The 𝐻¹(𝑅) Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

Abstract

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the space 𝐶([0,∞) × R)∩𝐿^∞([0,∞); 𝐻¹(R)) provided that its initial value 𝑢₀(𝑥) belongs to the space 𝐻¹(𝑅). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

1. Introduction

In [1], the author investigated the following weakly dissipative Camassa-Holm model 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥+𝜆𝑢−𝑢_𝑥𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (1.1) where 𝜆≥0. When 𝜆=0, (1.1) becomes the classical Camassa-Holm equation [2]. The authors in [1] obtained the local well-posedness of the solution for the model by using the Kato theorem. A necessary and sufficient condition of the blow-up of the solution and some criteria guaranteeing the blow-up of the solution are derived. The blow-up rate of the solution is discussed. It is also shown in [1] that the equation has global strong solutions, and these strong solutions decay to zero as time goes to infinite provided the potentials associated to their initial data are of one sign. However, the existence of global weak solutions in the space 𝐻¹(𝑅) is not discussed in paper [1]. This will constitute the objective of this work.

More relevant for the present paper, here we state several works on the global weak solution for the Camassa-Holm and Degasperis-Procesi equations. The existence and uniqueness results for global weak solutions to the Camassa-Holm equation have been proved by Constantin and Escher [3], Constantin and Molinet [4], and Danchin [5, 6]. Xin and Zhang [7] proved that the global existence of the weak solution for the Camassa-Holm equation in the energy space 𝐻¹(𝑅) without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under certain conditions on the solution [8]. Coclite et al. [9] investigated the global weak solutions for a generalized hyperelastic-rod wave equation or a generalized Camassa-Holm equation. The existence of a strongly continuous semigroup of global weak solutions for the generalized hyperelastic-rod equation with any initial value in the space 𝐻¹(𝑅) was established in [9]. Under the sign condition imposed on the initial value, Yin et al. [10] proved the existence and uniqueness results of global weak solution for a nonlinear shallow water equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases. For other dynamic properties about various generalized Camassa-Holm models and other partial differential equations, the reader is referred to [11–16].

The aim of this work is to study the existence of global weak solutions for (1.1) in the space 𝐶([0,∞)×𝑅)⋂𝐿^∞([0,∞);𝐻¹(𝑅)) under the assumption 𝑢₀(𝑥)∈𝐻¹(𝑅). The limits of viscous approximations for the equation are used to establish the existence of the global weak solution. Here, we should mention that up to now, there have been no global existence results for weak solutions to the generalized Camassa-Holm equation (1.1).

The rest of this paper is as follows. The main result is given in Section 2. In Section 3, we state the viscous problem and give a corresponding well-posedness result. An upper bound, higher integrability estimate, and basic compactness properties for the viscous approximations are also established in Section 3. Strong compactness of the derivative of the viscous approximations is obtained in Section 4, where the main result for the existence of (1.1) is proved.

2. Main Result

Consider the Cauchy problem for (1.1) 𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢²2_𝑥+𝜆𝑢−𝑢_𝑥𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,𝑢(0,𝑥)=𝑢₀(𝑥), (2.1) which is equivalent to 𝑢_𝑡+𝑢𝑢_𝑥+𝜕𝑃()𝜕𝑥=0,𝜕𝑃𝑡,𝑥𝜕𝑥=Λ⁻²𝜕_𝑥𝑢²+12𝑢²_𝑥𝑢()+𝜆𝑢,0,𝑥=𝑢₀(𝑥), (2.2) where operator Λ²=1−(𝜕²/𝜕𝑥²). For a fixed 1≤𝑝₀<∞, it has Λ⁻²𝑔(𝑥)=12_𝑅𝑒^{−|𝑥−𝑦|}𝑔𝑦𝑑𝑦𝑓𝑜𝑟𝑔(𝑥)∈𝐿^𝑝0(𝑅),1<𝑝₀<∞. (2.3)

In fact, problem (2.1) satisfies the following conservation law: _𝑅𝑢²+𝑢²_𝑥𝑑𝑥+2𝜆^𝑡_0𝑅𝑢²+𝑢²_𝑥𝑑𝑥𝑑𝑡=_𝑅𝑢²₀+𝑢²_0𝑥𝑑𝑥. (2.4)

Now, we introduce the definition of a weak solution to the Cauchy problem (2.1) or (2.2).

Definition 2.1. A continuous function 𝑢∶[0,∞)×𝑅→𝑅 is said to be a global weak solution to the Cauchy problem (2.2) if(i)𝑢∈𝐶([0,∞)×𝑅)⋂𝐿^∞([0,∞);𝐻¹(𝑅)); (ii)‖𝑢(𝑡,.)‖_𝐻¹_(𝑅)≤‖𝑢₀‖_𝐻¹_(𝑅); (iii)𝑢=𝑢(𝑡,𝑥) satisfies (2.2) in the sense of distributions and takes on the initial value pointwise.
The main result of this paper is stated as follows.

Theorem 2.2. Assume 𝑢₀(𝑥)∈𝐻¹(𝑅). Then, the Cauchy problem (2.1) or (2.2) has a global weak solution 𝑢(𝑡,𝑥) in the sense of Definition 2.1. Furthermore, this weak solution satisfies the following properties.(a)There exists a positive constant 𝑐₀ depending on ‖𝑢₀‖_𝐻¹_(𝑅) and 𝜆 such that the following one-sided 𝐿^∞ norm estimate on the first order spatial derivative holds 𝜕𝑢(𝑡,𝑥)≤4𝜕𝑥𝑡+𝑐₀,𝑓𝑜𝑟(𝑡,𝑥)∈[0,∞)×𝑅. (2.5)(b)Let 0<𝛾<1, 𝑇>0, and 𝑎,𝑏∈𝑅,𝑎<𝑏. Then, there exists a positive constant 𝑐₁ depending only on ‖𝑢₀‖_𝐻¹_(𝑅), 𝛾,𝑇,𝑎,𝑏 and 𝜆 such that the space higher integrability estimate holds 𝜕𝑢(^𝑡₀^𝑏_𝑎𝑡,𝑥)𝜕𝑥^2+𝛾𝑑𝑥≤𝑐₁. (2.6)

3. Viscous Approximations

Defining 𝜙(𝑥)=𝑒^{1/(𝑥2−1)},|𝑥||𝑥|<1,0,≥1, (3.1) and setting the mollifier 𝜙_𝜀(𝑥)=𝜀^−(1/4)𝜙(𝜀^−(1/4)𝑥) with 0<𝜀<1/4 and 𝑢_𝜀,0=𝜙_𝜀⋆𝑢₀, we know that 𝑢_𝜀,0∈𝐶^∞ for any 𝑢₀∈𝐻^𝑠,𝑠>0 (see [10]). In fact, suitably choosing the mollifier, we have 𝑢_{𝜀,0𝐻1(𝑅)}≤𝑢_0𝐻1(𝑅),𝑢_𝜀,0⟶𝑢₀in𝐻¹(𝑅). (3.2)

The existence of a weak solution to the Cauchy problem (2.2) will be established by proving compactness of a sequence of smooth functions {𝑢_𝜀}_𝜀>0 solving the following viscous problem: 𝜕𝑢_𝜀𝜕𝑡+𝑢_𝜀𝜕𝑢_𝜀+𝜕𝑥𝜕𝑃_𝜀𝜕𝜕𝑥=𝜀²𝑢_𝜀𝜕𝑥²,𝜕𝑃_𝜀(𝑡,𝑥)𝜕𝑥=Λ⁻²𝜕_𝑥𝑢²_𝜀+12𝜕𝑢_𝜀𝜕𝑥²+𝜆𝑢_𝜀,𝑢_𝜀(0,𝑥)=𝑢_𝜀,0(𝑥). (3.3)

The beginning point of our analysis is the following well-posedness result for problem (3.3).

Lemma 3.1. Provided that 𝑢₀∈𝐻¹(𝑅), then for any 𝜎≥3, there exists a unique solution 𝑢_𝜀∈𝐶([0,∞);𝐻^𝜎(𝑅)) to the Cauchy problem (3.3). Moreover, for any 𝑡>0, it holds that _𝑅𝑢²_𝜀+𝜕𝑢_𝜀𝜕𝑥²𝑑𝑥+2𝜆^𝑡_0𝑅𝑢²_𝜀+𝜕𝑢_𝜀𝜕𝑥²𝑑𝑥𝑑𝑡+2𝜀^𝑡_0𝑅𝜕𝑢_𝜀𝜕𝑥²+𝜕²𝑢_𝜀𝜕𝑥²²(𝑠,𝑥)𝑑𝑥𝑑𝑠=𝑢_𝜀,0²_𝐻1(𝑅), (3.4) or 𝑢_𝜀(𝑡,.)²_𝐻1(𝑅)+2𝜀^𝑡₀𝜕𝑢_𝜀(𝜕𝑥𝑠,.)²_𝐻1(𝑅)𝑑𝑠+2𝜆^𝑡₀𝑢_𝜀(𝑡,.)²_𝐻1(𝑅)𝑑𝑡=𝑢_𝜀,0²_𝐻1(𝑅). (3.5)

Proof. For any 𝜎≥3 and 𝑢₀∈𝐻¹(𝑅), we have 𝑢_𝜀,0∈𝐶([0,∞);𝐻^𝜎(𝑅)). From Theorem 2.1 in [9] or Theorem 2.3 in [11], we conclude that problem (3.3) has a unique solution 𝑢_𝜀∈𝐶([0,∞);𝐻^𝜎(𝑅)) for an arbitrary 𝜎>3.
We know that the first equation in system (3.3) is equivalent to the form 𝜕𝑢_𝜀−𝜕𝜕𝑡³𝑢_𝜀𝜕𝑡𝑥²+32𝜕𝑢²_𝜀𝑢𝜕𝑥+𝜆_𝜀−𝜕²𝑢_𝜀𝜕𝑥²=2𝜕𝑢_𝜀𝜕𝜕𝑥²𝑢_𝜀𝜕𝑥²+𝑢_𝜀𝜕³𝑢_𝜀𝜕𝑥³𝜕+𝜀²𝑢_𝜀𝜕𝑥²−𝜕⁴𝑢_𝜀𝜕𝑥⁴, (3.6) from which we derive that 12𝑑𝑑𝑡_𝑅𝑢²_𝜀+𝜕𝑢_𝜀𝜕𝑥²𝑑𝑥+𝜆_𝑅𝑢²_𝜀+𝜕𝑢_𝜀𝜕𝑥²𝑑𝑥+𝜀_𝑅𝜕𝑢_𝜀𝜕𝑥²+𝜕²𝑢_𝜀𝜕𝑥²²𝑑𝑥=0, (3.7) which completes the proof.

From Lemma 3.1, we have 𝑢_{𝜀𝐿∞(𝑅)}≤𝑢_{𝜀𝐻1(𝑅)}≤𝑢_{𝜀,0𝐻1(𝑅)}≤𝑢_0𝐻1(𝑅). (3.8) Differentiating the first equation of problem (3.3) with respect to the variable 𝑥 and writing 𝜕𝑢_𝜀/𝜕𝑥=𝑞_𝜀, we obtain 𝜕𝑞_𝜀𝜕𝑡+𝑢_𝜀𝜕𝑞_𝜀𝜕𝜕𝑥−𝜀²𝑞_𝜀𝜕𝑥²+12𝑞²_𝜀+𝜆𝑢_𝑥=𝑢²_𝜀−Λ⁻²𝑢²_𝜀+12𝑞²_𝜀=𝑄_𝜀(𝑡,𝑥). (3.9)

Lemma 3.2. Let 0<𝛾<1, 𝑇>0, and 𝑎,𝑏∈𝑅,𝑎<𝑏. Then, there exists a positive constant 𝑐₁ depending only on ‖𝑢₀‖_𝐻¹_(𝑅), 𝛾,𝑇,𝑎, 𝑏 and 𝜆, but independent of 𝜀, such that the space higher integrability estimate holds ^𝑇₀^𝑏_𝑎𝜕𝑢_𝜀(𝑡,𝑥)𝜕𝑥^2+𝛾𝑑𝑥≤𝑐₁, (3.10) where 𝑢_𝜀=𝑢_𝜀(𝑡,𝑥) is the unique solution of problem (3.3).

Proof. The proof is a variant of the proof presented by Xin and Zhang [7] (also see Coclite et al. [9]). Let 𝜒∈𝐶^∞(𝑅) be a cut-off function such that 0<𝜒<1 and 𝜙(𝑥)=1,if𝑥∈[𝑎,𝑏],0,if𝑥∈(−∞,𝑎−1]⋃[𝑏+1,∞). (3.11) Consider the map 𝜃(𝜉)∶=𝜉(1+|𝜉|)^𝛾, 𝜉∈𝑅,0<𝛾<1, and observe that 𝜃^′(𝜉)=1+1+𝛾𝜉1+𝜉^𝛾−1,𝜃^′′(𝜉)=𝛾sign(𝜉)1+𝜉^𝛾−2𝜉(𝜉)𝜉2+1+𝛾=𝛾1+𝛾sign1+^𝛾−1+1−𝛾𝛾sign(𝜉)1+𝜉^𝛾−2. (3.12) We have 𝜃(𝜉)≤𝜉+𝜉^1+𝛾,𝜃^′(𝜉)≤1+1+𝛾𝜉,𝜃^′′(𝜉)≤2𝛾,(𝜉)−1𝜉𝜃2𝜉²𝜃^′(𝜉)=1−𝛾2𝜉²1+𝜉^𝛾+𝛾2𝜉²1+𝜉^𝛾−1≥1−𝛾2𝜉²1+𝜉^𝛾. (3.13)(3.14)
Differentiating the first equation of problem (3.3) with respect to the variable 𝑥 and writing 𝑢=𝑢_𝜀 and 𝜕𝑢_𝜀/𝜕𝑥=𝑞_𝜀=𝑞 for simplicity, we obtain 𝑢_𝑡𝑥+𝑢𝑢_𝑥𝑥+12𝑢²_𝑥+𝜆𝑢_𝑥=𝑢²−Λ⁻²𝑢²+12𝑢²_𝑥=𝑄(𝑡,𝑥). (3.15) Multiplying (3.15) by 𝜒𝜃^′(𝑞), using the chain rule and integrating over Π_𝑇∶=[0,𝑇]×𝑅, we have _Π𝑇𝜒(𝑥)𝑞1𝑞𝜃𝑑𝑡𝑑𝑥−2_Π𝑇𝑞²𝜒(𝑥)𝜃^′𝑞=𝑑𝑡𝑑𝑥_𝑅𝜒(𝑥)𝜃𝑞(𝑡,𝑥)−𝜃𝑞(0,𝑥)𝑑𝑥−_Π𝑇𝑢𝜒^′(𝑥)𝜃𝑞𝑑𝑡𝑑𝑥+𝜀_Π𝑇𝜕𝑞𝜒𝜕𝑥^′(𝑥)𝜃^′𝑞𝑑𝑡𝑑𝑥+𝜀_Π𝑇𝜕𝑞𝜕𝑥²𝜒(𝑥)𝜃^′′𝑞𝑑𝑡𝑑𝑥+𝜆_Π𝑇𝑞𝜒(𝑥)𝜃^′𝑞𝑑𝑡𝑑𝑥−_Π𝑇𝑄(𝑡,𝑥)𝜒(𝑥)𝜃^′𝑞𝑑𝑡𝑑𝑥. (3.16) From (3.14), we get _Π𝑇𝜒(𝑥)𝑞1𝑞𝜃𝑑𝑡𝑑𝑥−2_Π𝑇𝑞²𝜒(𝑥)𝜃^′𝑞=𝑑𝑡𝑑𝑥_Π𝑇𝜒(𝑥)𝑞−1𝑞𝜃2𝑞²𝜃^′𝑞≥𝑑𝑡𝑑𝑥1−𝛾2_Π𝑇𝜒(𝑥)𝑞²1+𝑞^𝛾𝑑𝑡𝑑𝑥. (3.17) Using the Hölder inequality, (3.8) and (3.13), yields _𝑅𝜒(𝑥)𝜃𝑞𝑑𝑥≤_𝑅𝜒(𝑥)𝑞^1+𝛾+𝑞≤𝜒𝑑𝑥_{𝐿2/(1−𝛾)(𝑅)}𝑞_𝐿^1+𝛾_2(𝑅)+𝜒_𝐿2(𝑅)𝑞_𝐿2(𝑅)≤(𝑏−𝑎+2)^(1−𝛾)/2𝑢_0𝐻^1+𝛾_1(𝑅)+(𝑏−𝑎+2)^1/2𝑢_0𝐻1(𝑅),_Π𝑇𝑢𝜒^′(𝑥)𝜃𝑞𝑑𝑡𝑑𝑥≤_Π𝑇|𝑢|𝜒^′(𝑥)𝑞^1+𝛾+𝑞≤𝑢𝑑𝑡𝑑𝑥_{0𝐻1(𝑅)𝑇0}𝜒^′_{𝐿2/(1−𝛾)(𝑅)}𝑞_𝐿^1+𝛾_2(𝑅)+𝜒^′_𝐿2(𝑅)𝑞_𝐿2(𝑅)𝑢𝑑𝑡≤𝑇_0𝐻1(𝑅)𝜒^′_{𝐿2/(1−𝛾)(𝑅)}𝑢_0𝐻^1+𝛾_1(𝑅)+𝜒^′_𝐿2(𝑅)𝑢_0𝐻1(𝑅). (3.18)(3.19) Integration by parts gives rise to _Π𝑇𝜕𝑞𝜒𝜕𝑥^′(𝑥)𝜃^′𝑞𝑑𝑡𝑑𝑥=−_Π𝑇𝜃𝑞𝜒^′′(𝑥)𝑑𝑡𝑑𝑥. (3.20) From (3.13), (3.20), and the Hölder inequality, we have _Π𝑇𝜕𝑞𝜒𝜕𝑥^′(𝑥)𝜃^′𝑞𝑑𝑡𝑑𝑥≤𝜀_Π𝑇𝜃𝑞𝜒^′′(𝑥)𝑑𝑡𝑑𝑥≤𝜀_Π𝑇𝜒^′′(𝑥)𝑞^1+𝛾+𝑞𝜒𝑑𝑡𝑑𝑥≤𝜀𝑇^′′_{𝐿2/(1−𝛾)(𝑅)}𝑢_0𝐻^1+𝛾_1(𝑅)+𝜒^′′_𝐿2(𝑅)𝑢_0𝐻1(𝑅). (3.21) Using (3.13) and Lemma 3.1, we have 𝜀_Π𝑇𝜕𝑞𝜕𝑥²𝜒(𝑥)𝜃^′′𝑞𝑑𝑡𝑑𝑥≤2𝛾𝜀_Π𝑇𝜕𝑞𝜕𝑥²𝑑𝑡𝑑𝑥≤𝛾𝑢₀²_𝐻1(𝑅). (3.22) From (3.13), it has 𝜆_Π𝑇𝑞𝜒(𝑥)𝜃^′𝑞≤|𝜆|𝑑𝑡𝑑𝑥_Π𝑇𝜒(𝑥)1+𝛾𝑞²+𝑞≤|𝜆|𝜒𝑑𝑡𝑑𝑥1+𝛾_𝐿∞𝑢₀²_𝐻1(𝑅)+𝜒(𝑥)_𝐿2(𝑅)𝑢_0𝐻1(𝑅). (3.23) Applying (3.8), the Hölder inequality, Lemma 3.1 and ∫_𝑅𝑒^{−|𝑥−𝑦|}𝑑𝑦=2, we have |𝑄(𝑡,𝑥)|=𝑢²−Λ⁻²𝑢²+12𝑞²≤‖𝑢‖²_𝐿^∞_(𝑅)+12‖𝑢‖²_𝐻1(𝑅)≤32𝑢₀²_𝐻1(𝑅). (3.24) From (3.24), we obtain _Π𝑇𝑄(𝑡,𝑥)𝜒(𝑥)𝜃^′𝑞𝑑𝑡𝑑𝑥≤𝑐_Π𝑇𝜒(𝑥𝑞≤3)1+𝛾+1𝑑𝑡𝑑𝑥2𝑢₀²_𝐻1(𝑅)𝑇1+𝛾𝜒(𝑥)_𝐿2(𝑅)𝑢_0𝐻1(𝑅)+_𝑅𝜒(𝑥)𝑑𝑥. (3.25) The inequalities (3.15)–(3.23) and (3.25) derive the desired result (3.10).

Lemma 3.3. There exists a positive constant 𝐶 depending only on ‖𝑢₀‖_𝐻¹_(𝑅) and 𝜆 such that 𝑄_𝜀(𝑡,.)_{𝐿∞(𝑅)}≤𝐶,𝑄_𝜀(𝑡,.)_𝐿1(𝑅)≤𝐶,𝑄_𝜀(𝑡,.)_𝐿2(𝑅)≤𝐶,𝜕𝑃_𝜀(𝑡,.)𝜕𝑥_𝐿^∞_(𝑅)≤𝐶,𝜕𝑃_𝜀(𝑡,.)𝜕𝑥_𝐿²_(𝑅)≤𝐶, (3.26)(3.27)(3.28)(3.29)(3.30) where 𝑢_𝜀=𝑢_𝜀(𝑡,𝑥) is the unique solution of system (3.3).

Proof. For simplicity, setting 𝑢(𝑡,𝑥)=𝑢_𝜀(𝑡,𝑥), we have 𝑄_𝜀(𝑡,𝑥)=𝑢²−Λ⁻²𝑢²+12𝑢²_𝑥,𝜕𝑃_𝜀(𝑡,𝑥)𝜕𝑥=Λ⁻²𝜕_𝑥𝑢²+12𝜕𝑢𝜕𝑥²+𝜆𝑢. (3.31) The inequality (3.26) is proved in Lemma 3.2 (see (3.24)). Now, we prove (3.27). Using _𝑅Λ⁻²𝑢²+12𝑢²_𝑥=1𝑑𝑥2_𝑅𝑅𝑒^{−|𝑥−𝑦|}𝑢²+12𝑢²_𝑥≤‖𝑢‖𝑑𝑦𝑑𝑥_2𝐻1(𝑅)≤𝑢₀²_𝐻1(𝑅), (3.32) and (3.8) result in (3.27).
Applying the Tonelli theorem, (3.26) and (3.27), we get 𝑄_𝜀(𝑡,.)²_𝐿2(𝑅)≤𝑄_𝜀(𝑡,.)_𝐿∞𝑄_𝜀(𝑡,.)_𝐿1(𝑅)≤𝐶. (3.33) Since Λ⁻²𝜕_𝑥𝑢²+12𝜕𝑢𝜕𝑥²=12𝑒^−𝑥𝑥_−∞𝑒^𝑦𝜕_𝑦𝑢²+12𝜕𝑢𝜕𝑦²1𝑑𝑦+2𝑒^𝑥∞_𝑥𝑒^−𝑦𝜕_𝑦𝑢²+12𝜕𝑢𝜕𝑦²=−1𝑑𝑦2𝑒^−𝑥𝑥_−∞𝑒^𝑦𝑢²+12𝜕𝑢𝜕𝑦²1𝑑𝑦+2𝑒^−𝑥𝑥_−∞𝑒^𝑦𝑢²+12𝜕𝑢𝜕𝑦²≤𝑑𝑦_𝑅𝑒^{−|𝑥−𝑦|}𝑢²+12𝜕𝑢𝜕𝑦²≤‖𝑢‖𝑑𝑦_2𝐻1(𝑅)≤𝑢₀²_𝐻1(𝑅),_𝑅Λ⁻²𝜕_𝑥𝑢²+12𝜕𝑢𝜕𝑦²=𝑑𝑥_𝑅−12𝑒^−𝑥𝑥_−∞𝑒^𝑦𝑢²+12𝜕𝑢𝜕𝑦²1𝑑𝑦+2𝑒^−𝑥𝑥_−∞𝑒^𝑦𝑢²+12𝜕𝑢𝜕𝑦²≤𝑑𝑦𝑑𝑥_𝑅𝑅𝑒^{−|𝑥−𝑦|}𝑢²+12𝜕𝑢𝜕𝑦²‖𝑢‖𝑑𝑦𝑑𝑥≤2²_𝐻1(𝑅)≤2𝑢₀²_𝐻1(𝑅), (3.34) we have _𝑅Λ⁻²𝜕_𝑥𝑢²+12𝜕𝑢𝜕𝑥²²≤𝑢𝑑𝑥_{02𝐻1(𝑅)𝑅}Λ⁻²𝜕_𝑥𝑢²+12𝜕𝑢𝜕𝑥²𝑢𝑑𝑥≤2₀⁴_𝐻1(𝑅), (3.35) which, together with Lemma 3.1, we get (3.29) and (3.30). The proof of Lemma 3.3 is completed.

Lemma 3.4. Assume 𝑢_𝜀=𝑢_𝜀(𝑡,𝑥) is the unique solution of problem (3.3). There exists a positive constant 𝐶 depending only on ‖𝑢₀‖_𝐻¹_(𝑅) and 𝜆 such that the following one-sided 𝐿^∞ norm estimate on the first order spatial derivative holds 𝜕𝑢_𝜀(𝑡,𝑥)≤4𝜕𝑥𝑡+𝐶,𝑓𝑜𝑟(𝑡,𝑥)∈[0,∞)×𝑅. (3.36)

Proof. From (3.9) and Lemma 3.3, we know that there exists a positive constant 𝐶 depending only on ‖𝑢₀‖_𝐻¹_(𝑅) and 𝜆 such that ‖𝑄_𝜀(𝑡,𝑥)‖_𝐿^∞_(𝑅)≤𝐶. Therefore, 𝜕𝑞_𝜀𝜕𝑡+𝑢_𝜀𝜕𝑞_𝜀+1𝜕𝑥2𝑞²_𝜀+𝜆𝑞_𝜀=𝑄_𝜀(𝑡,𝑥)≤𝐶. (3.37)
Let 𝑓=𝑓(𝑡) be the solution of 𝑑𝑓+1𝑑𝑡2𝑓²+𝜆𝑓=𝐶,𝑡>0,𝑓(0)=𝜕𝑢_𝜀,0𝜕𝑥_𝐿^∞_(𝑅). (3.38) Letting sup_𝑥∈𝑅𝑞_𝜀(𝑡,𝑥)=𝑓(𝑡), due to the comparison principle for parabolic equations, we get 𝑞_𝜀(𝑡,𝑥)≤𝑓(𝑡). (3.39)
Using −𝜆𝑓≤𝜌²𝑓²+(1/4𝜌²)𝜆², we derive that 𝑑𝑓1𝑑𝑡=𝐶−2𝑓²1−𝜆𝑓≤𝐶−2𝑓²+𝜌²𝑓²+14𝜌²𝜆²1≤𝐶−4𝑓²+𝜆², (3.40) where (1/4𝜌²)𝜆²=𝜆² and 𝜌=1/2. Setting 𝑀₀=𝐶+𝜆², we obtain 𝑑𝑓+1𝑑𝑡4𝑓²≤𝑀₀. (3.41) Letting 𝐹(𝑡)=(4/𝑡)+2𝑀₀, we have (𝑑𝐹(𝑡)/𝑑𝑡)+(1/4)𝐹²(𝑡)−𝑀₀=(4𝑀₀/𝑡)>0. Due to the comparison principle for ordinary differential equations, we get 𝑓(𝑡)≤𝐹(𝑡) for all 𝑡>0. Therefore, by this and (3.39), the estimate (3.36) is proved.

Lemma 3.5. There exists a sequence {𝜀_𝑗}_𝑗∈𝑁 tending to zero and a function 𝑢∈𝐿^∞([0,∞);𝐻¹(𝑅))⋂𝐻¹([0,𝑇]×𝑅), such that 𝑢_𝜀𝑗⇀𝑢𝑖𝑛𝐻¹([0,𝑇]×𝑅),𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑇≥0,𝑢_𝜀𝑗→𝑢𝑖𝑛𝐿^∞_loc([0,∞)×𝑅), (3.42)(3.43) where 𝑢_𝜀=𝑢_𝜀(𝑡,𝑥) is the unique solution of (3.3).

Proof. For fixed 𝑇>0, using Lemmas 3.1 and 3.3, and 𝜕𝑢_𝜀𝜕𝑡+𝑢_𝜀𝜕𝑢_𝜀+𝜕𝑥𝜕𝑃_𝜀𝜕𝜕𝑥=𝜀²𝑢_𝜀𝜕𝑥², (3.44) we obtain 𝜕𝑢_𝜀𝜕𝑡_𝐿²_{([0,𝑇]×𝑅)}≤𝑐1+√𝜀𝑢_0𝐻1(𝑅), (3.45) where 𝑐 depends on 𝑇. Hence, {𝑢_𝜀} is uniformly bounded in 𝐿^∞([0,∞);𝐻¹(𝑅))⋂𝐻¹([0,𝑇]×𝑅) and (3.42) follows.
Observe that, for each 0≤𝑠,𝑡≤𝑇, 𝑢_𝜀(𝑡,.)−𝑢_𝜀(𝑠,.)²_𝐿2(𝑅)=_𝑅𝜕𝑢_𝜀()𝜕𝑡𝜏,𝑥𝑑𝜏²≤𝑑𝑥|𝑡−𝑠|_𝑅^𝑇₀𝜕𝑢_𝜀()𝜕𝑡𝜏,𝑥²𝑑𝜏𝑑𝑥. (3.46)
Moreover, {𝑢_𝜀} is uniformly bounded in 𝐿^∞([0,𝑇];𝐻¹(𝑅)) and 𝐻¹(𝑅)⊂𝐿^∞_loc⊂𝐿²_loc(𝑅). Then, (3.43) is valid.

Lemma 3.6. There exists a sequence {𝜀_𝑗}_𝑗∈𝑁 tending to zero and a function 𝑄∈𝐿^∞([0,∞);𝑊^1,∞(𝑅)) such that for each 1<𝑝<∞𝑄_𝜀𝑗→𝑄𝑠𝑡𝑟𝑜𝑛𝑔𝑙𝑦𝑖𝑛𝐿^𝑝_loc([0,∞)×𝑅). (3.47)

Proof. Using Lemma 3.3, we have the existence of pointwise convergence subsequence 𝑄_𝜀𝑗 which is uniformly bounded in 𝐿^∞([0,∞)×𝑅). This implies (3.47).

Throughout this paper we use overbars to denote weak limits (the space in which these weak limits are taken is 𝐿^𝑟_loc([0,∞)×𝑅) with 1<𝑟<(3/2)).

Lemma 3.7. There exists a sequence {𝜀_𝑗}_𝑗∈𝑁 tending to zero and two functions 𝑞∈𝐿^𝑝_loc([0,∞)×𝑅), 𝑞²∈𝐿^𝑟_loc([0,∞)×𝑅) such that 𝑞_𝜀𝑗⇀𝑞𝑖𝑛𝐿^𝑝_loc([0,∞)×𝑅),𝑞⇀_𝜀𝑗^⋆𝑞𝑖𝑛𝐿^∞_loc[0,∞);𝐿²(𝑅),𝑞²_𝜀𝑗⇀𝑞²in𝐿^𝑟_loc([0,∞)×𝑅), (3.48)(3.49) for each 1<𝑝<3 and 1<𝑟<(3/2). Moreover, 𝑞²(𝑡,𝑥)≤𝑞²(𝑡,𝑥),𝑓𝑜𝑟𝑎𝑙𝑚𝑜𝑠𝑡𝑒𝑣𝑒𝑟𝑦(𝑡,𝑥)∈[0,∞)×𝑅𝜕𝑢[)𝜕𝑥=𝑞𝑖𝑛𝑡ℎ𝑒𝑠𝑒𝑛𝑠𝑒𝑜𝑓𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠𝑜𝑛0,∞×𝑅. (3.50)(3.51)

Proof. (3.48) and (3.49) are direct consequence of Lemmas 3.1 and 3.2. Inequality (3.50) is valid because of the weak convergence in (3.49). Finally, (3.51) is a consequence of the definition of 𝑞_𝜀, Lemma 3.5 and (3.48).

In the following, for notational convenience, we replace the sequence {𝑢_𝜀𝑗}_𝑗∈𝑁, {𝑞_𝜀𝑗}_𝑗∈𝑁 and {𝑄_𝜀𝑗}_𝑗∈𝑁 by {𝑢_𝜀}_𝜀>0, {𝑞_𝜀}_𝜀>0 and {𝑄_𝜀}_𝜀>0, separately.

Using (3.48), we conclude that for any convex function 𝜂∈𝐶¹(𝑅) with 𝜂^′ bounded, Lipschitz continuous on 𝑅 and any 1<𝑝<3 we get 𝜂𝑞_𝜀⇀𝜂𝑞in𝐿^𝑝_loc)),𝜂𝑞([0,∞×𝑅_𝜀⋆⇀𝜂(𝑞)in𝐿^∞_loc[0,∞);𝐿²(𝑅). (3.52)

Multiplying (3.9) by 𝜂^′(𝑞_𝜀) yields 𝜕𝜂𝑞𝜕𝑡^𝜀+𝜕𝑢𝜕𝑥^𝜀𝜂𝑞_𝜀𝜕−𝜀²𝜕𝑥²𝜂𝑞_𝜀+𝜀𝜂^′′𝑞_𝜀𝜕𝑞_𝜀𝜕𝑥²=𝑞_𝜀𝜂𝑞_𝜀−12𝜂^′𝑞_𝜀𝑞²_𝜀−𝜆𝑞_𝜀𝜂^′𝑞_𝜀+𝑄_𝜀(𝑡,𝑥)𝜂^′𝑞_𝜀. (3.53)

Lemma 3.8. For any convex 𝜂∈𝐶¹(𝑅) with 𝜂^′ bounded, Lipschitz continuous on 𝑅, it holds that 𝜕𝜂𝑞+𝜕𝜕𝑡𝑢𝜕𝑥𝜂𝑞≤𝑞−1𝑞𝜂2𝜂^′𝑞𝑞²−𝜆𝑞𝜂^′𝑞+𝑄(𝑡,𝑥)𝜂^′𝑞 (3.54) in the sense of distributions on [0,∞)×𝑅. Here, 𝑞𝜂(𝑞) and 𝜂^′(𝑞)𝑞² denote the weak limits of 𝑞_𝜀𝜂(𝑞_𝜀) and 𝑞²_𝜀𝜂^′(𝑞_𝜀) in 𝐿^𝑟_loc([0,∞)×𝑅), 1<𝑟<(3/2), respectively.

Proof. In (3.53), by the convexity of 𝜂, (3.8), Lemmas 3.5, 3.6, and 3.7, sending 𝜀→0 gives rise to the desire result.

Remark 3.9. From (3.48) and (3.49), we know that 𝑞=𝑞₊+𝑞₋=𝑞₊+𝑞₋,𝑞²=𝑞₊²+𝑞₋²,𝑞²=𝑞₊²+𝑞₋², (3.55) almost everywhere in [0,∞)×𝑅, where 𝜉₊∶=𝜉_𝜒[0,+∞)(𝜉),𝜉₋∶=𝜉_{𝜒(−∞,0]}(𝜉) for 𝜉∈𝑅. From Lemma 3.4 and (3.48), we have 𝑞_𝜀()()≤4𝑡,𝑥,𝑞𝑡,𝑥𝑡+𝐶,for𝑡>0,𝑥∈𝑅, (3.56) where 𝐶 is a constant depending only on ‖𝑢₀‖_𝐻¹_(𝑅) and 𝜆.

Lemma 3.10. In the sense of distributions on [0,∞)×𝑅, it holds that 𝜕𝑞+𝜕𝜕𝑡=1𝜕𝑥𝑢𝑞2𝑞²−𝜆𝑞+𝑄(𝑡,𝑥). (3.57)

Proof. Using (3.9), Lemmas 3.5 and 3.6, (3.48), (3.49), and (3.51), the conclusion (3.57) holds by sending 𝜀→0 in (3.9).

The next lemma contains a generalized formulation of (3.57).

Lemma 3.11. For any 𝜂∈𝐶¹(𝑅) with 𝜂∈𝐿^∞(𝑅), it has 𝜕𝜂𝑞+𝜕𝜕𝑡𝑞𝑞+1𝜕𝑥𝑢𝜂=𝑞𝜂2𝑞²−𝑞²𝜂^′𝑞−𝜆𝑞𝜂^′𝑞+𝑄(𝑡,𝑥)𝜂^′𝑞, (3.58) in the sense of distributions on [0,∞)×𝑅.

Proof. Let {𝜔_𝛿}_𝛿 be a family of mollifiers defined on 𝑅. Denote 𝑞_𝛿(𝑡,𝑥)∶=(𝑞(𝑡,.)⋆𝜔_𝛿)(𝑥). The ⋆ is the convolution with respect to the 𝑥 variable. Multiplying (3.57) by 𝜂^′(𝑞_𝛿), it has 𝜕𝜂𝑞_𝛿𝜕𝑡=𝜂^′𝑞_𝛿𝜕𝑞_𝛿𝜕𝑡=𝜂^′𝑞_𝛿12𝑞²⋆𝜔_𝛿−𝜆𝑞_𝛿+𝑄(𝑡,𝑥)⋆𝜔_𝛿−𝑞²⋆𝜔_𝛿−𝑢𝜕𝑞𝜕𝑥⋆𝜔_𝛿,𝜕𝑞𝜕𝑥𝑢𝜂_𝛿=𝑞𝜂𝑞_𝛿+𝑢𝜂^′𝑞_𝛿𝜕𝑞_𝛿.𝜕𝑥 (3.59) Using the boundedness of 𝜂,𝜂^′ and letting 𝛿→0 in the above two equations, we obtain (3.58).

4. Strong Convergence of 𝑞_𝜀 and Existence for (1.1)

Following the ideas in [7] or [9], in this section, we improve the weak convergence of 𝑞_𝜀 in (3.48) to strong convergence, and then we have an existence result for problem (3.3). Generally speaking, we will derive a “transport equation” for the evolution of the defect measure (𝑞²−𝑞²)≥0. Namely, we will prove that the measure is zero initially then it will continue to be zero at all later times 𝑡>0.

Lemma 4.1. Assume 𝑢₀∈𝐻¹(𝑅). It holds that lim_𝑡→0𝑅𝑞²(𝑡,𝑥)𝑑𝑥=lim_𝑡→0𝑅𝑞²(𝑡,𝑥)𝑑𝑥=_𝑅𝜕𝑢₀𝜕𝑥²𝑑𝑥. (4.1)

Lemma 4.2. If 𝑢₀∈𝐻¹(𝑅), for each 𝑀>0, it has lim_𝑡→0𝑅𝜂^±_𝑀𝑞(𝑡,𝑥)−𝜂^±_𝑀𝑞(𝑡,𝑥)𝑑𝑥=0, (4.2) where 𝜂_𝑀(𝜉)1∶=2𝜉²𝜉𝑀𝜉−1,𝑖𝑓≤𝑀,2𝑀²,𝑖𝑓𝜉>𝑀, (4.3) and 𝜂⁺_𝑀(𝜉)∶=𝜂_𝑀(𝜉)𝜒_[0,+∞)(𝜉), 𝜂⁻_𝑀(𝜉)∶=𝜂_𝑀(𝜉)𝜒_(−∞,0](𝜉), 𝜉∈𝑅.
The proof of Lemmas 4.1 and 4.2 is similar to those of Lemmas 6.1 and 6.2 in [9]. Here, we omit their proofs.

Lemma 4.3 (Coclite et al. [9]). Let 𝑀>0. Then, for each 𝜉∈𝑅, it holds that 𝜂_𝑀(𝜉)=12𝜉²−12𝑀−𝜉²𝜒_{⋂(−∞,−𝑀)(𝑀,∞)}(𝜉),𝜂^′_𝑀(𝜉)𝜉=𝜉+𝑀−𝜉sign(𝜉)𝜒_{⋂(−∞,−𝑀)(𝑀,∞)}(𝜉),𝜂⁺_𝑀(𝜉)=12𝜉₊²−12(𝑀−𝜉)²𝜒_(𝑀,∞)(𝜉),𝜂⁺_𝑀^′(𝜉)=𝜉₊+(𝑀−𝜉)𝜒_(𝑀,∞)(𝜉),𝜂⁻_𝑀(𝜉)=12𝜉₋²−12(𝑀+𝜉)²𝜒_{(−∞,−𝑀)}(𝜉),𝜂⁻_𝑀^′(𝜉)=𝜉₋−(𝑀+𝜉)𝜒_{(−∞,−𝑀)}(𝜉). (4.4)

Lemma 4.4. Assume 𝑢₀∈𝐻¹(𝑅). Then, for almost all 𝑡>012_𝑅𝑞₊²−𝑞²₊(𝑡,𝑥)𝑑𝑥≤^𝑡_0𝑅𝑄(𝑠,𝑥)𝑞₊(𝑠,𝑥)−𝑞₊(𝑠,𝑥)𝑑𝑠𝑑𝑥. (4.5)

Proof. For an arbitrary 𝑇>0(0<𝑡<𝑇), we let 𝑀 be sufficiently large (see Lemma 3.4). Using (3.54) minus (3.58), and the entropy 𝜂⁺_𝑀 (see Lemma 4.2) results in 𝜕𝜕𝑡𝜂⁺_𝑀𝑞−𝜂⁺_𝑀𝑞+𝜕𝑢𝜕𝑥𝜂⁺_𝑀𝑞−𝜂⁺_𝑀𝑞≤𝑞𝜂⁺_𝑀𝑞−𝑞𝜂⁺_𝑀𝑞−12𝑞²𝜂⁺_𝑀^′𝑞−𝑞²𝜂⁺_𝑀^′𝑞−12𝑞²−𝑞²𝜂⁺_𝑀^′𝑞−𝜆𝑞𝜂⁺_𝑀^′𝑞−𝑞𝜂⁺_𝑀^′𝑞+𝑄(𝑡,𝑥)𝜂⁺_𝑀^′𝑞−𝜂⁺_𝑀^′𝑞. (4.6) By the increasing property of 𝜂⁺_𝑀 and the convexity of 𝑞(𝜂⁺_𝑀)^′(𝑞), from (3.50), we have −12𝑞²−𝑞²𝜂⁺_𝑀^′𝑞≤0,−𝜆𝑞𝜂⁺_𝑀^′𝑞−𝑞𝜂⁺_𝑀^′𝑞≤0. (4.7) It follows from Lemma 4.3 that 𝑞𝜂⁺_𝑀𝑞−12𝑞²𝜂⁺_𝑀^′𝑞𝑀=−2𝑞𝑀−𝑞𝜒_(𝑀,∞)𝑞,𝑞𝜂⁺_𝑀𝑞−12𝑞²𝜂⁺_𝑀^′𝑞𝑀=−2𝑞𝑀−𝑞𝜒_(𝑀,∞)𝑞. (4.8) In view of Remark 3.9. Let Ω_𝑀=(4/𝛼(𝑀−𝐶),∞)×𝑅. Applying (3.56) gives rise to 𝑞𝜂⁺_𝑀𝑞−12𝑞²𝜂⁺_𝑀^′𝑞=𝑞𝜂⁺_𝑀𝑞−12𝑞²𝜂⁺_𝑀^′𝑞=0,inΩ_𝑀. (4.9) In Ω_𝑀, it has 𝜂⁺_𝑀=12𝑞₊²,𝜂⁺_𝑀^′𝑞=𝑞₊,𝜂⁺_𝑀𝑞=12𝑞₊²,𝜂⁺_𝑀^′𝑞=𝑞₊. (4.10) From (4.6)–(4.10), we know that the following inequality holds in Ω_𝑀𝜕𝜕𝑡𝜂⁺_𝑀𝑞−𝜂⁺_𝑀𝑞+𝜕𝑢𝜕𝑥𝜂⁺_𝑀𝑞−𝜂⁺_𝑀𝑞≤𝑄(𝑡,𝑥)𝜂⁺_𝑀^′𝑞−𝜂⁺_𝑀^′𝑞. (4.11) Integrating the resultant inequality over (4/(𝑀−𝐶),𝑡)×𝑅 yields 12_𝑅𝑞₊²−𝑞²₊(𝑡,𝑥)𝑑𝑥≤lim_𝑡→0𝑅𝜂⁺_𝑀𝑞(𝑡,𝑥)−𝜂⁺_𝑀𝑞(+𝑡,𝑥)𝑑𝑥_{𝑡(𝛼(4/𝑀−𝐶))𝑅}𝑄(𝑠,𝑥)𝑞₊(𝑠,𝑥)−𝑞₊(𝑠,𝑥)𝑑𝑠𝑑𝑥, (4.12) for almost all 𝑡>4/(𝑀−𝐶). Sending 𝑀→∞ and using Lemma 4.2, we complete the proof.

Lemma 4.5. For any 𝑡>0 and 𝑀>0, it holds that _𝑅𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞()≤𝑀𝑡,𝑥𝑑𝑥₂2^𝑡_0𝑅𝑢𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞−𝑀𝑑𝑠𝑑𝑥₂2^𝑡_0𝑅𝑢𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞𝑑𝑠𝑑𝑥+𝑀^𝑡_0𝑅𝑢𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞+𝑀𝑑𝑠𝑑𝑥2^𝑡_0𝑅𝑢𝑞²₊−𝑞²₊𝑑𝑠𝑑𝑥+^𝑡_0𝑅𝑄(𝑡,𝑥)𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞𝑑𝑠𝑑𝑥. (4.13)

Proof. Let 𝑀>0. Subtracting (3.58) from (3.54) and using entropy 𝜂⁻_𝑀, we deduce 𝜕𝜕𝑡𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞+𝜕𝑢𝜕𝑥𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞≤𝑞𝜂⁻_𝑀𝑞−𝑞𝜂⁻_𝑀𝑞−12𝑞²𝜂⁻_𝑀^′𝑞−𝑞²𝜂⁻_𝑀^′𝑞−12𝑞²−𝑞²𝜂⁻_𝑀^′𝑞−𝜆𝑞𝜂⁻_𝑀^′𝑞−𝑞𝜂⁻_𝑀^′𝑞+𝑄(𝑡,𝑥)𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞. (4.14) Since −𝑀≤(𝜂⁻_𝑀)^′≤0, we get −12𝑞²−𝑞²𝜂⁻_𝑀^′𝑞≤𝑀2𝑞²−𝑞². (4.15) By the convexity of 𝑞(𝜂⁻_𝑀)^′(𝑞), it has −𝜆𝑞(𝜂⁻_𝑀)^′(𝑞)−𝑞𝜂⁻_𝑀^′𝑞≤0. (4.16) Using Remark 3.9 and Lemma 4.3 yields 𝑞𝜂⁻_𝑀𝑞−12𝑞²𝜂⁻_𝑀^′𝑞𝑀=−2𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞,𝑞𝜂⁻_𝑀𝑞−12𝑞²𝜂⁻_𝑀^′𝑞𝑀=−2𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞. (4.17)(4.18) Inserting the inequalities from (4.15) to (4.18) into (4.14) gives 𝜕𝜕𝑡𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞+𝜕𝑢𝜕𝑥𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞𝑀≤−2𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞+𝑀2𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞+𝑀2𝑞²−𝑞²+𝑄(𝑡,𝑥)𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞. (4.19) Integrating the above inequality over (0,𝑡)×𝑅, we obtain _𝑅𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞()𝑀𝑡,𝑥𝑑𝑥≤−2^𝑡_0𝑅𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞+𝑀𝑑𝑠𝑑𝑥2^𝑡_0𝑅𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞+𝑀𝑑𝑠𝑑𝑥2^𝑡_0𝑅𝑞²−𝑞²𝑑𝑠𝑑𝑥+^𝑡_0𝑅𝑄(𝑡,𝑥)𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞𝑑𝑠𝑑𝑥. (4.20) It follows from Lemma 4.3 that 𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞=12𝑞₋²−𝑞₋²+12𝑀+𝑞²𝜒_{(−∞,−𝑀)}𝑞−12𝑀+𝑞²𝜒_{(−∞,−𝑀)}𝑞. (4.21) Using Remark 3.9 and (4.20), we have _𝑅𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞()𝑀𝑡,𝑥𝑑𝑥≤−2^𝑡_0𝑅𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞+𝑀𝑑𝑠𝑑𝑥2^𝑡_0𝑅𝑞𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞𝑑𝑠𝑑𝑥+𝑀^𝑡_0𝑅𝜂⁻_𝑀𝑞−𝜂⁻_𝑀𝑞+𝑀𝑑𝑠𝑑𝑥2^𝑡_0𝑅𝑀+𝑞²𝜒_{(−∞,−𝑀)}𝑞−𝑀𝑑𝑠𝑑𝑥2^𝑡_0𝑅𝑀+𝑞²𝑢𝜒_{(−∞,−𝑀)}𝑞𝑀𝑑𝑠𝑑𝑥+2^𝑡_0𝑅𝑞²₊−𝑞²₊+𝑑𝑠𝑑𝑥_𝑡0𝑅𝑄(𝑡,𝑥)𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞𝑑𝑠𝑑𝑥. (4.22) Applying the identity 𝑀(𝑀+𝑞)²−𝑀𝑞(𝑀+𝑞)=𝑀²(𝑀+𝑞), we obtain (4.13).

Lemma 4.6. It holds that 𝑞²=𝑞²𝑎𝑙𝑚𝑜𝑠𝑡𝑒𝑣𝑒𝑟𝑦𝑤ℎ𝑒𝑟𝑒𝑖𝑛[0,∞)×(−∞,∞). (4.23)

Proof. Applying Lemmas 4.4 and 4.5 gives rise to _𝑅12𝑞₊²−𝑞₊²+𝜂⁻_𝑀−𝜂⁻_𝑀)≤𝑀(𝑡,𝑥𝑑𝑥₂2^𝑡_0𝑅𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞𝑀𝑑𝑠𝑑𝑥−²2^𝑡_0𝑅𝑀+𝑞𝜒_{(−∞,−𝑀)(𝑞)}𝑑𝑠𝑑𝑥+𝑀^𝑡_0𝑅𝜂⁻_𝑀−𝜂⁻_𝑀𝑀𝑑𝑠𝑑𝑥+2^𝑡_0𝑅𝑞₊²−𝑞₊²𝑑𝑠𝑑𝑥+^𝑡_0𝑅𝑄(𝑠,𝑥)𝑞₊−𝑞₊+𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞𝑑𝑠𝑑𝑥. (4.24) From Lemma 3.6, we know that there exists a constant 𝐿>0, depending only on ‖𝑢₀‖_𝐻¹_(𝑅), such that ‖𝑄(𝑡,𝑥)‖_𝐿^∞_{([0,∞)×𝑅)}≤𝐿. (4.25) By Remark 3.9 and Lemma 4.3, it has 𝑞₊+𝜂⁻_𝑀^′𝑞=𝑞−𝑀+𝑞𝜒_{(−∞,−𝑀)},𝑞₊+𝜂⁻_𝑀^′𝑞=𝑞−𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞. (4.26) Thus, by the convexity of the map 𝜉→𝜉₊+(𝜂⁻_𝑀)^′(𝜉), we get 0≤𝑞₊−𝑞₊+𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞=𝑀+𝑞𝜒_{(−∞,−𝑀)}−𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞. (4.27) Using (4.25) derives 𝑄(𝑠,𝑥)𝑞₊−𝑞₊+𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞≤−𝐿𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞−𝑀+𝑞𝜒_{(−∞,−𝑀)}. (4.28) Since 𝜉→(𝑀+𝜉)𝜒_{(−∞,−𝑀)} is concave and choosing 𝑀 large enough, we have 𝑀²2𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞−𝑀+𝑞𝜒_{(−∞,−𝑀)}+𝑄(𝑠,𝑥)𝑞₊−𝑞₊+𝜂⁻_𝑀^′𝑞−𝜂⁻_𝑀^′𝑞≤𝑀²2−𝐿𝑀+𝑞𝜒_{(−∞,−𝑀)}𝑞−𝑀+𝑞𝜒_{(−∞,−𝑀)}≤0. (4.29) Then, from (4.24) and (4.29), it has 0≤_𝑅12𝑞₊²−𝑞₊²+𝜂⁻_𝑀−𝜂⁻_𝑀(𝑡,𝑥)𝑑𝑥≤𝑐𝑀^𝑡_0𝑅12𝑞₊²−𝑞₊²+𝜂⁻_𝑀−𝜂⁻_𝑀(𝑡,𝑥)𝑑𝑠𝑑𝑥. (4.30) By making of the Gronwall inequality and Lemmas 4.1 and 4.2, for each 𝑡>0, we conclude that 0≤_𝑅12𝑞₊²−𝑞₊²+𝜂⁻_𝑀−𝜂⁻_𝑀(𝑡,𝑥)𝑑𝑥=0. (4.31) By the Fatou lemma, Remark 3.9, and (3.50), sending 𝑀→∞, we obtain 0≤_𝑅𝑞²−𝑞²(𝑡,𝑥)𝑑𝑥=0,for𝑡>0, (4.32) which completes the proof.

Proof of the main result. Using (3.2), (3.4), and Lemma 3.5, we know that the Conditions (i) and (ii) in Definition 2.1 are satisfied. We have to verify (iii). Due to Lemma 4.6, we have 𝑞_𝜀→𝑞in𝐿²_loc([0,∞)×𝑅). (4.33) From Lemma 3.5, (3.47), and (4.33), we know that 𝑢 is a distributional solution to problem (2.2). In addition, inequalities (2.5) and (2.6) are deduced from Lemmas 3.2 and 3.4. The proof of the main result is completed.

Acknowledgments

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