉𝑦(𝜉)𝑑𝜉. (4.25) From the two identities above, we infer that (𝑢_𝑥)²≤𝑢² on 𝑅 for all 𝑡≥0. This proves (i).
Due to 𝑦≥0, we obtain𝑢_𝑥(𝑡,𝑥)−^𝑥_−∞𝑢𝑑𝑥=−^𝑥_−∞𝑢−𝑢_𝑥𝑥𝑑𝑥=−^𝑥_−∞𝑦𝑑𝑥≤0. (4.26) From 𝑢≥0 and the inequality above, we get 𝑢_𝑥(𝑡,𝑥)≤^𝑥_−∞𝑢𝑑𝑥≤^∞_−∞𝑢𝑑𝑥=𝑒^{−𝜆𝑡∞}_−∞𝑢₀𝑑𝑥=𝑒^−𝜆𝑡𝑢_0𝐿1(𝑅). (4.27) On the other hand, from (4.23), we have that 𝑢_𝑥(𝑡,𝑥)≥−𝑒^−𝜆𝑡‖𝑢₀‖_𝐿1(𝑅). This proves (ii).
Multiplying the first equation of problem (2.1) by 𝑢 and integrating by parts, we find 12𝑑𝑑𝑡_𝑅𝑢²+𝑢²_𝑥𝑑𝑥=−(𝑎+𝑏)_𝑅𝑢²𝑢_𝑥𝑑𝑥+𝑎_𝑅𝑢𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+𝑏_𝑅𝑢²𝑢_𝑥𝑥𝑥𝑑𝑥−𝜆_𝑅𝑢²+𝑢²_𝑥𝑑𝑥=−𝑎−2𝑏2_𝑅𝑢³_𝑥𝑑𝑥−𝜆_𝑅𝑢²+𝑢²_𝑥𝑑𝑥, (4.28)
which yields _𝑅𝑢²+𝑢²_𝑥𝑑𝑥=−(𝑎−2𝑏)_𝑅𝑢³_𝑥𝑑𝑥−2𝜆_𝑅𝑢²+𝑢²_𝑥≤||𝑑𝑥𝑎−2𝑏_𝑅𝑢_𝑥³𝑑𝑥−2𝜆_𝑅𝑢²+𝑢²_𝑥≤|𝑢𝑑𝑥𝑎−2𝑏|_𝑥𝐿∞𝑢_𝑥²_𝐿2−2𝜆_𝑅𝑢²+𝑢²_𝑥≤|𝑢𝑑𝑥𝑎−2𝑏|_𝑥𝐿∞‖𝑢‖²_𝐻1−2𝜆_𝑅𝑢²+𝑢²_𝑥𝑑𝑥. (4.29) From Gronwall’s inequality, one has ‖𝑢‖²_𝐻1≤𝑢₀²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡𝑢_0𝐿1−2𝜆𝑡. (4.30) This proves (iii) and completes the proof of the theorem.

5. Global Weak Solution

Theorem 5.1. Let 𝑢₀∈𝐻¹(𝑅)⋂𝐿¹(𝑅) and 𝑦₀=(𝑢₀−𝑢_0𝑥𝑥)∈𝑀⁺(𝑅). Then equation (1.5) has a unique solution 𝑢∈𝑊^1,∞(𝑅₊×𝑅)⋂𝐿^∞_loc(𝑅₊;𝐻¹(𝑅)) with initial data 𝑢(0)=𝑢₀ and such that (𝑢−𝑢_𝑥𝑥)∈𝑀⁺,𝑎.𝑒.∈𝑅₊ is uniformly bounded on R.

Proof. We split the proof of Theorem 5.1 in two parts.
Let 𝑢₀∈𝐻¹(𝑅) and 𝑦₀=𝑢₀−𝑢_0,𝑥𝑥∈𝑀⁺(𝑅). Note that 𝑢₀=𝐺∗𝑦₀. Thus, for 𝜑∈𝐿^∞(𝑅), we have𝑢_0𝐿1(𝑅)=𝐺∗𝑦_0𝐿1(𝑅)=sup_{𝜑𝐿∞(𝑅)≤1𝑅}𝜑(𝑥)𝐺∗𝑦₀(𝑥)𝑑𝑥=sup_{𝜑𝐿∞(𝑅)≤1𝑅}𝜑(𝑥)_𝑅𝐺(𝑥−𝜉)𝑑𝑦₀(𝜉)𝑑𝑥=sup_{𝜑𝐿∞(𝑅)≤1𝑅}𝐺∗𝜑(𝜉)𝑑𝑦₀(𝜉)=sup_{𝜑𝐿∞(𝑅)≤1}𝐺∗𝜑_{𝐿∞(𝑅)}𝑦_0𝑀(𝑅)≤sup_{𝜑𝐿∞(𝑅)≤1}‖𝐺‖_𝐿1(𝑅)𝜑_{𝐿∞(𝑅)}𝑦_0𝑀(𝑅)=𝑦_0𝑀(𝑅). (5.1) Let us define 𝑢^𝑛₀∶=𝜌_𝑛∗𝑢₀∈𝐻^∞(𝑅) for 𝑛≥1. Obviously, we get 𝑢^𝑛₀⟶𝑢₀in𝐻¹(𝑅)𝑢for𝑛⟶∞,_{𝑛0𝐻1(𝑅)}=𝜌_𝑛∗𝑢_0𝐻1(𝑅)=𝜌_𝑛∗𝑢_0𝐿2+𝜌_𝑛∗𝑢^′_0𝐿2≤𝑢_0𝐻1(𝑅),‖𝑢^𝑛₀‖_𝐿1(𝑅)=‖𝜌_𝑛∗𝑢₀‖_𝐿1(𝑅)≤𝑢_0𝐿1(𝑅). (5.2) Note that, for all 𝑛≥1, 𝑦^𝑛₀∶=𝑢^𝑛₀−𝑢^𝑛_0,𝑥𝑥=𝜌_𝑛∗𝑦₀≥0. (5.3) Referring to the proof of (5.1), we have 𝑦^𝑛_0𝐿1(𝑅)≤𝑦_0𝑀(𝑅),𝑛≥1. (5.4) From the Theorem 4.4, we know that there exists a global strong solution 𝑢^𝑛=𝑢^𝑛⋅,𝑢^𝑛₀∈𝐶[0,∞);𝐻^𝑠(𝑅)∩𝐶¹[0,∞);𝐻^𝑠−1(𝑅3),𝑠≥2, (5.5) and 𝑢^𝑛(𝑡,𝑥)−𝑢^𝑛_𝑥𝑥(𝑡,𝑥)≥0 for all (𝑡,𝑥)∈𝑅₊×𝑅.
Note that for all (𝑡,𝑥)∈𝑅₊×𝑅𝑢^𝑛2=^𝑥_−∞2𝑢^𝑛𝑢^𝑛_𝑥𝑑𝜉≤_𝑅𝑢^𝑛2+𝑢^𝑛_𝑥²𝑑𝜉=𝑢^𝑛2_𝐻1. (5.6)
From Theorem 4.4 and (5.2), we obtain 𝑢^𝑛_𝑥²_{𝐿∞(𝑅)}≤𝑢^𝑛2_{𝐿∞(𝑅)}≤𝑢^𝑛2_𝐻1(𝑅)≤𝑢^𝑛₀²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡‖𝑢^𝑛₀‖_𝐿1≤𝑢−2𝜆𝑡_02𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡‖𝑢₀‖_𝐿1−2𝜆𝑡. (5.7) From the Hölder inequality, Theorem 4.4, and (5.2), for all 𝑡≥0 and 𝑛≥1, we have ‖𝑏𝑢^𝑛(𝑡)𝑢^𝑛_𝑥(𝑡)‖_𝐿2(𝑅)≤𝑏𝑢^𝑛(𝑡)_{𝐿∞(𝑅)}𝑢^𝑛_𝑥(𝑡)_𝐿2(𝑅)≤𝑏𝑢^𝑛2_𝐻1(𝑅)≤𝑏𝑢^𝑛₀²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡𝑢^𝑛_0𝐿1𝑢−2𝜆𝑡≤𝑏₀²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡𝑢_0𝐿1−2𝜆𝑡. (5.8) Using Young’s inequality, we get 𝜕_𝑥𝑎𝐺∗2𝑢^𝑛2+3𝑏−𝑎2𝑢^𝑛_𝑥²_𝐿2(𝑅)≤𝑎2𝜕_𝑥𝐺∗𝑢^𝑛2_𝐿2(𝑅)+3𝑏+𝑎2𝜕_𝑥𝐺∗𝑢^𝑛_𝑥²_𝐿2(𝑅)≤𝑎2𝜕_𝑥𝐺_𝐿2(𝑅)𝑢^𝑛2_𝐿1(𝑅)+3𝑏+𝑎2𝜕_𝑥𝐺_𝐿2(𝑅)𝑢^𝑛_𝑥²_𝐿1(𝑅)≤3𝑏+𝑎2𝜕_𝑥𝐺_𝐿2(𝑅)𝑢^𝑛2_𝐻1(𝑅)≤3𝑏+𝑎2𝜕_𝑥𝐺_𝐿2(𝑅)𝑢^𝑛₀²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡𝑢^𝑛_0𝐿1≤−2𝜆𝑡3𝑏+𝑎2𝜕_𝑥𝐺_𝐿2(𝑅)𝑢₀²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡𝑢_0𝐿1−2𝜆𝑡, (5.9) where ‖𝜕_𝑥𝐺‖_𝐿2(𝑅) is bounded, and 𝜆𝑢^𝑛_𝐿2≤𝜆𝑢^𝑛_𝐻1≤𝜆‖𝑢₀‖_𝐻1||exp𝑎−2𝑏2𝜆1−𝑒^−𝜆𝑡𝑢_0𝐿1−𝜆𝑡. (5.10) Applying (5.8)–(5.10) and problem (2.1), we have ‖𝑑𝑢𝑑𝑡^𝑛‖_𝐿2(𝑅)≤𝑏+3𝑏+𝑎2𝜕_𝑥𝐺_𝐿2(𝑅)‖𝑢₀‖²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡𝑢_0𝐿1−2𝜆𝑡+𝜆‖𝑢₀‖_𝐻1||exp𝑎−2𝑏2𝜆1−𝑒^−𝜆𝑡𝑢_0𝐿1−𝜆𝑡. (5.11) For fixed 𝑇>0, from (5.7) and (5.11), we deduce ^𝑇_0𝑅𝑢^𝑛(𝑡,𝑥)²+𝑢^𝑛_𝑥(𝑡,𝑥)²+𝑢^𝑛_𝑡(𝑡,𝑥)²𝑑𝑥𝑑𝑡≤𝑀, (5.12) where 𝑀 is a positive constant depending only on ‖𝐺_𝑥‖_𝐿2(𝑅), ‖𝑢₀‖_𝐻1(𝑅), ‖𝑢₀‖_𝐿1(𝑅), and 𝑇. It follows that the sequence {𝑢^𝑛}_𝑛≥1 is uniformly bounded in the space 𝐻¹((0,𝑇)×𝑅). Thus, we can extract a subsequence such that 𝑢^𝑛𝑘⇀𝑢,weaklyin𝐻¹((0,𝑇)×𝑅)for𝑛_𝑘𝑢⟶∞,_𝑛𝑘⟶𝑢,a.e.on(0,𝑇)×𝑅for𝑛_𝑘⟶∞, (5.13)(5.14) for some 𝑢∈𝐻¹((0,𝑇)×𝑅). From Theorem 4.4 and (5.2), for fixed 𝑡∈(0,𝑇), we have that the sequence 𝑢^𝑛𝑘_𝑥(𝑡,⋅)∈𝐵𝑉(𝑅) satisfies 𝑉𝑢^𝑛𝑘_𝑥(𝑡,𝑥)=𝑢^𝑛𝑘_𝑥𝑥(𝑡,⋅)_𝐿1(𝑅)≤𝑢^𝑛𝑘(𝑡,⋅)_𝐿1(𝑅)+𝑦^𝑛𝑘(𝑡,⋅)_𝐿1(𝑅)≤2𝑒^−𝜆𝑡𝑢^𝑛𝑘₀(𝑡,⋅)_𝐿1(𝑅)≤2𝑒^−𝜆𝑡𝑢₀(𝑡,⋅)_𝐿1(𝑅)≤2𝑒^−𝜆𝑡𝑦₀(𝑡,⋅)_𝑀(𝑅),𝑢^𝑛𝑘_𝑥(𝑡,⋅)_𝐿∞≤𝑒^−𝜆𝑡𝑢^𝑛𝑘_0𝐿1(𝑅)≤𝑒^−𝜆𝑡𝑢_0𝐿1(𝑅)≤𝑒^−𝜆𝑡𝑦_0𝑀(𝑅). (5.15) Applying Helly’s theorem [31], we infer that there exists a subsequence, denoted again by {𝑢^𝑛𝑘_𝑥(𝑡,⋅)}, which converges at every point to some function 𝜈(𝑡,⋅) of finite variation with 𝑉(𝜈(𝑡,⋅))≤2𝑒^−𝜆𝑡𝑦_0𝑀(𝑅). (5.16) From (5.14), we get that for almost all 𝑡∈(0,𝑇), 𝑢^𝑛𝑘_𝑥(𝑡,⋅)→𝑢_𝑥(𝑡,⋅) in 𝐷′(𝑅), it follows that 𝜈(𝑡,⋅)=𝑢_𝑥(𝑡,⋅) for a.e. 𝑡∈(0,𝑇). Therefore, we have 𝑢^𝑛𝑘_𝑥(𝑡,⋅)⟶𝑢_𝑥(𝑡,⋅)a.e.on(0,𝑇)×𝑅for𝑛_𝑘⟶∞, (5.17) and, for a.e. 𝑡∈(0,𝑇), 𝑉𝑢_𝑥(𝑡,⋅)=𝑢_𝑥𝑥(𝑡,⋅)_𝑀(𝑅)=2𝑒^−𝜆𝑡𝑢_0𝐿1≤2𝑒^−𝜆𝑡𝑦_0𝑀(𝑅). (5.18) By Theorem 4.4 and (5.7), we have 𝑎2𝑢^𝑛2+3𝑏−𝑎2𝑢^𝑛_𝑥²_𝐿2(𝑅)≤𝑎2𝑢^𝑛2_𝐿2(𝑅)+3𝑏+𝑎2𝑢^𝑛_𝑥²_𝐿2≤𝑎2𝑢^𝑛_𝐿∞𝑢^𝑛_𝐿2(𝑅)+3𝑏+𝑎2𝑢^𝑛_𝑥𝐿∞𝑢^𝑛_𝑥𝐿2≤𝑎2𝑢^𝑛2_𝐻1(𝑅)+3𝑏+𝑎2𝑢^𝑛2_𝐻1(𝑅)=𝑎+3𝑏2𝑢^𝑛2_𝐻1(𝑅)≤𝑎+3𝑏2𝑢₀²_𝐻1||exp𝑎−2𝑏𝜆1−𝑒^−𝜆𝑡x𝑢_0𝐿1−2𝜆𝑡. (5.19) Note that for fixed 𝑡∈(0,𝑇), the sequence {(𝑎/2)(𝑢^𝑛)²+((3𝑏−𝑎)/2)(𝑢^𝑛_𝑥)²}_𝑛≥1 is uniformly bounded in 𝐿²(𝑅). Therefore, it has a subsequence {(𝑎/2)(𝑢^𝑛𝑘)²+((3𝑏−𝑎)/2)(𝑢^𝑛𝑘_𝑥)²}_𝑛𝑘≥1, which converges weakly in 𝐿²(𝑅). From (5.14), we infer that the weak 𝐿²(𝑅)-limit is {(𝑎/2)(𝑢)²+((3𝑏−𝑎)/2)(𝑢_𝑥)²}. It follows from 𝐺_𝑥∈𝐿²(𝑅) that 𝜕_𝑥𝑎𝐺∗2𝑢^𝑛𝑘2+3𝑏−𝑎2𝑢^𝑛𝑘_𝑥²⟶𝜕_𝑥𝑎𝐺∗2(𝑢)²+3𝑏−𝑎2𝑢_𝑥²for𝑛_𝑘⟶∞. (5.20) From (5.14), (5.17), and(5.20), we have that 𝑢 solves (2.1) in 𝐷′((0,𝑇)×𝑅).
For fixed 𝑇>0, note that 𝑢^𝑛𝑘_𝑡 is uniformly bounded in 𝐿²(𝑅) as 𝑡∈[0,𝑇) and ‖𝑢^𝑛𝑘(𝑡)‖_𝐻1(𝑅) is uniformly bounded for all 𝑡∈[0,𝑇) and 𝑛≥1, and we infer that the family 𝑡→𝑢^𝑛𝑘∈𝐻¹(𝑅) is weakly equicontinous on [0,𝑇]. An application of the Arzela-Ascoli theorem yields that {𝑢^𝑛𝑘} has a subsequence, denoted again {𝑢^𝑛𝑘}, which converges weakly in 𝐻¹(𝑅), uniformly in 𝑡∈[0,𝑇). The limit function is 𝑢. 𝑇 being arbitrary, we have that 𝑢 is locally and weakly continuous from [0,∞) into 𝐻¹(𝑅), that is, 𝑢∈𝐶_𝑤,loc(𝑅₊;𝐻¹(𝑅)).
Since, for a.e. 𝑡∈𝑅₊, 𝑢^𝑛𝑘(𝑡,⋅)⇀𝑢(𝑡,⋅) weakly in 𝐻¹(𝑅), from Theorem 4.4, we get‖𝑢(𝑡,⋅)‖_{𝐿∞(𝑅)}≤‖𝑢(𝑡,⋅)‖_𝐻1(𝑅)≤liminf_{𝑛𝑘→∞}𝑢^𝑛𝑘(𝑡,⋅)_𝐻1(𝑅)≤𝑢_0𝐻1||exp𝑎−2𝑏2𝜆1−𝑒^−𝜆𝑡𝑢_0𝐿1(𝑅)−𝜆𝑡. (5.21) Inequality (5.21) shows that 𝑢∈𝐿^∞_loc𝑅₊×𝑅∩𝐿^∞_loc𝑅₊;𝐻¹(𝑅). (5.22) From Theorem 4.4, (5.1) and (5.2), for 𝑡∈𝑅₊, we obtain 𝑢^𝑛_𝑥(𝑡,⋅)_𝐿∞≤𝑒^−𝜆𝑡𝑢^𝑛_0𝐿1(𝑅)≤𝑒^−𝜆𝑡𝑢_0𝐿1(𝑅)≤𝑒^−𝜆𝑡𝑦_0𝑀(𝑅). (5.23) Combining with (5.14), we have 𝑢_𝑥∈𝐿^∞𝑅₊×𝑅. (5.24) Next, we will prove that ∫_𝑅𝑢(𝑡,⋅)𝑑𝑥=𝑒^−𝜆𝑡∫_𝑅𝑢(0,⋅)𝑑𝑥 by using a regularization approach.
Since 𝑢 satisfies (2.1) in distribution sense, convoluting (2.1) with 𝜌_𝑛, we have that, for a.e.𝑡∈𝑅₊, 𝜌_𝑛∗𝑢_𝑡+𝜌_𝑛∗𝑏𝑢𝑢_𝑥+𝜌_𝑛∗𝜕_𝑥𝑎𝑝∗2𝑢²+3𝑏−𝑎2𝑢_𝑥²+𝜆𝜌_𝑛∗𝑢=0. (5.25)
Integrating the above equation with respect to 𝑥 on 𝑅, we obtain 𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑢𝑑𝑥+_𝑅𝜌_𝑛∗𝑏𝑢𝑢_𝑥+𝑑𝑥_𝑅𝜌_𝑛∗𝜕_𝑥𝑎𝑝∗2𝑢²+3𝑏−𝑎2𝑢_𝑥²𝑑𝑥+𝜆_𝑅𝜌_𝑛∗𝑢𝑑𝑥=0. (5.26) Integration by parts gives rise to 𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑢𝑑𝑥=−𝜆_𝑅𝜌_𝑛∗𝑢𝑑𝑥,𝑡∈𝑅₊,𝑛≥1. (5.27) Utilizing Lemma 3.3, we obtain that _𝑅𝜌_𝑛∗𝑢(𝑡,⋅)𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝜌_𝑛∗𝑢₀𝑑𝑥. (5.28) Since lim_𝑛→∞𝜌_𝑛∗𝑢(𝑡,⋅)−𝑢(𝑡,⋅)_𝐿1(𝑅)=lim_𝑛→∞𝜌_𝑛∗𝑢₀−𝑢_0𝐿1(𝑅)=0. (5.29) it follows that, for a.e. 𝑡∈𝑅₊, _𝑅𝑢(𝑡,⋅)𝑑𝑥=lim_{𝑛→∞𝑅}𝜌_𝑛∗𝑢(𝑡,⋅)𝑑𝑥=lim_𝑛→∞𝑒^−𝜆𝑡_𝑅𝜌_𝑛∗𝑢₀𝑑𝑥=𝑒^−𝜆𝑡_𝑅𝑢₀𝑑𝑥. (5.30) Finally, we prove that (𝑢(𝑡,⋅)−𝑢_𝑥𝑥(𝑡,⋅))∈𝑀⁺ is uniformly bounded on 𝑅 and 𝑢(𝑡,𝑥)∈𝑊^1,∞(𝑅₊×𝑅).
Due to𝐿¹(𝑅)⊂𝐿^∞∗⊂𝐶₀(𝑅)^∗=𝑀(𝑅), (5.31) from (5.18), we get that, for a.e. 𝑡∈𝑅₊, 𝑢(𝑡,⋅)−𝑢_𝑥𝑥(𝑡,⋅)_𝑀(𝑅)≤‖𝑢(𝑡,⋅)‖_𝐿1(𝑅)+𝑢_𝑥𝑥(𝑡,⋅)_𝑀(𝑅)≤𝑒^−𝜆𝑡𝑢_0𝐿1(𝑅)+2𝑒^−𝜆𝑡𝑦_0𝑀(𝑅)≤3𝑒^−𝜆𝑡𝑦_0𝑀(𝑅). (5.32) The above inequality implies that, for a.e. 𝑡∈𝑅₊, (𝑢(𝑡,⋅)−𝑢_𝑥𝑥(𝑡,⋅))∈𝑀(𝑅) is uniformly bounded on 𝑅. For fixed 𝑇≥0, applying (5.13) and (5.14), we have 𝑢^𝑛𝑘(𝑡,⋅)−𝑢^𝑛𝑘_𝑥𝑥(𝑡,⋅)⟶𝑢(𝑡,⋅)−𝑢_𝑥𝑥(𝑡,⋅)in𝐷^′(𝑅)for𝑛⟶∞. (5.33) Since (𝑢^𝑛𝑘(𝑡,⋅)−𝑢^𝑛𝑘_𝑥𝑥(𝑡,⋅))≥0 for all (𝑡,𝑥)∈𝑅₊×𝑅, we obtain that for a.e. 𝑡∈𝑅₊, (𝑢(𝑡,⋅)−𝑢_𝑥𝑥(𝑡,⋅))∈𝑀⁺(𝑅).
Note that 𝑢(𝑡,𝑥)=𝐺∗(𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)). Then we get|𝑢(𝑡,𝑥)|=𝐺∗𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)≤‖𝐺‖_{𝐿∞(𝑅)}𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)_𝑀(𝑅)≤3𝑒^−𝜆𝑡𝑦_0𝑀(𝑅). (5.34) Combining with (5.24), it implies that 𝑢(𝑡,𝑥)∈𝑊^1,∞(𝑅₊×𝑅).
This completes the proof of the existence of Theorem 5.1.
Next, we present the uniqueness proof of the Theorem 5.1.
Let 𝑢,𝑣∈𝑊^1,∞(𝑅₊×𝑅)∩𝐿^∞_𝑙𝑜𝑐(𝑅₊;𝐻¹(𝑅)) be two global weak solutions of problem (2.1) with the same initial data 𝑢₀. Assume that (𝑢(𝑡,⋅)−𝑢_𝑥𝑥(𝑡,⋅))∈𝑀⁺(𝑅) and (𝑣(𝑡,⋅)−𝑣_𝑥𝑥(𝑡,⋅))∈𝑀⁺(𝑅) are uniformly bounded on 𝑅₊ and set 𝑁∶=sup_𝑡∈𝑅+𝑢(𝑡,⋅)−𝑢_𝑥𝑥(𝑡,⋅)_𝑀(𝑅)+𝑣(𝑡,⋅)−𝑣_𝑥𝑥(𝑡,⋅)_𝑀(𝑅). (5.35)
From assumption, we know that 𝑁<∞. Then, for all (𝑡,𝑥)∈𝑅₊×𝑅, |𝑢(𝑡,𝑥)|=𝐺∗𝑢(𝑡,𝑥)−𝑢_𝑥𝑥()≤‖𝐺‖𝑡,𝑥_{𝐿∞(𝑅)}𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)_𝑀(𝑅)≤𝑁2,𝑢_𝑥(𝑡,𝑥)=𝐺_𝑥∗𝑢(𝑡,𝑥)−𝑢_𝑥𝑥()≤𝐺𝑡,𝑥_{𝑥𝐿∞(𝑅)}𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)_𝑀(𝑅)≤𝑁2. (5.36)(5.37) Similarly, |𝑣(≤𝑁𝑡,𝑥)|2,𝑣_𝑥(≤𝑁𝑡,𝑥)2,(𝑡,𝑥)∈𝑅₊×𝑅. (5.38) Following the same procedure as in (5.1), we may also get that ‖𝑢(𝑡,𝑥)‖_𝐿1=𝐺∗𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)_𝐿1(𝑅)≤‖𝐺‖_𝐿1(𝑅)𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)_𝑀(𝑅)𝑢≤𝑁,_𝑥(𝑡,𝑥)_𝐿1(𝑅)=𝐺_𝑥∗𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)_𝐿1(𝑅)≤𝐺_{𝑥𝐿1(𝑅)}𝑢(𝑡,𝑥)−𝑢_𝑥𝑥(𝑡,𝑥)_𝑀(𝑅)≤𝑁 (5.39) and, for all (𝑡,𝑥)∈𝑅₊×𝑅, |𝑣(𝑡,𝑥)|≤𝑁,𝑣_𝑥(𝑡,𝑥)≤𝑁,(𝑡,𝑥)∈𝑅₊×𝑅. (5.40) We define 𝑤(𝑡,𝑥)∶=𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥),(𝑡,𝑥)∈𝑅₊×𝑅. (5.41) Convoluting (2.1) for 𝑢 and 𝑣 with 𝜌_𝑛, we get that for a.e. 𝑡∈𝑅₊ and all 𝑛≥1, 𝜌_𝑛∗𝑢_𝑡+𝜌_𝑛∗𝑏𝑢𝑢_𝑥+𝜌_𝑛∗𝜕_𝑥𝑎𝐺∗2𝑢²+3𝑏−𝑎2𝑢_𝑥²+𝜆𝜌_𝑛𝜌∗𝑢=0,_𝑛∗𝑣_𝑡+𝜌_𝑛∗𝑏𝑣𝑣_𝑥+𝜌_𝑛∗𝜕_𝑥𝑎𝐺∗2𝑣²+3𝑏−𝑎2𝑣_𝑥²+𝜆𝜌_𝑛∗𝑣=0. (5.42)(5.43) Subtracting (5.43) from (5.42) and using Lemma 3.4, integration by parts shows that, for a.e. 𝑡∈𝑅₊ and all𝑛≥1𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑤𝑑𝑥=_𝑅𝜌_𝑛∗𝑤_𝑡sgn𝜌_𝑛∗𝑤𝑑𝑥=−𝑏_𝑅𝜌_𝑛∗𝑤𝑢_𝑥sgn𝜌_𝑛∗𝑤𝑑𝑥−𝑏_𝑅𝜌_𝑛∗𝑣𝑤_𝑥sgn𝜌_𝑛−𝑎∗𝑤𝑑𝑥2_𝑅𝜌_𝑛∗𝜕_𝑥𝐺∗𝑤(𝑢+𝑣)sgn𝜌_𝑛−∗𝑤𝑑𝑥3𝑏−𝑎2_𝑅𝜌_𝑛∗𝜕_𝑥𝐺∗𝑤_𝑥𝑢_𝑥+𝑣_𝑥sgn𝜌_𝑛∗𝑤𝑑𝑥−𝜆_𝑅𝜌_𝑛∗𝑤sgn𝜌_𝑛∗𝑤𝑑𝑥. (5.44) Using (5.36)–(5.38) and Young’s inequality to the first term on the right-hand side of (5.44) yields, _𝑅𝜌_𝑛∗𝑤𝑢_𝑥sgn𝜌_𝑛≤∗𝑤𝑑𝑥_𝑅𝜌_𝑛∗𝑤𝑢_𝑥≤𝑑𝑥_𝑅𝜌_𝑛∗𝑤𝜌_𝑛∗𝑢_𝑥𝑑𝑥+_𝑅𝜌_𝑛∗𝑤𝑢_𝑥−𝜌_𝑛∗𝑤𝜌_𝑛∗𝑢_𝑥≤𝜌𝑑𝑥_𝑛∗𝑢_{𝑥𝐿∞𝑅}𝜌_𝑛∗𝑤𝑑𝑥+_𝑅𝜌_𝑛∗𝑤𝑢_𝑥−𝜌_𝑛∗𝑤𝜌_𝑛∗𝑢_𝑥≤𝜌𝑑𝑥_𝑛𝐿1𝑢_{𝑥𝐿∞𝑅}𝜌_𝑛∗𝑤𝑑𝑥+_𝑅𝜌_𝑛∗𝑤𝑢_𝑥−𝜌_𝑛∗𝑤𝜌_𝑛∗𝑢_𝑥≤𝑁𝑑𝑥2_𝑅𝜌_𝑛∗𝑤𝑑𝑥+_𝑅𝜌_𝑛∗𝑤𝑢_𝑥−𝜌_𝑛∗𝑤𝜌_𝑛∗𝑢_𝑥𝑑𝑥. (5.45) Similarly, for the second term and the third term on the right-hand side of (5.44), we have _𝑅𝜌_𝑛∗𝑤_𝑥𝑣sgn𝜌_𝑛≤∗𝑤𝑑𝑥_𝑅𝜌_𝑛∗𝑤_𝑥𝑣≤𝑑𝑥_𝑅𝜌_𝑛∗𝑤_𝑥𝜌_𝑛∗𝑣𝑑𝑥+_𝑅𝜌_𝑛∗𝑤_𝑥𝑣−𝜌_𝑛∗𝑤_𝑥𝜌_𝑛≤𝑁∗𝑣𝑑𝑥2_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+_𝑅𝜌_𝑛∗𝑤_𝑥𝑣−𝜌_𝑛∗𝑤_𝑥𝜌_𝑛∗𝑣𝑑𝑥,_𝑅𝜌_𝑛∗𝜕_𝑥𝐺∗[𝑤(𝑢+𝑣)]sgn𝜌_𝑛≤∗𝑤𝑑𝑥_𝑅𝜌_𝑛∗𝐺∗𝑤_𝑥(+𝑢+𝑣)𝑑𝑥_𝑅𝜌_𝑛∗𝐺∗𝑤(𝑢+𝑣)_𝑥≤1𝑑𝑥2_𝑅𝜌_𝑛∗𝑤_𝑥(1𝑢+𝑣)𝑑𝑥+2_𝑅𝜌_𝑛∗𝑤𝑢_𝑥+𝑣_𝑥≤𝑁𝑑𝑥2_𝑅𝜌_𝑛∗𝑤_𝑥𝑁𝑑𝑥+2_𝑅𝜌_𝑛+∗𝑤𝑑𝑥_𝑅𝜌_𝑛∗𝑤_𝑥(𝑢+𝑣)−𝜌_𝑛∗𝑤_𝑥𝜌_𝑛∗(+𝑢+𝑣)𝑑𝑥_𝑅𝜌_𝑛∗𝑤(𝑢+𝑣)_𝑥−𝜌_𝑛∗𝑤𝜌_𝑛∗(𝑢+𝑣)_𝑥𝑑𝑥. (5.46) For the last term on the right-hand side of (5.44), we have _𝑅𝜌_𝑛∗𝜕_𝑥𝐺∗𝑤_𝑥(𝑢+𝑣)_𝑥sgn𝜌_𝑛≤∗𝑤𝑑𝑥_𝑅𝜌_𝑛∗𝜕_𝑥𝐺∗𝑤_𝑥𝑢_𝑥+𝑣_𝑥𝑑𝑥≤𝑁_𝑅𝜌_𝑛∗𝜕_𝑥𝐺∗𝑤_𝑥𝜕𝑑𝑥≤𝑁_𝑥𝐺_{𝐿1(𝑅)𝑅}𝜌_𝑛∗𝑤_𝑥𝑑𝑥≤𝑁_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑁_𝑅𝜌_𝑛∗𝑤_𝑥−𝜌_𝑛∗𝑤_𝑥𝑑𝑥. (5.47) From (5.45)–(5.47), for a.e. 𝑡∈𝑅₊ and all 𝑛≥1, we find 𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑤𝑑𝑥≤𝑎+2𝑏4𝑁+𝜆_𝑅𝜌_𝑛+()𝑁∗𝑤𝑑𝑥𝑎+2𝑏_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛(𝑡), (5.48) where 𝑅_𝑛(𝑡)𝑅⟶0as𝑡⟶∞,_𝑛(𝑡)≤𝐾,𝑛≥1,𝑡∈𝑅₊, (5.49) where 𝐾 is a positive constant depending on 𝑁 and the 𝐻¹(𝑅)-norms of 𝑢(0) and 𝑣(0).
In the same way, convoluting (2.1) for 𝑢 and 𝑣 with 𝜌_𝑛,𝑥 and using Lemma 3.4, we get that, for a.e. 𝑡∈𝑅₊ and all 𝑛≥1, 𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥=_𝑅𝜌_𝑛∗𝑤_𝑥𝑡sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥=−𝑏_𝑅𝜌_𝑛∗𝑤_𝑥𝑢_𝑥+𝑣_𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥−𝑏_𝑅𝜌_𝑛∗𝑣_𝑥𝑥𝑤sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥−𝑏_𝑅𝜌_𝑛∗𝑢𝑤_𝑥𝑥sgn𝜌_𝑛,𝑥−∗𝑤𝑑𝑥_𝑅𝜌_𝑛∗𝜕_𝑥𝑥𝑎𝐺∗2𝑢₂−𝑣₂+3𝑏−𝑎2𝑢²_𝑥−𝑣²_𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥−𝜆_𝑅𝜌_𝑛∗𝑤_𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥. (5.50)
Using the identity 𝜕²_𝑥(𝐺∗𝑔)=𝐺∗𝑔−𝑔 for 𝑔∈𝐿²(𝑅) and Young’s inequality, we estimate the forth term of the right-hand side of (5.50) _𝑅𝜌_𝑛∗𝜕_𝑥𝑥𝑎𝐺∗2𝑢²−𝑣²+3𝑏−𝑎2𝑢²_𝑥−𝑣²_𝑥≤𝑑𝑥_𝑅𝜌_𝑛𝑎∗𝐺∗2𝑢²−𝑣²+3𝑏−𝑎2𝑢²_𝑥−𝑣²_𝑥+𝑑𝑥_𝑅𝜌_𝑛∗𝑎2𝑢²−𝑣²+3𝑏−𝑎2𝑢²_𝑥−𝑣²_𝑥≤𝐺‖𝑑𝑥‖_𝐿1(𝑅)+1_𝑅𝜌_𝑛∗𝑎2𝑤(𝑢+𝑣)+3𝑏−𝑎2𝑤_𝑥𝑢_𝑥+𝑣_𝑥𝑑𝑥≤𝑎_𝑅𝜌_𝑛∗[𝑤(𝑢+𝑣)]𝑑𝑥+(3𝑏+𝑎)_𝑅𝜌_𝑛∗𝑤_𝑥𝑢_𝑥+𝑣_𝑥𝑑𝑥≤𝑎𝑁_𝑅𝜌_𝑛∗𝑤𝑑𝑥+(3𝑏+𝑎)𝑁_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛. (5.51) Using (5.36)–(5.38) and Young’s inequality to the first term on the right-hand side of (5.50) gives rise to −𝑏_𝑅𝜌_𝑛∗𝑤_𝑥𝑢_𝑥+𝑣_𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥≤𝑏_𝑅𝜌_𝑛∗𝑤_𝑥𝑢_𝑥+𝑣_𝑥𝑑𝑥≤𝑏_𝑅𝜌_𝑛∗𝑤_𝑥𝜌_𝑛∗𝑢_𝑥+𝑣_𝑥𝑑𝑥+𝑏_𝑅𝜌_𝑛∗𝑤_𝑥𝑢_𝑥+𝑣_𝑥−𝜌_𝑛∗𝑤_𝑥𝜌_𝑛∗𝑢_𝑥+𝑣_𝑥𝑑𝑥≤𝑏𝑁_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛. (5.52) To treat the second term of the right-hand side of (5.50), we note that 𝑏_𝑅𝜌_𝑛∗𝑣_𝑥𝑥𝑤sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥≤𝑏_𝑅𝜌_𝑛∗𝑤𝜌_𝑛∗𝑣_𝑥𝑥𝑑𝑥+𝑏_𝑅𝜌_𝑛∗𝑣_𝑥𝑥𝑤−𝜌_𝑛∗𝑤𝜌_𝑛∗𝑣_𝑥𝑥𝑑𝑥. (5.53) Applying Lemma 3.1, the second expression of the right-hand side of (5.53) can be estimated by a function 𝑅_𝑛(𝑡) belonging to (5.49). Making use of the Hölder inequality and (5.1), for a.e. 𝑡∈𝑅₊ and all 𝑛≥1, we have _𝑅𝜌_𝑛∗𝑤𝜌_𝑛∗𝑣_𝑥𝑥𝑑𝑥≤𝜌_𝑛∗𝑤_{𝐿∞(𝑅)}𝜌_𝑛∗𝑣_{𝑥𝑥𝐿1(𝑅)}≤𝜌_𝑛∗𝑤_{𝑊1,1(𝑅)}𝑣_{𝑥𝑥𝑀(𝑅)}. (5.54) It follows from (5.53) and (5.54) that 𝑏_𝑅𝜌_𝑛∗𝑣_𝑥𝑥𝑤sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥≤bN_𝑅𝜌_𝑛∗𝑤𝑑𝑥+bN_𝑅𝜌_𝑛∗𝑤_𝑥+𝑅_𝑛(𝑡). (5.55) Now, we deal with the third term on the right-hand side of (5.50) −𝑏_𝑅𝜌_𝑛∗𝑢𝑤_𝑥𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥=−𝑏_𝑅𝜌_𝑛∗𝑢𝜌_𝑛∗𝑤_𝑥𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥−𝑏_𝑅𝜌_𝑛∗𝑢𝑤_𝑥𝑥−𝜌_𝑛∗𝑢𝜌_𝑛∗𝑤_𝑥𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥≤−𝑏_𝑅𝜌_𝑛𝜕∗𝑢𝜕_𝑥𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑏_𝑅𝜌_𝑛∗𝑢𝑤_𝑥𝑥−𝜌_𝑛∗𝑢𝜌_𝑛∗𝑤_𝑥𝑥𝑑𝑥=𝑏_𝑅𝜌_𝑛∗𝑢_𝑥𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛. (5.56) Therefore, (5.56) implies that, for a.e. 𝑡∈𝑅₊ and all 𝑛≥1, −𝑏_𝑅𝜌_𝑛∗𝑢𝑤_𝑥𝑥sgn𝜌_𝑛,𝑥∗𝑤𝑑𝑥≤𝑏𝑁_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛. (5.57) From (5.51), (5.52), (5.55), and (5.57), for a.e. 𝑡∈𝑅₊ and all 𝑛≥1, we deduce that 𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥≤(𝑎+𝑏)𝑁_𝑅𝜌_𝑛+)]∗𝑤𝑑𝑥[(6𝑏+𝑎𝑁+𝜆_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛. (5.58) Combining with (5.48) and (5.58), we find 𝑑𝑑𝑡_𝑅𝜌_𝑛∗𝑤+𝜌_𝑛∗𝑤_𝑥𝑑𝑥≤5𝑎+6𝑏4𝑁+𝜆_𝑅𝜌_𝑛+)]∗𝑤𝑑𝑥[(2𝑎+8𝑏𝑁+𝜆_𝑅𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛≤[(2𝑎+8𝑏)𝑁+𝜆]_𝑅𝜌_𝑛∗𝑤+𝜌_𝑛∗𝑤_𝑥𝑑𝑥+𝑅_𝑛. (5.59) It follows from Gronwall’ inequality that, for a.e. 𝑡∈𝑅₊ and all 𝑛≥1, _𝑅𝜌_𝑛∗𝑤+𝜌_𝑛∗𝑤_𝑥≤𝑑𝑥_𝑡0𝑅_𝑛(𝑠)𝑑𝑠+_𝑅𝜌_𝑛∗𝑤+𝜌_𝑛∗𝑤_𝑥(0,𝑥)𝑑𝑥𝑒^{[(2𝑎+8𝑏)𝑁+𝜆]𝑡}. (5.60) Fix 𝑡>0, and let 𝑛→∞ in (5.60). Since 𝑤=𝑢−𝑣∈𝑊^1,1(𝑅) and relation (5.49) holds, making use of Lebesgue’s dominated convergence theorem yields _𝑅|𝑤|+𝑤_𝑥𝑑𝑥≤_𝑅|𝑤|+𝑤_𝑥(0,𝑥)𝑑𝑥𝑒^{[(2𝑎+8𝑏)𝑁+𝜆]𝑡}. (5.61) Note that 𝑤(0)=𝑤_𝑥(0)=0; therefore, we obtain 𝑢(𝑡,𝑥)=𝑣(𝑡,𝑥) for a.e. (𝑡,𝑥)∈𝑅₊×𝑅. This completes the proof of the theorem.

Acknowledgments

This work was supported by RFSUSE (no. 2011KY12), RFSUSE (no. 2011RC10) and the project (no. 12ZB080). The authors thanks the referee for valuable comments.